diff --git a/.github/workflows/docs.yml b/.github/workflows/docs.yml index 6d9c782..a2bb1a9 100644 --- a/.github/workflows/docs.yml +++ b/.github/workflows/docs.yml @@ -20,7 +20,9 @@ jobs: - uses: actions/setup-python@v6 with: python-version: 3.x - - run: pipx run --with mkdocs-material mkdocs build --clean + - run: + pipx run --with mkdocs-material --with mkdocs-awesome-nav mkdocs build + --clean - uses: actions/upload-pages-artifact@v4 with: path: site diff --git a/.github/workflows/tests.yml b/.github/workflows/tests.yml index 1c9531a..4e1b53e 100644 --- a/.github/workflows/tests.yml +++ b/.github/workflows/tests.yml @@ -13,6 +13,8 @@ jobs: - uses: actions/checkout@v6 - name: Build in strict mode # The strict flag treats warnings as errors - run: pipx run --with mkdocs-material mkdocs build --clean --strict + run: + pipx run --with mkdocs-material --with mkdocs-awesome-nav mkdocs build + --clean --strict # TODO: test the code snippets diff --git a/.gitignore b/.gitignore index 9101956..a54bf23 100644 --- a/.gitignore +++ b/.gitignore @@ -317,4 +317,4 @@ TSWLatexianTemp* .cache/ .DS_Store __pycache__/* - +site/* diff --git a/docs/.nav.yml b/docs/.nav.yml new file mode 100644 index 0000000..22914c7 --- /dev/null +++ b/docs/.nav.yml @@ -0,0 +1,55 @@ +nav: + - Home: index.md + - Semigroups in GAP: + - gap/index.md + - "GAP: Installation": gap/install.md + - "GAP: First steps": gap/first-steps.md + - "GAP: Semigroups defined by generators": gap/generated.md + - "GAP: Finitely presented semigroups": gap/fp.md + - "GAP: Worksheet": gap/worksheet.md + - Semigroups in Python: + - py/index.md + - "Python: Installation": py/install.md + - "Python: First steps": py/first-steps.md + - "Python: Semigroups defined by generators": py/generated.md + - "Python: Finitely presented semigroups": py/fp.md + - "Python: Worksheet": py/worksheet.md + - Examples: + - examples/index.md + - examples/short.md + - "Verifying presentations for sporadic simple groups": + - examples/atlas/index.md + - "Mathieu groups": + - "Mathieu group M11": examples/atlas/mathieu/m11.md + - "Mathieu group M12": examples/atlas/mathieu/m12.md + - "Mathieu group M22": examples/atlas/mathieu/m22.md + - "Mathieu group M23": + - examples/atlas/mathieu/m23/index.md + - "M11, P1": examples/atlas/mathieu/m23/p1_m11.md + - "M22, P1": examples/atlas/mathieu/m23/p1_m22.md + - "M11, P2": examples/atlas/mathieu/m23/p2_m11.md + - "M22, P2": examples/atlas/mathieu/m23/p2_m22.md + - "Mathieu group M24": + - examples/atlas/mathieu/m24/index.md + - "Leech lattice groups": + - examples/atlas/leech-lattice/hs.md + - "Janko group J2, Hall-Janko group HJ": + - examples/atlas/leech-lattice/j2/index.md + - "P1": examples/atlas/leech-lattice/j2/j2-p1.md + - "P2": examples/atlas/leech-lattice/j2/j2-p2.md + - "Conway group Co2": + - examples/atlas/leech-lattice/co2/index.md + - "McLaughlin group McL": + - examples/atlas/leech-lattice/mcl/index.md + - "Monster sections": + - "Held group He": + - examples/atlas/monster-sections/he/index.md + - "Fischer group Fi22": + - examples/atlas/monster-sections/fi22/index.md + - "Pariahs": + - examples/atlas/pariahs/* + - "Janko group J4": + - examples/atlas/pariahs/j4/index.md + - "Miscellaneous": + - examples/atlas/misc/t.md + - "Finding complete rewriting systems for 1-relation monoids": examples/1-relation diff --git a/docs/examples/1-relation/000.md b/docs/examples/1-relation/000.md new file mode 100644 index 0000000..ca1dd6b --- /dev/null +++ b/docs/examples/1-relation/000.md @@ -0,0 +1,44 @@ +#\#000 ababbaaaaa=baaaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a^{5} = b a^{5} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d = c b a,\; c = b a^{5},\; d = b a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b a b c &\rightarrow d \\ + b a^{5} &\rightarrow c \\ + c b a &\rightarrow a d \\ + b a b a d &\rightarrow d b a \\ + a d b c &\rightarrow c d \\ + a d a^{4} &\rightarrow c^{2} \\ + b a^{4} c d &\rightarrow c d b c \\ + b a^{4} c^{2} &\rightarrow c d a^{4} \\ + a d b a d &\rightarrow c d b a \\ + c b c d &\rightarrow a d^{2} b c \\ + c b c^{2} &\rightarrow a d^{2} a^{4} \\ + c^{2} d b c &\rightarrow a d a^{3} c d \\ + c^{2} d a^{4} &\rightarrow a d a^{3} c^{2} \\ + b a^{4} c a d &\rightarrow c d a^{4} b a \\ + c b c a d &\rightarrow a d^{2} a^{4} b a \\ + c^{2} d b a d &\rightarrow a d a^{3} c d b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/001.md b/docs/examples/1-relation/001.md new file mode 100644 index 0000000..752164a --- /dev/null +++ b/docs/examples/1-relation/001.md @@ -0,0 +1,35 @@ +#\#001 aabaababba=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b a b^{2} a = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c d c = c a c,\; c = b a,\; d = a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c b &\rightarrow d \\ + a^{2} c d c &\rightarrow c a c \\ + b c a c &\rightarrow c a c d c \\ + b d &\rightarrow c^{2} b \\ + a c^{2} &\rightarrow d a \\ + b c d &\rightarrow c a c d c b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/002.md b/docs/examples/1-relation/002.md new file mode 100644 index 0000000..04bca26 --- /dev/null +++ b/docs/examples/1-relation/002.md @@ -0,0 +1,44 @@ +#\#002 abbabbaaaa=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a^{4} = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d a = b c a,\; c = a b^{2},\; d = a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b c a &\rightarrow c^{2} d a \\ + a b^{2} &\rightarrow c \\ + a^{3} &\rightarrow d \\ + b c d &\rightarrow c^{2} d^{2} \\ + a d &\rightarrow d a \\ + d b^{2} &\rightarrow a^{2} c \\ + b c^{2} &\rightarrow c^{2} d c \\ + a c^{2} a &\rightarrow d c^{2} d c d^{2} c d a \\ + a c^{2} d &\rightarrow d c^{2} d c d^{2} c d^{2} \\ + a c^{3} &\rightarrow d c^{2} d c d^{2} c d c \\ + d c^{2} d c d^{2} c d^{2} c d c &\rightarrow c^{3} \\ + d c^{2} d c d^{2} c d^{2} c d^{2} &\rightarrow c^{2} d \\ + d c^{2} d c d^{2} c d^{2} c d a &\rightarrow c^{2} a \\ + d c^{2} d c d^{2} c d^{2} c^{3} a &\rightarrow c^{4} d c d^{2} c d^{2} c d a \\ + d c^{2} d c d^{2} c d^{2} c^{3} d &\rightarrow c^{4} d c d^{2} c d^{2} c d^{2} \\ + d c^{2} d c d^{2} c d^{2} c^{4} &\rightarrow c^{4} d c d^{2} c d^{2} c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/003.md b/docs/examples/1-relation/003.md new file mode 100644 index 0000000..7cedaaf --- /dev/null +++ b/docs/examples/1-relation/003.md @@ -0,0 +1,38 @@ +#\#003 babbabaaba=abba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a b a^{2} b a = a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c d c = e c,\; c = b a,\; d = b c^{2} a,\; e = a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c^{2} a &\rightarrow d \\ + a b &\rightarrow e \\ + e c &\rightarrow c d c \\ + b e &\rightarrow c b \\ + b c^{2} e &\rightarrow d b \\ + a d &\rightarrow c d c^{2} a \\ + a c &\rightarrow e a \\ + b c d c &\rightarrow c b c \\ + b c^{3} d c &\rightarrow d b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/004.md b/docs/examples/1-relation/004.md new file mode 100644 index 0000000..9b021d9 --- /dev/null +++ b/docs/examples/1-relation/004.md @@ -0,0 +1,50 @@ +#\#004 baabbaaaba=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{3} b a = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d b a = a^{2} b^{2} a,\; c = b a^{2} b,\; d = b a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a^{2} b &\rightarrow c \\ + b a^{3} &\rightarrow d \\ + a^{2} b^{2} a &\rightarrow c d b a \\ + b a^{2} d &\rightarrow c a^{3} \\ + b a^{2} c &\rightarrow c a^{2} b \\ + b a c d b a &\rightarrow d b^{2} a \\ + b c d b a &\rightarrow c b a \\ + a^{2} b c &\rightarrow c d c \\ + a^{2} b d &\rightarrow c d^{2} \\ + d a b^{2} a &\rightarrow c^{2} d^{2} b a \\ + d a b d &\rightarrow c^{2} d^{3} \\ + d a b c &\rightarrow c^{2} d^{2} c \\ + b a c d^{2} &\rightarrow d b d \\ + b a c d c &\rightarrow d b c \\ + b c d^{2} &\rightarrow c d \\ + b c d c &\rightarrow c^{2} \\ + a^{2} c b a &\rightarrow c d c d b a \\ + d a c b a &\rightarrow c^{2} d^{2} c d b a \\ + a^{2} c d &\rightarrow c d c d^{2} \\ + a^{2} c^{2} &\rightarrow c d c d c \\ + d a c d &\rightarrow c^{2} d^{2} c d^{2} \\ + d a c^{2} &\rightarrow c^{2} d^{2} c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/005.md b/docs/examples/1-relation/005.md new file mode 100644 index 0000000..fce2987 --- /dev/null +++ b/docs/examples/1-relation/005.md @@ -0,0 +1,42 @@ +#\#005 baaaaabbba=aaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{3} a = a^{5} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c^{2} a,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{4} d a &\rightarrow a^{5} \\ + a b^{3} &\rightarrow d \\ + a b^{2} a^{5} &\rightarrow d a^{4} d a \\ + b a^{4} d^{2} &\rightarrow a^{4} d \\ + a b^{2} a^{4} d &\rightarrow d a^{4} d^{2} \\ + a b a^{4} d &\rightarrow d a^{4} d^{3} \\ + a b a^{5} &\rightarrow d a^{4} d^{2} a \\ + d a^{4} d^{4} &\rightarrow a^{5} d \\ + d a^{4} d^{3} a &\rightarrow a^{6} \\ + b a^{9} d &\rightarrow a^{8} d^{4} \\ + b a^{10} &\rightarrow a^{8} d^{3} a \\ + d a^{4} d^{2} a^{5} d &\rightarrow a^{9} d^{4} \\ + d a^{4} d^{2} a^{6} &\rightarrow a^{9} d^{3} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/007.md b/docs/examples/1-relation/007.md new file mode 100644 index 0000000..ab3ed89 --- /dev/null +++ b/docs/examples/1-relation/007.md @@ -0,0 +1,35 @@ +#\#007 abaaabaaba=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b a^{2} b a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a^{2} c a c = c^{2} a^{2},\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + b d a^{2} &\rightarrow d a d^{2} \\ + b d a d &\rightarrow d a d^{2} b a \\ + a b d &\rightarrow d b a \\ + d b a^{2} d &\rightarrow a d a d^{2} b a \\ + d b a^{3} &\rightarrow a d a d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/008.md b/docs/examples/1-relation/008.md new file mode 100644 index 0000000..e78df07 --- /dev/null +++ b/docs/examples/1-relation/008.md @@ -0,0 +1,37 @@ +#\#008 aabaaaabba=baaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{4} b^{2} a = b a^{3} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c d b a = c a c,\; c = b a^{2},\; d = a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a^{2} &\rightarrow c \\ + a^{2} b &\rightarrow d \\ + b a d &\rightarrow c a b \\ + b a c a c &\rightarrow c a c d b a \\ + b d &\rightarrow c b \\ + b c a c &\rightarrow c^{2} d b a \\ + a^{2} c &\rightarrow d a^{2} \\ + d a^{2} d c &\rightarrow c a c a \\ + d a^{2} d b a &\rightarrow c a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/010.md b/docs/examples/1-relation/010.md new file mode 100644 index 0000000..1c78e60 --- /dev/null +++ b/docs/examples/1-relation/010.md @@ -0,0 +1,115 @@ +#\#010 baaaaabbba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{3} a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a = a b a^{2},\; c = a^{2} b^{2},\; d = a^{2} c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} c b &\rightarrow d \\ + a^{2} b^{2} &\rightarrow c \\ + a b a^{2} &\rightarrow b a d a \\ + a b a d &\rightarrow b a d^{2} \\ + a b a c &\rightarrow b a d c \\ + b a d a c b &\rightarrow a b d \\ + b a d a b^{2} &\rightarrow a b c \\ + a b^{2} a d^{2} a &\rightarrow b a d b a d a \\ + a^{2} c a b d &\rightarrow d a d a c b \\ + a^{2} c a b c &\rightarrow d a d a b^{2} \\ + a^{2} b a b d &\rightarrow c a d a c b \\ + a^{2} b a b c &\rightarrow c a d a b^{2} \\ + a^{2} b d &\rightarrow b a d^{2} a c b \\ + a^{2} b c &\rightarrow b a d^{2} a b^{2} \\ + b a d^{2} b a d a &\rightarrow c a d^{2} a \\ + a b^{2} a d^{3} &\rightarrow b a d b a d^{2} \\ + a b^{2} a d^{2} c &\rightarrow b a d b a d c \\ + b a d a c a b d &\rightarrow a b d a d a c b \\ + b a d a c a b c &\rightarrow a b d a d a b^{2} \\ + b a d a b a b d &\rightarrow a b c a d a c b \\ + b a d a b a b c &\rightarrow a b c a d a b^{2} \\ + b a d b a d b a d a &\rightarrow a b c a d^{2} a \\ + a^{2} c^{2} a d^{2} a &\rightarrow d a d^{2} b a d a \\ + b a d^{2} b a d b a d a &\rightarrow c a d^{2} b a d a \\ + a c a d^{2} a &\rightarrow b a d^{3} b a d a \\ + b a d^{2} b a d^{2} &\rightarrow c a d^{3} \\ + b a d^{2} b a d c &\rightarrow c a d^{2} c \\ + b a d^{2} b a d b a d^{2} &\rightarrow c a d^{2} b a d^{2} \\ + b a d^{2} b a d b a d c &\rightarrow c a d^{2} b a d c \\ + b a d b a d b a d^{2} &\rightarrow a b c a d^{3} \\ + b a d b a d b a d c &\rightarrow a b c a d^{2} c \\ + b a d a c^{2} a d^{2} a &\rightarrow a b d a d^{2} b a d a \\ + b a d a b c a d^{2} a &\rightarrow a b c a d^{2} b a d a \\ + b a d^{2} a b d &\rightarrow c a d^{2} a c b \\ + b a d^{2} a b c &\rightarrow c a d^{2} a b^{2} \\ + b a d b a d a b d &\rightarrow a b c a d^{2} a c b \\ + b a d b a d a b c &\rightarrow a b c a d^{2} a b^{2} \\ + b a d a b c a d^{3} &\rightarrow a b c a d^{2} b a d^{2} \\ + b a d a b c a d^{2} c &\rightarrow a b c a d^{2} b a d c \\ + b a d a b c a d^{2} b a d^{2} &\rightarrow a b c a d^{2} b a d b a d^{2} \\ + b a d a b c a d^{2} b a d c &\rightarrow a b c a d^{2} b a d b a d c \\ + b a d a b c a d^{2} b a d b a d a &\rightarrow a b c a d^{2} a b c a d^{2} a \\ + b a d b a d c a d^{2} a &\rightarrow a b c a d^{3} b a d a \\ + a^{2} c^{2} a d^{3} &\rightarrow d a d^{2} b a d^{2} \\ + a^{2} c^{2} a d^{2} c &\rightarrow d a d^{2} b a d c \\ + a^{2} c^{2} a d^{2} b a d a &\rightarrow d a d^{2} b a d b a d a \\ + a^{2} c^{2} a d^{2} b a d^{2} &\rightarrow d a d^{2} b a d b a d^{2} \\ + a^{2} c^{2} a d^{2} b a d c &\rightarrow d a d^{2} b a d b a d c \\ + a^{2} c^{2} a d^{2} b a d b a d a &\rightarrow d a d^{2} a b c a d^{2} a \\ + a^{2} c^{2} a d^{2} b a d b a d^{2} &\rightarrow d a d^{2} a b c a d^{3} \\ + a^{2} c^{2} a d^{2} b a d b a d c &\rightarrow d a d^{2} a b c a d^{2} c \\ + a c a d^{3} &\rightarrow b a d^{3} b a d^{2} \\ + a c a d^{2} c &\rightarrow b a d^{3} b a d c \\ + a c a d^{2} b a d a &\rightarrow b a d^{3} b a d b a d a \\ + a c a d^{2} b a d^{2} &\rightarrow b a d^{3} b a d b a d^{2} \\ + a c a d^{2} b a d c &\rightarrow b a d^{3} b a d b a d c \\ + b a d a c^{2} a d^{2} b a d b a d^{2} &\rightarrow a b d a d^{2} a b c a d^{3} \\ + b a d a c^{2} a d^{2} b a d b a d c &\rightarrow a b d a d^{2} a b c a d^{2} c \\ + b a d a c^{2} a d^{3} &\rightarrow a b d a d^{2} b a d^{2} \\ + b a d a c^{2} a d^{2} c &\rightarrow a b d a d^{2} b a d c \\ + b a d a c^{2} a d^{2} b a d a &\rightarrow a b d a d^{2} b a d b a d a \\ + b a d a c^{2} a d^{2} b a d^{2} &\rightarrow a b d a d^{2} b a d b a d^{2} \\ + b a d a c^{2} a d^{2} b a d c &\rightarrow a b d a d^{2} b a d b a d c \\ + b a d a b c a d^{2} b a d a &\rightarrow a b c a d^{2} b a d b a d a \\ + a c a d^{2} b a d b a d a &\rightarrow b a d^{3} a b c a d^{2} a \\ + a c a d^{2} b a d b a d^{2} &\rightarrow b a d^{3} a b c a d^{3} \\ + a c a d^{2} b a d b a d c &\rightarrow b a d^{3} a b c a d^{2} c \\ + b a d^{2} c a d^{2} a &\rightarrow c a d^{3} b a d a \\ + b a d^{2} c a d^{3} &\rightarrow c a d^{3} b a d^{2} \\ + b a d^{2} c a d^{2} c &\rightarrow c a d^{3} b a d c \\ + b a d^{2} c a d^{2} b a d a &\rightarrow c a d^{3} b a d b a d a \\ + b a d^{2} c a d^{2} b a d^{2} &\rightarrow c a d^{3} b a d b a d^{2} \\ + b a d^{2} c a d^{2} b a d c &\rightarrow c a d^{3} b a d b a d c \\ + b a d b a d c a d^{3} &\rightarrow a b c a d^{3} b a d^{2} \\ + b a d b a d c a d^{2} c &\rightarrow a b c a d^{3} b a d c \\ + b a d b a d c a d^{2} b a d a &\rightarrow a b c a d^{3} b a d b a d a \\ + b a d b a d c a d^{2} b a d^{2} &\rightarrow a b c a d^{3} b a d b a d^{2} \\ + b a d b a d c a d^{2} b a d c &\rightarrow a b c a d^{3} b a d b a d c \\ + b a d a c^{2} a d^{2} b a d b a d a &\rightarrow a b d a d^{2} a b c a d^{2} a \\ + b a d a b c a d^{2} b a d b a d^{2} &\rightarrow a b c a d^{2} a b c a d^{3} \\ + b a d a b c a d^{2} b a d b a d c &\rightarrow a b c a d^{2} a b c a d^{2} c \\ + b a d b a d c a d^{2} b a d b a d a &\rightarrow a b c a d^{3} a b c a d^{2} a \\ + b a d b a d c a d^{2} b a d b a d^{2} &\rightarrow a b c a d^{3} a b c a d^{3} \\ + b a d b a d c a d^{2} b a d b a d c &\rightarrow a b c a d^{3} a b c a d^{2} c \\ + b a d^{2} c a d^{2} b a d b a d a &\rightarrow c a d^{3} a b c a d^{2} a \\ + b a d^{2} c a d^{2} b a d b a d^{2} &\rightarrow c a d^{3} a b c a d^{3} \\ + b a d^{2} c a d^{2} b a d b a d c &\rightarrow c a d^{3} a b c a d^{2} c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/011.md b/docs/examples/1-relation/011.md new file mode 100644 index 0000000..9db6f48 --- /dev/null +++ b/docs/examples/1-relation/011.md @@ -0,0 +1,36 @@ +#\#011 babaaababa=aabababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{3} b a b a = a^{2} b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a c^{2} a = a c^{3} a,\; c = a b,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b c a &\rightarrow d \\ + d a c^{2} a &\rightarrow a c^{3} a \\ + a b &\rightarrow c \\ + b c^{2} &\rightarrow d b \\ + a d &\rightarrow c^{2} a \\ + d a c^{3} &\rightarrow a c^{4} \\ + a^{2} c^{3} a &\rightarrow c^{2} a^{2} c^{2} a \\ + a^{2} c^{4} &\rightarrow c^{2} a^{2} c^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/012.md b/docs/examples/1-relation/012.md new file mode 100644 index 0000000..7a3d3a1 --- /dev/null +++ b/docs/examples/1-relation/012.md @@ -0,0 +1,51 @@ +#\#012 bababbba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b^{3} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c = a c,\; c = b a,\; d = c b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c &\rightarrow c d c \\ + c b^{2} &\rightarrow d \\ + b c d c &\rightarrow c^{2} \\ + c b c &\rightarrow d a \\ + a d &\rightarrow c d^{2} \\ + b c d^{2} &\rightarrow c d \\ + c b d a &\rightarrow d a b c \\ + d a b^{2} &\rightarrow c b d \\ + c^{3} &\rightarrow d c d^{2} c \\ + c d c d^{2} c &\rightarrow d c d^{2} c^{2} \\ + c b d c d c &\rightarrow d a b c^{2} \\ + c b d c d^{2} &\rightarrow d a b c d \\ + c^{2} d &\rightarrow d c d^{3} \\ + c d c d^{3} &\rightarrow d c d^{2} c d \\ + b d c d^{2} c^{2} &\rightarrow d c d^{4} c \\ + c b d^{2} c d^{2} c^{2} &\rightarrow d a b d c d^{4} c \\ + b d c d^{2} c d &\rightarrow d c d^{5} \\ + b d c d^{3} c d^{2} c &\rightarrow d c d^{4} c^{2} \\ + b d c d^{3} c d^{3} &\rightarrow d c d^{4} c d \\ + c b d^{2} c d^{3} c d^{2} c &\rightarrow d a b d c d^{4} c^{2} \\ + c b d^{2} c d^{3} c d^{3} &\rightarrow d a b d c d^{4} c d \\ + c b d^{2} c d^{2} c d &\rightarrow d a b d c d^{5} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/013.md b/docs/examples/1-relation/013.md new file mode 100644 index 0000000..70e81e7 --- /dev/null +++ b/docs/examples/1-relation/013.md @@ -0,0 +1,75 @@ +#\#013 ababaaabba=baaababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{3} b^{2} a = b a^{3} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b a b c b a = b c a b a,\; c = a^{3} b,\; d = c b a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + c b a b &\rightarrow d \\ + a b a b c b a &\rightarrow b c a b a \\ + a^{3} b &\rightarrow c \\ + b c a b a b &\rightarrow a b a b d \\ + c b^{2} c a b a &\rightarrow d a b c b a \\ + a b a b c b c &\rightarrow b c a b c \\ + a^{2} b c a b a &\rightarrow c a b c b a \\ + b c b c a b c &\rightarrow a b a b d c b c \\ + b c b c a b a &\rightarrow a b a b d c b a \\ + b c a b^{2} c a b c &\rightarrow a b a b d a b c b c \\ + b c a b^{2} c a b a &\rightarrow a b a b d a b c b a \\ + a b a b c b d &\rightarrow b c a b d \\ + b c a b a^{2} b a b d &\rightarrow a b a b d c a b a b \\ + c b^{2} c a b c &\rightarrow d a b c b c \\ + c b a^{2} b a b d &\rightarrow d c a b a b \\ + a^{2} b c a b c &\rightarrow c a b c b c \\ + a b a^{2} b a b d^{2} &\rightarrow b c a^{2} b a b d \\ + a c a b c b a &\rightarrow c^{2} a b a \\ + c^{2} a b a b &\rightarrow a c a b d \\ + b c b c a b d &\rightarrow a b a b d c b d \\ + b c b c a^{2} b a b d &\rightarrow a b a b d c a b a b d \\ + b c a b^{2} c a b d &\rightarrow a b a b d a b c b d \\ + b c a b^{2} c a^{2} b a b d &\rightarrow a b a b d a^{2} b a b d^{2} \\ + c b^{2} c a b d &\rightarrow d a b c b d \\ + c b^{2} c a^{2} b a b d &\rightarrow d a^{2} b a b d^{2} \\ + c^{2} a b a^{2} b a b d &\rightarrow a c a b d c a b a b \\ + a^{2} b c a b d &\rightarrow c a b c b d \\ + a^{2} b c a^{2} b a b d &\rightarrow c a^{2} b a b d^{2} \\ + a b a^{2} b a b d c b c &\rightarrow b c a b c a b c \\ + a b a^{2} b a b d c b a &\rightarrow b c a b c a b a \\ + a c a b c b c &\rightarrow c^{2} a b c \\ + c^{2} b c a b c &\rightarrow a c a b d c b c \\ + c^{2} b c a b a &\rightarrow a c a b d c b a \\ + c^{2} a b^{2} c a b c &\rightarrow a c a b d a b c b c \\ + c^{2} a b^{2} c a b a &\rightarrow a c a b d a b c b a \\ + a b a^{2} b a b d c b d &\rightarrow b c a b c a b d \\ + c^{2} b c a b d &\rightarrow a c a b d c b d \\ + c^{2} b c a^{2} b a b d &\rightarrow a c a b d c a b a b d \\ + c^{2} a b^{2} c a b d &\rightarrow a c a b d a b c b d \\ + c^{2} a b^{2} c a^{2} b a b d &\rightarrow a c a b d a^{2} b a b d^{2} \\ + a b a^{2} b a b d c a b a b d &\rightarrow b c a b c a^{2} b a b d \\ + a c a b c b d &\rightarrow c^{2} a b d \\ + a c a^{2} b a b d^{2} &\rightarrow c^{2} a^{2} b a b d \\ + a c a^{2} b a b d c b d &\rightarrow c^{2} a b c a b d \\ + a c a^{2} b a b d c b c &\rightarrow c^{2} a b c a b c \\ + a c a^{2} b a b d c b a &\rightarrow c^{2} a b c a b a \\ + a c a^{2} b a b d c a b a b d &\rightarrow c^{2} a b c a^{2} b a b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/014.md b/docs/examples/1-relation/014.md new file mode 100644 index 0000000..dd5e06d --- /dev/null +++ b/docs/examples/1-relation/014.md @@ -0,0 +1,85 @@ +#\#014 babbaababa=abba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a^{2} b a b a = a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b d b a = c,\; c = a b^{2} a,\; d = c a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < b < a\): + +\( +\begin{aligned} + a b^{2} a &\rightarrow c \\ + b d b a &\rightarrow c \\ + c a b a &\rightarrow d \\ + c b^{2} a &\rightarrow b d b c \\ + a b^{2} c &\rightarrow b d b c \\ + a b^{3} d b c &\rightarrow c b^{2} c \\ + b d b^{2} d b c &\rightarrow c b^{2} c \\ + a b^{2} d &\rightarrow b d b d \\ + c b d b d b c &\rightarrow d b^{2} c \\ + c b^{3} d b c &\rightarrow b d b c b^{2} c \\ + a b^{3} d b d &\rightarrow c b^{2} d \\ + a b c &\rightarrow b d c \\ + b d b^{2} d b d &\rightarrow c b^{2} d \\ + c b^{3} d b d &\rightarrow b d b c b^{2} d \\ + c b d b d b d &\rightarrow d b^{2} d \\ + d b^{3} d b c &\rightarrow c b d c b^{2} c \\ + d b^{2} a &\rightarrow c b d c \\ + a b^{3} d c &\rightarrow c b c \\ + b d b^{2} d c &\rightarrow c b c \\ + c b^{3} d c &\rightarrow b d b c b c \\ + a b d &\rightarrow b d^{2} \\ + d b^{3} d b d &\rightarrow c b d c b^{2} d \\ + d b^{3} d c &\rightarrow c b d c b c \\ + c b d b d c &\rightarrow d b c \\ + a b^{3} d^{2} &\rightarrow c b d \\ + a c &\rightarrow b d^{2} b a \\ + b d b^{2} d^{2} &\rightarrow c b d \\ + c b^{3} d^{2} &\rightarrow b d b c b d \\ + d b^{3} d^{2} &\rightarrow c b d c b d \\ + c b d b d^{2} &\rightarrow d b d \\ + b d^{2} b a^{2} b a &\rightarrow a d \\ + a d b^{2} d &\rightarrow b d^{2} b^{2} d^{2} b d b d \\ + a d b^{2} c &\rightarrow b d^{2} b^{2} d^{2} b d b c \\ + a d b d &\rightarrow b d^{2} b^{2} d^{2} b d^{2} \\ + a d b c &\rightarrow b d^{2} b^{2} d^{2} b d c \\ + a d c &\rightarrow b d^{2} b^{2} d^{2} b d^{2} b a \\ + a b a d &\rightarrow b d a d \\ + c b^{3} d a d &\rightarrow b d b c b a d \\ + a d b a d &\rightarrow b d^{2} b^{2} d^{2} b d a d \\ + b d^{3} b a^{2} b a &\rightarrow a^{2} d \\ + c b d b d a d &\rightarrow d b a d \\ + c b d a d &\rightarrow d^{2} \\ + a b^{3} d a d &\rightarrow c b a d \\ + b d b^{2} d a d &\rightarrow c b a d \\ + d b^{3} d a d &\rightarrow c b d c b a d \\ + a d a d &\rightarrow b d^{2} b^{2} d^{2} a^{2} d \\ + a d^{2} &\rightarrow b d^{2} b^{2} d^{2} a d \\ + a^{3} d &\rightarrow b d^{4} b a^{2} b a \\ + a b a^{2} d &\rightarrow b d a^{2} d \\ + a b^{3} d a^{2} d &\rightarrow c b a^{2} d \\ + a d a^{2} d &\rightarrow b d^{2} b^{2} d^{2} b d^{4} b a^{2} b a \\ + a d b a^{2} d &\rightarrow b d^{2} b^{2} d^{2} b d a^{2} d \\ + c b^{3} d a^{2} d &\rightarrow b d b c b a^{2} d \\ + c b d b d a^{2} d &\rightarrow d b a^{2} d \\ + c b d a^{2} d &\rightarrow d a d \\ + b d b^{2} d a^{2} d &\rightarrow c b a^{2} d \\ + d b^{3} d a^{2} d &\rightarrow c b d c b a^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/015.md b/docs/examples/1-relation/015.md new file mode 100644 index 0000000..a74b38b --- /dev/null +++ b/docs/examples/1-relation/015.md @@ -0,0 +1,34 @@ +#\#015 baabaabbaa=abbaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{2} a^{2} = a b^{2} a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c a b c a = a b c a c,\; c = b a,\; d = b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b a^{2} &\rightarrow d \\ + a b d b a &\rightarrow d^{2} b d \\ + b a d^{2} b d &\rightarrow d b d b a \\ + a b d b d^{2} b d &\rightarrow d^{2} b d b d b a \\ + a b d^{2} &\rightarrow d^{2} b d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/016.md b/docs/examples/1-relation/016.md new file mode 100644 index 0000000..072816e --- /dev/null +++ b/docs/examples/1-relation/016.md @@ -0,0 +1,42 @@ +#\#016 baaaabbbaa=aaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a^{2} = a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{3} c = c^{2},\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + b a^{3} d a^{2} &\rightarrow a^{4} \\ + a b^{2} a^{4} &\rightarrow d a^{3} d a^{2} \\ + b a^{3} d a d &\rightarrow a^{3} d \\ + a b^{2} a^{3} d &\rightarrow d a^{3} d a d \\ + a b a^{3} d &\rightarrow d a^{3} d a d a d \\ + a b a^{4} &\rightarrow d a^{3} d a d a^{2} \\ + d a^{3} d a d a d a d &\rightarrow a^{4} d \\ + d a^{3} d a d a d a^{2} &\rightarrow a^{5} \\ + b a^{7} d &\rightarrow a^{5} d a d a d a d \\ + b a^{8} &\rightarrow a^{5} d a d a d a^{2} \\ + d a^{3} d a d a^{5} d &\rightarrow a^{6} d a d a d a d \\ + d a^{3} d a d a^{6} &\rightarrow a^{6} d a d a d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/017.md b/docs/examples/1-relation/017.md new file mode 100644 index 0000000..574981b --- /dev/null +++ b/docs/examples/1-relation/017.md @@ -0,0 +1,38 @@ +#\#017 aababbabba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b^{2} a b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} d a = b a b a^{2},\; c = b a b^{2},\; d = c a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a b^{2} &\rightarrow c \\ + b a b a^{2} &\rightarrow a^{2} d a \\ + c a b^{2} &\rightarrow d \\ + b a b c &\rightarrow d \\ + c a b a^{2} &\rightarrow a^{2} d a d a \\ + d a b^{2} &\rightarrow c a b c \\ + d a b a^{2} &\rightarrow a^{2} d a d a d a \\ + b a b d &\rightarrow c a b c \\ + c a b d &\rightarrow d a b c \\ + c a b c a b c &\rightarrow d a b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/018.md b/docs/examples/1-relation/018.md new file mode 100644 index 0000000..206b9ee --- /dev/null +++ b/docs/examples/1-relation/018.md @@ -0,0 +1,42 @@ +#\#018 aaabaaabba=baaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a^{3} b^{2} a = b a^{3} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a = b a c a^{2},\; c = a^{2} b,\; d = a c a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a c a^{2} &\rightarrow d a \\ + a^{2} b &\rightarrow c \\ + a c a c b &\rightarrow d \\ + b a c a d &\rightarrow d^{2} \\ + b a c a c &\rightarrow d c \\ + b a c^{2} &\rightarrow d a b \\ + a^{2} d a &\rightarrow c a c a^{2} \\ + a c a c d a &\rightarrow d a c a^{2} \\ + b a c d &\rightarrow d c a c b \\ + b d &\rightarrow d c b \\ + a^{2} d^{2} &\rightarrow c a c a d \\ + a^{2} d c &\rightarrow c a c a c \\ + a c a c d^{2} &\rightarrow d a c a d \\ + a c a c d c &\rightarrow d a c a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/020.md b/docs/examples/1-relation/020.md new file mode 100644 index 0000000..1c7bde7 --- /dev/null +++ b/docs/examples/1-relation/020.md @@ -0,0 +1,44 @@ +#\#020 aaabaabbaa=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a^{2} b^{2} a^{2} = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a^{3} c d c a = c a^{2} c,\; c = b a,\; d = a b,\; e = d a d c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < e < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow d \\ + d a d c &\rightarrow e \\ + b d &\rightarrow c b \\ + a c &\rightarrow d a \\ + c a d^{2} &\rightarrow a^{2} e d \\ + c a d a &\rightarrow a^{2} e a \\ + b e &\rightarrow c^{2} d c \\ + c a e &\rightarrow a^{2} e a d c \\ + c a d e &\rightarrow a^{2} e^{2} \\ + a^{3} e a &\rightarrow d a^{2} d a \\ + a^{3} e d &\rightarrow d a^{2} d^{2} \\ + d a d a^{2} e a &\rightarrow e a d a \\ + d a d a^{2} e d &\rightarrow e a d^{2} \\ + a^{3} e^{2} &\rightarrow d a^{2} d e \\ + d a d a^{2} e^{2} &\rightarrow e a d e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/021.md b/docs/examples/1-relation/021.md new file mode 100644 index 0000000..e27e1c6 --- /dev/null +++ b/docs/examples/1-relation/021.md @@ -0,0 +1,36 @@ +#\#021 aaaabaabba=baabaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{4} b a^{2} b^{2} a = b a^{2} b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d a = b c b c a,\; c = a^{2},\; d = b c b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b c b^{2} &\rightarrow d \\ + b c b c a &\rightarrow c^{2} d a \\ + a^{2} &\rightarrow c \\ + b c b d &\rightarrow d c b^{2} \\ + b c b c^{2} &\rightarrow c^{2} d c \\ + a c &\rightarrow c a \\ + d c b c a &\rightarrow c^{2} d c d a \\ + d c b c^{2} &\rightarrow c^{2} d c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/023.md b/docs/examples/1-relation/023.md new file mode 100644 index 0000000..5b12863 --- /dev/null +++ b/docs/examples/1-relation/023.md @@ -0,0 +1,40 @@ +#\#023 baaabbbbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{5} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{3} d a = a^{2},\; c = b^{2},\; d = c^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + c^{2} b &\rightarrow d \\ + b c &\rightarrow c b \\ + d b &\rightarrow c^{3} \\ + c a^{2} &\rightarrow d a^{3} d a^{2} d a^{2} d a \\ + b d &\rightarrow c^{3} \\ + c d &\rightarrow d c \\ + d c b &\rightarrow c^{4} \\ + d a^{3} d a^{2} d a^{2} d a^{2} d a^{2} d a &\rightarrow a^{2} \\ + b a^{2} &\rightarrow d a^{3} d a^{2} d a^{2} d a^{2} d a \\ + c^{5} &\rightarrow d^{2} \\ + d a^{3} d a^{2} d a^{2} d a^{2} d a^{4} &\rightarrow a^{4} d a^{2} d a^{2} d a^{2} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/024.md b/docs/examples/1-relation/024.md new file mode 100644 index 0000000..514f82a --- /dev/null +++ b/docs/examples/1-relation/024.md @@ -0,0 +1,63 @@ +#\#024 baaabaaaba=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a^{3} b a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a c a c a = a d a,\; c = a^{2} b,\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} a &\rightarrow d \\ + b a c a c a &\rightarrow a d a \\ + a^{2} b &\rightarrow c \\ + b^{2} c &\rightarrow d a b \\ + b a d a &\rightarrow d c a c a \\ + b a c a c^{2} &\rightarrow a d c \\ + a^{3} d a &\rightarrow c a c a c a \\ + c b a &\rightarrow a^{2} d \\ + b a d c &\rightarrow d c a c^{2} \\ + b d c a c a &\rightarrow d^{2} a \\ + a^{3} d c &\rightarrow c a c a c^{2} \\ + a^{2} d^{2} a &\rightarrow c d c a c a \\ + a^{2} d c a c a &\rightarrow c a d a \\ + a^{2} d c a c^{2} &\rightarrow c a d c \\ + c b c &\rightarrow a^{2} d a b \\ + b d c a c^{2} &\rightarrow d^{2} c \\ + a^{2} d^{2} c &\rightarrow c d c a c^{2} \\ + c a c a c^{2} a c a &\rightarrow a c a d a \\ + c a c a c^{2} a c^{2} &\rightarrow a c a d c \\ + b a^{2} c a d a &\rightarrow a d a c^{2} a c a \\ + b a^{2} c a d c &\rightarrow a d a c^{2} a c^{2} \\ + b a c a^{2} c a d a &\rightarrow a d a c a c^{2} a c a \\ + b a c a^{2} c a d c &\rightarrow a d a c a c^{2} a c^{2} \\ + b d a c a d a &\rightarrow d^{2} a c^{2} a c a \\ + b d a c a d c &\rightarrow d^{2} a c^{2} a c^{2} \\ + b d c a^{2} c a d a &\rightarrow d^{2} a c a c^{2} a c a \\ + b d c a^{2} c a d c &\rightarrow d^{2} a c a c^{2} a c^{2} \\ + c a c a c a c a d a &\rightarrow a c a d a c^{2} a c a \\ + c a c a c a c a d c &\rightarrow a c a d a c^{2} a c^{2} \\ + a^{2} d a c a d a &\rightarrow c a d a c^{2} a c a \\ + a^{2} d a c a d c &\rightarrow c a d a c^{2} a c^{2} \\ + a^{2} d c a^{2} c a d a &\rightarrow c a d a c a c^{2} a c a \\ + a^{2} d c a^{2} c a d c &\rightarrow c a d a c a c^{2} a c^{2} \\ + c a c a c^{2} a^{2} c a d a &\rightarrow a c a d a c a c^{2} a c a \\ + c a c a c^{2} a^{2} c a d c &\rightarrow a c a d a c a c^{2} a c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/025.md b/docs/examples/1-relation/025.md new file mode 100644 index 0000000..588183e --- /dev/null +++ b/docs/examples/1-relation/025.md @@ -0,0 +1,34 @@ +#\#025 abbababaaa=baabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a b a^{3} = b a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a b a d = b a^{2} c a,\; c = b^{2},\; d = b a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a^{2} c a &\rightarrow a c a b a d \\ + b a^{3} &\rightarrow d \\ + b c &\rightarrow c b \\ + b d &\rightarrow c a^{3} \\ + b a c a b a d &\rightarrow c a^{2} c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/026.md b/docs/examples/1-relation/026.md new file mode 100644 index 0000000..cce2595 --- /dev/null +++ b/docs/examples/1-relation/026.md @@ -0,0 +1,39 @@ +#\#026 baaabaabaa=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a^{2} b a^{2} = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c e = d c,\; c = b a,\; d = a c,\; e = a d^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a d^{2} a &\rightarrow e \\ + a c &\rightarrow d \\ + d c &\rightarrow c e \\ + b d &\rightarrow c^{2} \\ + b e &\rightarrow c d^{2} a \\ + a d^{3} &\rightarrow e c \\ + a d^{2} e &\rightarrow e d^{2} a \\ + b c e &\rightarrow c^{3} \\ + d e^{3} &\rightarrow e c^{2} \\ + a d e c^{2} &\rightarrow e d^{2} a e^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/028.md b/docs/examples/1-relation/028.md new file mode 100644 index 0000000..b1b8a7c --- /dev/null +++ b/docs/examples/1-relation/028.md @@ -0,0 +1,37 @@ +#\#028 aaabababba=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a b a b^{2} a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{3} d a = c a^{3},\; c = b a b,\; d = c a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + c a b^{2} &\rightarrow d \\ + c a^{3} &\rightarrow a^{3} d a \\ + b a c &\rightarrow c a b \\ + c a b c &\rightarrow d a b \\ + b a^{4} d a &\rightarrow c a b a^{3} \\ + b a d &\rightarrow c a c b \\ + c a b a^{3} d a &\rightarrow d a b a^{3} \\ + c a b d &\rightarrow d a c b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/029.md b/docs/examples/1-relation/029.md new file mode 100644 index 0000000..8a042f2 --- /dev/null +++ b/docs/examples/1-relation/029.md @@ -0,0 +1,91 @@ +#\#029 abbabbaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a^{2} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a = b a b a,\; c = a b^{2} a,\; d = c b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b a b a &\rightarrow d a \\ + c b^{2} a &\rightarrow d \\ + a b^{2} a &\rightarrow c \\ + b a b c &\rightarrow d c \\ + b a d a &\rightarrow d a b a \\ + c b d a &\rightarrow d b a \\ + a b^{2} c &\rightarrow d \\ + a b d a &\rightarrow c b a \\ + b a d c &\rightarrow d a b c \\ + d b^{2} a &\rightarrow a b^{2} d \\ + a b d^{2} &\rightarrow c b d \\ + a b d c &\rightarrow c b c \\ + c b d c &\rightarrow d b c \\ + d b^{2} c &\rightarrow c b^{2} d \\ + c b d^{2} &\rightarrow d b d \\ + b a b d &\rightarrow d^{2} \\ + b a d^{2} &\rightarrow d a b d \\ + c b^{2} c &\rightarrow a b^{2} d \\ + b c b c &\rightarrow d^{2} c \\ + b c b a &\rightarrow d^{2} a \\ + b c b d &\rightarrow d^{3} \\ + b c d c &\rightarrow d^{2} a b c \\ + b c d a &\rightarrow d^{2} a b a \\ + b c d^{2} &\rightarrow d^{2} a b d \\ + b d b c &\rightarrow d^{3} c \\ + b d b a &\rightarrow d^{3} a \\ + b d b d &\rightarrow d^{4} \\ + b d^{2} c &\rightarrow d^{3} a b c \\ + b d^{2} a &\rightarrow d^{3} a b a \\ + b d^{3} &\rightarrow d^{3} a b d \\ + d b d^{2} &\rightarrow a d^{3} c b d \\ + a d^{3} a b c &\rightarrow d b c \\ + a d^{3} a b a &\rightarrow d b a \\ + a d^{3} a b d &\rightarrow d b d \\ + a d^{4} b c &\rightarrow d^{4} a b c \\ + a d^{4} b a &\rightarrow d^{4} a b a \\ + a d^{4} b d &\rightarrow d^{4} a b d \\ + c d^{3} a b d &\rightarrow a d^{3} c b d \\ + c d^{3} c b d &\rightarrow d^{4} a b d \\ + c d^{3} a d^{2} &\rightarrow a d^{3} c d^{2} \\ + c d^{3} a d a &\rightarrow a d^{3} c d a \\ + c d^{3} a d c &\rightarrow a d^{3} c d c \\ + c d^{5} c &\rightarrow d^{4} c d c \\ + c d^{5} a &\rightarrow d^{4} c d a \\ + c d^{6} &\rightarrow d^{4} c d^{2} \\ + c d^{4} b c &\rightarrow d^{4} c b c \\ + c d^{4} b a &\rightarrow d^{4} c b a \\ + a d^{3} a d c &\rightarrow d^{2} c \\ + a d^{3} a d a &\rightarrow d^{2} a \\ + a d^{3} a d^{2} &\rightarrow d^{3} \\ + c d^{3} a b c &\rightarrow a d^{3} c b c \\ + c d^{3} a b a &\rightarrow a d^{3} c b a \\ + a d^{5} c &\rightarrow d^{4} a d c \\ + a d^{5} a &\rightarrow d^{4} a d a \\ + a d^{6} &\rightarrow d^{4} a d^{2} \\ + d b d a &\rightarrow a d^{3} c b a \\ + d b d c &\rightarrow a d^{3} c b c \\ + c d^{3} c d^{2} &\rightarrow d^{4} a d^{2} \\ + c d^{3} c d a &\rightarrow d^{4} a d a \\ + c d^{3} c d c &\rightarrow d^{4} a d c \\ + c d^{4} b d &\rightarrow d^{4} c b d \\ + c d^{3} c b c &\rightarrow d^{4} a b c \\ + c d^{3} c b a &\rightarrow d^{4} a b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/030.md b/docs/examples/1-relation/030.md new file mode 100644 index 0000000..5a47ff3 --- /dev/null +++ b/docs/examples/1-relation/030.md @@ -0,0 +1,53 @@ +#\#030 baabbaabba=abababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{2} b^{2} a = a b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d^{2} c = a c^{3},\; c = b a,\; d = c a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{3} &\rightarrow d^{2} c \\ + c a b &\rightarrow d \\ + b d^{2} c &\rightarrow c^{4} \\ + c a c &\rightarrow d a \\ + a c^{2} d &\rightarrow d^{3} \\ + b d^{3} &\rightarrow c^{3} d \\ + d a^{2} b &\rightarrow c a d \\ + d a^{2} c &\rightarrow c a d a \\ + d a c^{2} &\rightarrow c d^{2} c \\ + d a c d &\rightarrow c d^{3} \\ + c d^{2} c d &\rightarrow d^{4} \\ + c d^{2} c^{2} &\rightarrow d^{3} c \\ + d a d^{2} c^{2} &\rightarrow c a d^{3} c \\ + d a d^{2} c d &\rightarrow c a d^{4} \\ + a c d^{3} c &\rightarrow d^{4} c^{2} \\ + a c d^{4} &\rightarrow d^{4} c d \\ + c d^{5} c &\rightarrow d^{5} c^{2} \\ + c d^{6} &\rightarrow d^{5} c d \\ + d a d^{3} c &\rightarrow c d^{4} c^{2} \\ + d a d^{4} &\rightarrow c d^{4} c d \\ + a d^{5} c^{2} &\rightarrow d^{4} c d^{2} c \\ + a d^{5} c d &\rightarrow d^{4} c d^{3} \\ + a d^{8} c &\rightarrow d^{4} c d^{4} c^{2} \\ + a d^{9} &\rightarrow d^{4} c d^{4} c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/031.md b/docs/examples/1-relation/031.md new file mode 100644 index 0000000..f4c00aa --- /dev/null +++ b/docs/examples/1-relation/031.md @@ -0,0 +1,36 @@ +#\#031 aaaabababa=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{4} b a b a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{3} c^{3} a = d a,\; c = a b,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c a &\rightarrow d \\ + d a &\rightarrow a^{3} c^{3} a \\ + a b &\rightarrow c \\ + b c^{2} &\rightarrow d b \\ + a d &\rightarrow c^{2} a \\ + d c &\rightarrow a^{3} c^{4} \\ + a^{4} c^{4} &\rightarrow c^{2} a c \\ + a^{4} c^{3} a &\rightarrow c^{2} a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/032.md b/docs/examples/1-relation/032.md new file mode 100644 index 0000000..c7f05f2 --- /dev/null +++ b/docs/examples/1-relation/032.md @@ -0,0 +1,38 @@ +#\#032 baaabababa=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b a b a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c a d = a e,\; c = b a,\; d = a c^{3},\; e = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b c a &\rightarrow e \\ + b a &\rightarrow c \\ + c a d &\rightarrow a e \\ + a c^{3} &\rightarrow d \\ + b c d &\rightarrow e c^{3} \\ + b d &\rightarrow c^{4} \\ + c e &\rightarrow e d \\ + a c^{2} a e &\rightarrow d a d \\ + a e d^{3} &\rightarrow d e \\ + a c^{2} d e &\rightarrow d a d^{4} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/034.md b/docs/examples/1-relation/034.md new file mode 100644 index 0000000..403a7a1 --- /dev/null +++ b/docs/examples/1-relation/034.md @@ -0,0 +1,84 @@ +#\#034 baaaabbbaa=abaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a^{2} = a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{3} c d = c a^{3},\; c = a b,\; d = b^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a b &\rightarrow c \\ + b a^{3} c d &\rightarrow c a^{3} \\ + b^{2} a^{2} &\rightarrow d \\ + a c a^{3} &\rightarrow c a^{3} c d \\ + c b a^{2} &\rightarrow a d \\ + b c a^{3} &\rightarrow d a c d \\ + b^{2} a c &\rightarrow d b \\ + c a^{3} c d b &\rightarrow a c a^{2} c \\ + c^{2} a^{3} &\rightarrow a d a c d \\ + c b a c &\rightarrow a d b \\ + b c a^{2} c &\rightarrow d a c d b \\ + d b a^{3} &\rightarrow b d a c d c d \\ + a^{2} c a^{2} c &\rightarrow c a^{3} c d c d b \\ + c a^{3} c b d a c d c d &\rightarrow a^{2} d a c d c d c d \\ + a c a^{2} c^{2} a^{2} c &\rightarrow c a^{3} c d^{2} a c d b \\ + a d a c d b &\rightarrow c^{2} a^{2} c \\ + b a^{3} c b d a c d c d &\rightarrow a d a c d c d c d \\ + c a c a^{2} c &\rightarrow a d a c d c d b \\ + d c a^{3} &\rightarrow b d a c d c d c d \\ + d b a^{2} c &\rightarrow b d a c d c d b \\ + b a c a^{2} c &\rightarrow d a c d c d b \\ + b a d a c d c d c d &\rightarrow d a c b d a c d c d \\ + d c a^{2} c &\rightarrow b d a c d c d c d b \\ + a^{3} d a c d c d c d &\rightarrow c a^{3} c d c b d a c d c d \\ + a c a^{2} c a d a c d c d c d &\rightarrow c a^{3} c d^{2} a c b d a c d c d \\ + a d a c b d a c d c d &\rightarrow c a d a c d c d c d \\ + c a^{2} d a c d c d c d &\rightarrow a d a c d c b d a c d c d \\ + c^{2} a^{2} c^{2} a^{2} c &\rightarrow a d a c d^{2} a c d b \\ + d a^{2} d a c d c d c d &\rightarrow b d a c d c d c d c b d a c d c d \\ + d a c a^{2} c &\rightarrow b d a c d c d c d c d b \\ + d a d a c d c d c d &\rightarrow b d a c d c d c b d a c d c d \\ + b a^{2} d a c d c d c d &\rightarrow d a c d c b d a c d c d \\ + c a^{3} c d^{2} a c d c d c d &\rightarrow a^{2} d a c d c d c b d a c d c d \\ + c^{2} a^{2} c a d a c d c d c d &\rightarrow a d a c d^{2} a c b d a c d c d \\ + a d a c d^{2} a c d c d c d &\rightarrow c a d a c d c d c b d a c d c d \\ + d a d a c d c d c b d a c d c d &\rightarrow b d a c d c d c d^{2} a c d c d c d \\ + b d a c d c b d a c d c d &\rightarrow d^{2} a c d c d c d \\ + b d a c d c d^{2} a c d c d c d &\rightarrow d^{2} a c d c d c b d a c d c d \\ + b a^{2} d a c d c d c b d a c d c d &\rightarrow d a c d c d^{2} a c d c d c d \\ + b a d a c d c d c b d a c d c d &\rightarrow d a c d^{2} a c d c d c d \\ + d a^{2} d a c d c d c b d a c d c d &\rightarrow b d a c d c d c d c d^{2} a c d c d c d \\ + c a^{2} d a c d c d c b d a c d c d &\rightarrow a d a c d c d^{2} a c d c d c d \\ + a c a^{2} c a d a c d c d c b d a c d c d &\rightarrow c a^{3} c d^{2} a c d^{2} a c d c d c d \\ + a^{3} d a c d c d c b d a c d c d &\rightarrow c a^{3} c d c d^{2} a c d c d c d \\ + c a^{3} c d^{2} a c d c d c b d a c d c d &\rightarrow a^{2} d a c d c d c d^{2} a c d c d c d \\ + a d^{2} a c d c d c d &\rightarrow c d a c d c b d a c d c d \\ + a c a^{2} c d a c d c d^{2} a c d c d c d &\rightarrow c a^{3} c d^{3} a c d c d c b d a c d c d \\ + a c a^{2} c d a c d c b d a c d c d &\rightarrow c a^{3} c d^{3} a c d c d c d \\ + a d a c d^{2} a c d c d c b d a c d c d &\rightarrow c a d a c d c d c d^{2} a c d c d c d \\ + c^{2} a^{2} c a d a c d c d c b d a c d c d &\rightarrow a d a c d^{2} a c d^{2} a c d c d c d \\ + a d^{2} a c d c d c b d a c d c d &\rightarrow c d a c d c d^{2} a c d c d c d \\ + c^{2} a^{2} c d a c d c d^{2} a c d c d c d &\rightarrow a d a c d^{3} a c d c d c b d a c d c d \\ + c^{2} a^{2} c d a c d c b d a c d c d &\rightarrow a d a c d^{3} a c d c d c d \\ + b d a c d c d^{2} a c d c d c b d a c d c d &\rightarrow d^{2} a c d c d c d^{2} a c d c d c d \\ + a c a^{2} c d a c d c d^{2} a c d c d c b d a c d c d &\rightarrow c a^{3} c d^{3} a c d c d c d^{2} a c d c d c d \\ + c^{2} a^{2} c d a c d c d^{2} a c d c d c b d a c d c d &\rightarrow a d a c d^{3} a c d c d c d^{2} a c d c d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/036.md b/docs/examples/1-relation/036.md new file mode 100644 index 0000000..d0e3d0d --- /dev/null +++ b/docs/examples/1-relation/036.md @@ -0,0 +1,69 @@ +#\#036 aababbaaaa=baaaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b^{2} a^{4} = b a^{5} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d a e a = b c d a,\; c = a^{3},\; d = a^{2} b,\; e = b^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b^{2} c &\rightarrow e \\ + b c d a &\rightarrow d a e a \\ + a^{2} b &\rightarrow d \\ + a^{3} &\rightarrow c \\ + b d a e a &\rightarrow e d a \\ + b c d^{2} &\rightarrow d a e d \\ + b c d c &\rightarrow d a e c \\ + a^{2} d a e a &\rightarrow d c d a \\ + a c &\rightarrow c a \\ + d b c &\rightarrow a^{2} e \\ + c b &\rightarrow a d \\ + c a b &\rightarrow a^{2} d \\ + b^{2} a^{2} d &\rightarrow e a b \\ + b^{2} a d &\rightarrow e b \\ + b d a e d &\rightarrow e d^{2} \\ + b d a e c &\rightarrow e d c \\ + a^{2} d a e d &\rightarrow d c d^{2} \\ + a^{2} d a e c &\rightarrow d c d c \\ + a^{2} e d a &\rightarrow d^{2} a e a \\ + a^{2} e d^{2} &\rightarrow d^{2} a e d \\ + a^{2} e d c &\rightarrow d^{2} a e c \\ + a d^{2} a e a &\rightarrow c e d a \\ + a d c d a &\rightarrow c d a e a \\ + a d c d^{2} &\rightarrow c d a e d \\ + a d c d c &\rightarrow c d a e c \\ + d b a^{2} d &\rightarrow a^{2} e a b \\ + d b a d &\rightarrow a^{2} e b \\ + b^{2} d c d a &\rightarrow e a b a e a \\ + b^{2} d c d^{2} &\rightarrow e a b a e d \\ + b^{2} d c d c &\rightarrow e a b a e c \\ + a^{2} e a b a e d &\rightarrow d b d c d^{2} \\ + a^{2} e a b a e c &\rightarrow d b d c d c \\ + a d^{2} a e d &\rightarrow c e d^{2} \\ + a d^{2} a e c &\rightarrow c e d c \\ + a^{2} e a b a e a &\rightarrow d b d c d a \\ + c a e a b a e a &\rightarrow a^{2} d b d c d a \\ + c a e a b a e d &\rightarrow a^{2} d b d c d^{2} \\ + c a e a b a e c &\rightarrow a^{2} d b d c d c \\ + c e a b a e a &\rightarrow a d b d c d a \\ + c e a b a e d &\rightarrow a d b d c d^{2} \\ + c e a b a e c &\rightarrow a d b d c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/037.md b/docs/examples/1-relation/037.md new file mode 100644 index 0000000..c89a338 --- /dev/null +++ b/docs/examples/1-relation/037.md @@ -0,0 +1,38 @@ +#\#037 baaaabbbaa=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a^{2} = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c = a b a,\; c = b a^{2},\; d = a^{2} b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} b^{2} &\rightarrow d \\ + a b a &\rightarrow c d c \\ + b a^{2} &\rightarrow c \\ + a^{2} b c &\rightarrow d a^{2} \\ + a c &\rightarrow c d c a \\ + a b c d c &\rightarrow c d c b a \\ + b a d &\rightarrow c a b^{2} \\ + b c d c a d c &\rightarrow c b a \\ + b d &\rightarrow c b^{2} \\ + d a^{2} d c &\rightarrow c d c a d c b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/038.md b/docs/examples/1-relation/038.md new file mode 100644 index 0000000..f9479f5 --- /dev/null +++ b/docs/examples/1-relation/038.md @@ -0,0 +1,47 @@ +#\#038 babaabbbba=abbbbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b^{4} a = a b^{4} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d e = a c d^{2},\; c = b^{3},\; d = b a,\; e = d a c d +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < e < d < c < b\): + +\( +\begin{aligned} + b^{3} &\rightarrow c \\ + b a &\rightarrow d \\ + d a c d &\rightarrow e \\ + a c d^{2} &\rightarrow d e \\ + b^{2} d &\rightarrow c a \\ + b c &\rightarrow c b \\ + b d e &\rightarrow d c d^{2} \\ + d^{2} e &\rightarrow e d \\ + e a c d &\rightarrow d a c e \\ + a c d e &\rightarrow d^{2} a c e \\ + b^{2} e &\rightarrow c a^{2} c d \\ + b d c d^{2} &\rightarrow c a e \\ + b d^{2} a c e &\rightarrow d c d e \\ + d^{3} a c e &\rightarrow e^{2} \\ + a c e d &\rightarrow d e^{2} \\ + b d c d e &\rightarrow c a d a c e \\ + b d c e d &\rightarrow c a e^{2} \\ + b d c e^{2} &\rightarrow c a e d a c e \\ + a c e^{2} &\rightarrow d e d a c e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/040.md b/docs/examples/1-relation/040.md new file mode 100644 index 0000000..f165f4d --- /dev/null +++ b/docs/examples/1-relation/040.md @@ -0,0 +1,44 @@ +#\#040 baabbaabba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{2} b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c b^{2} c b^{2} a = c,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a d a d a &\rightarrow a^{2} \\ + a b a^{2} &\rightarrow d a d a d a \\ + b a d a d^{2} &\rightarrow a d \\ + a b a d &\rightarrow d a d a d^{2} \\ + d a d a d^{2} a d^{2} &\rightarrow a^{2} d \\ + d a d a d^{2} a d a &\rightarrow a^{3} \\ + b a d a^{3} d &\rightarrow a^{2} d a d^{2} a d^{2} \\ + b a d a^{4} &\rightarrow a^{2} d a d^{2} a d a \\ + b a^{3} d &\rightarrow a^{2} d^{2} a d^{2} \\ + b a^{4} &\rightarrow a^{2} d^{2} a d a \\ + d a d a d^{2} a^{3} d &\rightarrow a^{3} d a d^{2} a d^{2} \\ + d a d a d^{2} a^{4} &\rightarrow a^{3} d a d^{2} a d a \\ + d a d a d a^{2} d &\rightarrow a^{3} d^{2} a d^{2} \\ + d a d a d a^{3} &\rightarrow a^{3} d^{2} a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/042.md b/docs/examples/1-relation/042.md new file mode 100644 index 0000000..ff54e6b --- /dev/null +++ b/docs/examples/1-relation/042.md @@ -0,0 +1,38 @@ +#\#042 baabababba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b a b^{2} a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c^{2} b c = a,\; c = b a,\; d = a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + b d c^{3} &\rightarrow d c^{2} b c \\ + a &\rightarrow b d c^{2} b c \\ + c b d c^{2} b c &\rightarrow b d \\ + b^{2} d c^{2} b c &\rightarrow c \\ + b d c^{2} b^{2} d &\rightarrow d \\ + b d c b d &\rightarrow d c^{3} \\ + b d c^{2} b d &\rightarrow d c^{2} b^{2} d \\ + b d b d &\rightarrow d c^{5} b c \\ + c d &\rightarrow d c^{5} b c^{3} b c \\ + b d^{2} &\rightarrow d c^{5} b c^{3} b^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/043.md b/docs/examples/1-relation/043.md new file mode 100644 index 0000000..467c9ed --- /dev/null +++ b/docs/examples/1-relation/043.md @@ -0,0 +1,38 @@ +#\#043 baaabbabba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{2} d = a^{2},\; c = b a,\; d = b c b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b c b c &\rightarrow d \\ + b a &\rightarrow c \\ + b c d &\rightarrow d b c \\ + c a &\rightarrow d a^{2} d a d^{2} \\ + b d a^{2} d a d^{2} a &\rightarrow d a^{2} d a d \\ + d a^{2} d a d^{2} a d &\rightarrow a^{2} \\ + b c b d a^{2} d a d^{2} &\rightarrow d a \\ + b d a^{2} d a d a^{2} &\rightarrow d a^{2} d a d a d a d^{2} a d \\ + d a^{2} d a d^{2} a^{3} &\rightarrow a^{4} d a d^{2} a d \\ + b d a^{2} d a^{3} &\rightarrow d a^{2} d a d a d a d^{2} a d^{2} a d^{2} a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/045.md b/docs/examples/1-relation/045.md new file mode 100644 index 0000000..09740db --- /dev/null +++ b/docs/examples/1-relation/045.md @@ -0,0 +1,90 @@ +#\#045 baabbaabaa=aabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{2} b a^{2} = a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d c a^{2} = c b a^{2},\; c = a^{2} b,\; d = b c b,\; e = b c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < e < d < c < a\): + +\( +\begin{aligned} + a^{2} b &\rightarrow c \\ + b c^{2} &\rightarrow e \\ + b c b &\rightarrow d \\ + a^{2} d &\rightarrow c^{2} b \\ + a^{2} e &\rightarrow c^{3} \\ + c b c &\rightarrow b c e \\ + a^{2} c b a^{2} &\rightarrow b c e^{2} a^{2} \\ + a^{2} c b a c &\rightarrow b c e^{2} a c \\ + c^{2} d &\rightarrow b c e^{2} b \\ + b c e^{2} c &\rightarrow c^{2} e \\ + d c &\rightarrow b^{2} c e \\ + b c e c &\rightarrow c e \\ + b c e b &\rightarrow c d \\ + e b c &\rightarrow b^{2} c e^{2} \\ + c b^{2} c e &\rightarrow d b c e^{2} \\ + b^{2} c e a b c e &\rightarrow c b a b c e \\ + b^{2} c e a c &\rightarrow c b a c \\ + b^{2} c e a^{2} &\rightarrow c b a^{2} \\ + a^{2} c^{2} e &\rightarrow c^{2} e^{2} c \\ + a^{2} c d &\rightarrow c^{2} e b \\ + a^{2} c e &\rightarrow c^{2} e c \\ + a^{2} c b a d b c e^{2} &\rightarrow b c e^{2} a d b c e^{2} \\ + a^{2} c b a b c e &\rightarrow b c e^{2} a b c e \\ + c^{3} e &\rightarrow b c e^{3} c \\ + c^{2} e b^{2} c e &\rightarrow b c e^{2} d b c e^{2} \\ + c^{2} b a^{2} &\rightarrow d b c e^{2} a^{2} \\ + c^{2} b a c &\rightarrow d b c e^{2} a c \\ + c^{2} b a b c e &\rightarrow d b c e^{2} a b c e \\ + d b c e^{3} c &\rightarrow c e^{2} \\ + d b c e^{2} b &\rightarrow b c e d \\ + e c^{2} e &\rightarrow b^{2} c e^{4} c \\ + e c d &\rightarrow b^{2} c e^{3} b \\ + b^{2} c e^{3} c &\rightarrow e c e \\ + b^{2} c e^{2} b &\rightarrow e d \\ + d b c e a^{2} &\rightarrow e b a^{2} \\ + d b c e a c &\rightarrow e b a c \\ + d b c e a b c e &\rightarrow e b a b c e \\ + d b c e^{4} c &\rightarrow c e c e \\ + d b c e^{3} b &\rightarrow c e d \\ + b d b c e^{2} &\rightarrow d b c e \\ + b^{2} c e a d b c e^{2} &\rightarrow c b a d b c e^{2} \\ + b c d b c e &\rightarrow d^{2} b c e^{2} \\ + d^{2} b c e^{3} &\rightarrow e b^{2} c e \\ + c^{2} e c b a^{2} &\rightarrow b c e^{2} d b c e^{2} a^{2} \\ + c^{2} e c b a c &\rightarrow b c e^{2} d b c e^{2} a c \\ + c^{2} e c b a b c e &\rightarrow b c e^{2} d b c e^{2} a b c e \\ + c^{2} b a d b c e^{2} &\rightarrow d b c e^{2} a d b c e^{2} \\ + e c e b^{2} c e &\rightarrow b^{2} c e^{3} d b c e^{2} \\ + d^{2} b c e^{2} a^{2} &\rightarrow c d a^{2} \\ + d^{2} b c e^{2} a c &\rightarrow c d a c \\ + d^{2} b c e^{2} a b c e &\rightarrow c d a b c e \\ + d b c e a d b c e^{2} &\rightarrow e b a d b c e^{2} \\ + e c e c b a^{2} &\rightarrow b^{2} c e^{3} d b c e^{2} a^{2} \\ + e c e c b a c &\rightarrow b^{2} c e^{3} d b c e^{2} a c \\ + e c e c b a b c e &\rightarrow b^{2} c e^{3} d b c e^{2} a b c e \\ + c^{2} e c b a d b c e^{2} &\rightarrow b c e^{2} d b c e^{2} a d b c e^{2} \\ + d b c e^{2} d b c e &\rightarrow b c e d^{2} b c e^{2} \\ + d b c e^{3} d b c e &\rightarrow c e d^{2} b c e^{2} \\ + b c e d b c e &\rightarrow c d^{2} b c e^{2} \\ + b^{2} c e^{2} d b c e &\rightarrow e d^{2} b c e^{2} \\ + e c e c b a d b c e^{2} &\rightarrow b^{2} c e^{3} d b c e^{2} a d b c e^{2} \\ + d^{2} b c e^{2} a d b c e^{2} &\rightarrow c d a d b c e^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/047.md b/docs/examples/1-relation/047.md new file mode 100644 index 0000000..e9e43b3 --- /dev/null +++ b/docs/examples/1-relation/047.md @@ -0,0 +1,34 @@ +#\#047 ababbababa=babababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b a b a = b a b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c^{3} = c^{4},\; c = b a,\; d = a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c b &\rightarrow d \\ + d c^{3} &\rightarrow c^{4} \\ + b d &\rightarrow c^{2} b \\ + a c^{2} &\rightarrow d a \\ + b c^{4} &\rightarrow c^{2} b c^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/048.md b/docs/examples/1-relation/048.md new file mode 100644 index 0000000..0ac0996 --- /dev/null +++ b/docs/examples/1-relation/048.md @@ -0,0 +1,48 @@ +#\#048 abbababbba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a b^{3} a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a b c^{2} d c = c^{2},\; c = b a,\; d = b^{2},\; e = d a c d +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < e < d < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + b d &\rightarrow d b \\ + b c &\rightarrow d a \\ + d a c &\rightarrow c e c \\ + e a c d &\rightarrow c e c e \\ + c e c d &\rightarrow e \\ + a e c &\rightarrow c^{2} \\ + d a e &\rightarrow c e^{2} \\ + c e c e a c &\rightarrow e a c^{2} e c \\ + c e c^{2} e c &\rightarrow e a c \\ + a e^{2} &\rightarrow c e \\ + d c e &\rightarrow c e^{3} \\ + d c^{2} &\rightarrow c e^{2} c \\ + c e c^{2} e^{2} &\rightarrow e a e \\ + c e c e a e &\rightarrow e a c^{2} e^{2} \\ + b e &\rightarrow c e^{2} c d \\ + d e &\rightarrow c e^{3} c d \\ + e a c^{2} e^{3} &\rightarrow c e c e c e \\ + e a c^{2} e^{2} c &\rightarrow c e c e c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/049.md b/docs/examples/1-relation/049.md new file mode 100644 index 0000000..9fa7f1b --- /dev/null +++ b/docs/examples/1-relation/049.md @@ -0,0 +1,37 @@ +#\#049 baabaabba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{2} a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c b c b^{2} a = a,\; c = a^{2},\; d = b a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{2} b &\rightarrow d \\ + d a^{2} b^{2} a &\rightarrow a \\ + b a^{2} d &\rightarrow d a^{2} b \\ + d a^{2} b d &\rightarrow a^{2} b \\ + d a^{2} b a &\rightarrow a^{2} d b a \\ + d a^{3} &\rightarrow a^{2} d a \\ + d a^{2} d &\rightarrow a^{2} d^{2} \\ + b a^{3} &\rightarrow a^{2} d^{2} b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/050.md b/docs/examples/1-relation/050.md new file mode 100644 index 0000000..c71212f --- /dev/null +++ b/docs/examples/1-relation/050.md @@ -0,0 +1,83 @@ +#\#050 babaabaaba=aabbabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a^{2} b a = a^{2} b^{2} a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d b c b a = c b d,\; c = a^{2},\; d = b a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b a b c &\rightarrow d \\ + d b c b a &\rightarrow c b d \\ + a^{2} &\rightarrow c \\ + d b c b c &\rightarrow c b d a \\ + a c &\rightarrow c a \\ + c b d b c &\rightarrow d b c d \\ + b a b d b c d &\rightarrow d b d b c \\ + c b d a b d b c &\rightarrow d b d b c d^{2} \\ + c b d b d b c d &\rightarrow d b c d b d b c \\ + c b c b d &\rightarrow d b c d b a \\ + c a b d b c &\rightarrow a d b c d \\ + b a b d b d b c d^{2} &\rightarrow d b d a b d b c \\ + b a b d b c^{2} b d &\rightarrow d b c b d a b a \\ + b a b a d b c d &\rightarrow d a b d b c \\ + d b d b c d b a &\rightarrow c b d a b d \\ + d b d b c d b d b c d &\rightarrow c b d a b d a b d b c \\ + c b d a b d b d b c d &\rightarrow d b d b c d^{2} b d b c \\ + c b d a b c b d &\rightarrow d b d b c d^{2} b a \\ + c b d b d b d b c d^{2} &\rightarrow d b c d b d a b d b c \\ + c b d b a d b c d &\rightarrow d b c d a b d b c \\ + c b d a b d b d b d b c d^{2} &\rightarrow d b d b c d^{2} b d a b d b c \\ + c b d a b d b a d b c d &\rightarrow d b d b c d^{2} a b d b c \\ + c a b d b d b d b c d^{2} &\rightarrow a d b c d b d a b d b c \\ + c a b d b d b c d &\rightarrow a d b c d b d b c \\ + c a b d b a d b c d &\rightarrow a d b c d a b d b c \\ + c a b d a b d b c &\rightarrow a d b d b c d^{2} \\ + c a b c b d &\rightarrow a d b c d b a \\ + b a b d b d b c d c b d &\rightarrow d b d a b c b d a b a \\ + b a b a d b c^{2} b d &\rightarrow d a b c b d a b a \\ + d b d b c d b c &\rightarrow c b d a b d a \\ + c b d b d b d b c d c b d &\rightarrow d b c d b d a b c b d a b a \\ + c b d b a d b c^{2} b d &\rightarrow d b c d a b c b d a b a \\ + c a b d a b d b d b d b c d^{2} &\rightarrow a d b d b c d^{2} b d a b d b c \\ + c a b d a b d b d b c d &\rightarrow a d b d b c d^{2} b d b c \\ + c a b d a b c b d &\rightarrow a d b d b c d^{2} b a \\ + b a b a d b d b c d^{2} &\rightarrow d a b d a b d b c \\ + c b d a b d b d b d b c d c b d &\rightarrow d b d b c d^{2} b d a b c b d a b a \\ + c b d a b d b a d b c^{2} b d &\rightarrow d b d b c d^{2} a b c b d a b a \\ + c a b d b d b d b c d c b d &\rightarrow a d b c d b d a b c b d a b a \\ + c a b d b a d b c^{2} b d &\rightarrow a d b c d a b c b d a b a \\ + c b d b a d b d b c d^{2} &\rightarrow d b c d a b d a b d b c \\ + c b d a b d b a d b d b c d^{2} &\rightarrow d b d b c d^{2} a b d a b d b c \\ + c a b d b a d b d b c d^{2} &\rightarrow a d b c d a b d a b d b c \\ + c a b d a b d b a d b d b c d^{2} &\rightarrow a d b d b c d^{2} a b d a b d b c \\ + c a b d a b d b a d b c d &\rightarrow a d b d b c d^{2} a b d b c \\ + d b d b c d b d b d b c d^{2} &\rightarrow c b d a b d a b d a b d b c \\ + d b d b c^{2} b d &\rightarrow c b d a b d a b a \\ + b a b a d b d b c d c b d &\rightarrow d a b d a b c b d a b a \\ + c a b d a b d b d b d b c d c b d &\rightarrow a d b d b c d^{2} b d a b c b d a b a \\ + c b d b a d b d b c d c b d &\rightarrow d b c d a b d a b c b d a b a \\ + c a b d a b d b a d b c^{2} b d &\rightarrow a d b d b c d^{2} a b c b d a b a \\ + c b d a b d b a d b d b c d c b d &\rightarrow d b d b c d^{2} a b d a b c b d a b a \\ + c a b d a b d b a d b d b c d c b d &\rightarrow a d b d b c d^{2} a b d a b c b d a b a \\ + c a b d b a d b d b c d c b d &\rightarrow a d b c d a b d a b c b d a b a \\ + d b d b c d b d b d b c d c b d &\rightarrow c b d a b d a b d a b c b d a b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/051.md b/docs/examples/1-relation/051.md new file mode 100644 index 0000000..a64f2e7 --- /dev/null +++ b/docs/examples/1-relation/051.md @@ -0,0 +1,40 @@ +#\#051 baaaaaabba=aaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{6} b^{2} a = a^{5} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{3} b^{2} a = c^{2} a,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{5} d a &\rightarrow a^{5} \\ + a b a^{5} &\rightarrow d a^{5} d a \\ + b a^{5} d^{2} &\rightarrow a^{4} d \\ + a b a^{4} d &\rightarrow d a^{5} d^{2} \\ + d a^{5} d a d^{2} &\rightarrow a^{5} d \\ + d a^{5} d a d a &\rightarrow a^{6} \\ + b a^{10} d &\rightarrow a^{9} d a d^{2} \\ + b a^{11} &\rightarrow a^{9} d a d a \\ + d a^{5} d a^{6} d &\rightarrow a^{10} d a d^{2} \\ + d a^{5} d a^{7} &\rightarrow a^{10} d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/052.md b/docs/examples/1-relation/052.md new file mode 100644 index 0000000..c6beae7 --- /dev/null +++ b/docs/examples/1-relation/052.md @@ -0,0 +1,68 @@ +#\#052 abaabbabba=babbaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{2} a b^{2} a = b a b^{2} a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c a = b c a^{2} b a,\; c = a b^{2},\; d = a b a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a b a c &\rightarrow d \\ + a b^{2} &\rightarrow c \\ + b c a^{2} b a &\rightarrow d c a \\ + c^{2} a^{2} b a &\rightarrow a b d c a \\ + b c a^{2} b d &\rightarrow d c d \\ + b c a^{2} b c &\rightarrow d c^{2} \\ + b c a d &\rightarrow d c a c \\ + a b a^{2} b d c a &\rightarrow d c a^{2} b a \\ + a b d c a c &\rightarrow c^{2} a d \\ + c^{2} a^{2} b d &\rightarrow a b d c d \\ + c^{2} a^{2} b c &\rightarrow a b d c^{2} \\ + b c a^{2} d c a &\rightarrow d a b d c a \\ + b c a^{2} d c d &\rightarrow d a b d c d \\ + b c a^{2} d c^{2} &\rightarrow d a b d c^{2} \\ + a b a^{2} b d c d &\rightarrow d c a^{2} b d \\ + a b a^{2} b d c^{2} &\rightarrow d c a^{2} b c \\ + a b d a b d c a &\rightarrow c^{2} a^{2} d c a \\ + a b d a b d c d &\rightarrow c^{2} a^{2} d c d \\ + a b d a b d c^{2} &\rightarrow c^{2} a^{2} d c^{2} \\ + a b d c a^{2} b d c a &\rightarrow c^{2} a d c a^{2} b a \\ + a b d c a^{2} b d c d &\rightarrow c^{2} a d c a^{2} b d \\ + a b d c a^{2} b d c^{2} &\rightarrow c^{2} a d c a^{2} b c \\ + c^{2} a^{2} d a b d c a &\rightarrow a b d c^{2} a^{2} d c a \\ + c^{2} a^{2} d a b d c d &\rightarrow a b d c^{2} a^{2} d c d \\ + c^{2} a^{2} d a b d c^{2} &\rightarrow a b d c^{2} a^{2} d c^{2} \\ + c^{2} a^{2} d c a c &\rightarrow a b d c^{2} a d \\ + c^{2} a c^{2} a d &\rightarrow a b d c d c a c \\ + b c a^{2} d a b d c a &\rightarrow d c^{2} a^{2} d c a \\ + b c a^{2} d a b d c d &\rightarrow d c^{2} a^{2} d c d \\ + b c a^{2} d a b d c^{2} &\rightarrow d c^{2} a^{2} d c^{2} \\ + b c a c^{2} a d &\rightarrow d c d c a c \\ + c^{2} a^{2} d c a^{2} b d c a &\rightarrow a b d c^{2} a d c a^{2} b a \\ + c^{2} a^{2} d c a^{2} b d c d &\rightarrow a b d c^{2} a d c a^{2} b d \\ + c^{2} a^{2} d c a^{2} b d c^{2} &\rightarrow a b d c^{2} a d c a^{2} b c \\ + c^{2} a c^{2} a^{2} d c a &\rightarrow a b d c d a b d c a \\ + c^{2} a c^{2} a^{2} d c d &\rightarrow a b d c d a b d c d \\ + c^{2} a c^{2} a^{2} d c^{2} &\rightarrow a b d c d a b d c^{2} \\ + b c a c^{2} a^{2} d c a &\rightarrow d c d a b d c a \\ + b c a c^{2} a^{2} d c d &\rightarrow d c d a b d c d \\ + b c a c^{2} a^{2} d c^{2} &\rightarrow d c d a b d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/053.md b/docs/examples/1-relation/053.md new file mode 100644 index 0000000..0ed38e9 --- /dev/null +++ b/docs/examples/1-relation/053.md @@ -0,0 +1,43 @@ +#\#053 baaababbaa=abababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b^{2} a^{2} = a b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} c d = c b c,\; c = a b a,\; d = b^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a b a &\rightarrow c \\ + b a^{2} c d &\rightarrow c b c \\ + b^{2} a^{2} &\rightarrow d \\ + a c b c &\rightarrow c a c d \\ + a b c &\rightarrow c b a \\ + b c b c &\rightarrow d c d \\ + b^{2} a c &\rightarrow d b a \\ + a c d c d &\rightarrow c a c d b c \\ + a d c d &\rightarrow c b c b a \\ + b c d c d &\rightarrow d c d b c \\ + b^{2} c a c d &\rightarrow d b c b a \\ + b a c a c d b c &\rightarrow c b c^{2} d \\ + b^{2} c^{2} a c d b c &\rightarrow d b c b a c d \\ + b a c a c d^{2} c d &\rightarrow c b c^{2} d b c \\ + b^{2} c^{2} a c d^{2} c d &\rightarrow d b c b a c d b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/054.md b/docs/examples/1-relation/054.md new file mode 100644 index 0000000..e7e718c --- /dev/null +++ b/docs/examples/1-relation/054.md @@ -0,0 +1,42 @@ +#\#054 bbaaabbaaa=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{3} b^{2} a^{3} = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{2} d a = d,\; c = b^{2},\; d = a c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a c a^{2} &\rightarrow d \\ + b c &\rightarrow c b \\ + a d &\rightarrow d^{2} a \\ + a c d^{2} a &\rightarrow d c a^{2} \\ + c d^{8} a^{2} &\rightarrow d c a^{2} \\ + c d^{4} a^{3} &\rightarrow d \\ + c d^{10} a &\rightarrow d c d^{2} a \\ + a c d^{3} &\rightarrow d c d^{2} a \\ + c b d^{4} a^{3} &\rightarrow b d \\ + c b d^{8} a^{2} &\rightarrow b d c a^{2} \\ + c b d^{10} a &\rightarrow b d c d^{2} a \\ + c d^{11} &\rightarrow d c d^{3} \\ + c b d^{11} &\rightarrow b d c d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/055.md b/docs/examples/1-relation/055.md new file mode 100644 index 0000000..c4f9a38 --- /dev/null +++ b/docs/examples/1-relation/055.md @@ -0,0 +1,41 @@ +#\#055 baabaabbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{3} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d = a^{2},\; c = b a^{2},\; d = b^{3} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{3} a &\rightarrow d \\ + a^{2} &\rightarrow c^{2} d \\ + a c^{2} d &\rightarrow c^{2} d a \\ + b c &\rightarrow d a c d \\ + b d a c d &\rightarrow d a \\ + d a c d c d &\rightarrow c \\ + b d a c^{2} &\rightarrow d c^{2} d c d c d \\ + c a c d c d &\rightarrow d a c d c^{2} \\ + a c^{3} &\rightarrow c^{2} d c^{2} d c d c d \\ + b d c^{2} d a &\rightarrow d c^{2} d c d c d^{2} \\ + b d c^{2} d c^{2} d &\rightarrow d c^{2} d c d c d^{2} a \\ + b d c^{3} &\rightarrow d c^{2} d c d c d^{2} c d c d \\ + b d c^{2} d c^{3} &\rightarrow d c^{2} d c d c d^{2} c^{2} d c d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/056.md b/docs/examples/1-relation/056.md new file mode 100644 index 0000000..d6a1c34 --- /dev/null +++ b/docs/examples/1-relation/056.md @@ -0,0 +1,43 @@ +#\#056 abbabbabaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a b a^{2} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a b d a = c^{2},\; c = b a,\; d = c b c^{2},\; e = b d c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < c < e < d < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + d &\rightarrow a e \\ + b c^{2} &\rightarrow c^{2} e a \\ + c e c a e &\rightarrow e c^{2} e a \\ + a c e a &\rightarrow c^{2} \\ + c^{3} e a &\rightarrow a e \\ + c e c^{2} &\rightarrow e \\ + b c a e &\rightarrow c^{2} e a c^{2} e a \\ + c e c e &\rightarrow e^{2} c^{2} \\ + c e a e &\rightarrow e c e a \\ + a e c e a &\rightarrow c^{2} e \\ + b c e &\rightarrow e^{2} c^{2} a c^{2} \\ + b e &\rightarrow e^{2} c^{2} a c^{4} \\ + c e^{2} &\rightarrow e^{2} c^{4} \\ + a e^{2} &\rightarrow e^{2} c^{6} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/057.md b/docs/examples/1-relation/057.md new file mode 100644 index 0000000..3980232 --- /dev/null +++ b/docs/examples/1-relation/057.md @@ -0,0 +1,43 @@ +#\#057 abaabbbaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{3} a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b a^{2} c a = b a b a^{2},\; c = b^{3} a b,\; d = a b a^{2} c a c a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{3} a b &\rightarrow c \\ + b a b a^{2} &\rightarrow a b a^{2} c a \\ + a b a^{2} c a c a c &\rightarrow d \\ + b^{3} a c &\rightarrow c b^{2} a b \\ + b^{3} a^{2} b a^{2} c a &\rightarrow c a b a^{2} \\ + c a^{2} &\rightarrow d a \\ + b a b a d &\rightarrow a b a^{2} c d \\ + b d &\rightarrow d a c \\ + b^{3} a d &\rightarrow c a b a^{2} c a c \\ + b^{3} a^{2} b a^{2} c d &\rightarrow c a b a d \\ + b^{3} a^{2} b a^{2} d a &\rightarrow c a b a^{3} \\ + c a d &\rightarrow d^{2} \\ + a b a^{2} d^{3} a &\rightarrow d a^{2} \\ + b^{3} a^{2} b a^{2} d^{2} &\rightarrow c a b a^{2} d \\ + a b a^{2} d^{4} &\rightarrow d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/058.md b/docs/examples/1-relation/058.md new file mode 100644 index 0000000..02bee51 --- /dev/null +++ b/docs/examples/1-relation/058.md @@ -0,0 +1,75 @@ +#\#058 baabbaaaba=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{3} b a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c b a c a = a d a,\; c = a^{2} b,\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b^{2} a &\rightarrow d \\ + a^{2} b &\rightarrow c \\ + b^{2} c &\rightarrow d a b \\ + c b a &\rightarrow a^{2} d \\ + c a^{2} d c a &\rightarrow a^{3} d a \\ + b a^{2} d c^{2} &\rightarrow a d c \\ + b a^{2} d c a &\rightarrow a d a \\ + b a d c &\rightarrow d a d c^{2} \\ + b a d a &\rightarrow d a d c a \\ + b a^{2} d a^{3} d a &\rightarrow a d a^{2} d c a \\ + c b c &\rightarrow a^{2} d a b \\ + c a^{2} d c^{2} &\rightarrow a^{3} d c \\ + a^{2} d a d c^{2} &\rightarrow c a d c \\ + a^{2} d a d c a &\rightarrow c a d a \\ + a^{3} d a^{2} d c a &\rightarrow c a^{2} d a^{3} d a \\ + b d a d c^{2} &\rightarrow d^{2} c \\ + b d a d c a &\rightarrow d^{2} a \\ + b a^{2} d a c a d c &\rightarrow a d a^{2} d c a d c^{2} \\ + b a^{2} d a c a d a &\rightarrow a d a^{2} d c a d c a \\ + b a^{2} d a^{3} d c &\rightarrow a d a^{2} d c^{2} \\ + c d a d c^{2} &\rightarrow a^{2} d^{2} c \\ + c d a d c a &\rightarrow a^{2} d^{2} a \\ + a^{3} d a^{2} d c^{2} &\rightarrow c a^{2} d a^{3} d c \\ + a^{2} d a d a^{3} d c &\rightarrow c a d a^{2} d c^{2} \\ + a^{2} d a d a^{3} d a &\rightarrow c a d a^{2} d c a \\ + b d a d a^{3} d c &\rightarrow d^{2} a^{2} d c^{2} \\ + b d a d a^{3} d a &\rightarrow d^{2} a^{2} d c a \\ + c a d a^{2} d c a d c^{2} &\rightarrow a^{2} d a d a c a d c \\ + c a d a^{2} d c a d c a &\rightarrow a^{2} d a d a c a d a \\ + a^{2} d^{2} a^{2} d c^{2} &\rightarrow c d a d a^{3} d c \\ + a^{2} d^{2} a^{2} d c a &\rightarrow c d a d a^{3} d a \\ + a^{2} d a d a c a d a^{2} d c^{2} &\rightarrow c a d a^{2} d c a d a^{3} d c \\ + a^{2} d a d a c a d a^{2} d c a &\rightarrow c a d a^{2} d c a d a^{3} d a \\ + b d a d a c a d c &\rightarrow d^{2} a^{2} d c a d c^{2} \\ + b d a d a c a d a &\rightarrow d^{2} a^{2} d c a d c a \\ + b d a d a^{2} d a d a c a d c &\rightarrow d^{2} a d a^{2} d c a d c^{2} \\ + b d a d a^{2} d a d a c a d a &\rightarrow d^{2} a d a^{2} d c a d c a \\ + b a^{2} d a^{2} d a d a c a d c &\rightarrow a d a d a^{2} d c a d c^{2} \\ + b a^{2} d a^{2} d a d a c a d a &\rightarrow a d a d a^{2} d c a d c a \\ + a^{2} d^{2} a d a^{2} d c a d c^{2} &\rightarrow c d a d a^{2} d a d a c a d c \\ + a^{2} d^{2} a d a^{2} d c a d c a &\rightarrow c d a d a^{2} d a d a c a d a \\ + a^{2} d a d a c a d a d a^{2} d c a d c^{2} &\rightarrow c a d a^{2} d c a d a^{2} d a d a c a d c \\ + a^{2} d a d a c a d a d a^{2} d c a d c a &\rightarrow c a d a^{2} d c a d a^{2} d a d a c a d a \\ + a^{2} d a d a^{2} d a d a c a d c &\rightarrow c a d a d a^{2} d c a d c^{2} \\ + a^{2} d a d a^{2} d a d a c a d a &\rightarrow c a d a d a^{2} d c a d c a \\ + a^{3} d a d a^{2} d c a d c^{2} &\rightarrow c a^{2} d a^{2} d a d a c a d c \\ + a^{3} d a d a^{2} d c a d c a &\rightarrow c a^{2} d a^{2} d a d a c a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/059.md b/docs/examples/1-relation/059.md new file mode 100644 index 0000000..5ca7499 --- /dev/null +++ b/docs/examples/1-relation/059.md @@ -0,0 +1,57 @@ +#\#059 abbabbbaba=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{3} a b a = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b d c = c b c,\; c = b a,\; d = c b^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c b^{2} c &\rightarrow d \\ + a b d c &\rightarrow c b c \\ + b c b c &\rightarrow c b d c \\ + c b^{2} d &\rightarrow d b^{2} c \\ + a b d^{2} &\rightarrow c b d \\ + b c b d &\rightarrow c b d^{2} \\ + b c^{2} b d c &\rightarrow c b d c b c \\ + c^{2} b d^{2} c &\rightarrow d b c \\ + b c^{2} b d^{2} &\rightarrow c b d c b d \\ + c^{2} b d^{3} &\rightarrow d b d \\ + c^{2} b d^{2} b d b c &\rightarrow d c b d^{2} c b d c \\ + c b d c b d^{2} &\rightarrow b d b d \\ + c b d c b d c &\rightarrow b d b c \\ + a b d b d b c &\rightarrow d c b d^{2} c \\ + b^{2} d b d &\rightarrow c b d^{2} c b d^{2} \\ + b^{2} d b c &\rightarrow c b d^{2} c b d c \\ + c b d b d b d &\rightarrow b d c b d^{3} \\ + c b d b d b c &\rightarrow b d c b d^{2} c \\ + c^{2} b d^{2} b d b d &\rightarrow d c b d^{2} c b d^{2} \\ + a b d b d b d &\rightarrow d c b d^{3} \\ + c^{2} b d^{2} b d c b d^{2} &\rightarrow d c b d^{2} c b d c b d \\ + c^{2} b d^{2} b d c b d c &\rightarrow d c b d^{2} c b d c b c \\ + a b d b d c b d^{2} &\rightarrow d c b d^{2} c b d \\ + a b d b d c b d c &\rightarrow d c b d^{2} c b c \\ + b^{2} d c b d^{2} &\rightarrow c b d^{2} c b d c b d \\ + b^{2} d c b d c &\rightarrow c b d^{2} c b d c b c \\ + c b d b d c b d^{2} &\rightarrow b d c b d^{2} c b d \\ + c b d b d c b d c &\rightarrow b d c b d^{2} c b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/060.md b/docs/examples/1-relation/060.md new file mode 100644 index 0000000..e0a435f --- /dev/null +++ b/docs/examples/1-relation/060.md @@ -0,0 +1,54 @@ +#\#060 baabbbabba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{3} a b^{2} a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a b^{2} c b c = a,\; c = b a,\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b^{3} &\rightarrow d \\ + a d a b^{2} a &\rightarrow d a^{2} b^{2} a \\ + a d a b^{2} d &\rightarrow d a^{2} b^{2} d \\ + a d a b a &\rightarrow d a^{2} b a \\ + a d a b d &\rightarrow d a^{2} b d \\ + a d a^{2} &\rightarrow d a^{3} \\ + a d a d &\rightarrow d a^{2} d \\ + b a^{2} d &\rightarrow d a^{2} b d a d \\ + b a^{3} &\rightarrow d a^{2} b d a^{2} \\ + d^{2} a^{2} d &\rightarrow a d \\ + d^{2} a^{3} &\rightarrow a^{2} \\ + b a^{2} b d &\rightarrow d a^{2} b d a b d \\ + b a^{2} b a &\rightarrow d a^{2} b d a b a \\ + b a^{2} b^{2} d &\rightarrow d a^{2} b d a b^{2} d \\ + b a^{2} b^{2} a &\rightarrow d a^{2} b d a b^{2} a \\ + b d a^{2} d &\rightarrow d a^{2} b d \\ + b d a^{3} &\rightarrow d a^{2} b a \\ + d^{2} a^{2} b d &\rightarrow a b d \\ + d^{2} a^{2} b a &\rightarrow a b a \\ + d^{2} a^{2} b^{2} d &\rightarrow a b^{2} d \\ + b d a^{2} b d &\rightarrow d a^{2} b^{2} d \\ + d^{2} a^{2} b^{2} a &\rightarrow a b^{2} a \\ + b d a^{2} b a &\rightarrow d a^{2} b^{2} a \\ + b d a^{2} b^{2} d &\rightarrow d \\ + b d a^{2} b^{2} a &\rightarrow a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/061.md b/docs/examples/1-relation/061.md new file mode 100644 index 0000000..0a992a7 --- /dev/null +++ b/docs/examples/1-relation/061.md @@ -0,0 +1,46 @@ +#\#061 baaababbba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b^{3} a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d a = c a,\; c = a b a,\; d = c b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a^{2} d a &\rightarrow c a \\ + a b a &\rightarrow c \\ + c b^{3} &\rightarrow d \\ + b a^{2} d c &\rightarrow c^{2} \\ + a c a &\rightarrow c a d a \\ + c b^{2} c a &\rightarrow d a^{2} d a \\ + c b a &\rightarrow a b c \\ + a b c a d a &\rightarrow c^{2} a \\ + a b c a d c &\rightarrow c^{3} \\ + a c^{2} &\rightarrow c a d c \\ + c b^{2} c^{2} &\rightarrow d a^{2} d c \\ + c b c a d a &\rightarrow a b c^{2} a \\ + b a^{2} d^{2} &\rightarrow c d \\ + a b c a d^{2} &\rightarrow c^{2} d \\ + c b c a d c &\rightarrow a b c^{3} \\ + a c d &\rightarrow c a d^{2} \\ + c b^{2} c d &\rightarrow d a^{2} d^{2} \\ + c b c a d^{2} &\rightarrow a b c^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/062.md b/docs/examples/1-relation/062.md new file mode 100644 index 0000000..8507b1a --- /dev/null +++ b/docs/examples/1-relation/062.md @@ -0,0 +1,40 @@ +#\#062 baabbbbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{5} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d a = a^{2},\; c = b^{2},\; d = c^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + c^{2} b &\rightarrow d \\ + b c &\rightarrow c b \\ + d b &\rightarrow c^{3} \\ + c a^{2} &\rightarrow d a^{2} d a d a d a \\ + b d &\rightarrow c^{3} \\ + c d &\rightarrow d c \\ + d c b &\rightarrow c^{4} \\ + d a^{2} d a d a d a d a d a &\rightarrow a^{2} \\ + b a^{2} &\rightarrow d a^{2} d a d a d a d a \\ + c^{5} &\rightarrow d^{2} \\ + d a^{2} d a d a d a d a^{3} &\rightarrow a^{3} d a d a d a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/064.md b/docs/examples/1-relation/064.md new file mode 100644 index 0000000..d0693a2 --- /dev/null +++ b/docs/examples/1-relation/064.md @@ -0,0 +1,31 @@ +#\#064 aaaabababa=babaaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{4} b a b a b a = b a b a^{5} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d^{3} = d b c^{2} a,\; c = a^{2},\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a &\rightarrow d \\ + d^{2} a^{4} &\rightarrow a^{4} d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/066.md b/docs/examples/1-relation/066.md new file mode 100644 index 0000000..b6f6c51 --- /dev/null +++ b/docs/examples/1-relation/066.md @@ -0,0 +1,45 @@ +#\#066 abababbba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b^{3} a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e c = c^{2},\; c = b a,\; d = b^{2},\; e = c^{2} d +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < e < d < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + c^{2} d &\rightarrow e \\ + a e c &\rightarrow c^{2} \\ + b d &\rightarrow d b \\ + b c &\rightarrow d a \\ + a e^{2} &\rightarrow c e \\ + d a c &\rightarrow c e c \\ + b e &\rightarrow c e c d \\ + d a e &\rightarrow c e^{2} \\ + e a c &\rightarrow c^{3} e c \\ + d c e &\rightarrow c e^{3} \\ + d e &\rightarrow c e^{2} c d \\ + d c^{2} &\rightarrow c e^{2} c \\ + e a e &\rightarrow c^{3} e^{2} \\ + c^{3} e^{3} &\rightarrow e c e \\ + c^{3} e^{2} c &\rightarrow e c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/067.md b/docs/examples/1-relation/067.md new file mode 100644 index 0000000..6df71ab --- /dev/null +++ b/docs/examples/1-relation/067.md @@ -0,0 +1,54 @@ +#\#067 baaaaabba=aabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a = a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a = c a^{2},\; c = a^{2} b,\; d = a^{2} c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} c b &\rightarrow d \\ + a^{2} b &\rightarrow c \\ + b a d a &\rightarrow c a^{2} \\ + a^{2} c a^{2} &\rightarrow c a d a \\ + b a d^{2} &\rightarrow c a d \\ + b a d c &\rightarrow c a c \\ + a^{2} c a d &\rightarrow c a d^{2} \\ + a^{2} c a c &\rightarrow c a d c \\ + a^{2} c d &\rightarrow c a d a c b \\ + a^{2} c^{2} &\rightarrow c a d a b \\ + c a d a b c &\rightarrow d a d a b \\ + c a d a b d &\rightarrow d a d a c b \\ + c a d a b a c &\rightarrow d a d c \\ + c a d a b a d &\rightarrow d a d^{2} \\ + c a d a c a^{2} &\rightarrow d a d^{2} a \\ + c a d a b a^{2} &\rightarrow d a d a \\ + a^{2} d a d a &\rightarrow c a d^{2} a b a^{2} \\ + a^{2} d a d^{2} &\rightarrow c a d^{2} a b a d \\ + a^{2} d a d c &\rightarrow c a d^{2} a b a c \\ + c a d a c d &\rightarrow d a d^{2} a c b \\ + c a d a c^{2} &\rightarrow d a d^{2} a b \\ + c a d a c a d &\rightarrow d a d^{3} \\ + c a d a c a c &\rightarrow d a d^{2} c \\ + c a d a d a d a &\rightarrow d a d^{3} a b a^{2} \\ + c a d a d a d^{2} &\rightarrow d a d^{3} a b a d \\ + c a d a d a d c &\rightarrow d a d^{3} a b a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/068.md b/docs/examples/1-relation/068.md new file mode 100644 index 0000000..991a870 --- /dev/null +++ b/docs/examples/1-relation/068.md @@ -0,0 +1,42 @@ +#\#068 baaaabbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{3} a = c,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + b a^{3} d a &\rightarrow a^{2} \\ + a b^{2} a^{2} &\rightarrow d a^{3} d a \\ + b a^{3} d^{2} &\rightarrow a d \\ + a b^{2} a d &\rightarrow d a^{3} d^{2} \\ + a b a d &\rightarrow d a^{3} d a^{2} d^{2} \\ + a b a^{2} &\rightarrow d a^{3} d a^{2} d a \\ + d a^{3} d a^{2} d a^{2} d^{2} &\rightarrow a^{2} d \\ + d a^{3} d a^{2} d a^{2} d a &\rightarrow a^{3} \\ + b a^{5} d &\rightarrow a^{4} d a^{2} d a^{2} d^{2} \\ + b a^{6} &\rightarrow a^{4} d a^{2} d a^{2} d a \\ + d a^{3} d a^{2} d a^{4} d &\rightarrow a^{5} d a^{2} d a^{2} d^{2} \\ + d a^{3} d a^{2} d a^{5} &\rightarrow a^{5} d a^{2} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/069.md b/docs/examples/1-relation/069.md new file mode 100644 index 0000000..dfb1635 --- /dev/null +++ b/docs/examples/1-relation/069.md @@ -0,0 +1,47 @@ +#\#069 abaabaabba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a^{2} b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d a = b c a,\; c = a b a,\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c a &\rightarrow c^{2} d a \\ + a b^{2} &\rightarrow d \\ + a b a &\rightarrow c \\ + b c^{2} &\rightarrow c^{2} d c \\ + c b^{2} &\rightarrow a b d \\ + b c d &\rightarrow c^{2} d^{2} \\ + a c^{2} d c d a &\rightarrow d c a \\ + a b c &\rightarrow c b a \\ + a c^{2} d c d^{2} &\rightarrow d c d \\ + a c^{2} d c d c &\rightarrow d c^{2} \\ + c b a d &\rightarrow a c^{2} d^{2} \\ + c b a c &\rightarrow a c^{2} d c \\ + c b a^{2} &\rightarrow a c^{2} d a \\ + a b d c d &\rightarrow c^{3} d c d^{2} \\ + a b d c^{2} &\rightarrow c^{3} d c d c \\ + a b d c a &\rightarrow c^{3} d c d a \\ + c b d c d &\rightarrow a c^{2} d c^{2} d c d^{2} \\ + c b d c^{2} &\rightarrow a c^{2} d c^{2} d c d c \\ + c b d c a &\rightarrow a c^{2} d c^{2} d c d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/070.md b/docs/examples/1-relation/070.md new file mode 100644 index 0000000..c69f4b8 --- /dev/null +++ b/docs/examples/1-relation/070.md @@ -0,0 +1,38 @@ +#\#070 abababbaaa=baababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b^{2} a^{3} = b a^{2} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d e = c d,\; c = b a,\; d = a c^{2},\; e = b c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b c a^{2} &\rightarrow e \\ + b a &\rightarrow c \\ + c d &\rightarrow d e \\ + a c^{2} &\rightarrow d \\ + b c a d &\rightarrow e c^{2} \\ + b d &\rightarrow c^{3} \\ + a d e^{2} &\rightarrow d^{2} \\ + c^{3} e d &\rightarrow e c^{2} e^{2} \\ + a c e c^{2} e^{2} &\rightarrow d c^{2} e d \\ + a e c^{2} e^{2} &\rightarrow d c e d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/071.md b/docs/examples/1-relation/071.md new file mode 100644 index 0000000..6784245 --- /dev/null +++ b/docs/examples/1-relation/071.md @@ -0,0 +1,51 @@ +#\#071 baababbba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{3} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c b^{2} c = a c,\; c = b a,\; d = a c a c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b d b^{2} c b c b d &\rightarrow d b^{3} d \\ + b d b^{2} c b^{2} d &\rightarrow d b^{3} d b^{2} c \\ + a b d &\rightarrow b d b^{2} c b^{3} d \\ + c b d b^{2} c &\rightarrow b d \\ + b d b^{2} c b c^{2} &\rightarrow d b^{2} c \\ + a c &\rightarrow b d b^{2} c b^{2} c \\ + b d b^{2} c b^{3} d b^{2} c &\rightarrow d \\ + a d &\rightarrow d b^{3} d b^{2} c \\ + b d b^{2} c b d &\rightarrow d b c^{2} b^{3} d b^{2} c \\ + b^{2} d b^{2} c b^{2} c &\rightarrow c^{2} \\ + b^{2} d b^{2} c b^{3} d &\rightarrow c b d \\ + b d b^{3} d &\rightarrow d b c^{2} b c^{2} \\ + c d &\rightarrow d b c^{2} b c^{2} b^{2} c \\ + b d b d &\rightarrow d b c^{2} b c^{2} b^{2} c b c^{2} b^{3} d b^{2} c \\ + b d b c b d &\rightarrow d b c^{2} b c^{2} b^{2} c b^{3} d \\ + b d b c^{2} &\rightarrow d b c^{2} b c^{2} b^{2} c b^{2} c \\ + b d^{2} &\rightarrow d b c^{2} b c^{2} b^{2} c b c^{2} b c^{2} b^{3} d b^{2} c \\ + b d b^{2} c^{3} &\rightarrow d b c^{2} b^{2} c \\ + b d b^{2} c^{2} b d &\rightarrow d b c^{2} b^{3} d \\ + b d b^{2} d &\rightarrow d b c^{2} b c^{2} b^{2} c b^{3} d b^{2} c \\ + b d c^{2} &\rightarrow d b c^{2} b c^{2} b^{2} c b c^{2} b^{2} c \\ + b d c b d &\rightarrow d b c^{2} b c^{2} b^{2} c b c^{2} b^{3} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/072.md b/docs/examples/1-relation/072.md new file mode 100644 index 0000000..8a34ebc --- /dev/null +++ b/docs/examples/1-relation/072.md @@ -0,0 +1,42 @@ +#\#072 baaaaabbba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{3} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c a,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{4} d a &\rightarrow a^{3} \\ + a b^{3} &\rightarrow d \\ + a b^{2} a^{3} &\rightarrow d a^{4} d a \\ + b a^{4} d^{2} &\rightarrow a^{2} d \\ + a b^{2} a^{2} d &\rightarrow d a^{4} d^{2} \\ + a b a^{2} d &\rightarrow d a^{4} d a^{2} d^{2} \\ + a b a^{3} &\rightarrow d a^{4} d a^{2} d a \\ + d a^{4} d a^{2} d a^{2} d^{2} &\rightarrow a^{3} d \\ + d a^{4} d a^{2} d a^{2} d a &\rightarrow a^{4} \\ + b a^{7} d &\rightarrow a^{6} d a^{2} d a^{2} d^{2} \\ + b a^{8} &\rightarrow a^{6} d a^{2} d a^{2} d a \\ + d a^{4} d a^{2} d a^{5} d &\rightarrow a^{7} d a^{2} d a^{2} d^{2} \\ + d a^{4} d a^{2} d a^{6} &\rightarrow a^{7} d a^{2} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/074.md b/docs/examples/1-relation/074.md new file mode 100644 index 0000000..6ec8d4f --- /dev/null +++ b/docs/examples/1-relation/074.md @@ -0,0 +1,78 @@ +#\#074 abaababaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a d = c a c,\; c = b a,\; d = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c^{2} a &\rightarrow d \\ + c a c &\rightarrow a c a d \\ + a c a d a d &\rightarrow d c \\ + c a^{2} c a d &\rightarrow a c a d a c \\ + a c a d c a &\rightarrow c a d \\ + b c a d &\rightarrow d^{2} c a \\ + b d c &\rightarrow d^{2} a d \\ + d c a d c a &\rightarrow c d^{2} \\ + c d c &\rightarrow d c a d \\ + a c a d a c a d &\rightarrow c a d c \\ + b c d^{2} &\rightarrow d^{2} a d a d c a \\ + b d^{2} &\rightarrow d^{2} a d c a \\ + b d a c a d &\rightarrow d^{2} a d a c \\ + d c a d a c a d &\rightarrow c d^{2} c \\ + c^{2} d^{2} &\rightarrow d c a d a d c a \\ + c d^{2} c a d &\rightarrow d c a d^{2} c \\ + c d a c a d &\rightarrow d c a d a c \\ + c a d c a d &\rightarrow a c a d^{2} c \\ + a c a d a c d^{2} &\rightarrow d^{3} c a \\ + c d^{3} c a &\rightarrow d a c a d^{3} \\ + c a d^{2} c a &\rightarrow a^{2} c a d^{3} \\ + c a^{3} c a d^{3} &\rightarrow d^{2} c a d a \\ + b d a^{2} c a d^{3} &\rightarrow d^{2} a d a d^{2} c a \\ + b d a^{3} c a d^{3} &\rightarrow d^{2} a d a d c a d a \\ + c d^{2} c d^{2} &\rightarrow d a c a d^{4} c a \\ + c d^{2} a c a d^{3} &\rightarrow d c a d^{4} c a \\ + c d a^{2} c a d^{3} &\rightarrow d c a d a d^{2} c a \\ + c a d c d^{2} &\rightarrow a^{2} c a d^{4} c a \\ + c a d a c a d^{3} &\rightarrow a c a d^{4} c a \\ + c a d^{2} a c a d &\rightarrow a^{2} c a d^{3} c \\ + c a d^{2} a^{2} c a d^{3} &\rightarrow a^{2} c a d^{5} c a \\ + c d^{2} a^{2} c a d^{3} &\rightarrow d c a d^{3} c a d a \\ + c d a^{3} c a d^{3} &\rightarrow d c a d a d c a d a \\ + d a c a d^{2} c &\rightarrow c d^{3} \\ + c a d a^{2} c a d^{3} &\rightarrow a c a d^{3} c a d a \\ + a^{2} c a d^{2} c &\rightarrow c a d^{2} \\ + d a c a d^{3} a c a d^{3} &\rightarrow c d^{6} c a \\ + d a c a d^{3} c a d &\rightarrow c d^{4} c \\ + d a c a d^{3} a^{2} c a d^{3} &\rightarrow c d^{5} c a d a \\ + c a d^{2} a^{3} c a d^{3} &\rightarrow a^{2} c a d^{4} c a d a \\ + a^{2} c a d^{3} a^{2} c a d^{3} &\rightarrow c a d^{4} c a d a \\ + a^{2} c a d^{3} a c a d^{3} &\rightarrow c a d^{5} c a \\ + a^{2} c a d^{3} c a d &\rightarrow c a d^{3} c \\ + c d^{3} a &\rightarrow d a^{3} c a d^{3} \\ + c d^{2} a^{3} c a d^{3} &\rightarrow d c a d^{4} a \\ + c a d a^{3} c a d^{3} &\rightarrow a c a d^{4} a \\ + d a c a d^{3} a^{3} c a d^{3} &\rightarrow c d^{6} a \\ + a^{2} c a d^{3} a^{3} c a d^{3} &\rightarrow c a d^{5} a \\ + a^{4} c a d^{3} &\rightarrow c a d^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/075.md b/docs/examples/1-relation/075.md new file mode 100644 index 0000000..da46cb8 --- /dev/null +++ b/docs/examples/1-relation/075.md @@ -0,0 +1,48 @@ +#\#075 baaababba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b^{2} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d b c = a c,\; c = b a,\; d = a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a c &\rightarrow c d b c \\ + b a &\rightarrow c \\ + a d &\rightarrow d^{2} b c \\ + c d b d &\rightarrow d^{2} b c \\ + b c d b c &\rightarrow c^{2} \\ + c d c^{2} &\rightarrow d \\ + c d c d &\rightarrow d^{2} c^{2} \\ + c^{2} d b c &\rightarrow b d \\ + b d^{2} b c &\rightarrow c d \\ + a b d &\rightarrow c d b^{2} d \\ + c d^{2} &\rightarrow d^{2} c^{4} \\ + c d c b d &\rightarrow d c d b c \\ + b d^{2} c d &\rightarrow d^{2} c^{4} b d \\ + b d^{2} b d &\rightarrow d^{2} c^{6} \\ + b d^{2} c^{2} &\rightarrow d^{2} c^{4} b c \\ + c^{2} d b^{2} d &\rightarrow b d c d b c \\ + b d^{2} b^{2} d &\rightarrow d^{2} c^{2} b c \\ + b c d b^{2} d &\rightarrow c b d \\ + b d^{2} c b d &\rightarrow d^{2} c^{4} b^{2} d \\ + b d^{3} &\rightarrow d^{2} c^{4} b d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/076.md b/docs/examples/1-relation/076.md new file mode 100644 index 0000000..66d2d55 --- /dev/null +++ b/docs/examples/1-relation/076.md @@ -0,0 +1,34 @@ +#\#076 baaaabbbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{4} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c,\; c = a^{2},\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{4} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a^{2} &\rightarrow d a^{4} d a^{3} d a^{3} d a \\ + d a^{4} d a^{3} d a^{3} d a^{3} d a &\rightarrow a^{2} \\ + d a^{4} d a^{3} d a^{3} d a^{5} &\rightarrow a^{5} d a^{3} d a^{3} d a^{3} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/077.md b/docs/examples/1-relation/077.md new file mode 100644 index 0000000..aaf2643 --- /dev/null +++ b/docs/examples/1-relation/077.md @@ -0,0 +1,38 @@ +#\#077 aabbabbaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{2} a b^{2} a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d a = c a^{2},\; c = b a b,\; d = a b c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + c a^{2} &\rightarrow a d a \\ + a b c^{2} &\rightarrow d \\ + b d &\rightarrow c^{3} \\ + b a c &\rightarrow c a b \\ + c a d &\rightarrow a d^{2} \\ + a b a d^{2} a &\rightarrow d a^{2} \\ + b a^{2} d^{2} &\rightarrow c a b a d \\ + b a^{2} d a &\rightarrow c a b a^{2} \\ + a b a d^{3} &\rightarrow d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/078.md b/docs/examples/1-relation/078.md new file mode 100644 index 0000000..7a24a07 --- /dev/null +++ b/docs/examples/1-relation/078.md @@ -0,0 +1,34 @@ +#\#078 baaaabbbba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{4} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c a,\; c = a^{2},\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{4} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a^{3} &\rightarrow d a^{4} d a^{2} d a^{2} d a \\ + d a^{4} d a^{2} d a^{2} d a^{2} d a &\rightarrow a^{3} \\ + d a^{4} d a^{2} d a^{2} d a^{5} &\rightarrow a^{6} d a^{2} d a^{2} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/080.md b/docs/examples/1-relation/080.md new file mode 100644 index 0000000..01f3ca3 --- /dev/null +++ b/docs/examples/1-relation/080.md @@ -0,0 +1,33 @@ +#\#080 aaaababbaa=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{4} b a b^{2} a^{2} = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{4} c b c a = c^{2} a^{2},\; c = b a,\; d = b a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b a b^{2} &\rightarrow d \\ + b a b a^{3} &\rightarrow a^{4} d a^{2} \\ + b a b d &\rightarrow d a b^{2} \\ + d a b a^{3} &\rightarrow a^{4} d a^{3} d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/081.md b/docs/examples/1-relation/081.md new file mode 100644 index 0000000..56f1c8b --- /dev/null +++ b/docs/examples/1-relation/081.md @@ -0,0 +1,64 @@ +#\#081 aabaababba=baabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b a b^{2} a = b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; e a c b c = c a c a,\; c = b a,\; d = a c b^{2},\; e = a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < e < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a^{2} c &\rightarrow e \\ + a c b^{2} &\rightarrow d \\ + b e &\rightarrow c a c \\ + a c b c &\rightarrow d a \\ + c^{2} b^{2} &\rightarrow b d \\ + e b^{2} &\rightarrow a d \\ + c a c a &\rightarrow e d a \\ + a^{2} e d a &\rightarrow e a c a \\ + c a e d a &\rightarrow e d a c a \\ + c^{2} b c &\rightarrow b d a \\ + e b c &\rightarrow a d a \\ + e c b^{2} &\rightarrow a^{2} b d \\ + c a c e &\rightarrow e d e \\ + c a c d &\rightarrow e d^{2} \\ + a c e d^{2} a &\rightarrow d e a \\ + b d e a &\rightarrow c^{2} e d^{2} a \\ + a^{2} b d a &\rightarrow e c b c \\ + a^{2} b d e &\rightarrow e c b c a c \\ + a^{2} e d e &\rightarrow e a c e \\ + a^{2} e d^{2} &\rightarrow e a c d \\ + a c e d^{2} e &\rightarrow d e^{2} \\ + a c e d^{3} &\rightarrow d e d \\ + a d e a &\rightarrow e^{2} d^{2} a \\ + c a e d e &\rightarrow e d a c e \\ + c a e d^{2} &\rightarrow e d a c d \\ + b d e^{2} &\rightarrow c^{2} e d^{2} e \\ + b d e d &\rightarrow c^{2} e d^{3} \\ + c d e a &\rightarrow e d e d^{2} a \\ + a d e^{2} &\rightarrow e^{2} d^{2} e \\ + a d e d &\rightarrow e^{2} d^{3} \\ + a^{2} b d^{2} e a &\rightarrow e c b c^{2} e d^{2} a \\ + a^{2} b d^{2} e^{2} &\rightarrow e c b c^{2} e d^{2} e \\ + a^{2} b d^{2} e d &\rightarrow e c b c^{2} e d^{3} \\ + c d e^{2} &\rightarrow e d e d^{2} e \\ + c d e d &\rightarrow e d e d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/082.md b/docs/examples/1-relation/082.md new file mode 100644 index 0000000..c15006a --- /dev/null +++ b/docs/examples/1-relation/082.md @@ -0,0 +1,38 @@ +#\#082 abbaaabbaa=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{3} b^{2} a^{2} = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c a^{2} b c a = c b c,\; c = b a,\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b^{2} &\rightarrow d \\ + b d a &\rightarrow d a^{2} d a^{2} \\ + a d a^{2} d a^{3} d a^{2} &\rightarrow d^{2} a \\ + b d^{2} &\rightarrow d a^{2} d a d \\ + a d a^{2} d a^{3} d a d &\rightarrow d^{3} \\ + a d a^{2} d a^{2} d^{2} a &\rightarrow d^{2} a d a^{3} d a^{2} \\ + a d a^{2} d a^{3} d^{3} a &\rightarrow d^{3} a^{2} d a^{3} d a^{2} \\ + a d a^{2} d a^{3} d^{4} &\rightarrow d^{3} a^{2} d a^{3} d a d \\ + a d a^{2} d a^{2} d^{3} &\rightarrow d^{2} a d a^{3} d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/083.md b/docs/examples/1-relation/083.md new file mode 100644 index 0000000..ef9e716 --- /dev/null +++ b/docs/examples/1-relation/083.md @@ -0,0 +1,44 @@ +#\#083 baabaabbaa=aabbaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{2} a^{2} = a^{2} b^{2} a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a c d a = d c a,\; c = a b,\; d = a c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a c b a &\rightarrow d \\ + a b &\rightarrow c \\ + b a c d a &\rightarrow d c a \\ + a d c a &\rightarrow c a c d a \\ + a c d c a &\rightarrow d c d a \\ + a c b d &\rightarrow d c b a \\ + a c b c &\rightarrow d b \\ + b a c d^{2} &\rightarrow d c d \\ + b a c d c &\rightarrow d c^{2} \\ + a d c d &\rightarrow c a c d^{2} \\ + a d c^{2} &\rightarrow c a c d c \\ + a c d c d &\rightarrow d c d^{2} \\ + a c d c^{2} &\rightarrow d c d c \\ + b d c d a &\rightarrow d c^{2} a \\ + b d c d^{2} &\rightarrow d c^{2} d \\ + b d c d c &\rightarrow d c^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/086.md b/docs/examples/1-relation/086.md new file mode 100644 index 0000000..48b42d3 --- /dev/null +++ b/docs/examples/1-relation/086.md @@ -0,0 +1,42 @@ +#\#086 aaabbbaaba=baabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b^{3} a^{2} b a = b a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{3} b d c = c a d,\; c = b a,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c a &\rightarrow d \\ + b a &\rightarrow c \\ + a^{3} b d c &\rightarrow c a d \\ + d a^{2} b d c &\rightarrow b c^{2} a d \\ + c a^{2} b d c &\rightarrow d^{2} \\ + b d^{2} &\rightarrow d a b d c \\ + b c^{2} a b c^{2} a d &\rightarrow d a^{2} d a b d c d \\ + c a^{2} d a b d c d &\rightarrow d b c^{2} a d \\ + a^{3} d a b d c d &\rightarrow c a b c^{2} a d \\ + b d b c^{2} a d &\rightarrow d a d a b d c d \\ + b c^{2} a d a^{2} d a b d c d &\rightarrow d a^{2} d a b d c b c^{2} a d \\ + c a^{2} d a b d c b c^{2} a d &\rightarrow d^{2} a^{2} d a b d c d \\ + a^{3} d a b d c b c^{2} a d &\rightarrow c a d a^{2} d a b d c d \\ + b c^{2} a d a^{2} d a b d c b c^{2} a d &\rightarrow d a^{2} d a b d c d a^{2} d a b d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/088.md b/docs/examples/1-relation/088.md new file mode 100644 index 0000000..fcdb037 --- /dev/null +++ b/docs/examples/1-relation/088.md @@ -0,0 +1,36 @@ +#\#088 bbaaabbbba=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{3} b^{4} a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d = a c,\; c = b^{2} a^{2},\; d = b^{4} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{4} a &\rightarrow d \\ + b^{2} a^{2} &\rightarrow c \\ + a c &\rightarrow c a d \\ + b^{2} c &\rightarrow d a \\ + d a^{2} d a d &\rightarrow c^{2} \\ + b^{2} d a^{2} d &\rightarrow d c \\ + b^{2} d c a d a d c &\rightarrow d c a^{2} d a d \\ + d a^{2} d c a d c &\rightarrow c^{2} a^{2} d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/089.md b/docs/examples/1-relation/089.md new file mode 100644 index 0000000..98d10f8 --- /dev/null +++ b/docs/examples/1-relation/089.md @@ -0,0 +1,39 @@ +#\#089 baabbaabba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{2} b^{2} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d^{2} c = a c,\; c = b a,\; d = c a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a b &\rightarrow d \\ + a c &\rightarrow d^{2} c \\ + b d^{2} c &\rightarrow c^{2} \\ + c d^{2} c &\rightarrow d a \\ + a d &\rightarrow d^{3} \\ + b d^{3} &\rightarrow c d \\ + c d^{3} a &\rightarrow d^{5} c \\ + d a^{2} b &\rightarrow c d^{3} \\ + c d^{5} c &\rightarrow d^{5} c^{2} \\ + c d^{6} &\rightarrow d^{5} c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/091.md b/docs/examples/1-relation/091.md new file mode 100644 index 0000000..968ea00 --- /dev/null +++ b/docs/examples/1-relation/091.md @@ -0,0 +1,56 @@ +#\#091 abaaabbaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b^{2} a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a c a^{2} b c^{2} = c^{2} a,\; c = b a,\; d = a b c b,\; e = c a d +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < e < c < d < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a d &\rightarrow e \\ + a b c b &\rightarrow d \\ + c b c b &\rightarrow b d \\ + a b c^{2} &\rightarrow d a \\ + c^{2} a &\rightarrow a e a \\ + c b^{2} d &\rightarrow b d c b \\ + c b c^{2} &\rightarrow b d a \\ + a b^{2} d &\rightarrow d c b \\ + a b c a e a &\rightarrow d a c a \\ + d a^{2} &\rightarrow a c e a \\ + c^{2} d &\rightarrow a e d \\ + a e a d &\rightarrow c e \\ + b c e &\rightarrow c e a d \\ + a b c a e d &\rightarrow d a c d \\ + c b c a e d &\rightarrow b d a c d \\ + c b c a e a &\rightarrow b d a c a \\ + c a^{2} c e a &\rightarrow e a^{2} \\ + a b c a e c e &\rightarrow d a c^{2} e \\ + d a d &\rightarrow a c e d \\ + d a c e &\rightarrow a c e c e \\ + c^{3} e &\rightarrow a e c e \\ + a e a^{2} c e a &\rightarrow c e a^{2} \\ + c b c a e c e &\rightarrow b d a c^{2} e \\ + a e a^{2} c e d &\rightarrow c e a d \\ + a e a^{2} c e c e &\rightarrow c e a c e \\ + c a^{2} c e d &\rightarrow e a d \\ + c a^{2} c e c e &\rightarrow e a c e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/092.md b/docs/examples/1-relation/092.md new file mode 100644 index 0000000..2198591 --- /dev/null +++ b/docs/examples/1-relation/092.md @@ -0,0 +1,56 @@ +#\#092 abbaababba=bababbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b a b^{2} a = b a b a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b^{2} d a = b c b^{2} a^{2},\; c = a b a,\; d = a c b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c b^{2} a^{2} &\rightarrow a b^{2} d a \\ + a b a &\rightarrow c \\ + a c b^{2} &\rightarrow d \\ + b c b^{2} a c &\rightarrow a b^{2} d c \\ + c^{2} b^{2} &\rightarrow a b d \\ + b c b^{2} a d &\rightarrow a b^{2} d^{2} \\ + a b c &\rightarrow c b a \\ + a c b a b^{2} d a &\rightarrow d c b^{2} a^{2} \\ + a c b a b^{2} d^{2} &\rightarrow d c b^{2} a d \\ + c b a b^{2} a d &\rightarrow a^{2} b^{2} d^{2} \\ + c b a b^{2} a c &\rightarrow a^{2} b^{2} d c \\ + c b a b^{2} a^{2} &\rightarrow a^{2} b^{2} d a \\ + c^{2} b a b^{2} d^{2} &\rightarrow a b d c b^{2} a d \\ + c^{2} b a b^{2} d c &\rightarrow a b d c b^{2} a c \\ + c^{2} b a b^{2} d a &\rightarrow a b d c b^{2} a^{2} \\ + a b^{2} d c b^{2} &\rightarrow b c b^{2} d \\ + a b^{2} d c b a b^{2} d a &\rightarrow b c b^{2} d c b^{2} a^{2} \\ + a c b a b^{2} d c &\rightarrow d c b^{2} a c \\ + a b^{2} d c b a b^{2} d b c b^{2} d &\rightarrow b c b^{2} d c b^{2} c b a b^{2} d \\ + a b^{2} d c b a b^{2} d c &\rightarrow b c b^{2} d c b^{2} a c \\ + a b^{2} d c b a b^{2} d^{2} &\rightarrow b c b^{2} d c b^{2} a d \\ + b c b^{2} c b a b^{2} d &\rightarrow a b^{2} d b c b^{2} d \\ + c b a b^{2} c b a b^{2} d &\rightarrow a^{2} b^{2} d b c b^{2} d \\ + a b^{2} c b^{2} d &\rightarrow c b^{2} d c b^{2} \\ + c^{2} b a b^{2} d b c b^{2} d &\rightarrow a b d c b^{2} c b a b^{2} d \\ + a c b a b^{2} d b c b^{2} d &\rightarrow d c b^{2} c b a b^{2} d \\ + b c b^{2} c b^{2} c b^{2} d &\rightarrow a b^{2} d b c b^{2} d c b^{2} \\ + c b a b^{2} c b^{2} c b^{2} d &\rightarrow a^{2} b^{2} d b c b^{2} d c b^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/093.md b/docs/examples/1-relation/093.md new file mode 100644 index 0000000..2fcd6c5 --- /dev/null +++ b/docs/examples/1-relation/093.md @@ -0,0 +1,41 @@ +#\#093 aabaaaabba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{4} b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c a^{3} b c = c^{2} a,\; c = b a,\; d = a^{2} b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a^{2} b^{2} &\rightarrow d \\ + b a b a^{2} &\rightarrow a^{2} b a^{2} d a \\ + a^{2} b a^{2} b a^{2} d a &\rightarrow d a b a^{2} \\ + a^{2} b a^{2} d a b^{2} &\rightarrow b a b d \\ + b a b a d &\rightarrow a^{2} b a^{2} d^{2} \\ + b a b^{2} a b d &\rightarrow a^{2} b a^{2} d a b a^{2} d a b^{2} \\ + b a b a b a b d &\rightarrow a^{2} b a^{2} d b a b d \\ + a^{2} b a^{2} b a^{2} d^{2} &\rightarrow d a b a d \\ + a^{2} b a^{2} b a^{2} d b a b d &\rightarrow d a b a b a b d \\ + a^{2} b a^{2} d a b a^{2} b a^{2} d^{2} &\rightarrow b a b d a b a d \\ + a^{2} b a^{2} d a b a^{2} b a^{2} d a &\rightarrow b a b d a b a^{2} \\ + a^{2} b a^{2} d a b a^{2} b a^{2} d b a b d &\rightarrow b a b d a b a b a b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/094.md b/docs/examples/1-relation/094.md new file mode 100644 index 0000000..f8836b2 --- /dev/null +++ b/docs/examples/1-relation/094.md @@ -0,0 +1,40 @@ +#\#094 baaaabba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{2} a = c,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{3} d a &\rightarrow a^{2} \\ + a b a^{2} &\rightarrow d a^{3} d a \\ + b a^{3} d^{2} &\rightarrow a d \\ + a b a d &\rightarrow d a^{3} d^{2} \\ + d a^{3} d a^{2} d^{2} &\rightarrow a^{2} d \\ + d a^{3} d a^{2} d a &\rightarrow a^{3} \\ + b a^{5} d &\rightarrow a^{4} d a^{2} d^{2} \\ + b a^{6} &\rightarrow a^{4} d a^{2} d a \\ + d a^{3} d a^{4} d &\rightarrow a^{5} d a^{2} d^{2} \\ + d a^{3} d a^{5} &\rightarrow a^{5} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/095.md b/docs/examples/1-relation/095.md new file mode 100644 index 0000000..1ba0b97 --- /dev/null +++ b/docs/examples/1-relation/095.md @@ -0,0 +1,54 @@ +#\#095 babbabbbba=abba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a b^{4} a = a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d b a = a b^{2} a,\; c = b a b,\; d = c b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + a b^{2} a &\rightarrow c d b a \\ + c b^{2} &\rightarrow d \\ + b a c &\rightarrow c a b \\ + b c d b a &\rightarrow c b a \\ + c b c &\rightarrow d a b \\ + a b c &\rightarrow c d c \\ + b c d c &\rightarrow c^{2} \\ + a c b a &\rightarrow c d c d b a \\ + d a b^{3} &\rightarrow c b d \\ + d a b^{2} c &\rightarrow c b d a b \\ + c a b^{3} &\rightarrow b a d \\ + c a b^{2} c &\rightarrow b a d a b \\ + a b d &\rightarrow c d^{2} \\ + d c d^{2} b a &\rightarrow c^{2} b a \\ + b c d^{2} &\rightarrow c d \\ + b c^{3} b a &\rightarrow c^{2} d^{2} b a \\ + a c^{2} &\rightarrow c d c d c \\ + d c d^{2} c &\rightarrow c^{3} \\ + b c^{4} &\rightarrow c^{2} d^{2} c \\ + a c d &\rightarrow c d c d^{2} \\ + d c d^{3} &\rightarrow c^{2} d \\ + d c d c^{2} b a &\rightarrow c^{3} d^{2} b a \\ + d c d c^{3} &\rightarrow c^{3} d^{2} c \\ + b c^{3} d &\rightarrow c^{2} d^{3} \\ + d c d c^{2} d &\rightarrow c^{3} d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/096.md b/docs/examples/1-relation/096.md new file mode 100644 index 0000000..5e21a16 --- /dev/null +++ b/docs/examples/1-relation/096.md @@ -0,0 +1,85 @@ +#\#096 baabaabbaa=abababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{2} a^{2} = a b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b a^{2} = a b a b a b a,\; c = a^{2} b,\; d = b a c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a c^{2} &\rightarrow d \\ + a b a b a b a &\rightarrow b c^{2} b a^{2} \\ + a^{2} b &\rightarrow c \\ + a b^{2} c^{2} b a^{2} &\rightarrow b c^{2} b c a \\ + a b c^{2} b a^{2} &\rightarrow c a b a b a \\ + a b a b a b c &\rightarrow b c^{2} b a c \\ + a b a b a d &\rightarrow b c^{2} b a^{2} c^{2} \\ + a^{2} d &\rightarrow c a c^{2} \\ + c a b a b a b &\rightarrow a b c^{2} b c \\ + a b c^{2} b a c &\rightarrow c a b a b c \\ + a b^{2} c^{2} b c &\rightarrow b c^{2} b c a b \\ + a b^{2} c^{2} b a c &\rightarrow b c^{2} b c^{2} \\ + a b c^{2} b c a &\rightarrow c b c^{2} b a^{2} \\ + a b a b c a b a b a &\rightarrow b c^{2} d b a^{2} \\ + a c a b a b a &\rightarrow c^{3} b a^{2} \\ + a b c^{2} b c b c^{2} b a^{2} &\rightarrow c b c^{2} b c a b a b a \\ + a b^{2} c^{2} b a b c^{2} b c &\rightarrow b c^{2} b c b c^{2} b c \\ + b a c a b c^{2} b c &\rightarrow d a b a b a b \\ + a b c^{2} b c b c^{2} b a c &\rightarrow c b c^{2} b c a b a b c \\ + a b c^{2} b a b c^{2} b c &\rightarrow c b c^{2} b c a b a b \\ + a b c^{2} b c^{2} &\rightarrow c b c^{2} b a c \\ + c a b c a b a b a &\rightarrow a b c^{2} d b a^{2} \\ + a b c^{2} b c b c^{2} b c &\rightarrow c b c^{2} b a b c^{2} b c \\ + c a b a b c^{2} &\rightarrow a b c^{2} d \\ + a b^{2} c^{2} d &\rightarrow b c^{2} b c^{3} \\ + a b a b c b c^{2} b a c &\rightarrow b c^{2} d b c^{2} \\ + a b a b c b c^{2} b a^{2} &\rightarrow b c^{2} d b c a \\ + a b a b c a b a b c &\rightarrow b c^{2} d b a c \\ + a b a b c a b a d &\rightarrow b c^{2} d b a^{2} c^{2} \\ + a c b c^{2} b a c &\rightarrow c^{3} b c^{2} \\ + a c b c^{2} b a^{2} &\rightarrow c^{3} b c a \\ + a c a b a b c &\rightarrow c^{3} b a c \\ + a c a b a d &\rightarrow c^{3} b a^{2} c^{2} \\ + a b c^{2} b c b c^{2} d &\rightarrow c b c^{2} b a b c^{2} d \\ + a b c^{2} b c b c^{2} b a b c^{2} b c &\rightarrow c b c^{2} b c b c^{2} b c a b a b \\ + c a b a b c a b c^{2} b c &\rightarrow a b c^{2} d a b a b a b \\ + a b^{2} c^{2} b a b c^{2} d &\rightarrow b c^{2} b c b c^{2} d \\ + b a c a b c^{2} d &\rightarrow d a b a b c^{2} \\ + a b c^{2} b c b c^{2} b a b c^{2} d &\rightarrow c b c^{2} b c b c^{2} b c a b c^{2} \\ + a b c^{2} b a b c^{2} d &\rightarrow c b c^{2} b c a b c^{2} \\ + c a b c b c^{2} b a b c^{2} b c &\rightarrow a b c^{2} d b c b c^{2} b c \\ + c a b c b c^{2} b a c &\rightarrow a b c^{2} d b c^{2} \\ + c a b c b c^{2} b a^{2} &\rightarrow a b c^{2} d b c a \\ + c a b c a b a b c &\rightarrow a b c^{2} d b a c \\ + c a b c a b a d &\rightarrow a b c^{2} d b a^{2} c^{2} \\ + c a b a b c a b c^{2} d &\rightarrow a b c^{2} d a b a b c^{2} \\ + a b a b c b c^{2} b a b c^{2} b c &\rightarrow b c^{2} d b c b c^{2} b c \\ + a c b c^{2} b a b c^{2} b c &\rightarrow c^{3} b c b c^{2} b c \\ + a b a b c b c^{2} b c &\rightarrow b c^{2} d b c a b \\ + a b a b c b c^{2} b a b c^{2} d &\rightarrow b c^{2} d b c b c^{2} d \\ + a b a b c b c^{2} d &\rightarrow b c^{2} d b c^{3} \\ + a c b c^{2} b c &\rightarrow c^{3} b c a b \\ + a c b c^{2} d &\rightarrow c^{3} b c^{3} \\ + c a b c b c^{2} b a b c^{2} d &\rightarrow a b c^{2} d b c b c^{2} d \\ + a c b c^{2} b a b c^{2} d &\rightarrow c^{3} b c b c^{2} d \\ + c a b c b c^{2} b c &\rightarrow a b c^{2} d b c a b \\ + c a b c b c^{2} d &\rightarrow a b c^{2} d b c^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/097.md b/docs/examples/1-relation/097.md new file mode 100644 index 0000000..d84a91c --- /dev/null +++ b/docs/examples/1-relation/097.md @@ -0,0 +1,36 @@ +#\#097 baababbbaa=aaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{3} a^{2} = a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c d = a^{2} c,\; c = b a,\; d = b^{2} c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} c a &\rightarrow d \\ + b a &\rightarrow c \\ + c a c &\rightarrow d c d^{2} \\ + b^{2} c d c d^{3} &\rightarrow d a c \\ + a^{2} c &\rightarrow d c d^{3} \\ + b d c d &\rightarrow d c \\ + a^{2} d c d^{2} &\rightarrow d c d^{3} a c \\ + c a d c d^{2} &\rightarrow d c d^{2} a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/098.md b/docs/examples/1-relation/098.md new file mode 100644 index 0000000..1c4bd72 --- /dev/null +++ b/docs/examples/1-relation/098.md @@ -0,0 +1,40 @@ +#\#098 baaaabba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{2} a = c a,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{3} d a &\rightarrow a^{3} \\ + a b a^{3} &\rightarrow d a^{3} d a \\ + b a^{3} d^{2} &\rightarrow a^{2} d \\ + a b a^{2} d &\rightarrow d a^{3} d^{2} \\ + d a^{3} d a d^{2} &\rightarrow a^{3} d \\ + d a^{3} d a d a &\rightarrow a^{4} \\ + b a^{6} d &\rightarrow a^{5} d a d^{2} \\ + b a^{7} &\rightarrow a^{5} d a d a \\ + d a^{3} d a^{4} d &\rightarrow a^{6} d a d^{2} \\ + d a^{3} d a^{5} &\rightarrow a^{6} d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/100.md b/docs/examples/1-relation/100.md new file mode 100644 index 0000000..8081bf9 --- /dev/null +++ b/docs/examples/1-relation/100.md @@ -0,0 +1,63 @@ +#\#100 aaabababaa=baaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a b a b a^{2} = b a^{4} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c d = b a^{3} c,\; c = a b a,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a^{3} c &\rightarrow a^{2} c d \\ + b c a &\rightarrow d \\ + a b a &\rightarrow c \\ + a^{3} c d &\rightarrow c a^{2} c \\ + c b a &\rightarrow a b c \\ + d b a &\rightarrow b c^{2} \\ + b a^{4} b c &\rightarrow a^{2} c b c^{2} \\ + a^{2} c d^{2} &\rightarrow d a c \\ + b c^{2} a^{2} c &\rightarrow d a^{2} c d \\ + a d a c d &\rightarrow c d a c \\ + c a^{2} c d &\rightarrow a d a c \\ + a^{3} c b c^{2} &\rightarrow c a^{3} b c \\ + b a^{4} d &\rightarrow a^{2} c b c^{2} a \\ + b a d a c &\rightarrow d a c d \\ + a^{2} c d b c^{2} &\rightarrow d a^{2} b c \\ + b c^{2} a^{3} b c &\rightarrow d a^{2} c b c^{2} \\ + b c^{2} a c d^{2} &\rightarrow d b d a c \\ + b c^{2} d a c &\rightarrow d^{2} a c d \\ + b c d a c &\rightarrow d a c d^{2} \\ + c a^{2} c b c^{2} &\rightarrow a d a^{2} b c \\ + a d a c b c^{2} &\rightarrow c d a^{2} b c \\ + c b d a c &\rightarrow a d c d^{2} \\ + a b d a c &\rightarrow c a c d^{2} \\ + b a d a^{2} b c &\rightarrow d a c b c^{2} \\ + b c^{2} a^{3} d &\rightarrow d a^{2} c b c^{2} a \\ + b c^{2} a c d b c^{2} &\rightarrow d b d a^{2} b c \\ + b c^{2} d a^{2} b c &\rightarrow d^{2} a c b c^{2} \\ + b c d a^{2} b c &\rightarrow d a c d b c^{2} \\ + c b d a^{2} b c &\rightarrow a d c d b c^{2} \\ + a b d a^{2} b c &\rightarrow c a c d b c^{2} \\ + b a d a^{2} d &\rightarrow d a c b c^{2} a \\ + b c^{2} d a^{2} d &\rightarrow d^{2} a c b c^{2} a \\ + b c d a^{2} d &\rightarrow d a c d b c^{2} a \\ + a b d a^{2} d &\rightarrow c a c d b c^{2} a \\ + c b d a^{2} d &\rightarrow a d c d b c^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/101.md b/docs/examples/1-relation/101.md new file mode 100644 index 0000000..323cb59 --- /dev/null +++ b/docs/examples/1-relation/101.md @@ -0,0 +1,42 @@ +#\#101 baaaaabbba=aaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{3} a = a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c^{2},\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{4} d a &\rightarrow a^{4} \\ + a b^{3} &\rightarrow d \\ + a b^{2} a^{4} &\rightarrow d a^{4} d a \\ + b a^{4} d^{2} &\rightarrow a^{3} d \\ + a b^{2} a^{3} d &\rightarrow d a^{4} d^{2} \\ + a b a^{3} d &\rightarrow d a^{4} d a d^{2} \\ + a b a^{4} &\rightarrow d a^{4} d a d a \\ + d a^{4} d a d a d^{2} &\rightarrow a^{4} d \\ + d a^{4} d a d a d a &\rightarrow a^{5} \\ + b a^{8} d &\rightarrow a^{7} d a d a d^{2} \\ + b a^{9} &\rightarrow a^{7} d a d a d a \\ + d a^{4} d a d a^{5} d &\rightarrow a^{8} d a d a d^{2} \\ + d a^{4} d a d a^{6} &\rightarrow a^{8} d a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/102.md b/docs/examples/1-relation/102.md new file mode 100644 index 0000000..a7e4540 --- /dev/null +++ b/docs/examples/1-relation/102.md @@ -0,0 +1,37 @@ +#\#102 baaabbaba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a b a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a d a = c,\; c = a^{2},\; d = b^{2} a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{2} a b &\rightarrow d \\ + b^{2} a d &\rightarrow d b a b \\ + b a^{2} &\rightarrow d a^{3} d a d a \\ + b d a^{3} d a d a^{2} &\rightarrow d a^{3} d a \\ + d a^{3} d a d a^{2} d a &\rightarrow a^{2} \\ + b d a^{3} d a^{3} &\rightarrow d a^{3} d a^{2} d a d a^{2} d a \\ + d a^{3} d a d a^{4} &\rightarrow a^{4} d a d a^{2} d a \\ + b d a^{5} &\rightarrow d a^{3} d a^{2} d a d a^{2} d a d a d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/103.md b/docs/examples/1-relation/103.md new file mode 100644 index 0000000..e11e9ed --- /dev/null +++ b/docs/examples/1-relation/103.md @@ -0,0 +1,36 @@ +#\#103 abbabbaa=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a^{2} = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a c a^{2} = b a c a,\; c = b^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b^{2} \\ + a b^{2} &\rightarrow d \\ + b d a &\rightarrow d^{2} a^{2} \\ + b d^{2} &\rightarrow d^{2} a d \\ + a d^{2} a d a^{2} &\rightarrow d^{2} a \\ + a d^{2} a d a d &\rightarrow d^{3} \\ + a d^{2} a d^{3} a &\rightarrow d^{4} a d a^{2} \\ + a d^{2} a d^{4} &\rightarrow d^{4} a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/104.md b/docs/examples/1-relation/104.md new file mode 100644 index 0000000..839346d --- /dev/null +++ b/docs/examples/1-relation/104.md @@ -0,0 +1,99 @@ +#\#104 babaaabbba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{3} b^{3} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a b d a = a b a,\; c = a^{2} b,\; d = a c b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a^{2} b &\rightarrow c \\ + a c b^{2} &\rightarrow d \\ + b a b d a &\rightarrow a b a \\ + a c a &\rightarrow c a b d a \\ + a c b a b a &\rightarrow d a b d a \\ + b a b d c &\rightarrow a b c \\ + b a b d^{2} &\rightarrow a b d \\ + a c^{2} &\rightarrow c a b d c \\ + a c d &\rightarrow c a b d^{2} \\ + d a b d a b d a &\rightarrow a c b c a \\ + d a b d a b d c &\rightarrow a c b c^{2} \\ + d a b d a b d^{2} &\rightarrow a c b c d \\ + a c b a b c &\rightarrow d a b d c \\ + d a b d^{2} a b d a &\rightarrow c a b d c a \\ + a c b a b d &\rightarrow d a b d^{2} \\ + c a b d a b a &\rightarrow d a b d^{2} a \\ + c a b d a b c &\rightarrow d a b d^{2} c \\ + c a b d a b d &\rightarrow d a b d^{3} \\ + d a b d a c b c a &\rightarrow c a b d c a b d a \\ + d a b d a c b c^{2} &\rightarrow c a b d c a b d c \\ + d a b d a c b c d &\rightarrow c a b d c a b d^{2} \\ + b a b a c b c a &\rightarrow c a b d a \\ + b a b a c b c^{2} &\rightarrow c a b d c \\ + b a b a c b c d &\rightarrow c a b d^{2} \\ + b a b c a b d c a &\rightarrow c d a b d a \\ + d a b d^{2} a b d c &\rightarrow c a b d c^{2} \\ + d a b d^{2} a b d^{2} &\rightarrow c a b d c d \\ + d a b a c b c a &\rightarrow a c b c a b d a \\ + d a b a c b c^{2} &\rightarrow a c b c a b d c \\ + d a b a c b c d &\rightarrow a c b c a b d^{2} \\ + d a b c a b d c a &\rightarrow a c b c d a b d a \\ + c a b a c b c a &\rightarrow d a b d^{3} a b d a \\ + c a b a c b c^{2} &\rightarrow d a b d^{3} a b d c \\ + c a b a c b c d &\rightarrow d a b d^{3} a b d^{2} \\ + c a b c a b d c a &\rightarrow d a b d^{4} a b d a \\ + a d a b d^{2} a &\rightarrow d a b d^{3} a b a \\ + a d a b d^{2} c &\rightarrow d a b d^{3} a b c \\ + a d a b d^{3} &\rightarrow d a b d^{3} a b d \\ + d a b d a c b d a b d^{2} a &\rightarrow c a b d^{2} a b d^{3} a b a \\ + d a b d a c b d a b d^{2} c &\rightarrow c a b d^{2} a b d^{3} a b c \\ + d a b d a c b d a b d^{3} &\rightarrow c a b d^{2} a b d^{3} a b d \\ + b a b a c b d a b d^{2} a &\rightarrow d a b d^{3} a b a \\ + b a b a c b d a b d^{2} c &\rightarrow d a b d^{3} a b c \\ + b a b a c b d a b d^{3} &\rightarrow d a b d^{3} a b d \\ + b a b c a b d^{2} a b d^{2} c &\rightarrow c a c b c a b c \\ + b a b c a b d^{2} a b d^{3} &\rightarrow c a c b c a b d \\ + d a b d c a b d c^{2} &\rightarrow c a b d c d a b d c \\ + d a b d c a b d c d &\rightarrow c a b d c d a b d^{2} \\ + d a b a c b d a b d^{2} a &\rightarrow a c b d a b d^{3} a b a \\ + d a b a c b d a b d^{2} c &\rightarrow a c b d a b d^{3} a b c \\ + d a b a c b d a b d^{3} &\rightarrow a c b d a b d^{3} a b d \\ + d a b c a b d^{2} a b d^{2} a &\rightarrow a c b c a c b c a b a \\ + d a b c a b d^{2} a b d^{2} c &\rightarrow a c b c a c b c a b c \\ + d a b c a b d^{2} a b d^{3} &\rightarrow a c b c a c b c a b d \\ + b a b c a b d c^{2} &\rightarrow c d a b d c \\ + b a b c a b d c d &\rightarrow c d a b d^{2} \\ + c a b c a b d c^{2} &\rightarrow d a b d^{4} a b d c \\ + c a b c a b d c d &\rightarrow d a b d^{4} a b d^{2} \\ + d a b c a b d c^{2} &\rightarrow a c b c d a b d c \\ + d a b c a b d c d &\rightarrow a c b c d a b d^{2} \\ + d a b d c a b d c a &\rightarrow c a b d c d a b d a \\ + b a b c a b d^{2} a b d^{2} a &\rightarrow c a c b c a b a \\ + c a b a c b d a b d^{2} a &\rightarrow d a b d^{2} a c b c a b a \\ + c a b a c b d a b d^{2} c &\rightarrow d a b d^{2} a c b c a b c \\ + c a b a c b d a b d^{3} &\rightarrow d a b d^{2} a c b c a b d \\ + c a b c a b d^{2} a b d^{2} a &\rightarrow d a b d^{3} a c b c a b a \\ + c a b c a b d^{2} a b d^{2} c &\rightarrow d a b d^{3} a c b c a b c \\ + c a b c a b d^{2} a b d^{3} &\rightarrow d a b d^{3} a c b c a b d \\ + d a b d c a b d^{2} a b d^{2} c &\rightarrow c a b d c a c b c a b c \\ + d a b d c a b d^{2} a b d^{3} &\rightarrow c a b d c a c b c a b d \\ + d a b d c a b d^{2} a b d^{2} a &\rightarrow c a b d c a c b c a b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/107.md b/docs/examples/1-relation/107.md new file mode 100644 index 0000000..94db900 --- /dev/null +++ b/docs/examples/1-relation/107.md @@ -0,0 +1,34 @@ +#\#107 baaaabbbba=aaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{4} a = a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c^{2},\; c = a^{2},\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{4} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a^{4} &\rightarrow d a^{4} d a d a d a \\ + d a^{4} d a d a d a d a &\rightarrow a^{4} \\ + d a^{4} d a d a d a^{5} &\rightarrow a^{7} d a d a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/108.md b/docs/examples/1-relation/108.md new file mode 100644 index 0000000..2fe055f --- /dev/null +++ b/docs/examples/1-relation/108.md @@ -0,0 +1,39 @@ +#\#108 baabbabba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a b^{2} a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a b c b c = a,\; c = b a,\; d = a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + b d b c^{2} &\rightarrow d b c b c \\ + a &\rightarrow b d b c b c \\ + c b d b c b c &\rightarrow b d \\ + b^{2} d b c b c &\rightarrow c \\ + b d b c b^{2} d &\rightarrow d \\ + b d b^{2} d &\rightarrow d b c^{2} \\ + b d b c b d &\rightarrow d b c b^{2} d \\ + b d c &\rightarrow d b c^{2} b c b c \\ + c d &\rightarrow d b c^{2} b c^{2} \\ + b d^{2} &\rightarrow d b c^{2} b c b d \\ + b d b d &\rightarrow d b c^{2} b c b^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/110.md b/docs/examples/1-relation/110.md new file mode 100644 index 0000000..4575d92 --- /dev/null +++ b/docs/examples/1-relation/110.md @@ -0,0 +1,38 @@ +#\#110 abbaababa=baabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b a b a = b a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c a c^{2} = c d c a,\; c = b a,\; d = a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow d \\ + b d &\rightarrow c b \\ + a c &\rightarrow d a \\ + c d c a &\rightarrow d c d^{2} a \\ + c d c d &\rightarrow d c d^{3} \\ + a d c d^{2} a &\rightarrow d a d c a \\ + c d^{2} c d^{3} &\rightarrow d c d^{3} c d \\ + c d^{2} c d^{2} a &\rightarrow d c d^{3} c a \\ + a d c d^{3} &\rightarrow d a d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/111.md b/docs/examples/1-relation/111.md new file mode 100644 index 0000000..3f28f56 --- /dev/null +++ b/docs/examples/1-relation/111.md @@ -0,0 +1,51 @@ +#\#111 baabbabaa=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a b a^{2} = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c d a = c a,\; c = a^{2} b^{2},\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} b^{2} &\rightarrow c \\ + a b a &\rightarrow d \\ + b c d a &\rightarrow c a \\ + a^{2} b c a &\rightarrow c^{2} d a \\ + d a b^{2} &\rightarrow a b c \\ + d b a &\rightarrow a b d \\ + b c d c &\rightarrow c^{2} \\ + b c d^{2} &\rightarrow c d \\ + a^{2} b c d &\rightarrow c^{2} d^{2} \\ + a^{2} b c^{2} &\rightarrow c^{2} d c \\ + d a b c a &\rightarrow a b c^{2} d a \\ + d a b c d &\rightarrow a b c^{2} d^{2} \\ + d a b c^{2} &\rightarrow a b c^{2} d c \\ + b c a b c &\rightarrow c a b^{2} \\ + a^{2} c a &\rightarrow c^{2} d^{2} a \\ + a^{2} c d &\rightarrow c^{2} d^{3} \\ + a^{2} c^{2} &\rightarrow c^{2} d^{2} c \\ + d a c a &\rightarrow a b c^{2} d^{2} a \\ + d a c d &\rightarrow a b c^{2} d^{3} \\ + d a c^{2} &\rightarrow a b c^{2} d^{2} c \\ + b c a c a &\rightarrow c a b^{2} d a \\ + b c a c d &\rightarrow c a b^{2} d^{2} \\ + b c a c^{2} &\rightarrow c a b^{2} d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/112.md b/docs/examples/1-relation/112.md new file mode 100644 index 0000000..e3e9890 --- /dev/null +++ b/docs/examples/1-relation/112.md @@ -0,0 +1,45 @@ +#\#112 baababba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{2} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c b c = a c,\; c = b a,\; d = a c a c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b d b c^{2} b d &\rightarrow d b^{2} d \\ + b d b c b d &\rightarrow d b^{2} d b c \\ + a b d &\rightarrow b d b c b^{2} d \\ + c b d b c &\rightarrow b d \\ + b d b c^{3} &\rightarrow d b c \\ + a c &\rightarrow b d b c b c \\ + b d b c b^{2} d b c &\rightarrow d \\ + a d &\rightarrow d b^{2} d b c \\ + b d b^{2} d &\rightarrow d c^{2} \\ + b d c^{2} &\rightarrow d c^{2} b c b c \\ + b d b d &\rightarrow d c^{2} b c b^{2} d b c \\ + c d &\rightarrow d c^{2} b c \\ + b^{2} d b c b c &\rightarrow c^{2} \\ + b^{2} d b c b^{2} d &\rightarrow c b d \\ + b d c b d &\rightarrow d c^{2} b c b^{2} d \\ + b d^{2} &\rightarrow d c^{2} b c^{3} b^{2} d b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/113.md b/docs/examples/1-relation/113.md new file mode 100644 index 0000000..9aac030 --- /dev/null +++ b/docs/examples/1-relation/113.md @@ -0,0 +1,35 @@ +#\#113 abbababbaa=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a b^{2} a^{2} = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c^{2} b c a = c b c,\; c = b a,\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b^{2} a &\rightarrow d \\ + a d b a d a &\rightarrow b a d \\ + b d^{2} &\rightarrow d^{2} b a d a \\ + b a d^{2} b a d a &\rightarrow a d b a d b a d \\ + b a d b a d b a d &\rightarrow d^{3} b a d a \\ + b a d^{3} &\rightarrow d^{3} b a d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/114.md b/docs/examples/1-relation/114.md new file mode 100644 index 0000000..9ec2e14 --- /dev/null +++ b/docs/examples/1-relation/114.md @@ -0,0 +1,35 @@ +#\#114 aababbabaa=babaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b^{2} a b a^{2} = b a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c b c^{2} a = c^{2} a c,\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + a d b^{2} d a &\rightarrow b d^{2} \\ + a b^{2} d^{2} &\rightarrow d^{2} b^{2} d a \\ + a b d &\rightarrow d b a \\ + a d b^{2} d b d^{2} &\rightarrow b d^{3} b^{2} d a \\ + a d b^{2} d^{2} &\rightarrow b d^{2} b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/115.md b/docs/examples/1-relation/115.md new file mode 100644 index 0000000..7ccf4d7 --- /dev/null +++ b/docs/examples/1-relation/115.md @@ -0,0 +1,56 @@ +#\#115 bbaabbbbaa=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{2} b^{4} a^{2} = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d a = a^{2} c a,\; c = b^{2},\; d = a c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a^{2} c a &\rightarrow c a d a \\ + a c^{2} a &\rightarrow d \\ + b c &\rightarrow c b \\ + c a d c a d a &\rightarrow a d^{2} a \\ + a c^{2} d &\rightarrow d c^{2} a \\ + a c^{3} a d a &\rightarrow d a c a \\ + a^{2} c d &\rightarrow c a d^{2} \\ + a^{3} d^{2} a &\rightarrow c a d a d c a d a \\ + c a d c a d^{2} &\rightarrow a d^{3} \\ + a c a d^{2} a &\rightarrow d^{2} c a d a \\ + a c^{3} a d^{2} &\rightarrow d a c d \\ + c b a d c a d a &\rightarrow b a d^{2} a \\ + c a d a d^{2} a &\rightarrow a d^{2} c a d a \\ + c a d a d^{3} &\rightarrow a d^{2} c a d^{2} \\ + c b a d c a d^{2} &\rightarrow b a d^{3} \\ + a^{3} d^{3} &\rightarrow c a d a d c a d^{2} \\ + a^{3} d^{2} c a d a &\rightarrow c a d a d a d^{2} a \\ + a^{3} d^{2} c a d^{2} &\rightarrow c a d a d a d^{3} \\ + c a d a d^{2} c a d a &\rightarrow a d^{2} a d^{2} a \\ + c a d a d^{2} c a d^{2} &\rightarrow a d^{2} a d^{3} \\ + a c a d^{3} &\rightarrow d^{2} c a d^{2} \\ + a c a d^{2} c a d a &\rightarrow d^{2} a d^{2} a \\ + a c a d^{2} c a d^{2} &\rightarrow d^{2} a d^{3} \\ + c b a d a d^{2} a &\rightarrow b a d^{2} c a d a \\ + c b a d a d^{3} &\rightarrow b a d^{2} c a d^{2} \\ + c b a d a d^{2} c a d a &\rightarrow b a d^{2} a d^{2} a \\ + c b a d a d^{2} c a d^{2} &\rightarrow b a d^{2} a d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/116.md b/docs/examples/1-relation/116.md new file mode 100644 index 0000000..b88b94e --- /dev/null +++ b/docs/examples/1-relation/116.md @@ -0,0 +1,36 @@ +#\#116 abababaaba=bababaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b a^{2} b a = b a b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c^{3} a c = c^{3} a,\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + b d b a^{2} &\rightarrow d b d^{2} \\ + b d b a d &\rightarrow d b d^{2} b a \\ + a b d &\rightarrow d b a \\ + a d b d^{2} &\rightarrow d b d a \\ + b d b d b d a &\rightarrow d b d^{3} b d^{2} b a \\ + b d b d b d^{2} &\rightarrow d b d^{3} b d^{2} b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/117.md b/docs/examples/1-relation/117.md new file mode 100644 index 0000000..6a2e4b8 --- /dev/null +++ b/docs/examples/1-relation/117.md @@ -0,0 +1,63 @@ +#\#117 baaaaabbba=abaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{3} a = a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d a = a b a^{3},\; c = a b^{3},\; d = a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} c &\rightarrow d \\ + a b a^{3} &\rightarrow b a^{2} d a \\ + a b^{3} &\rightarrow c \\ + b a^{2} d c &\rightarrow a b d \\ + a b a^{2} d &\rightarrow b a^{2} d^{2} \\ + b a^{2} d a c &\rightarrow a b a d \\ + b a^{2} d b a^{2} d a &\rightarrow a b^{2} a^{2} d^{3} a \\ + a^{2} b a d &\rightarrow b a^{2} d^{2} a c \\ + a^{2} b^{2} a^{2} d^{3} a &\rightarrow b a^{2} d^{2} b a^{2} d a \\ + b a^{2} d b a^{2} d^{2} &\rightarrow a b^{2} a^{2} d^{4} \\ + a b^{2} a b a d &\rightarrow c a^{2} d a c \\ + a b^{2} a b^{2} a^{2} d^{3} a &\rightarrow c a^{2} d b a^{2} d a \\ + c a^{2} d c &\rightarrow a b^{2} a b d \\ + b a^{2} d a b a d &\rightarrow a b^{2} a^{2} d^{3} a c \\ + b a^{2} d^{2} c &\rightarrow a^{2} b d \\ + a b^{2} a^{2} d^{3} c &\rightarrow b a^{2} d a b d \\ + b a^{2} d a b^{2} a^{2} d^{3} a &\rightarrow a b^{2} a^{2} d^{3} b a^{2} d a \\ + b a^{2} d a b^{2} a^{2} d^{4} &\rightarrow a b^{2} a^{2} d^{3} b a^{2} d^{2} \\ + a^{3} b d &\rightarrow b a^{2} d^{3} c \\ + a^{3} b^{2} a b d &\rightarrow d a^{2} d c \\ + a^{2} b d a^{2} d c &\rightarrow b a^{2} d^{2} a b^{2} a b d \\ + a^{2} b^{2} a^{2} d^{4} &\rightarrow b a^{2} d^{2} b a^{2} d^{2} \\ + a^{2} b^{2} a^{2} d^{3} b a^{2} d a &\rightarrow b a^{2} d^{2} a b^{2} a^{2} d^{3} a \\ + a^{2} b^{2} a^{2} d^{3} b a^{2} d^{2} &\rightarrow b a^{2} d^{2} a b^{2} a^{2} d^{4} \\ + b a^{2} d a b^{2} a^{2} d^{3} b a^{2} d a &\rightarrow a b^{2} a^{2} d^{3} a b^{2} a^{2} d^{3} a \\ + b a^{2} d a b^{2} a^{2} d^{3} b a^{2} d^{2} &\rightarrow a b^{2} a^{2} d^{3} a b^{2} a^{2} d^{4} \\ + a b^{2} a^{2} b d &\rightarrow c a^{2} d^{2} c \\ + a b^{2} a b d a^{2} d c &\rightarrow c a^{2} d a b^{2} a b d \\ + a b^{2} a b^{2} a^{2} d^{4} &\rightarrow c a^{2} d b a^{2} d^{2} \\ + a b^{2} a b^{2} a^{2} d^{3} b a^{2} d a &\rightarrow c a^{2} d a b^{2} a^{2} d^{3} a \\ + a b^{2} a b^{2} a^{2} d^{3} b a^{2} d^{2} &\rightarrow c a^{2} d a b^{2} a^{2} d^{4} \\ + b a^{2} d a^{2} b d &\rightarrow a b^{2} a^{2} d^{4} c \\ + b a^{2} d a^{2} b^{2} a b d &\rightarrow a b a d a^{2} d c \\ + b a^{2} d a b d a^{2} d c &\rightarrow a b^{2} a^{2} d^{3} a b^{2} a b d \\ + b a^{2} d a b^{2} a b d &\rightarrow a b d a^{2} d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/118.md b/docs/examples/1-relation/118.md new file mode 100644 index 0000000..a2922f9 --- /dev/null +++ b/docs/examples/1-relation/118.md @@ -0,0 +1,52 @@ +#\#118 ababaaaaba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{4} b a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d c = c^{2},\; c = b a,\; d = c a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a^{3} &\rightarrow d \\ + a c d c &\rightarrow c^{2} \\ + a c d^{2} &\rightarrow c d \\ + c a^{2} c^{2} &\rightarrow d c d c \\ + c a^{2} c d &\rightarrow d c d^{2} \\ + c a c^{2} &\rightarrow d c d^{2} c \\ + c a c d &\rightarrow d c d^{3} \\ + c^{3} &\rightarrow d c d^{3} c \\ + c^{2} d &\rightarrow d c d^{4} \\ + b c d &\rightarrow d c d^{5} \\ + b c^{2} &\rightarrow d c d^{4} c \\ + b d c d^{3} c &\rightarrow d c d^{4} c^{2} \\ + b d c d^{4} &\rightarrow d c d^{4} c d \\ + c d c d^{3} c &\rightarrow d c d^{3} c^{2} \\ + c d c d^{4} &\rightarrow d c d^{3} c d \\ + c a d c d^{3} c &\rightarrow d c d^{2} c^{2} \\ + c a d c d^{4} &\rightarrow d c d^{2} c d \\ + c a^{2} d c d^{3} c &\rightarrow d c d c^{2} \\ + c a^{2} d c d^{4} &\rightarrow d c d c d \\ + a d c d^{3} c^{2} &\rightarrow d c d^{6} c \\ + a d c d^{3} c d &\rightarrow d c d^{7} \\ + a d c d^{4} c d^{3} c &\rightarrow d c d^{6} c^{2} \\ + a d c d^{4} c d^{4} &\rightarrow d c d^{6} c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/119.md b/docs/examples/1-relation/119.md new file mode 100644 index 0000000..d72ad47 --- /dev/null +++ b/docs/examples/1-relation/119.md @@ -0,0 +1,39 @@ +#\#119 bbaabbaa=abba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{2} b^{2} a^{2} = a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d a = d,\; c = b^{2},\; d = a c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a c a &\rightarrow d \\ + b c &\rightarrow c b \\ + a d &\rightarrow d^{2} a \\ + a c d &\rightarrow d c a \\ + c d^{4} a &\rightarrow d c a \\ + c d^{2} a^{2} &\rightarrow d \\ + c d^{5} &\rightarrow d c d \\ + c b d^{2} a^{2} &\rightarrow b d \\ + c b d^{4} a &\rightarrow b d c a \\ + c b d^{5} &\rightarrow b d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/121.md b/docs/examples/1-relation/121.md new file mode 100644 index 0000000..ad48498 --- /dev/null +++ b/docs/examples/1-relation/121.md @@ -0,0 +1,40 @@ +#\#121 baaaaaabba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{6} b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{3} b^{2} a = c,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{5} d a &\rightarrow a^{2} \\ + a b a^{2} &\rightarrow d a^{5} d a \\ + b a^{5} d^{2} &\rightarrow a d \\ + a b a d &\rightarrow d a^{5} d^{2} \\ + d a^{5} d a^{4} d^{2} &\rightarrow a^{2} d \\ + d a^{5} d a^{4} d a &\rightarrow a^{3} \\ + b a^{7} d &\rightarrow a^{6} d a^{4} d^{2} \\ + b a^{8} &\rightarrow a^{6} d a^{4} d a \\ + d a^{5} d a^{6} d &\rightarrow a^{7} d a^{4} d^{2} \\ + d a^{5} d a^{7} &\rightarrow a^{7} d a^{4} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/122.md b/docs/examples/1-relation/122.md new file mode 100644 index 0000000..ab3cde9 --- /dev/null +++ b/docs/examples/1-relation/122.md @@ -0,0 +1,100 @@ +#\#122 babaababaa=abaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a b a^{2} = a b a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c a = a b a^{2} b^{2} a,\; c = b a b a,\; d = a c a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a c a b &\rightarrow d \\ + a b a^{2} b^{2} a &\rightarrow c a c a \\ + b a b a &\rightarrow c \\ + a c a c &\rightarrow d a b a \\ + a c^{2} a c a &\rightarrow d a^{2} b^{2} a \\ + a b a^{2} b c &\rightarrow c d a \\ + b a c &\rightarrow c b a \\ + b a b d &\rightarrow c^{2} a b \\ + b c a c a &\rightarrow c a b^{2} a \\ + a b a d b a &\rightarrow c d a^{2} c a \\ + d a^{2} b c &\rightarrow a c^{2} d a \\ + a c^{2} a c d &\rightarrow d a^{2} b^{2} d \\ + d a^{2} b^{2} a b &\rightarrow a c^{2} d \\ + d a b a^{2} c &\rightarrow a c d a b a \\ + d a b a^{2} b &\rightarrow a c d \\ + c a b c &\rightarrow b c d a \\ + c a b^{2} a b &\rightarrow b c d \\ + c b a^{2} c &\rightarrow b d a b a \\ + c b a^{2} b &\rightarrow b d \\ + b d a^{2} b^{2} a &\rightarrow c b d a b a^{2} \\ + b c a c d &\rightarrow c a b^{2} d \\ + a c d a^{2} b^{2} a &\rightarrow d a b d a b a^{2} \\ + a b a^{2} b^{2} d &\rightarrow c a c d \\ + a c d b a &\rightarrow d c a c a \\ + d a b a d a^{2} b^{2} a &\rightarrow a c d a b d a b a^{2} \\ + d a b a d a^{2} b^{2} d &\rightarrow a c d a b d a b a d \\ + a c d c &\rightarrow d c d a \\ + a c^{2} d a^{2} c a &\rightarrow d a d b a \\ + a c^{2} a c^{2} d &\rightarrow d a^{2} b^{2} c d \\ + a b a^{2} b^{2} c d &\rightarrow c a c^{2} d \\ + a b c d &\rightarrow d b a b \\ + d a^{2} b^{2} d b a &\rightarrow a c^{2} d c a c a \\ + a c d a^{2} b^{2} d &\rightarrow d a b d a b a d \\ + d a c b a d a^{2} c a &\rightarrow a c d a d b a \\ + c a b^{2} d b a &\rightarrow b c d c a c a \\ + a b a d b d &\rightarrow c d a^{2} c d \\ + a b a d c &\rightarrow c d a d a \\ + b d a^{2} b^{2} d &\rightarrow c b d a b a d \\ + c b a d b a &\rightarrow b d c a c a \\ + c b a d c &\rightarrow b d c d a \\ + c b a d a^{2} b^{2} a &\rightarrow b d a b d a b a^{2} \\ + c b a d a^{2} b^{2} d &\rightarrow b d a b d a b a d \\ + b c d a^{2} c a &\rightarrow c d b a \\ + b c a c^{2} d &\rightarrow c a b^{2} c d \\ + a c d b d &\rightarrow d c a c d \\ + a c^{2} d a^{2} c d &\rightarrow d a d b d \\ + d a^{2} b^{2} d b d &\rightarrow a c^{2} d c a c d \\ + c a b^{2} d b d &\rightarrow b c d c a c d \\ + c b a d b d &\rightarrow b d c a c d \\ + c^{2} b a d a^{2} c a &\rightarrow b d a d b a \\ + b c d a^{2} c d &\rightarrow c d b d \\ + d a b a d a^{2} b^{2} c d &\rightarrow a c d a b d a c b a d \\ + a c d b c d &\rightarrow d c a c^{2} d \\ + a c^{2} d a d &\rightarrow d a d b a b \\ + d a^{2} b^{2} d c &\rightarrow a c^{2} d c d a \\ + a c d a^{2} b^{2} c d &\rightarrow d a b d a c b a d \\ + c a b^{2} d c &\rightarrow b c d c d a \\ + b d a^{2} b^{2} c d &\rightarrow c b d a c b a d \\ + c b a d a^{2} b^{2} c d &\rightarrow b d a b d a c b a d \\ + b c d a d &\rightarrow c d b a b \\ + d a c b a d a^{2} c d &\rightarrow a c d a d b d \\ + d a c b a d a d &\rightarrow a c d a d b a b \\ + c^{2} b a d a^{2} c d &\rightarrow b d a d b d \\ + a b a d b c d &\rightarrow c d a^{2} c^{2} d \\ + c b a d b c d &\rightarrow b d c a c^{2} d \\ + a c^{2} d a^{2} c^{2} d &\rightarrow d a d b c d \\ + d a^{2} b^{2} d b c d &\rightarrow a c^{2} d c a c^{2} d \\ + c a b^{2} d b c d &\rightarrow b c d c a c^{2} d \\ + b c d a^{2} c^{2} d &\rightarrow c d b c d \\ + c^{2} b a d a d &\rightarrow b d a d b a b \\ + d a c b a d a^{2} c^{2} d &\rightarrow a c d a d b c d \\ + c^{2} b a d a^{2} c^{2} d &\rightarrow b d a d b c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/123.md b/docs/examples/1-relation/123.md new file mode 100644 index 0000000..6d5a413 --- /dev/null +++ b/docs/examples/1-relation/123.md @@ -0,0 +1,42 @@ +#\#123 baabbaaba=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{2} b a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d = a b c,\; c = b a^{2},\; d = b c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c b a &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a b c &\rightarrow c d \\ + b c b c d &\rightarrow d b c \\ + b c^{2} &\rightarrow d a \\ + b a c d &\rightarrow c b c \\ + c d b a &\rightarrow a d \\ + b c a d &\rightarrow d a d b a \\ + c d b c d &\rightarrow a d b c \\ + c d c &\rightarrow a d a \\ + a b a d &\rightarrow c d^{2} b a \\ + a d a d b c d &\rightarrow c d a d b c \\ + a d a d b a &\rightarrow c d a d \\ + a d a d c &\rightarrow c d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/124.md b/docs/examples/1-relation/124.md new file mode 100644 index 0000000..ba75e0e --- /dev/null +++ b/docs/examples/1-relation/124.md @@ -0,0 +1,62 @@ +#\#124 bbaabbbbaa=aabbbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{2} b^{4} a^{2} = a^{2} b^{4} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b^{2} a^{2} c a^{2} = a^{2} c a,\; c = b^{4},\; d = c a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{4} &\rightarrow c \\ + c a^{3} &\rightarrow d \\ + b c &\rightarrow c b \\ + a^{2} c a &\rightarrow c a^{2} d \\ + b^{2} a^{2} d &\rightarrow c a^{2} d a \\ + a^{2} c^{2} a^{2} d &\rightarrow c a^{2} d a c a \\ + c b a^{3} &\rightarrow b d \\ + c^{2} a^{2} d a d &\rightarrow d a c a \\ + c a^{2} d a^{2} &\rightarrow a^{2} d \\ + c a c a^{2} d &\rightarrow d c a \\ + c b^{2} a^{3} &\rightarrow b^{2} d \\ + c^{2} b a^{2} d a d &\rightarrow b d a c a \\ + c b a^{2} d a^{2} &\rightarrow b a^{2} d \\ + c b a c a^{2} d &\rightarrow b d c a \\ + c a^{2} d a c a^{2} d &\rightarrow a^{2} d a c a \\ + c a^{2} d a d a^{2} &\rightarrow a^{4} d \\ + c a^{2} d c a^{2} d &\rightarrow a^{2} d c a \\ + a^{6} d &\rightarrow c a^{2} d a d a d a^{2} \\ + c b^{3} a^{3} &\rightarrow b^{3} d \\ + b^{2} d a c^{2} a^{2} d &\rightarrow c^{2} a^{2} d^{2} a c a \\ + b^{2} d a c a &\rightarrow c^{2} a^{2} d^{2} \\ + c b^{2} a c a^{2} d &\rightarrow b^{2} d c a \\ + c b a^{2} d a c a^{2} d &\rightarrow b a^{2} d a c a \\ + c b a^{2} d a d a^{2} &\rightarrow b a^{4} d \\ + c b a^{2} d c a^{2} d &\rightarrow b a^{2} d c a \\ + c a^{2} d a d a c a^{2} d &\rightarrow a^{4} d a c a \\ + c a^{2} d a d c a^{2} d &\rightarrow a^{4} d c a \\ + b^{2} d a^{5} d &\rightarrow c^{2} a^{2} d^{2} a d a d a^{2} \\ + b^{2} d a^{3} d &\rightarrow c^{2} a^{2} d^{2} a d a^{2} \\ + b^{2} d a d &\rightarrow c^{2} a^{2} d^{2} a^{2} \\ + c b^{3} a c a^{2} d &\rightarrow b^{3} d c a \\ + b^{2} a^{4} d &\rightarrow c a^{2} d^{2} a^{2} \\ + c b a^{2} d a d a c a^{2} d &\rightarrow b a^{4} d a c a \\ + c b a^{2} d a d c a^{2} d &\rightarrow b a^{4} d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/126.md b/docs/examples/1-relation/126.md new file mode 100644 index 0000000..43d4f3a --- /dev/null +++ b/docs/examples/1-relation/126.md @@ -0,0 +1,41 @@ +#\#126 baaaababba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a b^{2} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{4} b a c a = a^{3},\; c = b^{2},\; d = a b a c a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a b a c a^{2} b &\rightarrow d \\ + b a^{3} &\rightarrow c a^{4} b a c a \\ + b c &\rightarrow c b \\ + a b a c a d &\rightarrow d a c a^{2} b \\ + a b a c a^{5} &\rightarrow d a^{4} b a c a \\ + a b a c a^{2} c &\rightarrow d b \\ + c a^{3} d a c d &\rightarrow a^{2} d \\ + c a^{3} d a c a &\rightarrow a^{3} \\ + a b a c a^{4} d &\rightarrow d a^{4} b a c d \\ + b a^{2} d &\rightarrow c a^{4} b a c d \\ + c a^{3} d a^{3} d &\rightarrow a^{5} d a c d \\ + c a^{3} d a^{4} &\rightarrow a^{5} d a c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/127.md b/docs/examples/1-relation/127.md new file mode 100644 index 0000000..61d5af0 --- /dev/null +++ b/docs/examples/1-relation/127.md @@ -0,0 +1,37 @@ +#\#127 baaabbabaa=abaaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a b a^{2} = a b a^{3} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c = a c d,\; c = b a^{2},\; d = a b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a c d &\rightarrow c d c \\ + a b^{2} a &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a b c &\rightarrow d a \\ + a b^{2} c d c &\rightarrow d c d \\ + a b^{2} d &\rightarrow d b^{2} a \\ + b c d c^{2} &\rightarrow c^{2} d \\ + b a d &\rightarrow c b^{2} a \\ + a c^{2} d &\rightarrow d a d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/128.md b/docs/examples/1-relation/128.md new file mode 100644 index 0000000..325ce67 --- /dev/null +++ b/docs/examples/1-relation/128.md @@ -0,0 +1,39 @@ +#\#128 aaaabbabba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{4} b^{2} a b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{3} d a = c a^{2},\; c = b a b,\; d = a b c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + c a^{2} &\rightarrow a^{3} d a \\ + a b c b &\rightarrow d \\ + b a c &\rightarrow c a b \\ + b d &\rightarrow c^{2} b \\ + a b c^{2} &\rightarrow d a b \\ + c a d &\rightarrow a^{3} d^{2} \\ + b a^{4} d a &\rightarrow c a b a^{2} \\ + b a^{4} d^{2} &\rightarrow c a b a d \\ + a b a^{3} d a^{2} d a &\rightarrow d a b a^{2} \\ + a b a^{3} d a^{2} d^{2} &\rightarrow d a b a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/129.md b/docs/examples/1-relation/129.md new file mode 100644 index 0000000..d4e9297 --- /dev/null +++ b/docs/examples/1-relation/129.md @@ -0,0 +1,36 @@ +#\#129 baaaaabba=abaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a = a b a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d = a c,\; c = b a^{4},\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{2} a &\rightarrow d \\ + b a^{4} &\rightarrow c \\ + a c &\rightarrow c a d \\ + b c &\rightarrow d a^{3} \\ + d a^{4} d a d a d a d &\rightarrow c^{2} \\ + b d a^{4} d &\rightarrow d c \\ + b d c a d a d a d a d c &\rightarrow d c a^{4} d a d a d a d \\ + d a^{4} d a d a d c a d c &\rightarrow c^{2} a^{4} d a d a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/130.md b/docs/examples/1-relation/130.md new file mode 100644 index 0000000..59fc347 --- /dev/null +++ b/docs/examples/1-relation/130.md @@ -0,0 +1,45 @@ +#\#130 baaaaabbaa=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a^{2} = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} a b^{2} c = c a,\; c = a^{2},\; d = a^{2} b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a^{2} b^{2} &\rightarrow d \\ + b a^{3} d a^{2} &\rightarrow a^{3} \\ + a^{2} b a^{3} &\rightarrow d a^{3} d a^{2} \\ + b a^{3} d^{2} &\rightarrow a d \\ + b a^{3} d a d &\rightarrow a^{2} d \\ + a^{2} b a d &\rightarrow d a^{3} d^{2} \\ + d a^{3} d a^{2} d^{2} &\rightarrow a^{3} d \\ + d a^{3} d a^{2} d a d &\rightarrow a^{4} d \\ + d a^{3} d a^{2} d a^{2} &\rightarrow a^{5} \\ + a^{2} b a^{2} d &\rightarrow d a^{3} d a d \\ + b a^{6} d &\rightarrow a^{4} d a^{2} d^{2} \\ + b a^{7} d &\rightarrow a^{4} d a^{2} d a d \\ + b a^{8} &\rightarrow a^{4} d a^{2} d a^{2} \\ + d a^{3} d a^{5} d &\rightarrow a^{6} d a^{2} d^{2} \\ + d a^{3} d a^{6} d &\rightarrow a^{6} d a^{2} d a d \\ + d a^{3} d a^{7} &\rightarrow a^{6} d a^{2} d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/131.md b/docs/examples/1-relation/131.md new file mode 100644 index 0000000..c8827ad --- /dev/null +++ b/docs/examples/1-relation/131.md @@ -0,0 +1,40 @@ +#\#131 babbaaabba=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a^{3} b^{2} a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c b c a^{2} b c = a b c a,\; c = b a,\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b^{2} &\rightarrow d \\ + b d a^{2} d a &\rightarrow d a^{2} \\ + b d a^{2} d^{2} &\rightarrow d a d \\ + a b d a^{2} &\rightarrow d^{2} a^{2} d a \\ + a b d a d &\rightarrow d^{2} a^{2} d^{2} \\ + d^{2} a^{2} d a d^{2} &\rightarrow a d a d \\ + d^{2} a^{2} d a d a &\rightarrow a d a^{2} \\ + b d a^{3} d a d &\rightarrow d a d a^{2} d a d^{2} \\ + b d a^{3} d a^{2} &\rightarrow d a d a^{2} d a d a \\ + d^{2} a^{2} d a^{2} d a d &\rightarrow a d a d a^{2} d a d^{2} \\ + d^{2} a^{2} d a^{2} d a^{2} &\rightarrow a d a d a^{2} d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/132.md b/docs/examples/1-relation/132.md new file mode 100644 index 0000000..d272bcf --- /dev/null +++ b/docs/examples/1-relation/132.md @@ -0,0 +1,46 @@ +#\#132 baabaababa=abaabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b a b a = a b a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c a d = a c a e a,\; c = b a,\; d = c a c^{2},\; e = b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b c &\rightarrow e \\ + b a &\rightarrow c \\ + c a c^{2} &\rightarrow d \\ + a c a e a &\rightarrow c a d \\ + c^{2} a e a &\rightarrow e a d \\ + b d &\rightarrow e a c^{2} \\ + c a c d &\rightarrow d a c^{2} \\ + c a d c a e a &\rightarrow a c a e c a d \\ + b e a d &\rightarrow e c a e a \\ + c^{2} a e c a d &\rightarrow e a d c a e a \\ + c a c e a d &\rightarrow d c a e a \\ + c a c a c a e c a d &\rightarrow d a d c a e a \\ + c a e a d &\rightarrow d a e a \\ + c d a e a &\rightarrow e a d^{2} \\ + c a d^{2} &\rightarrow a d a e a \\ + c d a e c a d &\rightarrow e a d^{2} c a e a \\ + c a c a d a e a &\rightarrow d a d^{2} \\ + c a c a d a e c a d &\rightarrow d a d^{2} c a e a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/133.md b/docs/examples/1-relation/133.md new file mode 100644 index 0000000..0e0a977 --- /dev/null +++ b/docs/examples/1-relation/133.md @@ -0,0 +1,43 @@ +#\#133 bbbaabbbaa=abbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{3} a^{2} b^{3} a^{2} = a b^{3} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c b a d a = d,\; c = b^{2},\; d = a c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a c b a &\rightarrow d \\ + b c &\rightarrow c b \\ + a d &\rightarrow d^{2} a \\ + b d &\rightarrow c^{2} d^{2} a^{2} \\ + c^{3} d^{18} a^{2} &\rightarrow d c^{3} d^{2} a^{2} \\ + c^{3} d^{6} a^{4} &\rightarrow d \\ + a c^{3} d^{2} a^{2} &\rightarrow c^{3} d^{14} a^{3} \\ + d c b a &\rightarrow c^{3} d^{14} a^{3} \\ + a c^{3} d^{4} a &\rightarrow d c^{3} d^{2} a^{2} \\ + c^{3} d^{20} a &\rightarrow d c^{3} d^{4} a \\ + c^{3} d^{21} &\rightarrow d c^{3} d^{5} \\ + a c^{3} d^{5} &\rightarrow d c^{3} d^{4} a \\ + a c^{3} d^{4} c^{3} d^{14} a^{3} &\rightarrow d c^{3} d^{5} \\ + c^{3} d^{20} c^{3} d^{14} a^{3} &\rightarrow d c^{3} d^{4} c^{3} d^{14} a^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/134.md b/docs/examples/1-relation/134.md new file mode 100644 index 0000000..54d869c --- /dev/null +++ b/docs/examples/1-relation/134.md @@ -0,0 +1,34 @@ +#\#134 baabbbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{4} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} c^{2} a = a^{2},\; c = b^{2},\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b^{2} \\ + b^{4} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a^{2} &\rightarrow d a^{2} d a d a d a \\ + d a^{2} d a d a d a d a &\rightarrow a^{2} \\ + d a^{2} d a d a d a^{3} &\rightarrow a^{3} d a d a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/135.md b/docs/examples/1-relation/135.md new file mode 100644 index 0000000..0dee88c --- /dev/null +++ b/docs/examples/1-relation/135.md @@ -0,0 +1,42 @@ +#\#135 baabbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{3} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c b^{3} a = c,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + b a d a &\rightarrow a^{2} \\ + a b^{2} a^{2} &\rightarrow d a d a \\ + b a d^{2} &\rightarrow a d \\ + a b^{2} a d &\rightarrow d a d^{2} \\ + a b a d &\rightarrow d a d^{3} \\ + a b a^{2} &\rightarrow d a d^{2} a \\ + d a d^{4} &\rightarrow a^{2} d \\ + d a d^{3} a &\rightarrow a^{3} \\ + b a^{3} d &\rightarrow a^{2} d^{4} \\ + b a^{4} &\rightarrow a^{2} d^{3} a \\ + d a d^{2} a^{2} d &\rightarrow a^{3} d^{4} \\ + d a d^{2} a^{3} &\rightarrow a^{3} d^{3} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/136.md b/docs/examples/1-relation/136.md new file mode 100644 index 0000000..ab78487 --- /dev/null +++ b/docs/examples/1-relation/136.md @@ -0,0 +1,80 @@ +#\#136 abaaababa=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b a b a = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e c = e,\; c = b a,\; d = c^{2} a^{2},\; e = c a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(e < d < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c^{2} a^{2} &\rightarrow d \\ + c a^{2} c &\rightarrow e \\ + a e c &\rightarrow e \\ + c e &\rightarrow d c \\ + e c a^{2} &\rightarrow a e d \\ + e a^{2} c &\rightarrow a e^{2} \\ + c a^{2} e &\rightarrow a e^{2} \\ + c a^{2} d &\rightarrow a e d \\ + b e &\rightarrow d c^{2} \\ + c d c &\rightarrow d e c^{2} \\ + a e a e d &\rightarrow e a^{2} d \\ + c a^{3} e^{2} &\rightarrow e a^{2} e \\ + a e a e^{2} &\rightarrow e a^{2} e \\ + a e d c &\rightarrow e^{2} \\ + c a e &\rightarrow a e^{2} c \\ + a^{2} e d &\rightarrow e a^{2} \\ + a e d^{2} &\rightarrow e a e d \\ + a e d e &\rightarrow e a e^{2} \\ + a e^{2} c d &\rightarrow d^{2} \\ + a e^{2} d c &\rightarrow d e \\ + a e^{2} a^{2} d &\rightarrow e a e a^{2} \\ + e a^{3} e^{2} &\rightarrow a e^{2} a^{2} e \\ + e a^{2} d c &\rightarrow a e^{3} \\ + b d^{2} &\rightarrow d^{2} c^{2} d \\ + b d e &\rightarrow d^{3} e c^{2} \\ + e a^{2} e a^{2} d &\rightarrow a e^{2} a e a^{2} \\ + e a^{2} d a e^{2} &\rightarrow a e^{3} a^{2} e \\ + c a d^{2} &\rightarrow a e^{3} c d \\ + c a d e &\rightarrow a e^{3} d c \\ + e a^{2} d^{2} &\rightarrow a e^{2} a e d \\ + e a^{2} d e &\rightarrow a e^{2} a e^{2} \\ + a e^{2} d^{2} &\rightarrow d a e d \\ + a e^{2} d e &\rightarrow d a e^{2} \\ + d c a^{2} &\rightarrow a e^{2} d \\ + a e d a e d &\rightarrow e^{2} a^{2} d \\ + a e d a e^{2} &\rightarrow e^{2} a^{2} e \\ + a e^{2} c^{2} &\rightarrow d c \\ + c d^{2} &\rightarrow d e c d \\ + c d e &\rightarrow d e d c \\ + a e^{2} d a e d &\rightarrow d e a^{2} d \\ + a e^{2} d a e^{2} &\rightarrow d e a^{2} e \\ + c d a e d &\rightarrow d e d^{2} \\ + c d a e^{2} &\rightarrow d e d e \\ + e a^{2} d a e d &\rightarrow a e^{3} a^{2} d \\ + b d a e d &\rightarrow d^{3} e c d \\ + b d a e^{2} &\rightarrow d^{3} e d c \\ + c a d a e d &\rightarrow a e^{3} d^{2} \\ + c a d a e^{2} &\rightarrow a e^{3} d e \\ + b d c &\rightarrow d^{2} c^{3} \\ + c a d c &\rightarrow a e^{3} c^{2} \\ + a e^{2} a e^{2} d &\rightarrow d e a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/137.md b/docs/examples/1-relation/137.md new file mode 100644 index 0000000..b83f3ac --- /dev/null +++ b/docs/examples/1-relation/137.md @@ -0,0 +1,38 @@ +#\#137 abaabbbbba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{5} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b a^{2} c^{2} b a = b a b a^{2},\; c = b^{2},\; d = a b c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a b c &\rightarrow d \\ + c b &\rightarrow b c \\ + b a b a d &\rightarrow a b a d c d \\ + b a b a^{2} &\rightarrow a b a d c a \\ + c a b a d &\rightarrow a b a d c d c d \\ + c a b a^{2} &\rightarrow a b a d c d c a \\ + d b &\rightarrow a c^{2} \\ + d a b a d &\rightarrow a^{2} b a d c d c d c d \\ + d a b a^{2} &\rightarrow a^{2} b a d c d c d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/138.md b/docs/examples/1-relation/138.md new file mode 100644 index 0000000..c8e6d7e --- /dev/null +++ b/docs/examples/1-relation/138.md @@ -0,0 +1,47 @@ +#\#138 aababbbaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b^{3} a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a^{2} c d = c a c,\; c = b a,\; d = b^{2} c a,\; e = a c d^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(e < c < a < d < b\): + +\( +\begin{aligned} + b^{2} c a &\rightarrow d \\ + b a &\rightarrow c \\ + a^{2} c d &\rightarrow c a c \\ + a c d^{2} &\rightarrow e \\ + b^{2} c^{2} a c &\rightarrow d a c d \\ + b e &\rightarrow c^{2} d^{2} \\ + c a c d &\rightarrow a e \\ + b c a c &\rightarrow a e \\ + b^{2} c^{2} a^{2} e &\rightarrow d a c d a c d \\ + b^{2} c^{2} e &\rightarrow d e d \\ + d c &\rightarrow c e \\ + a e d &\rightarrow c e \\ + b c a^{2} e &\rightarrow a e a c d \\ + b c e &\rightarrow c e d \\ + d a e &\rightarrow c e a c d \\ + a^{2} c^{2} e &\rightarrow c a c^{2} \\ + a c^{2} e^{2} &\rightarrow e c \\ + a e c e &\rightarrow c e c \\ + c a c^{2} e &\rightarrow a e c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/139.md b/docs/examples/1-relation/139.md new file mode 100644 index 0000000..deeddd7 --- /dev/null +++ b/docs/examples/1-relation/139.md @@ -0,0 +1,70 @@ +#\#139 abababbaaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b^{2} a^{3} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b a c a^{3} = b a^{2} b a,\; c = b a b^{2},\; d = c a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a b^{2} &\rightarrow c \\ + b a^{2} b a &\rightarrow a b a c a^{3} \\ + c a b^{2} &\rightarrow d \\ + b a b c &\rightarrow d \\ + b a b a b a c a^{3} &\rightarrow c a^{2} b a \\ + b a^{3} b a c a^{3} &\rightarrow a b a c a^{4} b a \\ + d a b^{2} &\rightarrow c a b c \\ + c a b a b a c a^{3} &\rightarrow d a^{2} b a \\ + a b a c a^{3} b^{2} &\rightarrow b a^{2} c \\ + b a b d &\rightarrow c a b c \\ + b a b a b a c a^{2} b a^{2} c &\rightarrow c a^{2} b^{2} a^{2} c \\ + a b a c a^{3} b c &\rightarrow b a^{2} d \\ + d a b a b a c a^{3} &\rightarrow c a b c a^{2} b a \\ + c a b a b a c a^{2} b a^{2} c &\rightarrow d a^{2} b^{2} a^{2} c \\ + c a b d &\rightarrow d a b c \\ + c a b c a b c &\rightarrow d a b d \\ + a b a c a^{3} b d &\rightarrow b a^{2} c a b c \\ + a b a c a^{3} b a b a c a^{3} &\rightarrow b a^{2} c a^{2} b a \\ + b a^{2} b^{2} a^{2} c &\rightarrow a b a c a^{2} b a^{2} c \\ + a b a c a^{3} c a^{3} b^{2} &\rightarrow b a b a^{2} c \\ + b a^{3} b a c a^{2} b a^{2} c &\rightarrow a b a c a^{4} b^{2} a^{2} c \\ + d a b a b a c a^{2} b a^{2} c &\rightarrow c a b c a^{2} b^{2} a^{2} c \\ + b a b a b a c a^{2} b a b a^{2} c &\rightarrow c a^{2} b^{2} a b a^{2} c \\ + b a b a b a c a^{2} b a^{2} d &\rightarrow c a^{2} b^{2} a^{2} d \\ + b a b a b a^{2} c &\rightarrow a b a c a^{3} c a^{3} c a^{3} b^{2} \\ + b a b a^{2} d &\rightarrow a b a c a^{3} c a^{3} b c \\ + b a b a^{2} c a b c &\rightarrow a b a c a^{3} c a^{3} b d \\ + b a^{2} b^{2} a b a^{2} c &\rightarrow a b a c a^{2} b a b a^{2} c \\ + b a^{2} b^{2} a^{2} d &\rightarrow a b a c a^{2} b a^{2} d \\ + a b a c a^{3} b a b a c a^{2} b a^{2} c &\rightarrow b a^{2} c a^{2} b^{2} a^{2} c \\ + c a b a b a c a^{2} b a^{2} d &\rightarrow d a^{2} b^{2} a^{2} d \\ + d a b a b a c a^{2} b a^{2} d &\rightarrow c a b c a^{2} b^{2} a^{2} d \\ + d a b a b a c a^{2} b a b a^{2} c &\rightarrow c a b c a^{2} b^{2} a b a^{2} c \\ + c a b a b a c a^{2} b a b a^{2} c &\rightarrow d a^{2} b^{2} a b a^{2} c \\ + a b a c a^{3} b a b a c a^{2} b a b a^{2} c &\rightarrow b a^{2} c a^{2} b^{2} a b a^{2} c \\ + a b a c a^{3} b a b a c a^{2} b a^{2} d &\rightarrow b a^{2} c a^{2} b^{2} a^{2} d \\ + a b a c a^{3} c a^{3} b a b a c a^{2} b a^{2} c &\rightarrow b a b a^{2} c a^{2} b^{2} a^{2} c \\ + a b a c a^{3} c a^{3} b a b a c a^{3} &\rightarrow b a b a^{2} c a^{2} b a \\ + b a^{3} b a c a^{2} b a^{2} d &\rightarrow a b a c a^{4} b^{2} a^{2} d \\ + b a^{3} b a c a^{2} b a b a^{2} c &\rightarrow a b a c a^{4} b^{2} a b a^{2} c \\ + a b a c a^{3} c a^{3} b a b a c a^{2} b a b a^{2} c &\rightarrow b a b a^{2} c a^{2} b^{2} a b a^{2} c \\ + a b a c a^{3} c a^{3} b a b a c a^{2} b a^{2} d &\rightarrow b a b a^{2} c a^{2} b^{2} a^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/140.md b/docs/examples/1-relation/140.md new file mode 100644 index 0000000..7cec696 --- /dev/null +++ b/docs/examples/1-relation/140.md @@ -0,0 +1,51 @@ +#\#140 abaaabaaba=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b a^{2} b a = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d a c = d,\; c = b a,\; d = c a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a^{2} c &\rightarrow d \\ + a d a c &\rightarrow d \\ + d a^{2} c &\rightarrow a d a d \\ + c a^{2} d &\rightarrow a d a d \\ + c a d &\rightarrow a d^{2} \\ + c a^{3} d a d &\rightarrow d a^{2} d \\ + a d a^{2} d a d &\rightarrow d a^{2} d \\ + a d a d a^{2} d &\rightarrow d a^{3} d a d \\ + c d &\rightarrow a d^{2} a c \\ + a d a d a^{3} d a d &\rightarrow d a^{2} d a^{2} d \\ + c a^{3} d^{2} &\rightarrow d a d \\ + a d a^{2} d^{2} &\rightarrow d a d \\ + a d a d a d &\rightarrow d a^{3} d^{2} \\ + d a^{3} d^{2} a^{3} d a d &\rightarrow a d^{2} a^{2} d a^{2} d \\ + d a^{2} d a^{3} d^{2} &\rightarrow a d^{2} a^{2} d \\ + b d &\rightarrow a d^{2} a c a c \\ + d a^{3} d a d^{2} &\rightarrow a d^{2} a d \\ + d a^{3} d^{2} a d &\rightarrow a d^{2} a^{3} d^{2} \\ + d a^{3} d^{2} a^{2} d &\rightarrow a d^{2} a^{3} d a d \\ + a d a d a^{3} d^{2} &\rightarrow d a^{2} d a d \\ + d a^{3} d^{2} a c &\rightarrow a d a d^{2} \\ + d a^{3} d^{2} a^{3} d^{2} &\rightarrow a d^{2} a^{2} d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/141.md b/docs/examples/1-relation/141.md new file mode 100644 index 0000000..72acd99 --- /dev/null +++ b/docs/examples/1-relation/141.md @@ -0,0 +1,51 @@ +#\#141 ababaaabba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{3} b^{2} a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d c = c^{2},\; c = b a,\; d = c a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a^{2} b &\rightarrow d \\ + a c d c &\rightarrow c^{2} \\ + a c d^{2} &\rightarrow c d \\ + c a^{2} c &\rightarrow d a \\ + c a c^{2} &\rightarrow d a d c \\ + c a c d &\rightarrow d a d^{2} \\ + c a^{2} d a &\rightarrow d a^{3} c \\ + d a^{3} b &\rightarrow c a^{2} d \\ + c^{3} &\rightarrow d a d^{2} c \\ + c^{2} d &\rightarrow d a d^{3} \\ + c a^{2} d c^{2} &\rightarrow d a^{3} d a d^{3} c \\ + c a^{2} d c d &\rightarrow d a^{3} d a d^{4} \\ + b c d &\rightarrow d a d^{4} \\ + b c^{2} &\rightarrow d a d^{3} c \\ + b d a d^{2} c &\rightarrow d a d^{3} c^{2} \\ + b d a d^{3} &\rightarrow d a d^{3} c d \\ + c d a d^{2} c &\rightarrow d a d^{2} c^{2} \\ + c d a d^{3} &\rightarrow d a d^{2} c d \\ + c a d a d^{2} c &\rightarrow d a d c^{2} \\ + c a d a d^{3} &\rightarrow d a d c d \\ + c a^{2} d^{2} a d^{2} c &\rightarrow d a^{3} d a d^{3} c^{2} \\ + c a^{2} d^{2} a d^{3} &\rightarrow d a^{3} d a d^{3} c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/142.md b/docs/examples/1-relation/142.md new file mode 100644 index 0000000..0781448 --- /dev/null +++ b/docs/examples/1-relation/142.md @@ -0,0 +1,47 @@ +#\#142 baabaaaaba=aabaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{4} b a = a^{2} b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b d a = c^{2} a,\; c = a^{2} b,\; d = c a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b d a &\rightarrow c^{2} a \\ + a^{2} b &\rightarrow c \\ + c a^{2} c &\rightarrow d \\ + b d c &\rightarrow c^{3} \\ + a^{2} c^{2} a &\rightarrow c d a \\ + d a^{2} c &\rightarrow c a^{2} d \\ + b c a^{2} d &\rightarrow c d \\ + b d^{2} &\rightarrow c^{2} d \\ + a^{2} c^{3} &\rightarrow c d c \\ + a^{2} c d &\rightarrow c^{2} a^{2} d \\ + c^{2} d a &\rightarrow d c a \\ + c^{3} a^{2} d &\rightarrow d^{2} \\ + a^{2} c^{2} d &\rightarrow c d^{2} \\ + c^{2} d c &\rightarrow d c^{2} \\ + a^{2} d c a &\rightarrow c d^{2} a \\ + a^{2} d c^{2} &\rightarrow c d^{2} c \\ + a^{2} d^{2} &\rightarrow c d c a^{2} d \\ + c^{2} d^{2} &\rightarrow d c d \\ + a^{2} d c d &\rightarrow c d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/143.md b/docs/examples/1-relation/143.md new file mode 100644 index 0000000..39346f7 --- /dev/null +++ b/docs/examples/1-relation/143.md @@ -0,0 +1,85 @@ +#\#143 baaaabbbba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{4} a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d b a = a c,\; c = b a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a b^{3} &\rightarrow d \\ + c a d b a &\rightarrow a c \\ + b a^{2} &\rightarrow c \\ + a c a &\rightarrow c a d c \\ + d a^{2} &\rightarrow a b^{2} c \\ + c a d b d &\rightarrow a c b^{3} \\ + b a d &\rightarrow c b^{3} \\ + a^{2} c &\rightarrow c a d c d b a \\ + a c^{2} a d c &\rightarrow c a d c^{2} a \\ + d a d &\rightarrow a b^{2} c b^{3} \\ + a b^{2} c^{2} a &\rightarrow d c a d c d c \\ + c a d c b^{3} &\rightarrow a c d \\ + c a d b c a d c &\rightarrow a c^{2} a \\ + b c a d c d c &\rightarrow c^{2} a \\ + a c d c a d c d c &\rightarrow c a d c b^{2} c^{2} a \\ + a b^{2} c a c &\rightarrow d c a d c d c d b a \\ + a b^{2} c^{2} d c a d c d c &\rightarrow d c a d c d c b^{2} c^{2} a \\ + a b^{2} c^{3} a d c &\rightarrow d c a d c d c^{2} a \\ + d c a d c d c b^{3} &\rightarrow a b^{2} c^{2} d \\ + d c a d c d b a &\rightarrow a b^{2} c^{2} \\ + c^{2} a^{2} d c b^{3} &\rightarrow b c a d c d a c d \\ + c^{2} a^{2} d b a &\rightarrow b c a d c d a c \\ + c^{2} a^{2} d b d &\rightarrow b c a d c d a c b^{3} \\ + c^{2} a^{2} d b c a d c &\rightarrow b c a d c d a c^{2} a \\ + b c a d c d b a &\rightarrow c^{2} \\ + a b c a d c d a c &\rightarrow c a d b c a d a c \\ + d c a d c d b d &\rightarrow a b^{2} c^{2} b^{3} \\ + d c a d c d b c a d c &\rightarrow a b^{2} c^{3} a \\ + c^{2} a^{2} d c d c b^{3} &\rightarrow b c a d c a b^{2} c^{2} d \\ + c^{2} a^{2} d c d b a &\rightarrow b c a d c a b^{2} c^{2} \\ + a b c a d c a b^{2} c^{2} &\rightarrow c a d b c a^{2} b^{2} c^{2} \\ + a c^{2} a d b c a d a c &\rightarrow c a d c b c a d c d a c \\ + a b^{2} c b c a d c a b^{2} c^{2} &\rightarrow d c a d c d b c a^{2} b^{2} c^{2} \\ + a b^{2} c b c a d c d a c &\rightarrow d c a d c d b c a d a c \\ + a b^{2} c^{3} a d b c a d a c &\rightarrow d c a d c d c b c a d c d a c \\ + d c a d c d b c a d b c a d a c &\rightarrow a b^{2} c^{2} b c a d c d a c \\ + c a d b c a d b c a d a c &\rightarrow a c b c a d c d a c \\ + c^{2} a^{2} d c d b d &\rightarrow b c a d c a b^{2} c^{2} b^{3} \\ + c^{2} a^{2} d c d b c a d c &\rightarrow b c a d c a b^{2} c^{3} a \\ + c^{2} a^{2} d c d b c a d b c a d a c &\rightarrow b c a d c a b^{2} c^{2} b c a d c d a c \\ + c^{2} a^{2} d b c a d b c a d a c &\rightarrow b c a d c d a c b c a d c d a c \\ + b c a d c d b c a d a c &\rightarrow c b c a d c d a c \\ + b c a d c d b d &\rightarrow c^{2} b^{3} \\ + b c a d c d b c a d c &\rightarrow c^{3} a \\ + b c a d c d b c a d b c a d a c &\rightarrow c^{2} b c a d c d a c \\ + a c d c a d c d b c a^{2} b^{2} c^{2} &\rightarrow c a d c b^{2} c b c a d c a b^{2} c^{2} \\ + a c d c a d c d b c a d a c &\rightarrow c a d c b^{2} c b c a d c d a c \\ + a c^{2} a d b c a^{2} b^{2} c^{2} &\rightarrow c a d c b c a d c a b^{2} c^{2} \\ + a b^{2} c^{2} d c a d c d b c a^{2} b^{2} c^{2} &\rightarrow d c a d c d c b^{2} c b c a d c a b^{2} c^{2} \\ + a b^{2} c^{2} d c a d c d b c a d a c &\rightarrow d c a d c d c b^{2} c b c a d c d a c \\ + a b^{2} c^{3} a d b c a^{2} b^{2} c^{2} &\rightarrow d c a d c d c b c a d c a b^{2} c^{2} \\ + d c a d c d b c a d b c a^{2} b^{2} c^{2} &\rightarrow a b^{2} c^{2} b c a d c a b^{2} c^{2} \\ + c a d b c a d b c a^{2} b^{2} c^{2} &\rightarrow a c b c a d c a b^{2} c^{2} \\ + c^{2} a^{2} d c d b c a d b c a^{2} b^{2} c^{2} &\rightarrow b c a d c a b^{2} c^{2} b c a d c a b^{2} c^{2} \\ + c^{2} a^{2} d b c a d b c a^{2} b^{2} c^{2} &\rightarrow b c a d c d a c b c a d c a b^{2} c^{2} \\ + b c a d c d b c a^{2} b^{2} c^{2} &\rightarrow c b c a d c a b^{2} c^{2} \\ + b c a d c d b c a d b c a^{2} b^{2} c^{2} &\rightarrow c^{2} b c a d c a b^{2} c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/144.md b/docs/examples/1-relation/144.md new file mode 100644 index 0000000..6cd6d2c --- /dev/null +++ b/docs/examples/1-relation/144.md @@ -0,0 +1,85 @@ +#\#144 baaaaababa=abaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b a b a = a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; b c e a = d c a,\; c = a^{2},\; d = a b,\; e = c d^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < e < c < a < b\): + +\( +\begin{aligned} + b c e a &\rightarrow d c a \\ + a b &\rightarrow d \\ + a^{2} &\rightarrow c \\ + c d^{2} &\rightarrow e \\ + b c e c &\rightarrow d c^{2} \\ + b c e d &\rightarrow d c d \\ + a c &\rightarrow c a \\ + a d c a &\rightarrow d c e a \\ + c b &\rightarrow a d \\ + b c e^{2} &\rightarrow d c e \\ + a e &\rightarrow c a d^{2} \\ + a d c^{2} &\rightarrow d c e c \\ + a d c d &\rightarrow d c e d \\ + a d c e &\rightarrow d c e^{2} \\ + a d e &\rightarrow d c e d^{2} \\ + c d c d &\rightarrow d c e^{2} d \\ + c d c^{2} &\rightarrow d c e^{2} c \\ + c d c e &\rightarrow d c e^{3} \\ + c d c a &\rightarrow d c e^{2} a \\ + d c e^{2} d^{2} &\rightarrow c d e \\ + a d^{2} c e^{2} a &\rightarrow d c e d c a \\ + a d^{2} c e^{2} c &\rightarrow d c e d c^{2} \\ + a d^{2} c e^{3} &\rightarrow d c e d c e \\ + a d^{2} c e^{2} d &\rightarrow d c e d c d \\ + d c e^{2} d c a &\rightarrow e c e^{2} a \\ + d c e^{2} d c^{2} &\rightarrow e c e^{2} c \\ + d c e^{2} d c e &\rightarrow e c e^{3} \\ + d c e^{2} d c d &\rightarrow e c e^{2} d \\ + c^{2} d e &\rightarrow d c e^{4} d^{2} \\ + c e c e^{2} a &\rightarrow d c e^{4} d c a \\ + c e c e^{2} c &\rightarrow d c e^{4} d c^{2} \\ + c e c e^{3} &\rightarrow d c e^{4} d c e \\ + c e c e^{2} d &\rightarrow d c e^{4} d c d \\ + d c e^{2} c d e &\rightarrow e c e^{4} d^{2} \\ + d c e^{3} c e^{2} a &\rightarrow e c e^{4} d c a \\ + d c e^{3} c e^{2} c &\rightarrow e c e^{4} d c^{2} \\ + d c e^{3} c e^{3} &\rightarrow e c e^{4} d c e \\ + d c e^{3} c e^{2} d &\rightarrow e c e^{4} d c d \\ + d c e^{2} d e &\rightarrow e c e^{2} d^{2} \\ + c d e c e^{2} a &\rightarrow e c e^{2} d c a \\ + c d e c e^{2} c &\rightarrow e c e^{2} d c^{2} \\ + c d e c e^{3} &\rightarrow e c e^{2} d c e \\ + c d e c e^{2} d &\rightarrow e c e^{2} d c d \\ + b d c e^{4} d c a &\rightarrow d c^{2} e^{2} a \\ + b d c e^{4} d c^{2} &\rightarrow d c^{2} e^{2} c \\ + b d c e^{4} d c e &\rightarrow d c^{2} e^{3} \\ + b d c e^{4} d c d &\rightarrow d c^{2} e^{2} d \\ + b d c e^{4} c d e &\rightarrow d c^{2} e^{4} d^{2} \\ + b d c e^{5} c e^{2} a &\rightarrow d c^{2} e^{4} d c a \\ + b d c e^{5} c e^{2} c &\rightarrow d c^{2} e^{4} d c^{2} \\ + b d c e^{5} c e^{3} &\rightarrow d c^{2} e^{4} d c e \\ + b d c e^{5} c e^{2} d &\rightarrow d c^{2} e^{4} d c d \\ + b d c e^{4} d e &\rightarrow d c^{2} e^{2} d^{2} \\ + b d c e^{4} d^{2} c e^{2} a &\rightarrow d c^{2} e^{2} d c a \\ + b d c e^{4} d^{2} c e^{2} c &\rightarrow d c^{2} e^{2} d c^{2} \\ + b d c e^{4} d^{2} c e^{3} &\rightarrow d c^{2} e^{2} d c e \\ + b d c e^{4} d^{2} c e^{2} d &\rightarrow d c^{2} e^{2} d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/145.md b/docs/examples/1-relation/145.md new file mode 100644 index 0000000..0294788 --- /dev/null +++ b/docs/examples/1-relation/145.md @@ -0,0 +1,35 @@ +#\#145 baababaaba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b a^{2} b a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c^{2} a c = a,\; c = b a,\; d = b a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b a^{2} b &\rightarrow d \\ + b a^{2} d &\rightarrow d a^{2} b \\ + a d a &\rightarrow d a^{2} \\ + d^{2} a^{2} &\rightarrow a \\ + b a^{3} &\rightarrow d a^{2} b d a^{2} \\ + b d a^{3} &\rightarrow d a^{2} b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/147.md b/docs/examples/1-relation/147.md new file mode 100644 index 0000000..2ec1a66 --- /dev/null +++ b/docs/examples/1-relation/147.md @@ -0,0 +1,45 @@ +#\#147 baaaaabbaa=aaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a^{2} = a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} a b^{2} c = c^{2},\; c = a^{2},\; d = a^{2} b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a^{2} b^{2} &\rightarrow d \\ + b a^{3} d a^{2} &\rightarrow a^{4} \\ + a^{2} b a^{4} &\rightarrow d a^{3} d a^{2} \\ + b a^{3} d^{2} &\rightarrow a^{2} d \\ + b a^{3} d a d &\rightarrow a^{3} d \\ + a^{2} b a^{2} d &\rightarrow d a^{3} d^{2} \\ + a^{2} b a^{3} d &\rightarrow d a^{3} d a d \\ + d a^{3} d a d^{2} &\rightarrow a^{4} d \\ + d a^{3} d a d a d &\rightarrow a^{5} d \\ + d a^{3} d a d a^{2} &\rightarrow a^{6} \\ + b a^{7} d &\rightarrow a^{5} d a d^{2} \\ + b a^{8} d &\rightarrow a^{5} d a d a d \\ + b a^{9} &\rightarrow a^{5} d a d a^{2} \\ + d a^{3} d a^{5} d &\rightarrow a^{7} d a d^{2} \\ + d a^{3} d a^{6} d &\rightarrow a^{7} d a d a d \\ + d a^{3} d a^{7} &\rightarrow a^{7} d a d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/148.md b/docs/examples/1-relation/148.md new file mode 100644 index 0000000..0cf89f3 --- /dev/null +++ b/docs/examples/1-relation/148.md @@ -0,0 +1,43 @@ +#\#148 ababaabbba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{2} b^{3} a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d c = c^{2},\; c = b a,\; d = c a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a b^{2} &\rightarrow d \\ + a c d c &\rightarrow c^{2} \\ + b c^{2} &\rightarrow c^{2} d c \\ + a c d^{2} &\rightarrow c d \\ + c a b c &\rightarrow d a \\ + b c d &\rightarrow c^{2} d^{2} \\ + d a c &\rightarrow c a c^{2} d c \\ + d a^{2} b^{2} &\rightarrow c a b d \\ + c a b d a &\rightarrow d a^{2} b c \\ + c a c^{2} d c d c &\rightarrow d c^{2} \\ + c a c^{2} d c d^{2} &\rightarrow d c d \\ + d a d &\rightarrow c a c^{2} d^{2} \\ + c a b d c d &\rightarrow d a^{2} c^{2} d c d^{2} \\ + c a b d c^{2} &\rightarrow d a^{2} c^{2} d c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/149.md b/docs/examples/1-relation/149.md new file mode 100644 index 0000000..d3fc75e --- /dev/null +++ b/docs/examples/1-relation/149.md @@ -0,0 +1,53 @@ +#\#149 abaababaa=baabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b a^{2} = b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} b a^{2} = b a c a,\; c = a b a,\; d = c a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + a b a &\rightarrow c \\ + c a b c &\rightarrow d \\ + c b a &\rightarrow a b c \\ + b a c^{2} &\rightarrow d c \\ + d a b c &\rightarrow c a b d \\ + b a d &\rightarrow c a b d \\ + a d a &\rightarrow c^{2} a \\ + b a c a &\rightarrow d a \\ + b a c d &\rightarrow d^{2} \\ + b c^{2} a &\rightarrow c a b d a \\ + a c a b d &\rightarrow c d \\ + a d c &\rightarrow c^{3} \\ + d b a &\rightarrow c^{2} b c \\ + b c^{3} &\rightarrow c a b d c \\ + b c d &\rightarrow d a b d \\ + a d^{2} &\rightarrow c^{2} d \\ + c^{2} d a &\rightarrow d c a \\ + c^{3} a b d &\rightarrow d^{2} \\ + c^{2} d c &\rightarrow d c^{2} \\ + b c^{2} d &\rightarrow d c^{2} a b d \\ + c^{2} d^{2} &\rightarrow d c d \\ + b d c a &\rightarrow d c^{2} a b d a \\ + b d c^{2} &\rightarrow d c^{2} a b d c \\ + b d c d &\rightarrow d c d c^{2} a b d \\ + b d^{2} &\rightarrow d c d a b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/151.md b/docs/examples/1-relation/151.md new file mode 100644 index 0000000..5b04ab6 --- /dev/null +++ b/docs/examples/1-relation/151.md @@ -0,0 +1,34 @@ +#\#151 ababaababa=bababaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{2} b a b a = b a b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c^{2} = c^{3} a,\; c = b a,\; d = a c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c^{3} a &\rightarrow d c^{2} \\ + a c^{2} a &\rightarrow d \\ + b d &\rightarrow d c^{2} \\ + c^{3} d &\rightarrow d c d c^{2} \\ + a c^{2} d &\rightarrow d c^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/152.md b/docs/examples/1-relation/152.md new file mode 100644 index 0000000..06ba9c4 --- /dev/null +++ b/docs/examples/1-relation/152.md @@ -0,0 +1,73 @@ +#\#152 babaaaabba=aaabbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{4} b^{2} a = a^{3} b^{2} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a b a^{2} c a = a d,\; c = a^{2} b^{2},\; d = c a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} b^{2} &\rightarrow c \\ + c a b a &\rightarrow d \\ + b a b a^{2} c a &\rightarrow a d \\ + d b a^{2} c a &\rightarrow c a^{2} d \\ + a^{2} b a d &\rightarrow d a c a \\ + c a b c &\rightarrow d a b^{2} \\ + b a b a^{2} d &\rightarrow a d b a \\ + b a b a^{2} c^{2} &\rightarrow a d a b^{2} \\ + a^{2} b a c a^{2} d &\rightarrow d a d a c a \\ + d a b^{2} a b a &\rightarrow c a b d \\ + d a b^{2} a b c &\rightarrow c a b d a b^{2} \\ + d b a^{2} d a b^{2} &\rightarrow c a^{2} d b c \\ + c a^{2} d a b a d &\rightarrow d b a^{2} c d a c a \\ + c a^{2} d a b^{2} &\rightarrow d b a^{2} c^{2} \\ + c a^{2} d b a &\rightarrow d b a^{2} d \\ + c a b d a c a &\rightarrow d a b a d \\ + b a b a^{2} c d &\rightarrow a c a b d \\ + a^{2} b a c a b d &\rightarrow d a c d \\ + d a b a d a b a d &\rightarrow c a b d a c d a c a \\ + d a b a d a b^{2} &\rightarrow c a b d a c^{2} \\ + d a b a d b a &\rightarrow c a b d a d \\ + d a b a c a^{2} d &\rightarrow c a b d a d a c a \\ + d a b a c a b d &\rightarrow c a b d a c d \\ + d a b^{2} a^{2} d a b a d &\rightarrow c a b d b a^{2} c d a c a \\ + d a b^{2} a^{2} d a b^{2} &\rightarrow c a b d b a^{2} c^{2} \\ + d a b^{2} a^{2} d b a &\rightarrow c a b d b a^{2} d \\ + d a b^{2} a^{2} c a b d &\rightarrow c a b d b a^{2} c d \\ + d a b^{2} a b d a c a &\rightarrow c a b d a b a d \\ + d b a^{2} d a b a d &\rightarrow c a^{2} d b d a c a \\ + d b a^{2} d b a^{2} d &\rightarrow c a^{2} d a d b a \\ + d b a^{2} d b a^{2} c d &\rightarrow c a^{2} d a c a b d \\ + d b a^{2} d b a^{2} c^{2} &\rightarrow c a^{2} d a d a b^{2} \\ + c a^{2} c a b d &\rightarrow d b a^{2} c d \\ + c a^{2} c a^{2} d &\rightarrow d b a^{2} d a c a \\ + c a b d a d a b a d &\rightarrow d a b a d b d a c a \\ + c a b d a d a b^{2} &\rightarrow d a b a d b c \\ + c a b d a d b a^{2} d &\rightarrow d a b a d a d b a \\ + c a b d a d b a^{2} c^{2} &\rightarrow d a b a d a d a b^{2} \\ + c a b d a d b a^{2} c d &\rightarrow d a b a d a c a b d \\ + d a b^{2} a^{2} c a^{2} d &\rightarrow c a b d b a^{2} d a c a \\ + d a b^{2} a b d a d a b a d &\rightarrow c a b d a b a d b d a c a \\ + d a b^{2} a b d a d a b^{2} &\rightarrow c a b d a b a d b c \\ + d a b^{2} a b d a d b a^{2} d &\rightarrow c a b d a b a d a d b a \\ + d a b^{2} a b d a d b a^{2} c^{2} &\rightarrow c a b d a b a d a d a b^{2} \\ + d a b^{2} a b d a d b a^{2} c d &\rightarrow c a b d a b a d a c a b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/153.md b/docs/examples/1-relation/153.md new file mode 100644 index 0000000..69a40cb --- /dev/null +++ b/docs/examples/1-relation/153.md @@ -0,0 +1,43 @@ +#\#153 baaaababaa=abaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a b a^{2} = a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c e = a c a c,\; c = b a,\; d = a^{2} c^{2},\; e = a d a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a^{2} c^{2} &\rightarrow d \\ + a d a &\rightarrow e \\ + a c a c &\rightarrow c e \\ + b e &\rightarrow c d a \\ + b d &\rightarrow c a c^{2} \\ + b c e &\rightarrow c^{2} a c \\ + a d e &\rightarrow e d a \\ + a d^{2} &\rightarrow e a c^{2} \\ + a d c e &\rightarrow e c a c \\ + a c^{2} e &\rightarrow c e a c \\ + a c e a c &\rightarrow d e \\ + a c e d e &\rightarrow d e^{2} a c \\ + a c e c e &\rightarrow d e a c \\ + a c d e &\rightarrow c e^{2} a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/154.md b/docs/examples/1-relation/154.md new file mode 100644 index 0000000..0e32791 --- /dev/null +++ b/docs/examples/1-relation/154.md @@ -0,0 +1,42 @@ +#\#154 aababaabba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b a^{2} b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c^{2} a b c = c^{2} a,\; c = b a,\; d = b a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b a^{2} b &\rightarrow d \\ + b a b a^{2} &\rightarrow a^{2} b a d b a \\ + b a^{2} d &\rightarrow d a^{2} b \\ + b a^{4} b a d b a &\rightarrow d a b a^{2} \\ + a^{2} b a d b a b &\rightarrow b a d \\ + b^{2} a d &\rightarrow d a d b a b \\ + b a^{4} b a d^{2} &\rightarrow d a b a^{3} b \\ + a^{2} b a d b a d &\rightarrow b a d a^{2} b \\ + b a b a b a d &\rightarrow a^{2} b a d^{2} a d b a b \\ + a^{2} b a d b a^{3} b a d b a &\rightarrow b a d a b a^{2} \\ + a^{2} b a d a^{2} b a d b a &\rightarrow b a d a^{2} \\ + a^{2} b a d b a^{3} b a d^{2} &\rightarrow b a d a b a^{3} b \\ + a^{2} b a d a^{2} b a d^{2} &\rightarrow b a d a^{3} b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/156.md b/docs/examples/1-relation/156.md new file mode 100644 index 0000000..5c8a313 --- /dev/null +++ b/docs/examples/1-relation/156.md @@ -0,0 +1,37 @@ +#\#156 ababaabaaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{2} b a^{3} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c^{2} d = c^{2},\; c = b a,\; d = a c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{2} d &\rightarrow c^{2} \\ + a c a^{2} &\rightarrow d \\ + b c^{2} &\rightarrow c^{3} d \\ + b d &\rightarrow c^{2} a^{2} \\ + a c a c^{2} &\rightarrow d c^{2} d \\ + a c a d &\rightarrow d c a^{2} \\ + a c^{3} &\rightarrow d c^{2} d^{2} \\ + a c d c^{2} d^{2} &\rightarrow d c^{2} d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/157.md b/docs/examples/1-relation/157.md new file mode 100644 index 0000000..a28502b --- /dev/null +++ b/docs/examples/1-relation/157.md @@ -0,0 +1,45 @@ +#\#157 bababaabba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a^{2} b^{2} a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d c = a c^{2},\; c = b a,\; d = c a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{2} &\rightarrow c^{2} d c \\ + c a b &\rightarrow d \\ + b c^{2} d c &\rightarrow c^{3} \\ + c a c &\rightarrow d a \\ + a c d &\rightarrow c^{2} d^{2} \\ + b c^{2} d^{2} &\rightarrow c^{2} d \\ + d a^{2} b &\rightarrow c a d \\ + d a^{2} c &\rightarrow c a d a \\ + d a c &\rightarrow c^{3} d c \\ + d a d &\rightarrow c^{3} d^{2} \\ + c^{3} d c d &\rightarrow d c^{2} d^{2} \\ + c^{3} d c^{2} &\rightarrow d c^{2} d c \\ + a d c^{2} d c &\rightarrow c^{2} d c^{2} d c^{2} \\ + a d c^{2} d^{2} &\rightarrow c^{2} d c^{2} d c d \\ + c^{3} d^{2} c^{2} d c &\rightarrow d c^{2} d c^{2} d c^{2} \\ + c^{3} d^{2} c^{2} d^{2} &\rightarrow d c^{2} d c^{2} d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/160.md b/docs/examples/1-relation/160.md new file mode 100644 index 0000000..7d0fe93 --- /dev/null +++ b/docs/examples/1-relation/160.md @@ -0,0 +1,99 @@ +#\#160 baaabbabba=abbbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a b^{2} a = a b^{3} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a = a b^{3} a^{2},\; c = b^{2} a b,\; d = a^{2} c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} c b &\rightarrow d \\ + a b^{3} a^{2} &\rightarrow b a d a \\ + b^{2} a b &\rightarrow c \\ + a^{2} c^{2} &\rightarrow d b a b \\ + b a d b a d a &\rightarrow a b c a d a \\ + a b^{3} a d &\rightarrow b a d^{2} \\ + b a d a c b &\rightarrow a b^{3} d \\ + b^{2} a c &\rightarrow c b a b \\ + b^{3} a d a &\rightarrow c b^{2} a^{2} \\ + a^{2} c a b c a d a &\rightarrow d a d b a d a \\ + a^{2} c a b^{3} d &\rightarrow d a d a c b \\ + a c b^{2} a^{2} &\rightarrow b a d^{2} a \\ + b a d b a d^{2} &\rightarrow a b c a d^{2} \\ + a b c a d a c b &\rightarrow b a d a b^{3} d \\ + a b c a d b a d a &\rightarrow b a d a b c a d a \\ + a b c a d b a d^{2} &\rightarrow b a d a b c a d^{2} \\ + b a d^{2} b a d a &\rightarrow a c^{2} a d a \\ + b a d^{2} a c b &\rightarrow a c b^{2} d \\ + b^{3} a d^{2} &\rightarrow c b^{2} a d \\ + b a d a c a b c a d a &\rightarrow a b^{3} d a d b a d a \\ + b a d a c a b^{3} d &\rightarrow a b^{3} d a d a c b \\ + b a d a c^{2} &\rightarrow a b^{3} d b a b \\ + b^{2} a^{2} b c a d a &\rightarrow c a d b a d a \\ + b^{2} a^{2} b^{3} d &\rightarrow c a d a c b \\ + a^{2} c a c^{2} a d a &\rightarrow d a d^{2} b a d a \\ + a^{2} c a c b^{2} d &\rightarrow d a d^{2} a c b \\ + a^{2} c a b c a d^{2} &\rightarrow d a d b a d^{2} \\ + b a d^{2} b a d^{2} &\rightarrow a c^{2} a d^{2} \\ + a c^{2} a d a c a b c a d a &\rightarrow b a d^{2} a b^{3} d a d b a d a \\ + a c^{2} a d a c b &\rightarrow b a d^{2} a b^{3} d \\ + a c^{2} a d b a d a &\rightarrow b a d^{2} a b c a d a \\ + a c^{2} a d b a d^{2} &\rightarrow b a d^{2} a b c a d^{2} \\ + a c b^{2} a d &\rightarrow b a d^{3} \\ + a b a d^{2} a &\rightarrow d b a^{2} \\ + a b c a d a c a c^{2} a d a &\rightarrow b a d a b^{3} d a d^{2} b a d a \\ + a b c a d a c a c b^{2} d &\rightarrow b a d a b^{3} d a d^{2} a c b \\ + a b c a d a c a b c a d a &\rightarrow b a d a b^{3} d a d b a d a \\ + a b c a d a c a b c a d^{2} &\rightarrow b a d a b^{3} d a d b a d^{2} \\ + a b c a d a c a b^{3} d &\rightarrow b a d a b^{3} d a d a c b \\ + a b c a d a c^{2} &\rightarrow b a d a b^{3} d b a b \\ + a c b a b c a d a &\rightarrow b a d^{3} b a d a \\ + a c b a b^{3} d &\rightarrow b a d^{3} a c b \\ + b a d a c a c^{2} a d a &\rightarrow a b^{3} d a d^{2} b a d a \\ + b a d a c a c b^{2} d &\rightarrow a b^{3} d a d^{2} a c b \\ + b a d a c a b c a d^{2} &\rightarrow a b^{3} d a d b a d^{2} \\ + b a d^{2} a c a c^{2} a d a &\rightarrow a c b^{2} d a d^{2} b a d a \\ + b a d^{2} a c a c b^{2} d &\rightarrow a c b^{2} d a d^{2} a c b \\ + b a d^{2} a c a b c a d a &\rightarrow a c b^{2} d a d b a d a \\ + b a d^{2} a c a b c a d^{2} &\rightarrow a c b^{2} d a d b a d^{2} \\ + b a d^{2} a c a b^{3} d &\rightarrow a c b^{2} d a d a c b \\ + b a d^{2} a c^{2} &\rightarrow a c b^{2} d b a b \\ + c a d^{2} b a d a &\rightarrow b^{2} d b a b a d a \\ + b^{2} a^{2} b c a d^{2} &\rightarrow c a d b a d^{2} \\ + a^{2} c a c^{2} a d^{2} &\rightarrow d a d^{2} b a d^{2} \\ + a c^{2} a d a c a b^{3} d &\rightarrow b a d^{2} a b^{3} d a d a c b \\ + a c^{2} a d a c^{2} &\rightarrow b a d^{2} a b^{3} d b a b \\ + a b c a d a c a c^{2} a d^{2} &\rightarrow b a d a b^{3} d a d^{2} b a d^{2} \\ + a c b a b c a d^{2} &\rightarrow b a d^{3} b a d^{2} \\ + b a d a c a c^{2} a d^{2} &\rightarrow a b^{3} d a d^{2} b a d^{2} \\ + b a d^{2} a c a c^{2} a d^{2} &\rightarrow a c b^{2} d a d^{2} b a d^{2} \\ + c a d^{2} b a d^{2} &\rightarrow b^{2} d b a b a d^{2} \\ + a c^{2} a d a c a c^{2} a d a &\rightarrow b a d^{2} a b^{3} d a d^{2} b a d a \\ + a c^{2} a d a c a c^{2} a d^{2} &\rightarrow b a d^{2} a b^{3} d a d^{2} b a d^{2} \\ + a c^{2} a d a c a c b^{2} d &\rightarrow b a d^{2} a b^{3} d a d^{2} a c b \\ + a c^{2} a d a c a b c a d^{2} &\rightarrow b a d^{2} a b^{3} d a d b a d^{2} \\ + a c b a c^{2} a d a &\rightarrow b a d^{4} b a d a \\ + a c b a c^{2} a d^{2} &\rightarrow b a d^{4} b a d^{2} \\ + a c b a c b^{2} d &\rightarrow b a d^{4} a c b \\ + a b a d^{3} &\rightarrow d b a d \\ + c a d^{2} a &\rightarrow b^{2} d b a^{2} \\ + c a d^{3} &\rightarrow b^{2} d b a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/161.md b/docs/examples/1-relation/161.md new file mode 100644 index 0000000..85f63eb --- /dev/null +++ b/docs/examples/1-relation/161.md @@ -0,0 +1,60 @@ +#\#161 baaaababba=abaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a b^{2} a = a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d a = a b a^{3},\; c = b a b,\; d = a^{2} c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} c b &\rightarrow d \\ + a b a^{3} &\rightarrow b a^{2} d a \\ + b a b &\rightarrow c \\ + a^{2} c^{2} &\rightarrow d a b \\ + a b a^{2} d &\rightarrow b a^{2} d^{2} \\ + b a^{2} d a c b &\rightarrow a b a d \\ + b a c &\rightarrow c a b \\ + b^{2} a^{2} d a &\rightarrow c a^{3} \\ + b a^{2} d b a^{2} d a &\rightarrow a c a^{2} d^{2} a \\ + a^{2} c a^{2} d^{2} a &\rightarrow b a^{2} d^{2} b a^{2} d a \\ + a^{2} c a c a^{2} d^{2} a &\rightarrow d a^{2} d b a^{2} d a \\ + a^{2} c a b a d &\rightarrow d a^{2} d a c b \\ + a^{2} b a d &\rightarrow b a^{2} d^{2} a c b \\ + b a^{2} d b a^{2} d^{2} &\rightarrow a c a^{2} d^{3} \\ + b^{2} a^{2} d^{2} &\rightarrow c a^{2} d \\ + b a^{2} d a c a c a^{2} d^{2} a &\rightarrow a b a d a^{2} d b a^{2} d a \\ + b a^{2} d a c a b a d &\rightarrow a b a d a^{2} d a c b \\ + b a^{2} d a c^{2} &\rightarrow a b a d a b \\ + b a^{2} d a b a d &\rightarrow a c a^{2} d^{2} a c b \\ + b a^{2} d a c a^{2} d^{2} a &\rightarrow a c a^{2} d^{2} b a^{2} d a \\ + b a^{2} d a c a^{2} d^{3} &\rightarrow a c a^{2} d^{2} b a^{2} d^{2} \\ + a^{2} c a^{2} d^{3} &\rightarrow b a^{2} d^{2} b a^{2} d^{2} \\ + a^{2} c a^{2} d^{2} b a^{2} d a &\rightarrow b a^{2} d^{2} a c a^{2} d^{2} a \\ + a^{2} c a^{2} d^{2} b a^{2} d^{2} &\rightarrow b a^{2} d^{2} a c a^{2} d^{3} \\ + a^{2} c a c a^{2} d^{3} &\rightarrow d a^{2} d b a^{2} d^{2} \\ + a^{2} c a c a^{2} d^{2} b a^{2} d a &\rightarrow d a^{2} d a c a^{2} d^{2} a \\ + a^{2} c a c a^{2} d^{2} b a^{2} d^{2} &\rightarrow d a^{2} d a c a^{2} d^{3} \\ + b a^{2} d a c a^{2} d^{2} b a^{2} d a &\rightarrow a c a^{2} d^{2} a c a^{2} d^{2} a \\ + b a^{2} d a c a^{2} d^{2} b a^{2} d^{2} &\rightarrow a c a^{2} d^{2} a c a^{2} d^{3} \\ + b a^{2} d a c a c a^{2} d^{3} &\rightarrow a b a d a^{2} d b a^{2} d^{2} \\ + b a^{2} d a c a c a^{2} d^{2} b a^{2} d a &\rightarrow a b a d a^{2} d a c a^{2} d^{2} a \\ + b a^{2} d a c a c a^{2} d^{2} b a^{2} d^{2} &\rightarrow a b a d a^{2} d a c a^{2} d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/163.md b/docs/examples/1-relation/163.md new file mode 100644 index 0000000..4c8acb2 --- /dev/null +++ b/docs/examples/1-relation/163.md @@ -0,0 +1,38 @@ +#\#163 aabbaababa=baaabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{2} a^{2} b a b a = b a^{3} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d c a c^{2} = c a d c a,\; c = b a,\; d = a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow d \\ + b d &\rightarrow c b \\ + a c &\rightarrow d a \\ + c a d c a &\rightarrow a d c d^{2} a \\ + c a d a d c d^{2} a &\rightarrow a d c d^{2} a d c a \\ + c a d c d &\rightarrow a d c d^{3} \\ + a^{2} d c d^{2} a &\rightarrow d a^{2} d c a \\ + c a d a d c d^{3} &\rightarrow a d c d^{2} a d c d \\ + a^{2} d c d^{3} &\rightarrow d a^{2} d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/166.md b/docs/examples/1-relation/166.md new file mode 100644 index 0000000..ceba756 --- /dev/null +++ b/docs/examples/1-relation/166.md @@ -0,0 +1,52 @@ +#\#166 abbabaaaba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a^{3} b a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d a^{2} c = c^{2},\; c = b a,\; d = b c^{2},\; e = d a^{2} d +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < c < e < d < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + d &\rightarrow a e a^{2} c \\ + c a e a^{2} c &\rightarrow a e \\ + a e a^{2} c a^{2} e &\rightarrow e a^{3} e a^{2} c \\ + a e a^{2} c a c^{2} &\rightarrow e a^{2} c \\ + a^{2} e a^{2} c a^{2} c &\rightarrow c^{2} \\ + a e a^{2} c a^{3} e a^{2} c &\rightarrow e \\ + b c^{2} &\rightarrow a e a^{2} c \\ + a e a^{2} c^{3} &\rightarrow e a c^{2} a^{2} c \\ + a e a^{2} c a e &\rightarrow e a c^{2} a^{3} e a^{2} c \\ + a e a^{2} c a c a e &\rightarrow e a^{3} e \\ + a^{2} e a^{2} c a^{3} e &\rightarrow c a e \\ + a e a^{3} e &\rightarrow e a c^{2} a c^{2} \\ + c e &\rightarrow e a c^{2} a c^{2} a^{2} c \\ + a e c^{2} &\rightarrow e a c^{2} a c^{2} a^{2} c a c^{2} a^{2} c \\ + a e a e &\rightarrow e a c^{2} a c^{2} a^{2} c a c^{2} a^{3} e a^{2} c \\ + a e a c a e &\rightarrow e a c^{2} a c^{2} a^{2} c a^{3} e \\ + a e^{2} &\rightarrow e a c^{2} a c^{2} a^{2} c a c^{2} a c^{2} a^{3} e a^{2} c \\ + a e a^{2} c^{2} a e &\rightarrow e a c^{2} a^{3} e \\ + a e a^{2} e &\rightarrow e a c^{2} a c^{2} a^{2} c a^{3} e a^{2} c \\ + a e a c^{2} &\rightarrow e a c^{2} a c^{2} a^{2} c a^{2} c \\ + b c a e &\rightarrow e a c^{2} a c^{2} \\ + b e &\rightarrow e a c^{2} a c^{2} a^{2} c a^{2} c a^{3} e a^{2} c \\ + a e c a e &\rightarrow e a c^{2} a c^{2} a^{2} c a c^{2} a^{3} e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/168.md b/docs/examples/1-relation/168.md new file mode 100644 index 0000000..662aff3 --- /dev/null +++ b/docs/examples/1-relation/168.md @@ -0,0 +1,34 @@ +#\#168 baaabbbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{4} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a d a = c,\; c = a^{2},\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{4} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a^{2} &\rightarrow d a^{3} d a^{2} d a^{2} d a \\ + d a^{3} d a^{2} d a^{2} d a^{2} d a &\rightarrow a^{2} \\ + d a^{3} d a^{2} d a^{2} d a^{4} &\rightarrow a^{4} d a^{2} d a^{2} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/169.md b/docs/examples/1-relation/169.md new file mode 100644 index 0000000..50dc279 --- /dev/null +++ b/docs/examples/1-relation/169.md @@ -0,0 +1,47 @@ +#\#169 abbaababaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b a b a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b d b c = d,\; c = a^{2},\; d = b c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b c b a &\rightarrow d \\ + a b d b c &\rightarrow d \\ + a^{2} &\rightarrow c \\ + b c b d &\rightarrow d b d b c \\ + b c b c &\rightarrow d a \\ + a b d^{2} &\rightarrow d b a \\ + a c &\rightarrow c a \\ + c b d b c &\rightarrow a d \\ + b c d &\rightarrow d a b a \\ + d b d^{2} a &\rightarrow b a d \\ + a b d b a d &\rightarrow d b d b c \\ + c b d b a d &\rightarrow a d b d b c \\ + c b d^{2} &\rightarrow a d b a \\ + b a d b d b a d &\rightarrow d b d^{3} b d b c \\ + b a d b d b c &\rightarrow d b d^{3} \\ + b a d b d^{2} &\rightarrow d b d^{3} b a \\ + b c b^{2} a d &\rightarrow d b d b a d b c \\ + d b d^{2} c &\rightarrow b a d a \\ + b a b a d &\rightarrow d b d^{3} b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/170.md b/docs/examples/1-relation/170.md new file mode 100644 index 0000000..c5424e7 --- /dev/null +++ b/docs/examples/1-relation/170.md @@ -0,0 +1,42 @@ +#\#170 baaabbbaa=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{3} a^{2} = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c d c = c a,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{2} d a^{2} &\rightarrow a^{3} \\ + a b^{3} &\rightarrow d \\ + a b^{2} a^{3} &\rightarrow d a^{2} d a^{2} \\ + b a^{2} d a d &\rightarrow a^{2} d \\ + a b^{2} a^{2} d &\rightarrow d a^{2} d a d \\ + a b a^{2} d &\rightarrow d a^{2} d a d a d \\ + a b a^{3} &\rightarrow d a^{2} d a d a^{2} \\ + d a^{2} d a d a d a d &\rightarrow a^{3} d \\ + d a^{2} d a d a d a^{2} &\rightarrow a^{4} \\ + b a^{5} d &\rightarrow a^{3} d a d a d a d \\ + b a^{6} &\rightarrow a^{3} d a d a d a^{2} \\ + d a^{2} d a d a^{4} d &\rightarrow a^{4} d a d a d a d \\ + d a^{2} d a d a^{5} &\rightarrow a^{4} d a d a d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/171.md b/docs/examples/1-relation/171.md new file mode 100644 index 0000000..c0dff01 --- /dev/null +++ b/docs/examples/1-relation/171.md @@ -0,0 +1,42 @@ +#\#171 baaabbabaa=aaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a b a^{2} = a^{3} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a c a b a^{2} = a c a,\; c = a^{2} b^{2},\; d = b a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a^{2} b &\rightarrow d \\ + b a c a b a^{2} &\rightarrow a c a \\ + a^{2} b^{2} &\rightarrow c \\ + b a^{2} d &\rightarrow d a^{2} b \\ + b a c a b a c &\rightarrow a c^{2} \\ + b a c a d &\rightarrow a c a b \\ + b c &\rightarrow d b \\ + a^{2} b a c a &\rightarrow c a c a b a^{2} \\ + a^{2} b d &\rightarrow c a^{2} b \\ + b a c a^{2} c a &\rightarrow a c^{2} a b a^{2} \\ + b a c a^{2} c^{2} &\rightarrow a c^{2} a b a c \\ + a^{2} b a c^{2} &\rightarrow c a c a b a c \\ + a^{3} c a &\rightarrow c a c a d a^{2} \\ + a^{3} c^{2} &\rightarrow c a c a d a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/172.md b/docs/examples/1-relation/172.md new file mode 100644 index 0000000..f54c412 --- /dev/null +++ b/docs/examples/1-relation/172.md @@ -0,0 +1,34 @@ +#\#172 baaabbbba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{4} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a d a = c a,\; c = a^{2},\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{4} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a^{3} &\rightarrow d a^{3} d a d a d a \\ + d a^{3} d a d a d a d a &\rightarrow a^{3} \\ + d a^{3} d a d a d a^{4} &\rightarrow a^{5} d a d a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/173.md b/docs/examples/1-relation/173.md new file mode 100644 index 0000000..1bca07a --- /dev/null +++ b/docs/examples/1-relation/173.md @@ -0,0 +1,41 @@ +#\#173 baabbabbba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a b^{3} a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d c = a,\; c = b^{2} a,\; d = c b +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + c b &\rightarrow d \\ + b a &\rightarrow c a d c \\ + d c a d c &\rightarrow c^{2} \\ + c a c^{2} &\rightarrow a \\ + b c a d c &\rightarrow c \\ + d a &\rightarrow c^{2} a d c \\ + b c a d^{2} &\rightarrow d \\ + c a c a &\rightarrow a^{2} c^{2} \\ + d c a d^{2} &\rightarrow c d \\ + a b &\rightarrow c a c d \\ + b c a c d &\rightarrow c a d^{2} \\ + d c a c d &\rightarrow c^{2} a d^{2} \\ + c a^{2} &\rightarrow a^{2} c^{4} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/174.md b/docs/examples/1-relation/174.md new file mode 100644 index 0000000..f25f653 --- /dev/null +++ b/docs/examples/1-relation/174.md @@ -0,0 +1,40 @@ +#\#174 ababaaaba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{3} b a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c^{2} a^{2} c = c^{2},\; c = b a,\; d = a^{3} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a^{3} b &\rightarrow d \\ + a d a b d a &\rightarrow d a b a \\ + a d a b d^{2} &\rightarrow d a b d \\ + b a b d &\rightarrow d a b d^{4} \\ + b a b a &\rightarrow d a b d^{3} a \\ + b d a b d^{3} &\rightarrow d a b d^{3} a b d \\ + b d a b d^{2} a &\rightarrow d a b d^{3} a b a \\ + a d^{2} a b d^{2} a b d &\rightarrow d^{2} a b d^{5} \\ + a d^{2} a b d^{2} a b a &\rightarrow d^{2} a b d^{4} a \\ + a d^{2} a b d^{3} a b d^{3} &\rightarrow d^{2} a b d^{4} a b d \\ + a d^{2} a b d^{3} a b d^{2} a &\rightarrow d^{2} a b d^{4} a b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/175.md b/docs/examples/1-relation/175.md new file mode 100644 index 0000000..8fcd28b --- /dev/null +++ b/docs/examples/1-relation/175.md @@ -0,0 +1,44 @@ +#\#175 baaabaabaa=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a^{2} b a^{2} = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a = a^{2} c a,\; c = b^{2},\; d = a^{2} b a^{2} b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a d a &\rightarrow a^{2} c a \\ + a^{2} b a^{2} b a &\rightarrow d \\ + b c &\rightarrow c b \\ + b a^{2} c a &\rightarrow c a d a \\ + a^{2} b d &\rightarrow d a b a \\ + a^{2} b a^{4} c a &\rightarrow d^{2} a \\ + b a d^{2} &\rightarrow a^{2} c d \\ + b a^{2} c d &\rightarrow c a d^{2} \\ + a^{2} b a^{4} c d &\rightarrow d^{3} \\ + a^{2} c a d a d a &\rightarrow d a c a \\ + b d a c a &\rightarrow c a d a d a d a \\ + a^{2} c a d a d^{2} &\rightarrow d a c d \\ + a^{2} b a^{2} d a c a &\rightarrow d^{2} a d a d a \\ + b d a c d &\rightarrow c a d a d a d^{2} \\ + a^{2} b a^{2} d a c d &\rightarrow d^{2} a d a d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/176.md b/docs/examples/1-relation/176.md new file mode 100644 index 0000000..25139f7 --- /dev/null +++ b/docs/examples/1-relation/176.md @@ -0,0 +1,36 @@ +#\#176 bababbbba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b^{4} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d = a c,\; c = b a,\; d = b^{3} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{3} c &\rightarrow d \\ + b a &\rightarrow c \\ + a c &\rightarrow c^{2} d \\ + b d c d &\rightarrow d c \\ + b c^{2} &\rightarrow d c d^{2} \\ + d c d^{3} &\rightarrow c^{2} \\ + b d c^{3} &\rightarrow d c^{2} d^{3} \\ + d c d^{2} c^{2} &\rightarrow c^{3} d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/177.md b/docs/examples/1-relation/177.md new file mode 100644 index 0000000..c322fb3 --- /dev/null +++ b/docs/examples/1-relation/177.md @@ -0,0 +1,38 @@ +#\#177 abbababbaa=bababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a b^{2} a^{2} = b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e = c^{3},\; c = b a,\; d = b c,\; e = d c d a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < c < e < d < b\): + +\( +\begin{aligned} + b c &\rightarrow d \\ + b a &\rightarrow c \\ + d c d a &\rightarrow e \\ + a e &\rightarrow c^{3} \\ + d c^{2} &\rightarrow c e \\ + c e^{2} c &\rightarrow e^{2} \\ + d c e^{2} &\rightarrow c e^{3} c \\ + c e^{4} &\rightarrow e^{4} c \\ + d e^{2} &\rightarrow c e^{3} c^{2} \\ + b e^{2} &\rightarrow c e^{3} c^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/179.md b/docs/examples/1-relation/179.md new file mode 100644 index 0000000..f640ab8 --- /dev/null +++ b/docs/examples/1-relation/179.md @@ -0,0 +1,38 @@ +#\#179 babababbba=abbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a b^{3} a = a b^{2} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c^{3} e = a d,\; c = b a,\; d = b c^{2},\; e = b^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(e < d < c < a < b\): + +\( +\begin{aligned} + b^{2} c &\rightarrow e \\ + b a &\rightarrow c \\ + b c^{2} &\rightarrow d \\ + a d &\rightarrow c^{3} e \\ + c d &\rightarrow d c e \\ + b d &\rightarrow e c \\ + b e c^{2} e &\rightarrow e d \\ + e c^{2} e c e &\rightarrow d^{2} \\ + b e d c e c e d &\rightarrow e d c^{2} e c e \\ + e c^{2} e d c e d &\rightarrow d^{2} c^{2} e c e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/180.md b/docs/examples/1-relation/180.md new file mode 100644 index 0000000..b28b7d2 --- /dev/null +++ b/docs/examples/1-relation/180.md @@ -0,0 +1,36 @@ +#\#180 babababbba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a b^{3} a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{3} d = a c^{2},\; c = b a,\; d = b^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} c &\rightarrow d \\ + b a &\rightarrow c \\ + a c^{2} &\rightarrow c^{3} d \\ + b c^{3} &\rightarrow d c^{2} d \\ + d c^{2} d^{2} &\rightarrow c^{3} \\ + b d c^{2} d &\rightarrow d c^{2} \\ + d c^{2} d c^{3} &\rightarrow c^{5} d^{2} \\ + b d c^{5} &\rightarrow d c^{4} d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/181.md b/docs/examples/1-relation/181.md new file mode 100644 index 0000000..cff152d --- /dev/null +++ b/docs/examples/1-relation/181.md @@ -0,0 +1,36 @@ +#\#181 aabbaabaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{2} a^{2} b a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} d a = c a c,\; c = b a,\; d = b c a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a c &\rightarrow a^{2} d a \\ + c a^{3} d a &\rightarrow a^{2} d a^{2} c \\ + c a d &\rightarrow a^{2} d a^{2} d a \\ + a^{2} d a^{2} d a^{2} &\rightarrow d \\ + b d &\rightarrow d a d a^{2} \\ + a^{2} d a^{2} d a d &\rightarrow d a d a^{2} d a^{2} \\ + a^{2} d^{2} &\rightarrow d^{2} a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/182.md b/docs/examples/1-relation/182.md new file mode 100644 index 0000000..e02b54b --- /dev/null +++ b/docs/examples/1-relation/182.md @@ -0,0 +1,42 @@ +#\#182 baaabbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{3} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c d a = c,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{2} d a &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + a b^{2} a^{2} &\rightarrow d a^{2} d a \\ + b a^{2} d^{2} &\rightarrow a d \\ + a b^{2} a d &\rightarrow d a^{2} d^{2} \\ + a b a d &\rightarrow d a^{2} d a d^{2} \\ + a b a^{2} &\rightarrow d a^{2} d a d a \\ + d a^{2} d a d a d^{2} &\rightarrow a^{2} d \\ + d a^{2} d a d a d a &\rightarrow a^{3} \\ + b a^{4} d &\rightarrow a^{3} d a d a d^{2} \\ + b a^{5} &\rightarrow a^{3} d a d a d a \\ + d a^{2} d a d a^{3} d &\rightarrow a^{4} d a d a d^{2} \\ + d a^{2} d a d a^{4} &\rightarrow a^{4} d a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/183.md b/docs/examples/1-relation/183.md new file mode 100644 index 0000000..135b91c --- /dev/null +++ b/docs/examples/1-relation/183.md @@ -0,0 +1,39 @@ +#\#183 abbabbabba=bbabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a b^{2} a = b^{2} a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{3} a = d a,\; c = a b^{2},\; d = b^{2} c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b^{2} c a &\rightarrow d \\ + d a &\rightarrow c^{3} a \\ + a b^{2} &\rightarrow c \\ + b^{2} c^{2} &\rightarrow d b^{2} \\ + d c &\rightarrow c^{4} \\ + a b d &\rightarrow c b c a \\ + a d &\rightarrow c^{2} a \\ + a b c^{4} &\rightarrow c b c a c \\ + a b c^{3} a &\rightarrow c b c a^{2} \\ + a c^{4} &\rightarrow c^{2} a c \\ + a c^{3} a &\rightarrow c^{2} a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/184.md b/docs/examples/1-relation/184.md new file mode 100644 index 0000000..4d5a213 --- /dev/null +++ b/docs/examples/1-relation/184.md @@ -0,0 +1,61 @@ +#\#184 abababbaba=bababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b^{2} a b a = b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d = c^{3},\; c = b a,\; d = c b c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c b c^{2} &\rightarrow d \\ + c^{3} &\rightarrow a c d \\ + a c d^{2} &\rightarrow d c \\ + c a c d &\rightarrow a c d c \\ + d b c^{2} &\rightarrow c b c d \\ + a c^{2} b c d &\rightarrow c^{2} d \\ + b c^{2} d &\rightarrow a c d b c d \\ + b d c &\rightarrow c^{2} d^{2} \\ + a c d c b c d &\rightarrow d^{2} \\ + a c d c d &\rightarrow c d c \\ + b c d c &\rightarrow c^{2} d c d \\ + b d a c d &\rightarrow c^{2} d^{2} c^{2} \\ + c^{2} d c b c d &\rightarrow b d^{2} \\ + a c d c^{2} d &\rightarrow c^{2} d c \\ + b c d b d^{2} &\rightarrow c^{2} d c d c d c b c d \\ + b c d a c d &\rightarrow c^{2} d c d c^{2} \\ + b d b d^{2} &\rightarrow c^{2} d^{2} c d c b c d \\ + b d^{2} b d^{2} &\rightarrow c^{2} d^{2} c^{2} d c d c b c d \\ + b d^{2} c &\rightarrow c^{2} d^{2} c^{2} d \\ + b d^{2} a c d &\rightarrow c^{2} d^{2} c^{2} d c^{2} \\ + b^{2} d^{2} &\rightarrow c^{2} d c^{2} d b c d \\ + d c^{2} b c d &\rightarrow c b d^{2} \\ + b c d^{2} &\rightarrow a c d b d^{2} \\ + a c^{2} b d^{2} &\rightarrow c d^{2} \\ + d a c d b c d &\rightarrow a c d c b d^{2} \\ + b d^{3} &\rightarrow c^{2} d a c d c b d^{2} \\ + a^{2} c d c^{2} b d^{2} &\rightarrow c^{2} d^{2} \\ + b c b d^{2} &\rightarrow c^{2} d^{2} c b c d \\ + b c^{2} b d^{2} &\rightarrow c^{2} d c d c b c d \\ + a c d a c d a c d b d^{2} &\rightarrow c^{2} d^{3} \\ + a c d a c d c b d^{2} &\rightarrow d^{3} \\ + a c d a c d c^{2} b d^{2} &\rightarrow c d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/185.md b/docs/examples/1-relation/185.md new file mode 100644 index 0000000..effc0dc --- /dev/null +++ b/docs/examples/1-relation/185.md @@ -0,0 +1,34 @@ +#\#185 baaabbba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{3} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a d a = c a,\; c = a^{2},\; d = b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{3} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a^{3} &\rightarrow d a^{3} d a d a \\ + d a^{3} d a d a d a &\rightarrow a^{3} \\ + d a^{3} d a d a^{4} &\rightarrow a^{5} d a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/186.md b/docs/examples/1-relation/186.md new file mode 100644 index 0000000..1a54051 --- /dev/null +++ b/docs/examples/1-relation/186.md @@ -0,0 +1,61 @@ +#\#186 baabaaaba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{3} b a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d c = a c a,\; c = b a,\; d = c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c a &\rightarrow c a d c \\ + c a^{2} &\rightarrow d \\ + b c a d c &\rightarrow c^{2} a \\ + a c^{2} a d c &\rightarrow c a d c^{2} a \\ + c a d c a &\rightarrow a d \\ + c^{2} a d c d c &\rightarrow d c a \\ + b c a d^{2} &\rightarrow c d a \\ + a^{2} d &\rightarrow c a d c d c a \\ + a d a &\rightarrow c a d^{2} \\ + c^{2} a d c d^{2} &\rightarrow d^{2} a \\ + c^{2} a d c d a d &\rightarrow d^{3} c a \\ + a d^{2} c a &\rightarrow c^{2} a d^{3} \\ + a c^{2} a d^{2} &\rightarrow c a d c d a \\ + a c^{3} a d^{3} &\rightarrow c a d c d^{2} c a \\ + c a d c^{2} a d c &\rightarrow a d c a \\ + b c^{2} a d^{3} &\rightarrow c d^{2} c a \\ + b c^{3} a d^{3} &\rightarrow c d c a d c \\ + a d^{2} a d &\rightarrow c^{2} a d^{4} c a \\ + a d^{2} c^{2} a d^{2} &\rightarrow c^{2} a d^{4} a \\ + c^{2} a d^{3} c^{3} a d^{3} &\rightarrow a d^{4} c a d c \\ + a d^{2} c^{2} a d c &\rightarrow c^{2} a d^{3} c a \\ + a d^{2} c^{3} a d^{3} &\rightarrow c^{2} a d^{5} c a \\ + c a d c^{2} a d^{2} &\rightarrow a d^{2} a \\ + a d c^{2} a d^{3} &\rightarrow c a d^{4} c a \\ + a d c a d^{2} &\rightarrow c a d^{3} a \\ + a d c a d c &\rightarrow c^{3} a d^{3} \\ + a d c^{3} a d^{3} &\rightarrow c a d^{3} c a d c \\ + c^{2} a d^{3} a &\rightarrow a d^{3} \\ + c^{2} a d^{3} c a d c &\rightarrow a d^{3} c a \\ + c^{2} a d^{3} c^{2} a d^{3} &\rightarrow a d^{5} c a \\ + c^{2} a d^{3} c a d^{2} &\rightarrow a d^{4} a \\ + c^{4} a d^{3} &\rightarrow a d^{2} c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/187.md b/docs/examples/1-relation/187.md new file mode 100644 index 0000000..11af1d7 --- /dev/null +++ b/docs/examples/1-relation/187.md @@ -0,0 +1,47 @@ +#\#187 babbababa=abba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a b a b a = a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d b a = a b^{2} a,\; c = b a b,\; d = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + a b^{2} a &\rightarrow d b a \\ + c^{2} a &\rightarrow d \\ + b a c &\rightarrow c a b \\ + b d b a &\rightarrow c b a \\ + d b^{2} a &\rightarrow c^{2} d b a \\ + a b c &\rightarrow d c \\ + b d c &\rightarrow c^{2} \\ + a b d &\rightarrow d^{2} \\ + d b c &\rightarrow c^{2} d c \\ + b d^{2} &\rightarrow c d \\ + a c b a &\rightarrow d^{2} b a \\ + a c^{2} &\rightarrow d^{2} c \\ + d b d &\rightarrow c^{2} d^{2} \\ + a d &\rightarrow d^{2} c a \\ + a c d &\rightarrow d^{3} \\ + c^{2} d^{2} b a &\rightarrow d c b a \\ + c^{2} d^{3} &\rightarrow d c d \\ + c^{2} d^{2} c &\rightarrow d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/188.md b/docs/examples/1-relation/188.md new file mode 100644 index 0000000..a10fa1c --- /dev/null +++ b/docs/examples/1-relation/188.md @@ -0,0 +1,38 @@ +#\#188 abbaaababa=baabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{3} b a b a = b a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c a^{2} c^{2} = c d c a,\; c = b a,\; d = a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow d \\ + b d &\rightarrow c b \\ + a c &\rightarrow d a \\ + c d c a &\rightarrow d c a d^{2} a \\ + c d c d &\rightarrow d c a d^{3} \\ + a d c a d^{2} a &\rightarrow d a d c a \\ + c d^{2} c a d^{3} &\rightarrow d c a d^{3} c d \\ + c d^{2} c a d^{2} a &\rightarrow d c a d^{3} c a \\ + a d c a d^{3} &\rightarrow d a d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/189.md b/docs/examples/1-relation/189.md new file mode 100644 index 0000000..adf8145 --- /dev/null +++ b/docs/examples/1-relation/189.md @@ -0,0 +1,42 @@ +#\#189 ababaabbaa=baabaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{2} b^{2} a^{2} = b a^{2} b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e a = c a c a c,\; c = b a,\; d = a b,\; e = c^{2} d c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(e < c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow d \\ + c a c a c &\rightarrow a e a \\ + c^{2} d c &\rightarrow e \\ + d a &\rightarrow a c \\ + c b &\rightarrow b d \\ + c a^{2} e a &\rightarrow a e a^{2} c \\ + c a c a e &\rightarrow a e a c d c \\ + c^{2} a c e a &\rightarrow e a c a c \\ + c^{2} d e &\rightarrow e c d c \\ + c a^{2} e d &\rightarrow a e a d^{2} \\ + c a c d^{2} &\rightarrow a e d \\ + c^{2} d b d &\rightarrow e b \\ + c^{2} a c e d &\rightarrow e a c d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/190.md b/docs/examples/1-relation/190.md new file mode 100644 index 0000000..81fb81a --- /dev/null +++ b/docs/examples/1-relation/190.md @@ -0,0 +1,34 @@ +#\#190 abaaabbaba=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b^{2} a b a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d b c^{2} = c^{2} a^{2},\; c = b a,\; d = a c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < b < a\): + +\( +\begin{aligned} + a c a^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + c^{2} a^{2} &\rightarrow d b c^{2} \\ + a c a d &\rightarrow d c a^{2} \\ + c^{2} a d &\rightarrow d b c d b c^{2} \\ + b d &\rightarrow d b c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/191.md b/docs/examples/1-relation/191.md new file mode 100644 index 0000000..2dd58af --- /dev/null +++ b/docs/examples/1-relation/191.md @@ -0,0 +1,40 @@ +#\#191 baaaaaabba=aaaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{6} b^{2} a = a^{6} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{3} b^{2} a = c^{3},\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{5} d a &\rightarrow a^{6} \\ + a b a^{6} &\rightarrow d a^{5} d a \\ + b a^{5} d^{2} &\rightarrow a^{5} d \\ + a b a^{5} d &\rightarrow d a^{5} d^{2} \\ + d a^{5} d^{3} &\rightarrow a^{6} d \\ + d a^{5} d^{2} a &\rightarrow a^{7} \\ + b a^{11} d &\rightarrow a^{10} d^{3} \\ + b a^{12} &\rightarrow a^{10} d^{2} a \\ + d a^{5} d a^{6} d &\rightarrow a^{11} d^{3} \\ + d a^{5} d a^{7} &\rightarrow a^{11} d^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/192.md b/docs/examples/1-relation/192.md new file mode 100644 index 0000000..d8c0dd3 --- /dev/null +++ b/docs/examples/1-relation/192.md @@ -0,0 +1,34 @@ +#\#192 baaabbbbaa=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{4} a^{2} = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a d c = c a,\; c = a^{2},\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{4} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a^{3} &\rightarrow d a^{3} d a^{2} d a^{2} d a^{2} \\ + d a^{3} d a^{2} d a^{2} d a^{2} d a^{2} &\rightarrow a^{3} \\ + d a^{3} d a^{2} d a^{2} d a^{5} &\rightarrow a^{4} d a^{2} d a^{2} d a^{2} d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/194.md b/docs/examples/1-relation/194.md new file mode 100644 index 0000000..526858f --- /dev/null +++ b/docs/examples/1-relation/194.md @@ -0,0 +1,35 @@ +#\#194 baaabbaaba=aabbbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a^{2} b a = a^{2} b^{3} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{2} b c a c = a^{2} d a,\; c = b a,\; d = b^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b^{2} c &\rightarrow d \\ + b a &\rightarrow c \\ + c a^{2} b c a c &\rightarrow a^{2} d a \\ + b c a d a &\rightarrow d a^{2} b c a c \\ + c a^{2} b c a^{3} d a &\rightarrow a^{2} d a^{3} b c a c \\ + b d a^{2} b c a c &\rightarrow d a d a \\ + b d a^{2} b c a^{3} d a &\rightarrow d a d a^{3} b c a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/195.md b/docs/examples/1-relation/195.md new file mode 100644 index 0000000..3deb97d --- /dev/null +++ b/docs/examples/1-relation/195.md @@ -0,0 +1,44 @@ +#\#195 baabbababa=abbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a b a b a = a b^{2} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a b a = d a,\; c = b a b,\; d = a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + a b c &\rightarrow d \\ + b d &\rightarrow c^{2} \\ + b a c &\rightarrow c a b \\ + b a d a &\rightarrow c a d a b a \\ + c a d a c a &\rightarrow d a \\ + b a d^{2} &\rightarrow c a d a c^{2} \\ + c a d a d a &\rightarrow d a d a c a \\ + c a d a c d &\rightarrow d^{2} \\ + a c^{2} a &\rightarrow d a d a c a \\ + c a d^{2} a d a d a &\rightarrow d^{2} a \\ + c a d a d^{2} &\rightarrow d a d a c d \\ + a c^{2} d &\rightarrow d a d a c d \\ + a c d^{2} &\rightarrow d a d a d^{2} \\ + a c d a &\rightarrow d a d a d a \\ + c a d^{2} a d a d^{2} &\rightarrow d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/196.md b/docs/examples/1-relation/196.md new file mode 100644 index 0000000..b611619 --- /dev/null +++ b/docs/examples/1-relation/196.md @@ -0,0 +1,43 @@ +#\#196 babbababba=abbabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a b a b^{2} a = a b^{2} a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d a = a b c b a,\; c = b a b,\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + a b^{2} &\rightarrow d \\ + b a c &\rightarrow c a b \\ + b d &\rightarrow c b \\ + a b c &\rightarrow d a b \\ + c^{2} d^{2} &\rightarrow d^{3} \\ + c^{2} d a &\rightarrow d^{2} a \\ + b a d^{2} a &\rightarrow c d a c b a \\ + b a d^{3} &\rightarrow c d a c^{2} b \\ + a c^{2} b a &\rightarrow d^{2} a c b a \\ + a c^{3} &\rightarrow d^{2} a c^{2} \\ + a d^{3} a &\rightarrow d^{4} a d^{2} a \\ + a d^{4} &\rightarrow d^{4} a d^{3} \\ + a c d^{2} a &\rightarrow d^{2} a d^{2} a \\ + a c d^{3} &\rightarrow d^{2} a d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/198.md b/docs/examples/1-relation/198.md new file mode 100644 index 0000000..5b6c396 --- /dev/null +++ b/docs/examples/1-relation/198.md @@ -0,0 +1,36 @@ +#\#198 baaaaabbaa=abaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a^{2} = a b a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d = a c,\; c = b a^{4},\; d = b^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{2} a^{2} &\rightarrow d \\ + b a^{4} &\rightarrow c \\ + a c &\rightarrow c a d \\ + b c &\rightarrow d a^{2} \\ + d a^{3} d a d a d a d &\rightarrow c^{2} \\ + b d a^{3} d a d &\rightarrow d c \\ + b d a^{3} d c a d c &\rightarrow d c a^{3} d a d a d a d \\ + d a^{3} d a d a d c a d c &\rightarrow c^{2} a^{3} d a d a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/199.md b/docs/examples/1-relation/199.md new file mode 100644 index 0000000..96d0240 --- /dev/null +++ b/docs/examples/1-relation/199.md @@ -0,0 +1,51 @@ +#\#199 ababbababa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b a b a = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e c = d c,\; c = b a,\; d = c a,\; e = c b c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c b c^{2} &\rightarrow e \\ + c a &\rightarrow d \\ + a e c &\rightarrow d c \\ + b d c &\rightarrow c e c \\ + c b c e &\rightarrow e b c^{2} \\ + c b c d &\rightarrow e a \\ + c d c &\rightarrow d e c \\ + a e^{2} &\rightarrow d e \\ + a e d &\rightarrow d^{2} \\ + b d e &\rightarrow c e^{2} \\ + b d^{2} &\rightarrow c e d \\ + c^{2} e^{2} c &\rightarrow e a c \\ + c d e &\rightarrow d e^{2} \\ + c d^{2} &\rightarrow d e d \\ + c b e a c &\rightarrow e^{3} c \\ + c^{2} e^{3} &\rightarrow e a e \\ + c^{2} e^{2} d &\rightarrow e a d \\ + c b e a e &\rightarrow e^{4} \\ + c b e a d &\rightarrow e^{3} d \\ + c b e d c &\rightarrow e^{4} c \\ + c b e d e &\rightarrow e^{5} \\ + c b e d^{2} &\rightarrow e^{4} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/200.md b/docs/examples/1-relation/200.md new file mode 100644 index 0000000..45998ce --- /dev/null +++ b/docs/examples/1-relation/200.md @@ -0,0 +1,33 @@ +#\#200 ababaaabaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{3} b a^{2} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c^{2} a^{2} c a = c^{2},\; c = d,\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow d \\ + b a &\rightarrow d \\ + a d^{2} a^{2} d a &\rightarrow d^{2} \\ + b d^{2} &\rightarrow d^{3} a^{2} d a \\ + a d^{2} a^{2} d^{3} &\rightarrow d^{4} a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/203.md b/docs/examples/1-relation/203.md new file mode 100644 index 0000000..35f669f --- /dev/null +++ b/docs/examples/1-relation/203.md @@ -0,0 +1,53 @@ +#\#203 baabbabbaa=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a b^{2} a^{2} = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b d a = a b a,\; c = a b^{2},\; d = a c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a c^{2} a &\rightarrow d \\ + a b a &\rightarrow b d a \\ + a b^{2} &\rightarrow c \\ + a c^{2} d &\rightarrow d c^{2} a \\ + a c^{3} &\rightarrow d b^{2} \\ + a c^{2} b d a &\rightarrow d b a \\ + a b c &\rightarrow b d c \\ + b d b d a &\rightarrow c d a \\ + a b d &\rightarrow b d^{2} \\ + a c d a &\rightarrow b d^{2} b d a \\ + a c^{2} b d^{2} &\rightarrow d b d \\ + a c^{2} b d c &\rightarrow d b c \\ + b d c d a &\rightarrow c d b d a \\ + b d b d^{2} &\rightarrow c d^{2} \\ + b d b d c &\rightarrow c d c \\ + b d c d^{2} &\rightarrow c d b d^{2} \\ + b d c d c &\rightarrow c d b d c \\ + a c d^{2} &\rightarrow b d^{2} b d^{2} \\ + a c d c &\rightarrow b d^{2} b d c \\ + a c d b d a &\rightarrow b d^{2} c d a \\ + a c d b d^{2} &\rightarrow b d^{2} c d^{2} \\ + a c d b d c &\rightarrow b d^{2} c d c \\ + b d c d b d a &\rightarrow c d c d a \\ + b d c d b d^{2} &\rightarrow c d c d^{2} \\ + b d c d b d c &\rightarrow c d c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/204.md b/docs/examples/1-relation/204.md new file mode 100644 index 0000000..af46ba7 --- /dev/null +++ b/docs/examples/1-relation/204.md @@ -0,0 +1,38 @@ +#\#204 aabaaababa=baabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{3} b a b a = b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c d = c a c a,\; c = b a,\; d = a^{2} c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a c a &\rightarrow a^{2} c d \\ + a^{2} c^{2} &\rightarrow d \\ + b d &\rightarrow c a c^{2} \\ + c a^{3} c d &\rightarrow a^{2} c d c a \\ + a^{2} c d a c^{2} &\rightarrow c a c d \\ + a^{2} c a^{2} c d &\rightarrow d a c a \\ + b c a c d &\rightarrow c a c d a c^{2} \\ + c a c^{2} a c d &\rightarrow a^{2} c d a c d a c^{2} \\ + a^{2} c d a c a^{2} c d &\rightarrow c a c d a c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/205.md b/docs/examples/1-relation/205.md new file mode 100644 index 0000000..95c3264 --- /dev/null +++ b/docs/examples/1-relation/205.md @@ -0,0 +1,36 @@ +#\#205 baababbaa=aaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{2} a^{2} = a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c d = a^{2} c,\; c = b a,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow d \\ + c a c &\rightarrow d c d \\ + b c d c d^{2} &\rightarrow d a c \\ + a^{2} c &\rightarrow d c d^{2} \\ + b d c d &\rightarrow d c \\ + a^{2} d c d &\rightarrow d c d^{2} a c \\ + c a d c d &\rightarrow d c d a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/206.md b/docs/examples/1-relation/206.md new file mode 100644 index 0000000..a4f33a6 --- /dev/null +++ b/docs/examples/1-relation/206.md @@ -0,0 +1,38 @@ +#\#206 abababbaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b^{2} a^{2} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c^{2} b c a = c^{2},\; c = b a,\; d = b^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b^{2} a^{2} &\rightarrow d \\ + a b a b a d &\rightarrow b a b a \\ + b^{2} a b a b a &\rightarrow d b a b a d \\ + b^{3} a b a &\rightarrow d b a b a d^{2} \\ + b^{2} a b^{2} a b a &\rightarrow d b a b a d b a d \\ + b^{2} a b a b^{2} a b a &\rightarrow d b a b a d b a b a d \\ + b^{2} a b a d b a b a d &\rightarrow d b a b a d b a b a d b a \\ + b^{2} a d b a b a d &\rightarrow d b a b a d b a d b a \\ + b d b a b a d &\rightarrow d b a b a d^{2} b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/207.md b/docs/examples/1-relation/207.md new file mode 100644 index 0000000..d32193c --- /dev/null +++ b/docs/examples/1-relation/207.md @@ -0,0 +1,43 @@ +#\#207 aababbaaaa=baaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b^{2} a^{4} = b a^{4} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d b c d a = b d^{2} b a,\; c = a b^{2} a,\; d = a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < b < a\): + +\( +\begin{aligned} + a^{2} &\rightarrow d \\ + a b^{2} a &\rightarrow c \\ + b d^{2} b a &\rightarrow d b c d a \\ + a b^{2} d &\rightarrow c a \\ + a b^{2} c &\rightarrow c b^{2} a \\ + a d &\rightarrow d a \\ + a c &\rightarrow d b^{2} a \\ + b d^{2} b d &\rightarrow d b c d^{2} \\ + b d^{2} b c &\rightarrow d b c d c \\ + a b d b c d a &\rightarrow c d a b a \\ + a b d b c d^{2} &\rightarrow c d a b d \\ + a b d b c d c &\rightarrow c d a b c \\ + b d^{3} b c d a &\rightarrow d b c d^{3} b a \\ + b d^{3} b c d^{2} &\rightarrow d b c d^{3} b d \\ + b d^{3} b c d c &\rightarrow d b c d^{3} b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/208.md b/docs/examples/1-relation/208.md new file mode 100644 index 0000000..4fecfd5 --- /dev/null +++ b/docs/examples/1-relation/208.md @@ -0,0 +1,36 @@ +#\#208 abbbabbbaa=babbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{3} a b^{3} a^{2} = b a b^{3} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c b a c b a^{2} = b a c b a,\; c = b^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b^{2} \\ + a b^{3} &\rightarrow d \\ + b d a &\rightarrow d^{2} a^{2} \\ + b d^{2} &\rightarrow d^{2} a d \\ + a d^{2} a d a d a^{2} &\rightarrow d^{2} a \\ + a d^{2} a d a d a d &\rightarrow d^{3} \\ + a d^{2} a d a d^{3} a &\rightarrow d^{4} a d a d a^{2} \\ + a d^{2} a d a d^{4} &\rightarrow d^{4} a d a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/209.md b/docs/examples/1-relation/209.md new file mode 100644 index 0000000..890012e --- /dev/null +++ b/docs/examples/1-relation/209.md @@ -0,0 +1,50 @@ +#\#209 abbaabbbaa=baabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b^{3} a^{2} = b a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d a = b a c a,\; c = a b^{2},\; d = a c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a c a &\rightarrow c d a \\ + a b^{2} &\rightarrow c \\ + a c b a &\rightarrow d \\ + b a c d &\rightarrow c d^{2} \\ + b a c^{2} &\rightarrow c d c \\ + d b^{2} &\rightarrow a c b c \\ + a b c d a &\rightarrow c a c a \\ + a c b d &\rightarrow d c b a \\ + a c^{2} d a &\rightarrow d c a \\ + b d c a &\rightarrow c d c d a \\ + a b c d c &\rightarrow c a c^{2} \\ + a c^{2} d c &\rightarrow d c^{2} \\ + a c^{2} d^{2} &\rightarrow d c d \\ + a b c d^{2} &\rightarrow c a c d \\ + a c b c a c^{2} &\rightarrow d b c d c \\ + a c b c a c d &\rightarrow d b c d^{2} \\ + a c b c a c a &\rightarrow d b c d a \\ + b d c^{2} &\rightarrow c d c d c \\ + b d c d &\rightarrow c d c d^{2} \\ + d b c d c d c &\rightarrow a c b c d c^{2} \\ + d b c d c d^{2} &\rightarrow a c b c d c d \\ + d b c d c d a &\rightarrow a c b c d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/210.md b/docs/examples/1-relation/210.md new file mode 100644 index 0000000..7dc74d5 --- /dev/null +++ b/docs/examples/1-relation/210.md @@ -0,0 +1,42 @@ +#\#210 baaabbaaba=abbabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a^{2} b a = a b^{2} a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d c = a b c^{2} a,\; c = b a,\; d = a b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b c^{2} a &\rightarrow c a d c \\ + a b c a &\rightarrow d \\ + b d &\rightarrow c b c a \\ + b c a d c &\rightarrow c b c^{2} a \\ + a b c d &\rightarrow d b c a \\ + a b c^{2} d &\rightarrow c a d c b c a \\ + a b c^{3} a d c &\rightarrow c a d c b c^{2} a \\ + d b c^{2} a &\rightarrow c a d c d c \\ + d b c^{2} d &\rightarrow c a d c d c b c a \\ + d b c^{3} a d c &\rightarrow c a d c d c b c^{2} a \\ + a c b c^{2} a &\rightarrow d^{2} c \\ + a c b c^{2} d &\rightarrow d^{2} c b c a \\ + a c b c^{3} a d c &\rightarrow d^{2} c b c^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/211.md b/docs/examples/1-relation/211.md new file mode 100644 index 0000000..bc5210e --- /dev/null +++ b/docs/examples/1-relation/211.md @@ -0,0 +1,42 @@ +#\#211 baaaabbba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{3} a = c a,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + b a^{3} d a &\rightarrow a^{3} \\ + a b^{2} a^{3} &\rightarrow d a^{3} d a \\ + b a^{3} d^{2} &\rightarrow a^{2} d \\ + a b^{2} a^{2} d &\rightarrow d a^{3} d^{2} \\ + a b a^{2} d &\rightarrow d a^{3} d a d^{2} \\ + a b a^{3} &\rightarrow d a^{3} d a d a \\ + d a^{3} d a d a d^{2} &\rightarrow a^{3} d \\ + d a^{3} d a d a d a &\rightarrow a^{4} \\ + b a^{6} d &\rightarrow a^{5} d a d a d^{2} \\ + b a^{7} &\rightarrow a^{5} d a d a d a \\ + d a^{3} d a d a^{4} d &\rightarrow a^{6} d a d a d^{2} \\ + d a^{3} d a d a^{5} &\rightarrow a^{6} d a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/213.md b/docs/examples/1-relation/213.md new file mode 100644 index 0000000..ad7abec --- /dev/null +++ b/docs/examples/1-relation/213.md @@ -0,0 +1,33 @@ +#\#213 abbaabbbba=babbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b^{4} a = b a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a^{2} c^{2} a = b a c a^{2},\; c = b^{2},\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b^{2} \\ + b^{4} &\rightarrow d \\ + b a b^{2} a^{2} &\rightarrow a b^{2} a^{2} d a \\ + d a b^{2} a^{2} &\rightarrow a b^{2} a^{2} d a d a d a d a \\ + d b &\rightarrow b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/214.md b/docs/examples/1-relation/214.md new file mode 100644 index 0000000..40b7d7f --- /dev/null +++ b/docs/examples/1-relation/214.md @@ -0,0 +1,113 @@ +#\#214 ababbaaaba=baaababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a^{3} b a = b a^{3} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e = c a^{2} c^{2},\; c = b a,\; d = c b c a,\; e = d a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(e < c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + d a c &\rightarrow e \\ + c b c a &\rightarrow d \\ + c a^{2} c^{2} &\rightarrow a e \\ + c^{2} e &\rightarrow e c \\ + e b c a &\rightarrow d a d \\ + e a^{2} c^{2} &\rightarrow d a^{2} e \\ + c a^{2} c a e &\rightarrow a d a^{2} e \\ + c a^{2} c d &\rightarrow a d a d \\ + c a^{2} c e &\rightarrow a d a e \\ + d a^{2} d a^{2} e &\rightarrow e a^{2} c a e \\ + d a^{2} d a d &\rightarrow e a^{2} c d \\ + d a^{2} e a^{2} c a e &\rightarrow e a^{2} c a d a^{2} e \\ + d a^{2} e a^{2} c d &\rightarrow e a^{2} c a d a d \\ + d a e c &\rightarrow e c e \\ + c^{2} d a^{2} e &\rightarrow e a e \\ + c^{2} d a d &\rightarrow e d \\ + c a^{2} e c &\rightarrow a e^{2} \\ + d a e a e &\rightarrow e c d a^{2} e \\ + d a e d &\rightarrow e c d a d \\ + e a^{2} c e c &\rightarrow d a^{2} e c e \\ + e a^{2} e c &\rightarrow d a^{2} e^{2} \\ + c a^{2} c a d a^{2} e &\rightarrow a e a^{2} c a e \\ + c a^{2} c a d a d &\rightarrow a e a^{2} c d \\ + c a^{2} e d &\rightarrow a e d a d \\ + c a^{2} e a e &\rightarrow a e d a^{2} e \\ + d a^{2} d a e &\rightarrow e a^{2} c e \\ + d a^{2} e a^{2} c a d a^{2} e &\rightarrow e a^{2} c a e a^{2} c a e \\ + d a^{2} e a^{2} c a d a d &\rightarrow e a^{2} c a e a^{2} c d \\ + d a^{2} e a^{2} c e &\rightarrow e a^{2} c a d a e \\ + d a^{2} e d a^{2} e &\rightarrow e a^{2} e a e \\ + d a^{2} e d a d &\rightarrow e a^{2} e d \\ + d a^{2} e c d a^{2} e &\rightarrow e a^{2} c e a e \\ + d a^{2} e c d a d &\rightarrow e a^{2} c e d \\ + d a^{2} e c e a^{2} c a e &\rightarrow e a^{2} c e a d a^{2} e \\ + d a^{2} e c e a^{2} c d &\rightarrow e a^{2} c e a d a d \\ + d a^{2} e c e a^{2} c e &\rightarrow e a^{2} c e a d a e \\ + d a^{2} e c e c d a^{2} e &\rightarrow e a^{2} c e^{2} a e \\ + d a^{2} e c e c d a d &\rightarrow e a^{2} c e^{2} d \\ + d a^{2} e c e c e &\rightarrow e a^{2} c e^{2} c \\ + d a^{2} e^{2} a^{2} c a e &\rightarrow e a^{2} e a d a^{2} e \\ + d a^{2} e^{2} a^{2} c d &\rightarrow e a^{2} e a d a d \\ + d a^{2} e^{2} a^{2} c e &\rightarrow e a^{2} e a d a e \\ + d a^{2} e^{2} c d a^{2} e &\rightarrow e a^{2} e^{2} a e \\ + d a^{2} e^{2} c d a d &\rightarrow e a^{2} e^{2} d \\ + d a^{2} e^{2} c e &\rightarrow e a^{2} e^{2} c \\ + d a e a d a^{2} e &\rightarrow e c e a^{2} c a e \\ + d a e a d a d &\rightarrow e c e a^{2} c d \\ + d a e a d a e &\rightarrow e c e a^{2} c e \\ + c^{2} d a e &\rightarrow e^{2} \\ + c a^{2} c a d a e &\rightarrow a e a^{2} c e \\ + c a^{2} e a d a^{2} e &\rightarrow a e^{2} a^{2} c a e \\ + c a^{2} e a d a d &\rightarrow a e^{2} a^{2} c d \\ + c a^{2} e a d a e &\rightarrow a e^{2} a^{2} c e \\ + d a e^{2} &\rightarrow e c d a e \\ + c a^{2} e^{2} &\rightarrow a e d a e \\ + d a^{2} e a^{2} c a d a e &\rightarrow e a^{2} c a e a^{2} c e \\ + d a^{2} e a^{2} e a e &\rightarrow e a^{2} c a e d a^{2} e \\ + d a^{2} e a^{2} e d &\rightarrow e a^{2} c a e d a d \\ + d a^{2} e c e a^{2} c a d a^{2} e &\rightarrow e a^{2} c e a e a^{2} c a e \\ + d a^{2} e c e a^{2} c a d a d &\rightarrow e a^{2} c e a e a^{2} c d \\ + d a^{2} e c e a^{2} e a e &\rightarrow e a^{2} c e a e d a^{2} e \\ + d a^{2} e c e a^{2} e d &\rightarrow e a^{2} c e a e d a d \\ + d a^{2} e^{2} a^{2} c a d a^{2} e &\rightarrow e a^{2} e a e a^{2} c a e \\ + d a^{2} e^{2} a^{2} c a d a d &\rightarrow e a^{2} e a e a^{2} c d \\ + d a^{2} e^{2} a^{2} e a e &\rightarrow e a^{2} e a e d a^{2} e \\ + d a^{2} e^{2} a^{2} e d &\rightarrow e a^{2} e a e d a d \\ + d a^{2} e c e a^{2} c a d a e &\rightarrow e a^{2} c e a e a^{2} c e \\ + d a^{2} e^{2} a^{2} c a d a e &\rightarrow e a^{2} e a e a^{2} c e \\ + d a^{2} e c d a e &\rightarrow e a^{2} c e^{2} \\ + d a^{2} e d a e &\rightarrow e a^{2} e^{2} \\ + d a^{2} e c e c d a e &\rightarrow e a^{2} c e^{3} \\ + d a^{2} e a^{2} e a d a^{2} e &\rightarrow e a^{2} c a e^{2} a^{2} c a e \\ + d a^{2} e a^{2} e a d a d &\rightarrow e a^{2} c a e^{2} a^{2} c d \\ + d a^{2} e a^{2} e a d a e &\rightarrow e a^{2} c a e^{2} a^{2} c e \\ + d a^{2} e c e a^{2} e a d a^{2} e &\rightarrow e a^{2} c e a e^{2} a^{2} c a e \\ + d a^{2} e c e a^{2} e a d a d &\rightarrow e a^{2} c e a e^{2} a^{2} c d \\ + d a^{2} e c e a^{2} e a d a e &\rightarrow e a^{2} c e a e^{2} a^{2} c e \\ + d a^{2} e^{2} a^{2} e a d a^{2} e &\rightarrow e a^{2} e a e^{2} a^{2} c a e \\ + d a^{2} e^{2} a^{2} e a d a d &\rightarrow e a^{2} e a e^{2} a^{2} c d \\ + d a^{2} e^{2} a^{2} e a d a e &\rightarrow e a^{2} e a e^{2} a^{2} c e \\ + d a^{2} e^{2} c d a e &\rightarrow e a^{2} e^{3} \\ + d a^{2} e a^{2} e^{2} &\rightarrow e a^{2} c a e d a e \\ + d a^{2} e c e a^{2} e^{2} &\rightarrow e a^{2} c e a e d a e \\ + d a^{2} e^{2} a^{2} e^{2} &\rightarrow e a^{2} e a e d a e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/215.md b/docs/examples/1-relation/215.md new file mode 100644 index 0000000..23b127c --- /dev/null +++ b/docs/examples/1-relation/215.md @@ -0,0 +1,34 @@ +#\#215 bababababa=ababbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a b a b a = a b a b^{2} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{5} = d c^{2},\; c = b a,\; d = a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c b &\rightarrow d \\ + d c^{2} &\rightarrow c^{5} \\ + b d &\rightarrow c^{2} b \\ + a c^{2} &\rightarrow d a \\ + b c^{5} &\rightarrow c^{2} b c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/216.md b/docs/examples/1-relation/216.md new file mode 100644 index 0000000..f83455d --- /dev/null +++ b/docs/examples/1-relation/216.md @@ -0,0 +1,42 @@ +#\#216 abbababbaa=bababba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a b^{2} a^{2} = b a b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c b d a^{2} = b d a,\; c = a b,\; d = c^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + c^{2} b &\rightarrow d \\ + a b &\rightarrow c \\ + b d c &\rightarrow d^{2} a^{2} c \\ + b d a &\rightarrow d^{2} a^{3} \\ + b d^{2} &\rightarrow d^{2} a^{2} d \\ + c d c &\rightarrow a d^{2} a^{2} c \\ + c d a &\rightarrow a d^{2} a^{3} \\ + c d^{2} &\rightarrow a d^{2} a^{2} d \\ + a d^{2} a^{2} d a d &\rightarrow d^{3} \\ + a d^{2} a^{2} d a c &\rightarrow d^{2} c \\ + a d^{2} a^{2} d a^{2} &\rightarrow d^{2} a \\ + a d^{2} a^{2} d^{3} c &\rightarrow d^{4} a^{2} d a c \\ + a d^{2} a^{2} d^{3} a &\rightarrow d^{4} a^{2} d a^{2} \\ + a d^{2} a^{2} d^{4} &\rightarrow d^{4} a^{2} d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/217.md b/docs/examples/1-relation/217.md new file mode 100644 index 0000000..26a5ebe --- /dev/null +++ b/docs/examples/1-relation/217.md @@ -0,0 +1,45 @@ +#\#217 babaaabba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{3} b^{2} a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c = a c^{2},\; c = b a,\; d = c a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{2} &\rightarrow c d c \\ + c a^{2} b &\rightarrow d \\ + b c d c &\rightarrow c^{3} \\ + c a^{2} c &\rightarrow d a \\ + a c d &\rightarrow c d^{2} \\ + b c d^{2} &\rightarrow c^{2} d \\ + d a c &\rightarrow c^{2} d^{2} c \\ + d a d &\rightarrow c^{2} d^{3} \\ + d a^{3} b &\rightarrow c a^{2} d \\ + d a^{3} c &\rightarrow c a^{2} d a \\ + c^{2} d^{2} c d &\rightarrow d c d^{2} \\ + c^{2} d^{2} c^{2} &\rightarrow d c d c \\ + a d c d c &\rightarrow c d c d^{2} c^{2} \\ + a d c d^{2} &\rightarrow c d c d^{2} c d \\ + c^{2} d^{3} c d c &\rightarrow d c d c d^{2} c^{2} \\ + c^{2} d^{3} c d^{2} &\rightarrow d c d c d^{2} c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/218.md b/docs/examples/1-relation/218.md new file mode 100644 index 0000000..e0bb1c8 --- /dev/null +++ b/docs/examples/1-relation/218.md @@ -0,0 +1,44 @@ +#\#218 abaababba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c d a = e a,\; c = a b,\; d = a c^{2} b,\; e = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < e < b\): + +\( +\begin{aligned} + b c a &\rightarrow e \\ + e a &\rightarrow c d a \\ + a b &\rightarrow c \\ + a c^{2} b &\rightarrow d \\ + b c^{2} &\rightarrow e b \\ + a e &\rightarrow c^{2} a \\ + a c^{2} e &\rightarrow d c a \\ + e c &\rightarrow c d c \\ + e d &\rightarrow c d^{2} \\ + b c d &\rightarrow c d c^{2} b \\ + a c d a &\rightarrow c^{2} a^{2} \\ + a c d^{2} &\rightarrow c^{2} a d \\ + a c d c &\rightarrow c^{2} a c \\ + a c^{3} d a &\rightarrow d c a^{2} \\ + a c^{3} d^{2} &\rightarrow d c a d \\ + a c^{3} d c &\rightarrow d c a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/219.md b/docs/examples/1-relation/219.md new file mode 100644 index 0000000..88c31f0 --- /dev/null +++ b/docs/examples/1-relation/219.md @@ -0,0 +1,39 @@ +#\#219 bababaaaba=aababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a^{3} b a = a^{2} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c = a^{2} c^{2},\; c = b a,\; d = c^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a^{2} c^{2} &\rightarrow c d c \\ + c^{2} a^{2} &\rightarrow d \\ + b c d c &\rightarrow c a c^{2} \\ + a^{2} c d &\rightarrow c d^{2} \\ + a^{2} d &\rightarrow c d c a^{2} \\ + c^{2} a c d c &\rightarrow d a c^{2} \\ + c^{3} d c &\rightarrow d c^{2} \\ + b c d^{2} &\rightarrow c a c d \\ + c^{2} a c d^{2} &\rightarrow d a c d \\ + c^{3} d^{2} &\rightarrow d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/220.md b/docs/examples/1-relation/220.md new file mode 100644 index 0000000..2b9192f --- /dev/null +++ b/docs/examples/1-relation/220.md @@ -0,0 +1,34 @@ +#\#220 baaabbaaba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a^{2} b a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d = a c,\; c = b a^{2},\; d = b c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a^{2} &\rightarrow c \\ + b c b a &\rightarrow d \\ + a c &\rightarrow c a d \\ + b c a d a d &\rightarrow c^{2} \\ + b c b c a d &\rightarrow d c \\ + b c^{2} &\rightarrow d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/221.md b/docs/examples/1-relation/221.md new file mode 100644 index 0000000..a9f2e28 --- /dev/null +++ b/docs/examples/1-relation/221.md @@ -0,0 +1,43 @@ +#\#221 aaababbaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a b^{2} a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c d = b c a,\; c = a b a,\; d = b^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} c &\rightarrow d \\ + b c a &\rightarrow a^{2} c d \\ + a b a &\rightarrow c \\ + b c^{2} &\rightarrow a^{2} c d b a \\ + b a^{2} c d &\rightarrow d a \\ + a b c &\rightarrow c b a \\ + c b a c &\rightarrow a^{3} c d b a \\ + c b a^{2} &\rightarrow a^{3} c d \\ + c a c d &\rightarrow a d a \\ + b^{2} a^{3} c d &\rightarrow d b a^{2} \\ + b a d a &\rightarrow a^{2} c d c d \\ + a^{3} c d c d &\rightarrow c d a \\ + b a d c &\rightarrow a^{2} c d c d b a \\ + c b c d a &\rightarrow a^{3} c d a c d c d \\ + c b c d c &\rightarrow a^{3} c d a c d c d b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/222.md b/docs/examples/1-relation/222.md new file mode 100644 index 0000000..af23efb --- /dev/null +++ b/docs/examples/1-relation/222.md @@ -0,0 +1,35 @@ +#\#222 aabaaababa=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{3} b a b a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c a^{2} c^{2} = c^{2} a^{2},\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + b d a^{2} &\rightarrow a d a d b a \\ + a b d &\rightarrow d b a \\ + b d a d &\rightarrow a d a d b d \\ + d b a^{2} d &\rightarrow a^{2} d a d b d \\ + d b a^{3} &\rightarrow a^{2} d a d b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/224.md b/docs/examples/1-relation/224.md new file mode 100644 index 0000000..ab2cd33 --- /dev/null +++ b/docs/examples/1-relation/224.md @@ -0,0 +1,36 @@ +#\#224 babababbba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a b^{3} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{3} d = a c,\; c = b a,\; d = b^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} c &\rightarrow d \\ + b a &\rightarrow c \\ + a c &\rightarrow c^{3} d \\ + b c^{2} &\rightarrow d c^{2} d \\ + d c^{2} d c d &\rightarrow c^{2} \\ + b d c^{2} d &\rightarrow d c \\ + d c^{2} d c^{3} &\rightarrow c^{4} d c d \\ + b d c^{4} &\rightarrow d c^{3} d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/225.md b/docs/examples/1-relation/225.md new file mode 100644 index 0000000..5c051a9 --- /dev/null +++ b/docs/examples/1-relation/225.md @@ -0,0 +1,37 @@ +#\#225 abbaabbaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b^{2} a^{2} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d^{2} = c^{2},\; c = b a,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow d \\ + a d^{2} &\rightarrow c^{2} \\ + b c^{2} &\rightarrow c d^{2} \\ + c d^{2} c &\rightarrow d^{3} \\ + b c d^{3} &\rightarrow c d^{4} c \\ + c d^{5} &\rightarrow d^{5} c \\ + b d^{5} c &\rightarrow c d^{4} c d^{2} \\ + b d^{8} &\rightarrow c d^{4} c d^{4} c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/226.md b/docs/examples/1-relation/226.md new file mode 100644 index 0000000..c1fb18e --- /dev/null +++ b/docs/examples/1-relation/226.md @@ -0,0 +1,104 @@ +#\#226 aabaabbaaa=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b^{2} a^{3} = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c b c b^{2} c a = b c a d,\; c = a^{2},\; d = b a,\; e = a b d c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < a < e < d < b\): + +\( +\begin{aligned} + b a &\rightarrow d \\ + a b d c &\rightarrow e \\ + a^{2} &\rightarrow c \\ + b c &\rightarrow d a \\ + d b d c &\rightarrow b e \\ + c b d c &\rightarrow a e \\ + c a &\rightarrow a c \\ + c d a b d a e &\rightarrow d c b e \\ + d c d &\rightarrow c d e \\ + b d c b e &\rightarrow d a d a b d a e \\ + d b d a c &\rightarrow b e a \\ + b e d &\rightarrow d^{2} a d e \\ + d^{2} c b e &\rightarrow c d e a b d a e \\ + e b d c &\rightarrow a b d a e \\ + a b d a c &\rightarrow e a \\ + a d a d e &\rightarrow e d \\ + c b d a c &\rightarrow a e a \\ + c d a d e &\rightarrow a e d \\ + c d e c d &\rightarrow d c^{2} d e \\ + b d c^{2} d e &\rightarrow d a d e c d \\ + b e b e &\rightarrow d^{2} a d a b d a e \\ + b e c d e &\rightarrow d^{2} a d e c d \\ + d b d a d c b e &\rightarrow b e a d a b d a e \\ + d b d a d c^{2} d e &\rightarrow b e a d e c d \\ + d^{2} c^{2} d e &\rightarrow c d e^{2} c d \\ + e b d a c &\rightarrow a b d a e a \\ + a b d a d c b e &\rightarrow e a d a b d a e \\ + a b d a d c^{2} d e &\rightarrow e a d e c d \\ + a d a d a b d a e &\rightarrow e b e \\ + c b d a d c b e &\rightarrow a e a d a b d a e \\ + c b d a d c^{2} d e &\rightarrow a e a d e c d \\ + c d e d c b e &\rightarrow d c^{2} d e a b d a e \\ + c d e d c^{2} d e &\rightarrow d c^{2} d e^{2} c d \\ + c d e a d e &\rightarrow d a e d \\ + c d a d a b d a e &\rightarrow a e b e \\ + e b d a d c b e &\rightarrow a b d a e a d a b d a e \\ + e b d a d c^{2} d e &\rightarrow a b d a e a d e c d \\ + b d a e d &\rightarrow d a d e a d e \\ + d b d a d a e d &\rightarrow b e a d e a d e \\ + d^{2} a e d &\rightarrow c d e^{2} a d e \\ + e b d a d a e d &\rightarrow a b d a e a d e a d e \\ + a b d a d a e d &\rightarrow e a d e a d e \\ + c b d a d a e d &\rightarrow a e a d e a d e \\ + c d e d a e d &\rightarrow d c^{2} d e^{2} a d e \\ + c d e a d a b d a e &\rightarrow d a e b e \\ + b d a e b e &\rightarrow d a d e a d a b d a e \\ + b d a e c d e &\rightarrow d a d e a d e c d \\ + d b d a d a e b e &\rightarrow b e a d e a d a b d a e \\ + d b d a d a e c d e &\rightarrow b e a d e a d e c d \\ + e b d a d a e b e &\rightarrow a b d a e a d e a d a b d a e \\ + e b d a d a e c d e &\rightarrow a b d a e a d e a d e c d \\ + a b d a d a e b e &\rightarrow e a d e a d a b d a e \\ + a b d a d a e c d e &\rightarrow e a d e a d e c d \\ + c b d a d a e b e &\rightarrow a e a d e a d a b d a e \\ + c b d a d a e c d e &\rightarrow a e a d e a d e c d \\ + c d e^{2} a d a b d a e &\rightarrow d^{2} a e b e \\ + c d e^{2} a d e d a e d &\rightarrow d^{2} a e c d e^{2} a d e \\ + c d e^{2} a d e d c b e &\rightarrow d^{2} a e c d e a b d a e \\ + c d e^{2} a d e d c^{2} d e &\rightarrow d^{2} a e c d e^{2} c d \\ + c d e^{2} a d e c d &\rightarrow d^{2} a e c d e \\ + b d^{2} a e b e &\rightarrow d a d e^{2} a d a b d a e \\ + d b d a d^{2} a e b e &\rightarrow b e a d e^{2} a d a b d a e \\ + d^{3} a e b e &\rightarrow c d e^{3} a d a b d a e \\ + e b d a d^{2} a e b e &\rightarrow a b d a e a d e^{2} a d a b d a e \\ + a b d a d^{2} a e b e &\rightarrow e a d e^{2} a d a b d a e \\ + c b d a d^{2} a e b e &\rightarrow a e a d e^{2} a d a b d a e \\ + c d e d^{2} a e b e &\rightarrow d c^{2} d e^{3} a d a b d a e \\ + b d^{2} a e c d e &\rightarrow d a d e^{2} a d e c d \\ + d b d a d^{2} a e c d e &\rightarrow b e a d e^{2} a d e c d \\ + d^{3} a e c d e &\rightarrow c d e^{3} a d e c d \\ + e b d a d^{2} a e c d e &\rightarrow a b d a e a d e^{2} a d e c d \\ + a b d a d^{2} a e c d e &\rightarrow e a d e^{2} a d e c d \\ + c b d a d^{2} a e c d e &\rightarrow a e a d e^{2} a d e c d \\ + c d e d^{2} a e c d e &\rightarrow d c^{2} d e^{3} a d e c d \\ + c d e^{2} a d e d^{2} a e b e &\rightarrow d^{2} a e c d e^{3} a d a b d a e \\ + c d e^{2} a d e d^{2} a e c d e &\rightarrow d^{2} a e c d e^{3} a d e c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/227.md b/docs/examples/1-relation/227.md new file mode 100644 index 0000000..726d70a --- /dev/null +++ b/docs/examples/1-relation/227.md @@ -0,0 +1,42 @@ +#\#227 baaaabbba=aaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a = a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{3} a = c^{2},\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + b a^{3} d a &\rightarrow a^{4} \\ + a b^{2} a^{4} &\rightarrow d a^{3} d a \\ + b a^{3} d^{2} &\rightarrow a^{3} d \\ + a b^{2} a^{3} d &\rightarrow d a^{3} d^{2} \\ + a b a^{3} d &\rightarrow d a^{3} d^{3} \\ + a b a^{4} &\rightarrow d a^{3} d^{2} a \\ + d a^{3} d^{4} &\rightarrow a^{4} d \\ + d a^{3} d^{3} a &\rightarrow a^{5} \\ + b a^{7} d &\rightarrow a^{6} d^{4} \\ + b a^{8} &\rightarrow a^{6} d^{3} a \\ + d a^{3} d^{2} a^{4} d &\rightarrow a^{7} d^{4} \\ + d a^{3} d^{2} a^{5} &\rightarrow a^{7} d^{3} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/229.md b/docs/examples/1-relation/229.md new file mode 100644 index 0000000..5dc2fe1 --- /dev/null +++ b/docs/examples/1-relation/229.md @@ -0,0 +1,48 @@ +#\#229 abbabbabaa=bababba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a b a^{2} = b a b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c d = c^{2} b c,\; c = b a,\; d = b c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c^{2} a &\rightarrow d \\ + b a &\rightarrow c \\ + a b c d &\rightarrow c^{2} b c \\ + b c^{2} b c &\rightarrow c b c d \\ + b c^{4} b c &\rightarrow d b c d \\ + b c^{2} d b c d &\rightarrow c b c d c^{3} b c \\ + b c^{3} b c d &\rightarrow c b c d c b c \\ + b c^{4} d b c d &\rightarrow d b c d c^{3} b c \\ + b c^{5} b c d &\rightarrow d b c d c b c \\ + d b c d c a &\rightarrow b c^{4} d \\ + c b c d c a &\rightarrow b c^{2} d \\ + b c b c^{2} d &\rightarrow c b c d^{2} c a \\ + b c^{2} b^{2} c^{2} d &\rightarrow c b c b c^{4} d \\ + b c^{3} b c^{2} d &\rightarrow d b c d^{2} c a \\ + b c^{4} b^{2} c^{2} d &\rightarrow d b c b c^{4} d \\ + b c^{5} b c b c^{4} d &\rightarrow d b c d c b^{2} c^{2} d \\ + b c^{4} d b c b c^{4} d &\rightarrow d b c d c^{3} b^{2} c^{2} d \\ + b c^{3} b c b c^{4} d &\rightarrow c b c d c b^{2} c^{2} d \\ + b c^{2} d b c b c^{4} d &\rightarrow c b c d c^{3} b^{2} c^{2} d \\ + a b c b c^{4} d &\rightarrow c^{2} b^{2} c^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/231.md b/docs/examples/1-relation/231.md new file mode 100644 index 0000000..2bbc687 --- /dev/null +++ b/docs/examples/1-relation/231.md @@ -0,0 +1,37 @@ +#\#231 abbaababba=babbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b a b^{2} a = b a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a b c = b c a,\; c = a b^{2} a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a b a &\rightarrow d \\ + a b^{2} a &\rightarrow c \\ + b c a &\rightarrow c a b c \\ + c b a &\rightarrow a b^{2} d \\ + c b^{2} a &\rightarrow a b^{2} c \\ + d b a &\rightarrow a b d \\ + d b^{2} a &\rightarrow a b c \\ + b c^{2} &\rightarrow c d b^{2} c \\ + b c d &\rightarrow c d b^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/232.md b/docs/examples/1-relation/232.md new file mode 100644 index 0000000..184a27b --- /dev/null +++ b/docs/examples/1-relation/232.md @@ -0,0 +1,36 @@ +#\#232 baaabbbbaa=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{4} a^{2} = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d = a b a,\; c = b^{4},\; d = a c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b^{4} &\rightarrow c \\ + a b a &\rightarrow b a^{2} d \\ + a c a^{2} &\rightarrow d \\ + c b &\rightarrow b c \\ + a b^{2} a^{2} d &\rightarrow b a^{2} d b a \\ + a b d &\rightarrow b a^{2} d c a^{2} \\ + a b c a^{2} d a d &\rightarrow d b a \\ + a c a d &\rightarrow d c a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/233.md b/docs/examples/1-relation/233.md new file mode 100644 index 0000000..b06ef30 --- /dev/null +++ b/docs/examples/1-relation/233.md @@ -0,0 +1,48 @@ +#\#233 abaababaaa=baaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b a^{3} = b a^{3} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e = d c a,\; c = b a,\; d = c a^{2},\; e = c a c d +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < e < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a c d &\rightarrow e \\ + c a^{2} &\rightarrow d \\ + d c a &\rightarrow a e \\ + c a c a e &\rightarrow e c a \\ + a e c d &\rightarrow d e \\ + a e a &\rightarrow d^{2} \\ + b d e &\rightarrow c e c d \\ + b d^{2} &\rightarrow c e a \\ + c a d e &\rightarrow d e c d \\ + c a d^{2} &\rightarrow d e a \\ + a e c a e &\rightarrow d e c a \\ + d c d e &\rightarrow a e^{2} c d \\ + d c d^{2} &\rightarrow a e^{2} a \\ + d^{2} e c d &\rightarrow a e d e \\ + d^{2} e a &\rightarrow a e d^{2} \\ + b d a e &\rightarrow c e a c a \\ + c a d a e &\rightarrow d e a c a \\ + d c d a e &\rightarrow a e^{2} a c a \\ + d^{2} e c a e &\rightarrow a e d e c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/234.md b/docs/examples/1-relation/234.md new file mode 100644 index 0000000..6a495c4 --- /dev/null +++ b/docs/examples/1-relation/234.md @@ -0,0 +1,39 @@ +#\#234 baaababbaa=abbaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b^{2} a^{2} = a b^{2} a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d b c = a b c d,\; c = b a^{2},\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a b a &\rightarrow d \\ + a b c d &\rightarrow c d b c \\ + b a^{2} &\rightarrow c \\ + a c &\rightarrow d a \\ + a b d &\rightarrow d b a \\ + d b c d &\rightarrow c d b c b c \\ + b a d &\rightarrow c b a \\ + b d a d b c &\rightarrow c b c d \\ + a b c^{2} d b c b c &\rightarrow c d b c b c d \\ + d b c^{2} d b c b c &\rightarrow c d b c b c b c d \\ + b d^{2} a d b c b c &\rightarrow c b c d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/235.md b/docs/examples/1-relation/235.md new file mode 100644 index 0000000..1905819 --- /dev/null +++ b/docs/examples/1-relation/235.md @@ -0,0 +1,36 @@ +#\#235 abaabababa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b a b a = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c^{3} a = d c a,\; c = a b,\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a &\rightarrow d \\ + d c a &\rightarrow c a c^{3} a \\ + a b &\rightarrow c \\ + b c &\rightarrow d b \\ + d c^{2} &\rightarrow c a c^{4} \\ + a d &\rightarrow c a \\ + a c a c^{4} &\rightarrow c a c^{2} \\ + a c a c^{3} a &\rightarrow c a c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/236.md b/docs/examples/1-relation/236.md new file mode 100644 index 0000000..9c08943 --- /dev/null +++ b/docs/examples/1-relation/236.md @@ -0,0 +1,58 @@ +#\#236 babaaaaaba=ababaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{5} b a = a b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a b a = a b a c,\; c = b a^{2},\; d = c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + c a^{2} &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + b a d a b a &\rightarrow a b a c \\ + a b a c a &\rightarrow b a d a c \\ + a b a c b a d a c &\rightarrow b a d a c b a c a \\ + b a d a c a &\rightarrow a b a d \\ + b a d a b^{2} a d a c &\rightarrow a b a c b a c a \\ + b a d b a d a c &\rightarrow a b a c^{2} a \\ + a b a d a &\rightarrow b a d a d \\ + b a d b a d a d &\rightarrow a b a c d a \\ + b a d a b^{2} a d a d &\rightarrow a b a c b a d a \\ + b a d a c b a d a c &\rightarrow a b a d b a c a \\ + a b a c b a d a d &\rightarrow b a d a c b a d a \\ + b^{2} a d a d^{2} &\rightarrow c b a d a \\ + c b a d a d^{2} &\rightarrow d b a d a \\ + b^{2} a d a d c &\rightarrow c b a c a \\ + c b a d a d c &\rightarrow d b a c a \\ + a^{2} b a d &\rightarrow b a d a d c a \\ + a^{2} b a c &\rightarrow b a d a d b a \\ + b a d a c b a d a d &\rightarrow a b a d b a d a \\ + c b a d a d b a &\rightarrow d b a c \\ + b^{2} a d a d b a &\rightarrow c b a c \\ + c b a d a d a b a d &\rightarrow d b a c d a c a \\ + c b a d a d a b a c &\rightarrow d b a c d a b a \\ + c b a d a d b^{2} a d a d &\rightarrow d b a c b a d a \\ + c b a d a d b^{2} a d a c &\rightarrow d b a c b a c a \\ + b^{2} a d a d a b a d &\rightarrow c b a c d a c a \\ + b^{2} a d a d a b a c &\rightarrow c b a c d a b a \\ + b^{2} a d a d b^{2} a d a d &\rightarrow c b a c b a d a \\ + b^{2} a d a d b^{2} a d a c &\rightarrow c b a c b a c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/237.md b/docs/examples/1-relation/237.md new file mode 100644 index 0000000..396322f --- /dev/null +++ b/docs/examples/1-relation/237.md @@ -0,0 +1,60 @@ +#\#237 ababbaba=bababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b a = b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d = c^{3},\; c = b a,\; d = c b c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a d &\rightarrow c^{3} \\ + c b c^{2} &\rightarrow d \\ + b c^{3} &\rightarrow c d \\ + d b c^{2} &\rightarrow c b c d \\ + c^{2} d &\rightarrow d c \\ + a c b c d &\rightarrow d c \\ + c^{2} b c d &\rightarrow b d c \\ + b d c^{2} &\rightarrow c d c d \\ + b c d c &\rightarrow c d^{2} \\ + c d b c d &\rightarrow b d^{2} \\ + c b d c &\rightarrow d^{2} \\ + c b^{2} d c &\rightarrow d b c d \\ + d b^{2} d c &\rightarrow c b^{2} d^{2} \\ + b c^{2} b d^{2} &\rightarrow c d^{2} b c d \\ + b c d b d^{2} &\rightarrow c d^{3} b c d \\ + b d^{2} b c d &\rightarrow c d b^{2} d^{2} \\ + b d c b d^{2} &\rightarrow c d c d^{2} b c d \\ + b d^{2} c &\rightarrow c d c d^{2} \\ + c^{3} b^{2} d^{2} &\rightarrow d^{2} b c d \\ + d c b c d &\rightarrow c b d^{2} \\ + a c b^{2} d^{2} &\rightarrow c b d^{2} \\ + b d^{3} &\rightarrow c d c^{2} b d^{2} \\ + b c b d^{2} &\rightarrow c^{2} b^{2} d^{2} \\ + b c d^{2} &\rightarrow c^{2} b d^{2} \\ + b d^{2} b d^{2} &\rightarrow c d c d^{3} b c d \\ + b d b d^{2} &\rightarrow c d c^{2} b^{2} d^{2} \\ + d b d c &\rightarrow c^{3} b d^{2} \\ + a c^{3} b d^{2} &\rightarrow d c d \\ + a c b d^{2} &\rightarrow c d^{2} \\ + c^{5} b d^{2} &\rightarrow d^{3} \\ + b d c d &\rightarrow c^{4} b d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/238.md b/docs/examples/1-relation/238.md new file mode 100644 index 0000000..747dfc4 --- /dev/null +++ b/docs/examples/1-relation/238.md @@ -0,0 +1,36 @@ +#\#238 baabaabaa=aabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b a^{2} = a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{3} = d c,\; c = b a^{2},\; d = a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a^{2} &\rightarrow c \\ + a^{2} b &\rightarrow d \\ + d c &\rightarrow c^{3} \\ + b a d &\rightarrow c a b \\ + b d &\rightarrow c b \\ + a^{2} c &\rightarrow d a^{2} \\ + b a c^{3} &\rightarrow c a b c \\ + b c^{3} &\rightarrow c b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/239.md b/docs/examples/1-relation/239.md new file mode 100644 index 0000000..f8fb4e8 --- /dev/null +++ b/docs/examples/1-relation/239.md @@ -0,0 +1,52 @@ +#\#239 abaabaabba=baababaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a^{2} b^{2} a = b a^{2} b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b a d a = b a c a,\; c = a b a b a,\; d = a b a^{2} b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a c a &\rightarrow a b a d a \\ + a b a b a &\rightarrow c \\ + a b a^{2} b^{2} &\rightarrow d \\ + c b a^{2} b^{2} &\rightarrow a b a b d \\ + c a b^{2} &\rightarrow a b d \\ + a b c &\rightarrow c b a \\ + b a c^{2} &\rightarrow a b a d c \\ + b a c d &\rightarrow a b a d^{2} \\ + a^{2} b a d^{2} a &\rightarrow d a c a \\ + a b a^{2} b a d a &\rightarrow c^{2} a \\ + a b a^{2} b a d^{2} &\rightarrow c^{2} d \\ + a b d a c^{2} &\rightarrow c^{2} d c \\ + a b d a c d &\rightarrow c^{2} d^{2} \\ + a b d a c a &\rightarrow c^{2} d a \\ + a^{2} b a d^{2} c &\rightarrow d a c^{2} \\ + a^{2} b a d^{3} &\rightarrow d a c d \\ + a b a d a b^{2} &\rightarrow b a^{2} b d \\ + a b a^{2} b a d c &\rightarrow c^{3} \\ + a b^{2} a^{2} b d &\rightarrow c d a b^{2} \\ + a b d a c b a^{2} b d &\rightarrow c^{2} d b a^{2} b d \\ + a b d a^{2} b d &\rightarrow c^{2} d a b^{2} \\ + b a c b a^{2} b d &\rightarrow a b a d b a^{2} b d \\ + a b a^{2} b a d b a^{2} b d &\rightarrow c^{2} b a^{2} b d \\ + a^{2} b a d^{2} b a^{2} b d &\rightarrow d a c b a^{2} b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/242.md b/docs/examples/1-relation/242.md new file mode 100644 index 0000000..ddd604f --- /dev/null +++ b/docs/examples/1-relation/242.md @@ -0,0 +1,85 @@ +#\#242 abbabbbaaa=bababba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{3} a^{3} = b a b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c b c d a = b c^{2} b a,\; c = a b,\; d = b^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} a^{2} &\rightarrow d \\ + b c^{2} b a &\rightarrow c b c d a \\ + a b &\rightarrow c \\ + b^{2} a c &\rightarrow d b \\ + b c^{2} b c &\rightarrow c b c d c \\ + c b a^{2} &\rightarrow a d \\ + a c b c d a &\rightarrow c^{3} b a \\ + b^{2} c^{3} b a &\rightarrow d b^{2} c d a \\ + b c^{3} b c d c &\rightarrow c b c d c^{2} b c \\ + b c^{3} b c d a &\rightarrow c b c d c^{2} b a \\ + c b c d a^{2} &\rightarrow b c a d \\ + c b a c &\rightarrow a d b \\ + a c b c d c &\rightarrow c^{3} b c \\ + b^{2} c^{3} b c &\rightarrow d b^{2} c d c \\ + b^{2} c^{2} a d &\rightarrow d b^{2} c d a^{2} \\ + b c b c a d &\rightarrow c b c d c d a^{2} \\ + b c^{2} b^{2} c a d &\rightarrow c b c d b c a d \\ + b c a d b &\rightarrow c b c d a c \\ + b c a d c b c d c &\rightarrow c b c d a c^{3} b c \\ + b c a d c b c d a &\rightarrow c b c d a c^{3} b a \\ + b c^{3} b c d b c a d &\rightarrow c b c d c^{2} b^{2} c a d \\ + a c b c d b c a d &\rightarrow c^{3} b^{2} c a d \\ + a d b^{2} c d c &\rightarrow c b c^{3} b c \\ + a d b^{2} c d a &\rightarrow c b c^{3} b a \\ + d b^{2} c d c d a^{2} &\rightarrow b c b c d c a d \\ + b^{2} c^{3} b^{2} c a d &\rightarrow d b^{2} c d b c a d \\ + b^{2} c^{4} b c d b c a d &\rightarrow d b^{2} c d c^{2} b^{2} c a d \\ + b^{2} c^{4} b c d c &\rightarrow d b^{2} c d c^{2} b c \\ + b^{2} c^{4} b c d a &\rightarrow d b^{2} c d c^{2} b a \\ + b c b c d c^{2} b^{2} c a d &\rightarrow d b^{2} c d c d b c a d \\ + b c b c d c^{2} b c &\rightarrow d b^{2} c d c d c \\ + b c b c d c^{2} b a &\rightarrow d b^{2} c d c d a \\ + b c a d c b c d b c a d &\rightarrow c b c d a c^{3} b^{2} c a d \\ + b c a d^{2} b^{2} c d c &\rightarrow c b c d a c b c^{3} b c \\ + b c a d^{2} b^{2} c d a &\rightarrow c b c d a c b c^{3} b a \\ + a d b^{2} c d b c a d &\rightarrow c b c^{3} b^{2} c a d \\ + b^{2} c^{3} d b^{2} c d c d b c a d &\rightarrow d b^{2} c d c b c d c^{2} b^{2} c a d \\ + b^{2} c^{3} d b^{2} c d c d c &\rightarrow d b^{2} c d c b c d c^{2} b c \\ + b^{2} c^{3} d b^{2} c d c d a &\rightarrow d b^{2} c d c b c d c^{2} b a \\ + b c b c d c b c a d &\rightarrow d b^{2} c d c d c d a^{2} \\ + b c b c d c^{3} b c d b c a d &\rightarrow d b^{2} c d c d c^{2} b^{2} c a d \\ + b c b c d c^{3} b c d c &\rightarrow d b^{2} c d c d c^{2} b c \\ + b c b c d c^{3} b c d a &\rightarrow d b^{2} c d c d c^{2} b a \\ + b c b c d c^{2} d b^{2} c d c d b c a d &\rightarrow d b^{2} c d c d c b c d c^{2} b^{2} c a d \\ + b c b c d c^{2} d b^{2} c d c d c &\rightarrow d b^{2} c d c d c b c d c^{2} b c \\ + b c b c d c^{2} d b^{2} c d c d a &\rightarrow d b^{2} c d c d c b c d c^{2} b a \\ + b c b c d c a d b &\rightarrow d b^{2} c d c d a c \\ + b c b c d c a d c b c d b c a d &\rightarrow d b^{2} c d c d a c^{3} b^{2} c a d \\ + b c b c d c a d c b c d c &\rightarrow d b^{2} c d c d a c^{3} b c \\ + b c b c d c a d c b c d a &\rightarrow d b^{2} c d c d a c^{3} b a \\ + b c b c d c a d^{2} b^{2} c d c &\rightarrow d b^{2} c d c d a c b c^{3} b c \\ + b c b c d c a d^{2} b^{2} c d a &\rightarrow d b^{2} c d c d a c b c^{3} b a \\ + b c^{2} d b^{2} c d c d b c a d &\rightarrow c b c d c b c d c^{2} b^{2} c a d \\ + b c^{2} d b^{2} c d c d c &\rightarrow c b c d c b c d c^{2} b c \\ + b c^{2} d b^{2} c d c d a &\rightarrow c b c d c b c d c^{2} b a \\ + b c a d^{2} b^{2} c d b c a d &\rightarrow c b c d a c b c^{3} b^{2} c a d \\ + b c b c d c a d^{2} b^{2} c d b c a d &\rightarrow d b^{2} c d c d a c b c^{3} b^{2} c a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/243.md b/docs/examples/1-relation/243.md new file mode 100644 index 0000000..9b59ddf --- /dev/null +++ b/docs/examples/1-relation/243.md @@ -0,0 +1,34 @@ +#\#243 baaaabbbaa=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a^{2} = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d c = a,\; c = a^{2},\; d = b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{3} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a &\rightarrow d a^{4} d a^{5} d a^{2} \\ + d a^{4} d a^{5} d a^{5} d a^{2} &\rightarrow a \\ + d a^{4} d a^{5} d a^{6} &\rightarrow a^{3} d a^{5} d a^{5} d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/244.md b/docs/examples/1-relation/244.md new file mode 100644 index 0000000..0d73b2d --- /dev/null +++ b/docs/examples/1-relation/244.md @@ -0,0 +1,40 @@ +#\#244 baaaabba=aaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a = a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{2} a = c^{2},\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{3} d a &\rightarrow a^{4} \\ + a b a^{4} &\rightarrow d a^{3} d a \\ + b a^{3} d^{2} &\rightarrow a^{3} d \\ + a b a^{3} d &\rightarrow d a^{3} d^{2} \\ + d a^{3} d^{3} &\rightarrow a^{4} d \\ + d a^{3} d^{2} a &\rightarrow a^{5} \\ + b a^{7} d &\rightarrow a^{6} d^{3} \\ + b a^{8} &\rightarrow a^{6} d^{2} a \\ + d a^{3} d a^{4} d &\rightarrow a^{7} d^{3} \\ + d a^{3} d a^{5} &\rightarrow a^{7} d^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/245.md b/docs/examples/1-relation/245.md new file mode 100644 index 0000000..f00b2d6 --- /dev/null +++ b/docs/examples/1-relation/245.md @@ -0,0 +1,35 @@ +#\#245 abaabaaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a^{2} b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a c a c = c^{2} a,\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + b d a &\rightarrow d^{3} \\ + b d^{2} &\rightarrow d^{3} b a \\ + a b d &\rightarrow d b a \\ + d b a d &\rightarrow a d^{3} b a \\ + d b a^{2} &\rightarrow a d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/247.md b/docs/examples/1-relation/247.md new file mode 100644 index 0000000..4281b44 --- /dev/null +++ b/docs/examples/1-relation/247.md @@ -0,0 +1,65 @@ +#\#247 abaaabbaaa=baaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b^{2} a^{3} = b a^{4} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; e c b^{2} c = b c e a,\; c = a^{3},\; d = b^{2} a^{2},\; e = a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + b^{2} a^{2} &\rightarrow d \\ + a b &\rightarrow e \\ + a^{3} &\rightarrow c \\ + b^{2} a e &\rightarrow d b \\ + b^{2} c &\rightarrow d a \\ + a c &\rightarrow c a \\ + e b a^{2} &\rightarrow a d \\ + c b &\rightarrow a^{2} e \\ + a e c d a &\rightarrow e c e a \\ + e b a e &\rightarrow a d b \\ + e b c &\rightarrow a d a \\ + b c e^{2} &\rightarrow e c d e \\ + b c e c &\rightarrow e c d c \\ + a^{2} e c e a &\rightarrow c e c d a \\ + b e c d a &\rightarrow d a e a \\ + b c e a &\rightarrow e c d a \\ + b^{2} e c e a &\rightarrow d b c d a \\ + b e c d e &\rightarrow d a e^{2} \\ + b e c d c &\rightarrow d a e c \\ + a^{2} e^{2} c d a &\rightarrow c d a e a \\ + a^{2} e c e^{2} &\rightarrow c e c d e \\ + a^{2} e c e c &\rightarrow c e c d c \\ + a e c d e &\rightarrow e c e^{2} \\ + a e c d c &\rightarrow e c e c \\ + a d a e a &\rightarrow e^{2} c d a \\ + a d a e^{2} &\rightarrow e^{2} c d e \\ + a d a e c &\rightarrow e^{2} c d c \\ + e b e c e a &\rightarrow a d b c d a \\ + a d e c e a &\rightarrow e^{2} c d c d a \\ + b^{2} e c e^{2} &\rightarrow d b c d e \\ + b^{2} e c e c &\rightarrow d b c d c \\ + a^{2} e^{2} c d e &\rightarrow c d a e^{2} \\ + a^{2} e^{2} c d c &\rightarrow c d a e c \\ + a d e c e^{2} &\rightarrow e^{2} c d c d e \\ + a d e c e c &\rightarrow e^{2} c d c d c \\ + e b e c e^{2} &\rightarrow a d b c d e \\ + e b e c e c &\rightarrow a d b c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/248.md b/docs/examples/1-relation/248.md new file mode 100644 index 0000000..c23c1d7 --- /dev/null +++ b/docs/examples/1-relation/248.md @@ -0,0 +1,34 @@ +#\#248 baaaaabbaa=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a^{2} = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} a d c = a,\; c = a^{2},\; d = b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{2} &\rightarrow d \\ + b d &\rightarrow d b \\ + b a &\rightarrow d a^{5} d a^{2} \\ + d a^{5} d a^{6} d a^{2} &\rightarrow a \\ + d a^{5} d a^{7} &\rightarrow a^{4} d a^{6} d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/249.md b/docs/examples/1-relation/249.md new file mode 100644 index 0000000..3678a8f --- /dev/null +++ b/docs/examples/1-relation/249.md @@ -0,0 +1,36 @@ +#\#249 baaaabaaaa=aaaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a^{4} = a^{5} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d^{2} = a^{5} c,\; c = b a,\; d = c a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a^{3} &\rightarrow d \\ + a^{5} c &\rightarrow d^{2} \\ + b d^{2} &\rightarrow d a c \\ + c a^{2} d^{2} &\rightarrow d a^{4} c \\ + c a d^{2} &\rightarrow d a^{3} c \\ + c d^{2} &\rightarrow d a^{2} c \\ + a^{5} d &\rightarrow d^{2} a^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/250.md b/docs/examples/1-relation/250.md new file mode 100644 index 0000000..9e58ea6 --- /dev/null +++ b/docs/examples/1-relation/250.md @@ -0,0 +1,42 @@ +#\#250 ababbbbaaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{4} a^{3} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b a d = b a b a,\; c = b^{2} a^{2},\; d = b^{2} c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b^{2} c a &\rightarrow d \\ + b^{2} a^{2} &\rightarrow c \\ + b a b a &\rightarrow a b a d \\ + b a^{2} b a d &\rightarrow a b a d b a \\ + a b a d^{2} b a &\rightarrow c b a d \\ + b^{2} c^{2} b a d &\rightarrow d b a d^{2} b a \\ + b^{2} a c b a d &\rightarrow c b a d^{2} b a \\ + b c b a d &\rightarrow a b a d^{3} b a \\ + a b a d b a d b a &\rightarrow b a c b a d \\ + a b a d^{2} a b a d &\rightarrow c b a d b a \\ + b^{2} c b a c b a d &\rightarrow d b a d b a d b a \\ + c b a d b a d b a &\rightarrow a b a d^{2} c b a d \\ + c b a d b a d a b a d &\rightarrow a b a d^{2} c b a d b a \\ + a b a d b a d a b a d &\rightarrow b a c b a d b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/251.md b/docs/examples/1-relation/251.md new file mode 100644 index 0000000..c4fc602 --- /dev/null +++ b/docs/examples/1-relation/251.md @@ -0,0 +1,87 @@ +#\#251 baabaabaa=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b a^{2} = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a = a b a b a,\; c = a^{2} b a^{2} b a,\; d = c a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + c a b a &\rightarrow d \\ + a b a b a &\rightarrow b c a \\ + a^{2} b a^{2} b a &\rightarrow c \\ + a b a b c &\rightarrow b c^{2} \\ + d a b a^{2} b a &\rightarrow c a b c \\ + c b c a &\rightarrow d b a \\ + a b^{2} c a &\rightarrow b d \\ + a b^{2} d &\rightarrow b d b a \\ + a^{2} b c &\rightarrow d \\ + a b c^{2} a &\rightarrow c b a \\ + a^{2} b d &\rightarrow d a b a \\ + d b a b a &\rightarrow c b d \\ + a b^{2} c b d &\rightarrow b c b d b a \\ + a b^{2} c^{2} &\rightarrow b c a b c \\ + d b^{2} c^{2} &\rightarrow c b d b c \\ + d b^{2} c a &\rightarrow c b d b a \\ + d b^{2} c b d &\rightarrow c b d b^{2} d \\ + c b c b d &\rightarrow d b^{2} d \\ + d b a b d &\rightarrow c b d a b c \\ + c b c d &\rightarrow d b d \\ + c b c^{2} &\rightarrow d b c \\ + c a b c a b c &\rightarrow d a b d a b a \\ + c a b d &\rightarrow d a b c \\ + a b c^{2} b d &\rightarrow c b^{2} d \\ + c b^{2} c^{2} &\rightarrow a b c d b c \\ + c b^{2} c a &\rightarrow a b c d b a \\ + c b a b a &\rightarrow a b c d \\ + a b c^{2} d &\rightarrow c b d \\ + a b c^{3} &\rightarrow c b c \\ + a b^{2} c d &\rightarrow b d a b c \\ + a b c b a &\rightarrow b c^{3} a \\ + a b a b d &\rightarrow b c d \\ + a c b a &\rightarrow d c a \\ + d b^{2} c d &\rightarrow c b d b d \\ + d b a b c a b c &\rightarrow c b d a b d a b a \\ + d a b d a b a c a &\rightarrow c a b c^{2} b a \\ + c b^{2} c b d &\rightarrow a b c d b^{2} d \\ + c b a b d &\rightarrow a b c d a b c \\ + c b a b c a b c &\rightarrow a b c d a b d a b a \\ + a b c b^{2} d &\rightarrow b c^{3} b d \\ + c b^{2} c d &\rightarrow a b c d b d \\ + a c b^{2} d &\rightarrow d c b d \\ + a b c b c &\rightarrow b c^{4} \\ + d a b d a b a c d &\rightarrow c^{2} b^{2} d \\ + d a b d a b a c^{2} &\rightarrow c a b c^{2} b c \\ + a b c b d &\rightarrow b c^{3} d \\ + a c b d &\rightarrow d c d \\ + a c b c &\rightarrow d c^{2} \\ + c a b c^{2} b^{2} d &\rightarrow d a b d a b d c d \\ + d b a b c^{2} b^{2} d &\rightarrow c b d a b d a b d c d \\ + c b a b c^{2} b^{2} d &\rightarrow a b c d a b d a b d c d \\ + a b d b^{2} d &\rightarrow b c^{4} b d \\ + a b d b d &\rightarrow b c^{4} d \\ + a b d b c &\rightarrow b c^{5} \\ + a b d b a &\rightarrow b c^{4} a \\ + a d b^{2} d &\rightarrow d c^{2} b d \\ + a d b d &\rightarrow d c^{2} d \\ + a d b c &\rightarrow d c^{3} \\ + a d b a &\rightarrow d c^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/252.md b/docs/examples/1-relation/252.md new file mode 100644 index 0000000..33e9b91 --- /dev/null +++ b/docs/examples/1-relation/252.md @@ -0,0 +1,40 @@ +#\#252 baaaaaabba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{6} b^{2} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{3} b^{2} a = c a,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{5} d a &\rightarrow a^{3} \\ + a b a^{3} &\rightarrow d a^{5} d a \\ + b a^{5} d^{2} &\rightarrow a^{2} d \\ + a b a^{2} d &\rightarrow d a^{5} d^{2} \\ + d a^{5} d a^{3} d^{2} &\rightarrow a^{3} d \\ + d a^{5} d a^{3} d a &\rightarrow a^{4} \\ + b a^{8} d &\rightarrow a^{7} d a^{3} d^{2} \\ + b a^{9} &\rightarrow a^{7} d a^{3} d a \\ + d a^{5} d a^{6} d &\rightarrow a^{8} d a^{3} d^{2} \\ + d a^{5} d a^{7} &\rightarrow a^{8} d a^{3} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/255.md b/docs/examples/1-relation/255.md new file mode 100644 index 0000000..e1d5b68 --- /dev/null +++ b/docs/examples/1-relation/255.md @@ -0,0 +1,73 @@ +#\#255 baabababaa=abbaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b a b a^{2} = a b^{2} a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d = a b^{2} a^{2} b a,\; c = a b a^{2},\; d = b a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a b c &\rightarrow d \\ + a b^{2} a^{2} b a &\rightarrow b a^{2} d \\ + a b a^{2} &\rightarrow c \\ + c b^{2} a^{2} b a &\rightarrow a b c d \\ + a b^{2} a c &\rightarrow b a^{2} d a \\ + a b^{2} a^{2} b c &\rightarrow b a^{2} d b a^{2} \\ + a b^{2} a^{2} d &\rightarrow b a^{2} d b c \\ + a b a c &\rightarrow c b a^{2} \\ + d b^{2} a^{2} b a &\rightarrow b a d^{2} \\ + c b^{2} a c &\rightarrow a b c d a \\ + c b^{2} c d b c &\rightarrow a b c b a d^{2} \\ + c b^{2} a^{2} b c &\rightarrow a b c d b a^{2} \\ + c b^{2} a^{2} d &\rightarrow a b c d b c \\ + a b^{2} c d b c &\rightarrow b a^{2} b a d^{2} \\ + d b^{2} c d b c &\rightarrow b a d b a d^{2} \\ + d b^{2} a^{2} d &\rightarrow b a d^{2} b c \\ + d b^{2} a^{2} b c &\rightarrow b a d^{2} b a^{2} \\ + d b^{2} a c &\rightarrow b a d^{2} a \\ + b a^{2} b a d b a d^{2} &\rightarrow a b^{2} c d^{3} \\ + b a^{2} d b^{2} c d^{2} b a d^{2} &\rightarrow a b^{2} c d^{2} b a d b a d^{2} \\ + b a^{2} d b^{2} c d^{3} &\rightarrow a b^{2} c d^{2} b a d^{2} \\ + a b a b^{2} c d^{3} &\rightarrow b a^{2} d^{2} b a d^{2} \\ + b a d b a d b a d^{2} &\rightarrow d b^{2} c d^{3} \\ + a b c d b^{2} c d^{3} &\rightarrow c b^{2} c d^{2} b a d^{2} \\ + a b c b a d b a d^{2} &\rightarrow c b^{2} c d^{3} \\ + a^{2} b^{2} c d^{3} &\rightarrow c b a d b a d^{2} \\ + c b a b^{2} c d^{3} &\rightarrow a b c d^{2} b a d^{2} \\ + b c b^{2} c d^{2} b a d^{2} &\rightarrow d^{2} b^{2} c d^{3} \\ + b c b^{2} c d^{3} &\rightarrow d b a d b a d^{2} \\ + d b a b^{2} c d^{2} b a d b a d^{2} &\rightarrow b a d^{4} b^{2} c d^{3} \\ + d b a b^{2} c d^{2} b a d^{2} &\rightarrow b a d^{3} b a d b a d^{2} \\ + a b a b^{2} c d^{2} b a d b a d^{2} &\rightarrow b a^{2} d^{3} b^{2} c d^{3} \\ + a b a b^{2} c d^{2} b a d^{2} &\rightarrow b a^{2} d^{2} b a d b a d^{2} \\ + b a d^{2} b^{2} c d^{3} &\rightarrow d b^{2} c d^{2} b a d^{2} \\ + d b a b^{2} c d^{3} &\rightarrow b a d^{3} b a d^{2} \\ + a b c d b^{2} c d^{2} b a d b a d^{2} &\rightarrow c b^{2} c d^{3} b^{2} c d^{3} \\ + a b c d b^{2} c d^{2} b a d^{2} &\rightarrow c b^{2} c d^{2} b a d b a d^{2} \\ + a^{2} b^{2} c d^{2} b a d b a d^{2} &\rightarrow c d b^{2} c d^{2} b a d^{2} \\ + a^{2} b^{2} c d^{2} b a d^{2} &\rightarrow c d b^{2} c d^{3} \\ + b a^{2} d b^{2} c d^{2} b a d b a d^{2} &\rightarrow a b^{2} c d^{3} b^{2} c d^{3} \\ + c b a b^{2} c d^{2} b a d b a d^{2} &\rightarrow a b c d^{3} b^{2} c d^{3} \\ + c b a b^{2} c d^{2} b a d^{2} &\rightarrow a b c d^{2} b a d b a d^{2} \\ + b a d^{2} b^{2} c d^{2} b a d^{2} &\rightarrow d b^{2} c d^{2} b a d b a d^{2} \\ + b c b^{2} c d^{2} b a d b a d^{2} &\rightarrow d^{2} b^{2} c d^{2} b a d^{2} \\ + b a d^{2} b^{2} c d^{2} b a d b a d^{2} &\rightarrow d b^{2} c d^{3} b^{2} c d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/256.md b/docs/examples/1-relation/256.md new file mode 100644 index 0000000..45a676f --- /dev/null +++ b/docs/examples/1-relation/256.md @@ -0,0 +1,39 @@ +#\#256 ababbaabaa=baabaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a^{2} b a^{2} = b a^{2} b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e a = c a c a c,\; c = b a,\; d = b c a,\; e = c d c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(e < c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow d \\ + c a c a c &\rightarrow a e a \\ + c d c &\rightarrow e \\ + d c a c &\rightarrow c e a \\ + c a^{2} e a &\rightarrow a e a^{2} c \\ + c a c a e &\rightarrow a e a d c \\ + c d e &\rightarrow e d c \\ + d a e a &\rightarrow c e a^{2} c \\ + d c a e &\rightarrow c e a d c \\ + c^{2} e a &\rightarrow e a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/257.md b/docs/examples/1-relation/257.md new file mode 100644 index 0000000..fbf9637 --- /dev/null +++ b/docs/examples/1-relation/257.md @@ -0,0 +1,38 @@ +#\#257 baaabbbaba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{3} a b a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{2} d = a^{2},\; c = b a,\; d = b^{2} c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} c^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + b d a^{2} d^{2} a &\rightarrow d a^{2} d \\ + c a &\rightarrow d a^{2} d^{2} a d \\ + d a^{2} d^{2} a d a d &\rightarrow a^{2} \\ + b^{2} c d a^{2} d^{2} a d &\rightarrow d a \\ + b d a^{2} d a^{2} &\rightarrow d a^{2} d a d^{2} a d a d \\ + d a^{2} d^{2} a d a^{3} &\rightarrow a^{4} d^{2} a d a d \\ + b^{2} c d a^{2} d^{2} a^{3} &\rightarrow d a^{3} d^{2} a d a d \\ + b d a^{4} &\rightarrow d a^{2} d a d^{2} a d a d^{3} a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/258.md b/docs/examples/1-relation/258.md new file mode 100644 index 0000000..bccfaa7 --- /dev/null +++ b/docs/examples/1-relation/258.md @@ -0,0 +1,38 @@ +#\#258 aabbaabbaa=baabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{2} a^{2} b^{2} a^{2} = b a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c a^{2} c a^{2} = b a^{2} c a,\; c = b^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b^{2} \\ + a b^{2} &\rightarrow d \\ + b a d a &\rightarrow a d a d a^{2} \\ + a^{2} d a d a^{2} d a^{2} &\rightarrow d a d a \\ + b a d^{2} &\rightarrow a d a d a d \\ + a^{2} d a d a^{2} d a d &\rightarrow d a d^{2} \\ + a^{2} d a d a^{2} d^{2} a d a &\rightarrow d a d a d a d a^{2} d a^{2} \\ + a^{2} d a d a^{2} d^{2} a d^{2} &\rightarrow d a d a d a d a^{2} d a d \\ + a^{2} d a d^{2} a d a &\rightarrow d a d^{2} a^{2} d a^{2} \\ + a^{2} d a d^{2} a d^{2} &\rightarrow d a d^{2} a^{2} d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/259.md b/docs/examples/1-relation/259.md new file mode 100644 index 0000000..9fa252d --- /dev/null +++ b/docs/examples/1-relation/259.md @@ -0,0 +1,108 @@ +#\#259 bababbbaba=ababbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b^{3} a b a = a b a b^{3} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; e = a d c,\; c = b a,\; d = c b^{2},\; e = c d c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(e < d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a d c &\rightarrow e \\ + c b^{2} &\rightarrow d \\ + c d c^{2} &\rightarrow e \\ + b e &\rightarrow c d c \\ + a d e &\rightarrow e d c^{2} \\ + c b c &\rightarrow d a \\ + c d c e &\rightarrow e d c^{2} \\ + e b^{2} &\rightarrow c d c d \\ + a d^{2} &\rightarrow c d c d \\ + b c d c d &\rightarrow c d^{2} \\ + c b d a &\rightarrow d a b c \\ + c^{2} d e &\rightarrow d e^{2} \\ + c^{2} d c &\rightarrow d e c \\ + d a b^{2} &\rightarrow c b d \\ + e b c &\rightarrow c d c d a \\ + b c d e &\rightarrow c d^{2} c^{2} \\ + c d c d e c &\rightarrow e c d c \\ + c d c d e^{2} &\rightarrow e c d e \\ + c b d c d c d &\rightarrow d a b c d^{2} \\ + c b d e c &\rightarrow d a c d c \\ + c b d e^{2} &\rightarrow d a c d e \\ + c d c d e d &\rightarrow e c d^{2} \\ + c d^{2} e c &\rightarrow e d c \\ + c d^{2} e^{2} &\rightarrow e d e \\ + d a b c d c &\rightarrow c b d e \\ + c^{2} d^{2} &\rightarrow d e d \\ + d e c^{2} &\rightarrow c e \\ + e b d a &\rightarrow c d c d a b c \\ + e b d e c &\rightarrow c d c d a c d c \\ + e b d e^{2} &\rightarrow c d c d a c d e \\ + c d^{2} e d &\rightarrow e d^{2} \\ + c b d c d e d c^{2} &\rightarrow d a b c d^{2} c e \\ + c e d c &\rightarrow d e d e c \\ + c e d^{2} &\rightarrow d e d e d \\ + c e d e &\rightarrow d e d e^{2} \\ + c e c d c &\rightarrow d e c d e c \\ + c e c d^{2} &\rightarrow d e c d e d \\ + c e c d e &\rightarrow d e c d e^{2} \\ + d a b c d^{2} c^{2} &\rightarrow c b d c d e \\ + d a b c d^{2} c d e c &\rightarrow c b d c d e c d c \\ + d a b c d^{2} c d e d &\rightarrow c b d c d e c d^{2} \\ + d a b c d^{2} c d e^{2} &\rightarrow c b d c d e c d e \\ + c b d e d &\rightarrow d a c d^{2} \\ + e b d c d c d &\rightarrow c d c d a b c d^{2} \\ + b c^{2} e &\rightarrow c d^{2} c^{4} \\ + a c e &\rightarrow e d c^{4} \\ + c^{3} e &\rightarrow d e^{2} c^{2} \\ + d a b c d^{3} e c &\rightarrow c b d c d e d c \\ + d a b c d^{3} e^{2} &\rightarrow c b d c d e d e \\ + e b d e d &\rightarrow c d c d a c d^{2} \\ + c b d c d e d c d e d e c &\rightarrow d a b c d^{2} c e^{2} d c \\ + c b d c d e d c d e d e d &\rightarrow d a b c d^{2} c e^{2} d^{2} \\ + c b d c d e d c d e d e^{2} &\rightarrow d a b c d^{2} c e^{2} d e \\ + c e c e &\rightarrow d e d e^{2} c^{2} \\ + d a b c d^{3} e d &\rightarrow c b d c d e d^{2} \\ + d e c d e d e c &\rightarrow c e^{2} d c \\ + d e c d e d e d &\rightarrow c e^{2} d^{2} \\ + d e c d e d e^{2} &\rightarrow c e^{2} d e \\ + e b d c d e d c^{2} &\rightarrow c d c d a b c d^{2} c e \\ + c b d c d e d c d e c d e c &\rightarrow d a b c d^{2} c e^{2} c d c \\ + c b d c d e d c d e c d e d &\rightarrow d a b c d^{2} c e^{2} c d^{2} \\ + c b d c d e d c d e c d e^{2} &\rightarrow d a b c d^{2} c e^{2} c d e \\ + c e c^{2} e &\rightarrow d e c d e^{2} c^{2} \\ + c e^{2} d c^{2} &\rightarrow d e c d e c e \\ + d e c d e c d e c &\rightarrow c e^{2} c d c \\ + d e c d e c d e d &\rightarrow c e^{2} c d^{2} \\ + d e c d e c d e^{2} &\rightarrow c e^{2} c d e \\ + e b d c d e d c d e d e c &\rightarrow c d c d a b c d^{2} c e^{2} d c \\ + e b d c d e d c d e d e d &\rightarrow c d c d a b c d^{2} c e^{2} d^{2} \\ + e b d c d e d c d e d e^{2} &\rightarrow c d c d a b c d^{2} c e^{2} d e \\ + c e^{2} d c d e d e c &\rightarrow d e c d e c e^{2} d c \\ + c e^{2} d c d e d e d &\rightarrow d e c d e c e^{2} d^{2} \\ + c e^{2} d c d e d e^{2} &\rightarrow d e c d e c e^{2} d e \\ + e b d c d e d c d e c d e c &\rightarrow c d c d a b c d^{2} c e^{2} c d c \\ + e b d c d e d c d e c d e d &\rightarrow c d c d a b c d^{2} c e^{2} c d^{2} \\ + e b d c d e d c d e c d e^{2} &\rightarrow c d c d a b c d^{2} c e^{2} c d e \\ + c e^{2} d c d e c d e c &\rightarrow d e c d e c e^{2} c d c \\ + c e^{2} d c d e c d e d &\rightarrow d e c d e c e^{2} c d^{2} \\ + c e^{2} d c d e c d e^{2} &\rightarrow d e c d e c e^{2} c d e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/260.md b/docs/examples/1-relation/260.md new file mode 100644 index 0000000..9d866a8 --- /dev/null +++ b/docs/examples/1-relation/260.md @@ -0,0 +1,36 @@ +#\#260 baaabbaa=abaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a^{2} = a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d = a c,\; c = b a^{3},\; d = b^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{2} a^{2} &\rightarrow d \\ + b a^{3} &\rightarrow c \\ + a c &\rightarrow c d \\ + b c &\rightarrow d a \\ + d a d^{3} &\rightarrow c^{2} \\ + b d a d^{2} &\rightarrow d c \\ + b d a d c^{2} &\rightarrow d c a d^{3} \\ + d a d^{2} c^{2} &\rightarrow c^{2} a d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/262.md b/docs/examples/1-relation/262.md new file mode 100644 index 0000000..e33f208 --- /dev/null +++ b/docs/examples/1-relation/262.md @@ -0,0 +1,36 @@ +#\#262 baabaaabba=aabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{3} b^{2} a = a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} a d = a^{2} c,\; c = b a^{2},\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{2} a &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a^{2} c &\rightarrow c^{2} a d \\ + b a c^{2} a d &\rightarrow c a c \\ + b c &\rightarrow d a \\ + d a c a d &\rightarrow c^{2} \\ + b a c^{2} a c^{2} &\rightarrow c a c a c a d \\ + d a c a c^{2} &\rightarrow c^{2} a c a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/263.md b/docs/examples/1-relation/263.md new file mode 100644 index 0000000..64acf37 --- /dev/null +++ b/docs/examples/1-relation/263.md @@ -0,0 +1,35 @@ +#\#263 aabaabaaba=baabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b a^{2} b a = b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c b c b c b a = b c b c,\; c = a^{2},\; d = a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a^{2} b &\rightarrow d \\ + b d a^{2} &\rightarrow d^{3} a \\ + b d^{2} &\rightarrow d^{3} a b \\ + b d a d &\rightarrow d^{4} \\ + a^{2} d^{3} a &\rightarrow d^{2} a^{2} \\ + a^{2} d^{4} &\rightarrow d^{2} a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/264.md b/docs/examples/1-relation/264.md new file mode 100644 index 0000000..99dc4f3 --- /dev/null +++ b/docs/examples/1-relation/264.md @@ -0,0 +1,41 @@ +#\#264 abaabbaaba=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{2} a^{2} b a = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a d = c a c,\; c = b a,\; d = b c a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a c &\rightarrow a c a d \\ + c^{2} a d &\rightarrow d \\ + c a^{2} c a d &\rightarrow a c a d a c \\ + a c a d c a d &\rightarrow c a d \\ + b c a d &\rightarrow d c a d \\ + a c a d a d c a d &\rightarrow d \\ + b d &\rightarrow d a d c a d \\ + a c a d a d a d c a d &\rightarrow c d \\ + b c d &\rightarrow d a d a d c a d \\ + a c a d a d a d a d c a d &\rightarrow c^{2} d \\ + b c^{2} d &\rightarrow d a d a d a d c a d \\ + c^{3} d &\rightarrow a c a d a d a d a d a d c a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/265.md b/docs/examples/1-relation/265.md new file mode 100644 index 0000000..c9b8e4a --- /dev/null +++ b/docs/examples/1-relation/265.md @@ -0,0 +1,37 @@ +#\#265 baaaabbaba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a b a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c,\; c = a^{2},\; d = b^{2} a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{2} a b &\rightarrow d \\ + b^{2} a d &\rightarrow d b a b \\ + b a^{2} &\rightarrow d a^{4} d a^{2} d a \\ + b d a^{4} d a^{2} d a^{2} &\rightarrow d a^{4} d a \\ + d a^{4} d a^{2} d a^{3} d a &\rightarrow a^{2} \\ + b d a^{4} d a^{4} &\rightarrow d a^{4} d a^{3} d a^{2} d a^{3} d a \\ + d a^{4} d a^{2} d a^{5} &\rightarrow a^{5} d a^{2} d a^{3} d a \\ + b d a^{6} &\rightarrow d a^{4} d a^{3} d a^{2} d a^{3} d a d a^{2} d a^{3} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/266.md b/docs/examples/1-relation/266.md new file mode 100644 index 0000000..0e5e280 --- /dev/null +++ b/docs/examples/1-relation/266.md @@ -0,0 +1,97 @@ +#\#266 ababbabbaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b^{2} a^{2} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b d a = b a b a,\; c = a b^{2} a,\; d = c b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b a b a &\rightarrow a b d a \\ + c b^{2} a &\rightarrow d \\ + a b^{2} a &\rightarrow c \\ + b a^{2} b d a &\rightarrow a b d a b a \\ + c b^{2} c &\rightarrow d b^{2} a \\ + a b^{2} c &\rightarrow d \\ + b a b c &\rightarrow a b d c \\ + b a^{2} b d c &\rightarrow a b d a b c \\ + c b^{2} d &\rightarrow d b^{2} c \\ + a b^{2} d &\rightarrow d b^{2} a \\ + b a b d &\rightarrow a b d^{2} \\ + a b d a b d a &\rightarrow b c b a \\ + b a^{2} b d^{2} &\rightarrow a b d a b d \\ + a^{2} b d^{2} c &\rightarrow c b c \\ + a^{2} b d^{3} &\rightarrow c b d \\ + c a b d^{2} c &\rightarrow d b c \\ + c a b d^{3} &\rightarrow d b d \\ + a^{2} b d^{2} a &\rightarrow c b a \\ + c a b d^{2} a &\rightarrow d b a \\ + b^{2} c b a &\rightarrow a b d^{2} a b d a \\ + a b d a b d^{2} &\rightarrow b c b d \\ + a b d a b d c &\rightarrow b c b c \\ + a b d b c b a &\rightarrow b d b a \\ + c a b d^{2} b c b a &\rightarrow d a b d^{2} a b d a \\ + a^{2} b d^{2} b c b a &\rightarrow d a b d^{2} a \\ + a b d b d b a &\rightarrow b d a b d^{2} a \\ + b^{2} d b a &\rightarrow a b d^{2} b c b a \\ + b^{2} c b d &\rightarrow a b d^{2} a b d^{2} \\ + b^{2} c b c &\rightarrow a b d^{2} a b d c \\ + b^{2} c a b d^{2} &\rightarrow a b d^{2} a b d a b d \\ + b^{2} c a b d c &\rightarrow a b d^{2} a b d a b c \\ + b^{2} c a b d a &\rightarrow a b d^{2} a b d a b a \\ + a b d b c b d &\rightarrow b d b d \\ + a b d b c b c &\rightarrow b d b c \\ + a b d b d b d &\rightarrow b d a b d^{3} \\ + a b d b d b c &\rightarrow b d a b d^{2} c \\ + c a b d^{2} b d b a &\rightarrow d a b d^{2} b c b a \\ + c a b d^{2} b c b d &\rightarrow d a b d^{2} a b d^{2} \\ + c a b d^{2} b c b c &\rightarrow d a b d^{2} a b d c \\ + a^{2} b d^{2} b d b a &\rightarrow d a b d^{2} a b d a \\ + a^{2} b d^{2} b c b d &\rightarrow d a b d^{3} \\ + a^{2} b d^{2} b c b c &\rightarrow d a b d^{2} c \\ + a b d b c a b d^{2} &\rightarrow b d a b d^{2} \\ + a b d b c a b d c &\rightarrow b d a b d c \\ + a b d b c a b d a &\rightarrow b d a b d a \\ + c a b d^{2} b c a b d^{2} &\rightarrow d a b d^{2} a b d a b d \\ + c a b d^{2} b c a b d c &\rightarrow d a b d^{2} a b d a b c \\ + c a b d^{2} b c a b d a &\rightarrow d a b d^{2} a b d a b a \\ + a b d b d a b d^{2} &\rightarrow b d a b d^{2} a b d \\ + a b d b d a b d c &\rightarrow b d a b d^{2} a b c \\ + a b d b d a b d a &\rightarrow b d a b d^{2} a b a \\ + b^{2} d b d &\rightarrow a b d^{2} b c b d \\ + b^{2} d b c &\rightarrow a b d^{2} b c b c \\ + b^{2} d a b d^{2} &\rightarrow a b d^{2} b c a b d^{2} \\ + b^{2} d a b d c &\rightarrow a b d^{2} b c a b d c \\ + b^{2} d a b d a &\rightarrow a b d^{2} b c a b d a \\ + a^{2} b d^{2} b c a b d^{2} &\rightarrow d a b d^{2} a b d \\ + a^{2} b d^{2} b c a b d c &\rightarrow d a b d^{2} a b c \\ + a^{2} b d^{2} b c a b d a &\rightarrow d a b d^{2} a b a \\ + c a b d^{2} b d b d &\rightarrow d a b d^{2} b c b d \\ + c a b d^{2} b d b c &\rightarrow d a b d^{2} b c b c \\ + c a b d^{2} b d a b d^{2} &\rightarrow d a b d^{2} b c a b d^{2} \\ + a^{2} b d^{2} b d b d &\rightarrow d a b d^{2} a b d^{2} \\ + a^{2} b d^{2} b d b c &\rightarrow d a b d^{2} a b d c \\ + c a b d^{2} b d a b d c &\rightarrow d a b d^{2} b c a b d c \\ + c a b d^{2} b d a b d a &\rightarrow d a b d^{2} b c a b d a \\ + a^{2} b d^{2} b d a b d^{2} &\rightarrow d a b d^{2} a b d a b d \\ + a^{2} b d^{2} b d a b d c &\rightarrow d a b d^{2} a b d a b c \\ + a^{2} b d^{2} b d a b d a &\rightarrow d a b d^{2} a b d a b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/267.md b/docs/examples/1-relation/267.md new file mode 100644 index 0000000..8861503 --- /dev/null +++ b/docs/examples/1-relation/267.md @@ -0,0 +1,52 @@ +#\#267 ababbabbba=babbbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b^{3} a = b a b^{3} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c^{2} b a = b d a,\; c = a b^{2},\; d = c b a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + c b a b &\rightarrow d \\ + a b^{2} &\rightarrow c \\ + a b c^{2} b a &\rightarrow b d a \\ + b d a b &\rightarrow a b c d \\ + c b^{2} d a &\rightarrow d c^{2} b a \\ + c b c &\rightarrow d b \\ + a b c d b &\rightarrow b d c \\ + b d b d c &\rightarrow a b c d c d b \\ + b d b d a &\rightarrow a b c d c^{2} b a \\ + b d a^{2} b c d &\rightarrow a b c d^{2} a b \\ + d b^{3} d a &\rightarrow c b d c^{2} b a \\ + d b^{2} c &\rightarrow c b d b \\ + d b^{2} a b &\rightarrow c b d \\ + c b^{2} d c &\rightarrow d c d b \\ + c b a^{2} b c d &\rightarrow d^{2} a b \\ + a b c d a b c d &\rightarrow b d c d a b \\ + a b a b c d &\rightarrow c d a b \\ + a b c^{2} b d &\rightarrow b d^{2} \\ + d b^{3} d c &\rightarrow c b d c d b \\ + d b^{2} a^{2} b c d &\rightarrow c b d^{2} a b \\ + c b^{2} d^{2} &\rightarrow d c^{2} b d \\ + b d b d^{2} &\rightarrow a b c d c^{2} b d \\ + d b^{3} d^{2} &\rightarrow c b d c^{2} b d \\ + a b c^{2} a b c d &\rightarrow b d^{2} a b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/268.md b/docs/examples/1-relation/268.md new file mode 100644 index 0000000..a417aa8 --- /dev/null +++ b/docs/examples/1-relation/268.md @@ -0,0 +1,33 @@ +#\#268 aababaabaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b a^{2} b a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c^{2} a c a = c a c,\; c = d,\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow d \\ + b a &\rightarrow d \\ + a^{2} d^{2} a d a &\rightarrow d a d \\ + b d a d &\rightarrow d a d^{2} a d a \\ + a^{2} d^{2} a d^{2} a d &\rightarrow d a d a d^{2} a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/269.md b/docs/examples/1-relation/269.md new file mode 100644 index 0000000..a726442 --- /dev/null +++ b/docs/examples/1-relation/269.md @@ -0,0 +1,40 @@ +#\#269 baaaaaabba=aaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{6} b^{2} a = a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{3} b^{2} a = c^{2},\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{5} d a &\rightarrow a^{4} \\ + a b a^{4} &\rightarrow d a^{5} d a \\ + b a^{5} d^{2} &\rightarrow a^{3} d \\ + a b a^{3} d &\rightarrow d a^{5} d^{2} \\ + d a^{5} d a^{2} d^{2} &\rightarrow a^{4} d \\ + d a^{5} d a^{2} d a &\rightarrow a^{5} \\ + b a^{9} d &\rightarrow a^{8} d a^{2} d^{2} \\ + b a^{10} &\rightarrow a^{8} d a^{2} d a \\ + d a^{5} d a^{6} d &\rightarrow a^{9} d a^{2} d^{2} \\ + d a^{5} d a^{7} &\rightarrow a^{9} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/270.md b/docs/examples/1-relation/270.md new file mode 100644 index 0000000..7eb3783 --- /dev/null +++ b/docs/examples/1-relation/270.md @@ -0,0 +1,37 @@ +#\#270 baaaabbaba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a b a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c a,\; c = a^{2},\; d = b^{2} a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{2} a b &\rightarrow d \\ + b^{2} a d &\rightarrow d b a b \\ + b a^{3} &\rightarrow d a^{4} d a d a \\ + b d a^{4} d a d a^{2} &\rightarrow d a^{4} d a \\ + d a^{4} d a d a^{2} d a &\rightarrow a^{3} \\ + b d a^{4} d a^{4} &\rightarrow d a^{4} d a^{3} d a d a^{2} d a \\ + d a^{4} d a d a^{5} &\rightarrow a^{6} d a d a^{2} d a \\ + b d a^{7} &\rightarrow d a^{4} d a^{3} d a d a^{2} d a d a d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/271.md b/docs/examples/1-relation/271.md new file mode 100644 index 0000000..04f78d7 --- /dev/null +++ b/docs/examples/1-relation/271.md @@ -0,0 +1,36 @@ +#\#271 baaaaabbba=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{3} a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d b a = d c,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{4} d b a &\rightarrow d a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{4} d b d &\rightarrow d a d \\ + b a^{4} d^{2} a^{2} &\rightarrow d a^{5} d b a \\ + a b d a^{2} &\rightarrow d a^{4} d b a \\ + b a^{4} d^{2} a d &\rightarrow d a^{5} d b d \\ + a b d a d &\rightarrow d a^{4} d b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/272.md b/docs/examples/1-relation/272.md new file mode 100644 index 0000000..f88ef4e --- /dev/null +++ b/docs/examples/1-relation/272.md @@ -0,0 +1,36 @@ +#\#272 ababababa=babaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b a b a = b a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{4} a = d c a,\; c = a b,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c a &\rightarrow d \\ + d c a &\rightarrow c^{4} a \\ + a b &\rightarrow c \\ + b c^{2} &\rightarrow d b \\ + d c^{2} &\rightarrow c^{5} \\ + a d &\rightarrow c^{2} a \\ + a c^{5} &\rightarrow c^{2} a c^{2} \\ + a c^{4} a &\rightarrow c^{2} a c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/273.md b/docs/examples/1-relation/273.md new file mode 100644 index 0000000..586731a --- /dev/null +++ b/docs/examples/1-relation/273.md @@ -0,0 +1,31 @@ +#\#273 aaabababa=babaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a b a b a = b a b a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d^{3} = d b c^{2},\; c = a^{2},\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a &\rightarrow d \\ + d^{2} a^{3} &\rightarrow a^{3} d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/274.md b/docs/examples/1-relation/274.md new file mode 100644 index 0000000..368ae38 --- /dev/null +++ b/docs/examples/1-relation/274.md @@ -0,0 +1,44 @@ +#\#274 baababaaba=abaaca + +## Original presentation + +\( +\left\langle + a, b, c \;\middle|\; b a^{2} b a b a^{2} b a = a b a^{2} c a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d = c b a,\; c = b a^{5},\; d = b a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b a b c &\rightarrow d \\ + b a^{5} &\rightarrow c \\ + c b a &\rightarrow a d \\ + b a b a d &\rightarrow d b a \\ + a d b c &\rightarrow c d \\ + a d a^{4} &\rightarrow c^{2} \\ + b a^{4} c d &\rightarrow c d b c \\ + b a^{4} c^{2} &\rightarrow c d a^{4} \\ + a d b a d &\rightarrow c d b a \\ + c b c d &\rightarrow a d^{2} b c \\ + c b c^{2} &\rightarrow a d^{2} a^{4} \\ + c^{2} d b c &\rightarrow a d a^{3} c d \\ + c^{2} d a^{4} &\rightarrow a d a^{3} c^{2} \\ + b a^{4} c a d &\rightarrow c d a^{4} b a \\ + c b c a d &\rightarrow a d^{2} a^{4} b a \\ + c^{2} d b a d &\rightarrow a d a^{3} c d b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/275.md b/docs/examples/1-relation/275.md new file mode 100644 index 0000000..868c001 --- /dev/null +++ b/docs/examples/1-relation/275.md @@ -0,0 +1,40 @@ +#\#275 baaabba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c d a = c,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{2} d a &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + a b a^{2} &\rightarrow d a^{2} d a \\ + b a^{2} d^{2} &\rightarrow a d \\ + a b a d &\rightarrow d a^{2} d^{2} \\ + d a^{2} d a d^{2} &\rightarrow a^{2} d \\ + d a^{2} d a d a &\rightarrow a^{3} \\ + b a^{4} d &\rightarrow a^{3} d a d^{2} \\ + b a^{5} &\rightarrow a^{3} d a d a \\ + d a^{2} d a^{3} d &\rightarrow a^{4} d a d^{2} \\ + d a^{2} d a^{4} &\rightarrow a^{4} d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/276.md b/docs/examples/1-relation/276.md new file mode 100644 index 0000000..569fde1 --- /dev/null +++ b/docs/examples/1-relation/276.md @@ -0,0 +1,33 @@ +#\#276 aabaababaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b a b a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c a c^{2} a = c a c,\; c = d,\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow d \\ + b a &\rightarrow d \\ + a^{2} d a d^{2} a &\rightarrow d a d \\ + b d a d &\rightarrow d a d a d^{2} a \\ + a^{2} d a d^{3} a d &\rightarrow d a d a d a d^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/277.md b/docs/examples/1-relation/277.md new file mode 100644 index 0000000..4cffbd3 --- /dev/null +++ b/docs/examples/1-relation/277.md @@ -0,0 +1,38 @@ +#\#277 baaaababaa=aaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a b a^{2} = a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a^{2} c d = a^{3} c,\; c = b a,\; d = c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a^{3} c &\rightarrow d a^{2} c d \\ + c a &\rightarrow d \\ + b d a^{2} c d &\rightarrow d a c \\ + a^{3} d &\rightarrow d a^{2} c d a \\ + c d a^{2} c d &\rightarrow d a^{2} c \\ + b d a^{2} d a^{2} c &\rightarrow d a d a c d \\ + c d a^{2} d a^{2} c &\rightarrow d a^{2} d a c d \\ + b d a^{2} d a^{2} d &\rightarrow d a d a c d a \\ + c d a^{2} d a^{2} d &\rightarrow d a^{2} d a c d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/278.md b/docs/examples/1-relation/278.md new file mode 100644 index 0000000..19f7ca3 --- /dev/null +++ b/docs/examples/1-relation/278.md @@ -0,0 +1,39 @@ +#\#278 baababbba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{3} a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c b^{2} c = a,\; c = b a,\; d = a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + b d c b c &\rightarrow d c b^{2} c \\ + a &\rightarrow b d c b^{2} c \\ + c b d c b^{2} c &\rightarrow b d \\ + b^{2} d c b^{2} c &\rightarrow c \\ + b d c b^{3} d &\rightarrow d \\ + b d c^{2} &\rightarrow d c b c \\ + b d c b^{2} d &\rightarrow d c b^{3} d \\ + b d c b d &\rightarrow d c b^{2} d \\ + b d b d &\rightarrow d c^{2} \\ + c d &\rightarrow d c^{3} b^{2} c \\ + b d^{2} &\rightarrow d c^{3} b^{3} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/279.md b/docs/examples/1-relation/279.md new file mode 100644 index 0000000..9796ef3 --- /dev/null +++ b/docs/examples/1-relation/279.md @@ -0,0 +1,47 @@ +#\#279 abbabaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a^{2} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d = c^{2},\; c = b a,\; d = b c^{2} a,\; e = d c d +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < a < e < d < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a e &\rightarrow e c^{2} \\ + c a e a &\rightarrow a e \\ + c e &\rightarrow e c^{2} a \\ + a e c^{2} &\rightarrow e c^{2} a c^{2} a^{2} \\ + a e a c^{2} &\rightarrow e c^{2} a^{2} \\ + a e a^{2} c a e &\rightarrow e a^{2} e a \\ + a e a^{2} c^{2} &\rightarrow e a \\ + d &\rightarrow a e a^{3} \\ + b c^{2} &\rightarrow a e a^{2} \\ + a e a^{4} e a^{2} &\rightarrow e \\ + a e a^{3} e &\rightarrow e a^{2} e a^{2} \\ + a^{2} e a^{3} &\rightarrow c^{2} \\ + b e &\rightarrow e c^{2} a^{5} e a^{2} \\ + a e a^{2} e &\rightarrow e c^{2} a^{3} e a^{2} \\ + a e a c a e &\rightarrow e c^{2} a^{3} e a \\ + a e c a e &\rightarrow e c^{2} a c^{2} a^{3} e a \\ + a e^{2} &\rightarrow e c^{2} a c^{2} a c^{2} a^{3} e a^{2} \\ + a e a e &\rightarrow e c^{2} a c^{2} a^{3} e a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/280.md b/docs/examples/1-relation/280.md new file mode 100644 index 0000000..dd66e77 --- /dev/null +++ b/docs/examples/1-relation/280.md @@ -0,0 +1,51 @@ +#\#280 abaabbabaa=babaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{2} a b a^{2} = b a b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a c d c^{2} a = c^{2} a^{2} c,\; c = b a,\; d = a b,\; e = d a d c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow d \\ + d a d c &\rightarrow e \\ + b d &\rightarrow c b \\ + a c &\rightarrow d a \\ + d a d e c a &\rightarrow e c a d a \\ + c^{2} a^{2} e c a &\rightarrow e c d a^{2} d a \\ + c^{2} a d a &\rightarrow e c a \\ + c^{2} a d^{2} &\rightarrow e c d \\ + d a d e c d &\rightarrow e c a d^{2} \\ + c^{2} a^{2} e c d &\rightarrow e c d a^{2} d^{2} \\ + d^{2} a^{3} e c a &\rightarrow a e c d a^{2} d a \\ + d^{2} a^{2} d a &\rightarrow a e c a \\ + b e &\rightarrow c^{2} d c \\ + c^{2} a e &\rightarrow e c a d c \\ + d^{2} a^{2} d^{2} &\rightarrow a e c d \\ + d^{2} a^{3} e c d &\rightarrow a e c d a^{2} d^{2} \\ + d a d e c e &\rightarrow e c a d e \\ + c^{2} a^{2} e c e &\rightarrow e c d a^{2} d e \\ + d^{2} a^{2} e &\rightarrow a e c a d c \\ + c^{2} a d e &\rightarrow e c e \\ + d^{2} a^{3} e c e &\rightarrow a e c d a^{2} d e \\ + d^{2} a^{2} d e &\rightarrow a e c e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/282.md b/docs/examples/1-relation/282.md new file mode 100644 index 0000000..00625a4 --- /dev/null +++ b/docs/examples/1-relation/282.md @@ -0,0 +1,34 @@ +#\#282 abababaaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b a^{2} b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d c = d,\; c = b a,\; d = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c^{2} a &\rightarrow d \\ + a c d c &\rightarrow d \\ + c^{2} d &\rightarrow d c d c \\ + a c d^{2} &\rightarrow d c a \\ + b d &\rightarrow d c d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/283.md b/docs/examples/1-relation/283.md new file mode 100644 index 0000000..f46a3e8 --- /dev/null +++ b/docs/examples/1-relation/283.md @@ -0,0 +1,42 @@ +#\#283 baabbbaaba=aabbbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{3} a^{2} b a = a^{2} b^{3} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c a c = a d c a,\; c = b a,\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a d c a &\rightarrow c d c a c \\ + a b^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + a d c d &\rightarrow c d c a c b^{2} \\ + a b c &\rightarrow d a \\ + b d &\rightarrow c b^{2} \\ + b c d c a c &\rightarrow c d c a \\ + a c d c a &\rightarrow d c d c a c^{2} \\ + a d c^{2} d c a c &\rightarrow c d c d c d c a c^{2} \\ + a c d c d &\rightarrow d c d c a c^{2} b^{2} \\ + a c d c^{2} d c a c &\rightarrow d c d c a c^{2} d c a \\ + a c d c^{2} d c d c d c a c^{2} &\rightarrow d c d c a c^{2} d c^{2} d c a c \\ + b c d c d c d c a c^{2} &\rightarrow c d c^{2} d c a c \\ + a d c^{2} d c d c d c a c^{2} &\rightarrow c d c d c d c a c^{2} d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/285.md b/docs/examples/1-relation/285.md new file mode 100644 index 0000000..ab965f1 --- /dev/null +++ b/docs/examples/1-relation/285.md @@ -0,0 +1,42 @@ +#\#285 baaabaaaba=aaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a^{3} b a = a^{3} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} b a = d b a,\; c = b a^{3},\; d = a^{3} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a^{3} &\rightarrow c \\ + a^{3} b &\rightarrow d \\ + d b a &\rightarrow c^{2} b a \\ + b a^{2} d &\rightarrow c a^{2} b \\ + b a d &\rightarrow c a b \\ + b d &\rightarrow c b \\ + a^{3} c &\rightarrow d a^{3} \\ + d c &\rightarrow c^{3} \\ + b a^{2} c^{2} b a &\rightarrow c a^{2} b^{2} a \\ + b a^{2} c^{3} &\rightarrow c a^{2} b c \\ + b a c^{2} b a &\rightarrow c a b^{2} a \\ + b a c^{3} &\rightarrow c a b c \\ + b c^{2} b a &\rightarrow c b^{2} a \\ + b c^{3} &\rightarrow c b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/286.md b/docs/examples/1-relation/286.md new file mode 100644 index 0000000..9ea1f66 --- /dev/null +++ b/docs/examples/1-relation/286.md @@ -0,0 +1,36 @@ +#\#286 bababaaaba=aabababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a^{3} b a = a^{2} b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a c a = a c^{3} a,\; c = a b,\; d = b c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b c^{2} a &\rightarrow d \\ + d a c a &\rightarrow a c^{3} a \\ + a b &\rightarrow c \\ + b c^{3} &\rightarrow d b \\ + a d &\rightarrow c^{3} a \\ + d a c^{2} &\rightarrow a c^{4} \\ + a^{2} c^{3} a &\rightarrow c^{3} a^{2} c a \\ + a^{2} c^{4} &\rightarrow c^{3} a^{2} c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/288.md b/docs/examples/1-relation/288.md new file mode 100644 index 0000000..fd00c34 --- /dev/null +++ b/docs/examples/1-relation/288.md @@ -0,0 +1,38 @@ +#\#288 abbaabbaa=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b^{2} a^{2} = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c a b c a = c b c,\; c = b a,\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b^{2} &\rightarrow d \\ + b d a &\rightarrow d a d a^{2} \\ + a d a d a^{2} d a^{2} &\rightarrow d^{2} a \\ + b d^{2} &\rightarrow d a d a d \\ + a d a d a^{2} d a d &\rightarrow d^{3} \\ + a d a d a^{2} d^{3} a &\rightarrow d^{3} a d a^{2} d a^{2} \\ + a d a d a^{2} d^{4} &\rightarrow d^{3} a d a^{2} d a d \\ + a d a d a d^{2} a &\rightarrow d^{3} a^{2} d a^{2} \\ + a d a d a d^{3} &\rightarrow d^{3} a^{2} d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/289.md b/docs/examples/1-relation/289.md new file mode 100644 index 0000000..5767468 --- /dev/null +++ b/docs/examples/1-relation/289.md @@ -0,0 +1,53 @@ +#\#289 abbabbbaba=bababba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{3} a b a = b a b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a b c d c^{2} = c^{2} b c,\; c = b a,\; d = b^{2},\; e = d a d c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < e < d < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + d a d c &\rightarrow e \\ + b d &\rightarrow d b \\ + b c &\rightarrow d a \\ + d a c d a &\rightarrow c e c \\ + c^{2} d a &\rightarrow a e c \\ + b e &\rightarrow d c d c \\ + d a d a e c &\rightarrow e c d a \\ + d a c e &\rightarrow c e c d c \\ + d a c^{2} e c &\rightarrow c e a e c \\ + a e c d c &\rightarrow c^{2} e \\ + a e a e c &\rightarrow c^{3} e c \\ + d a c d c^{2} e &\rightarrow c e c e c d c \\ + d a c d c^{3} e c &\rightarrow c e c e a e c \\ + d a c^{2} e a e c &\rightarrow c e c^{3} e c \\ + c^{2} d c^{2} e &\rightarrow a e c e c d c \\ + c^{2} d c^{3} e c &\rightarrow a e c e a e c \\ + c^{3} e c d c &\rightarrow a e c^{2} e \\ + a e c^{3} e c &\rightarrow c^{3} e a e c \\ + a e c d a e c &\rightarrow c^{2} e c d a \\ + d a c d c^{3} e a e c &\rightarrow c e c e c^{3} e c \\ + c^{2} d c^{3} e a e c &\rightarrow a e c e c^{3} e c \\ + c^{3} e c d a e c &\rightarrow a e c^{2} e c d a \\ + a e c^{3} e a e c &\rightarrow c^{3} e c^{3} e c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/290.md b/docs/examples/1-relation/290.md new file mode 100644 index 0000000..3c80e3d --- /dev/null +++ b/docs/examples/1-relation/290.md @@ -0,0 +1,71 @@ +#\#290 abaabbaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{2} a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d a = b a c a,\; c = a b,\; d = a c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a c a &\rightarrow c d a \\ + a b &\rightarrow c \\ + a c b a &\rightarrow d \\ + b a c^{2} &\rightarrow c d c \\ + b a c d &\rightarrow c d^{2} \\ + c a c a &\rightarrow a c d a \\ + a c b c &\rightarrow d b \\ + a c b d &\rightarrow d c b a \\ + a c^{2} d a &\rightarrow d c a \\ + b d c a &\rightarrow c d c d a \\ + b a^{2} c d a &\rightarrow c d a c a \\ + c a c^{2} &\rightarrow a c d c \\ + c a c d &\rightarrow a c d^{2} \\ + c a^{2} c d a &\rightarrow a c d a c a \\ + a c^{2} d c &\rightarrow d c^{2} \\ + a c^{2} d^{2} &\rightarrow d c d \\ + b d c^{2} &\rightarrow c d c d c \\ + b d c d &\rightarrow c d c d^{2} \\ + b d a c d c &\rightarrow c d c d a c^{2} \\ + b d a c d^{2} &\rightarrow c d c d a c d \\ + b d a c d a &\rightarrow c d c d a c a \\ + b a^{2} c d c &\rightarrow c d a c^{2} \\ + b a^{2} c d^{2} &\rightarrow c d a c d \\ + c a^{2} c d c &\rightarrow a c d a c^{2} \\ + c a^{2} c d^{2} &\rightarrow a c d a c d \\ + a c d c d c &\rightarrow c d c^{2} \\ + a c d c d^{2} &\rightarrow c d c d \\ + a c d c d a &\rightarrow c d c a \\ + b c d c^{2} &\rightarrow c d^{2} c d c \\ + b c d c d &\rightarrow c d^{2} c d^{2} \\ + c^{2} d c^{2} &\rightarrow a c d^{2} c d c \\ + c^{2} d c d &\rightarrow a c d^{2} c d^{2} \\ + b c d c a &\rightarrow c d^{2} c d a \\ + c^{2} d c a &\rightarrow a c d^{2} c d a \\ + a^{2} c d^{2} c d c &\rightarrow d c^{3} \\ + a^{2} c d^{2} c d^{2} &\rightarrow d c^{2} d \\ + b c d a c d c &\rightarrow c d^{2} c d a c^{2} \\ + b c d a c d^{2} &\rightarrow c d^{2} c d a c d \\ + b c d a c d a &\rightarrow c d^{2} c d a c a \\ + c^{2} d a c d c &\rightarrow a c d^{2} c d a c^{2} \\ + c^{2} d a c d^{2} &\rightarrow a c d^{2} c d a c d \\ + a^{2} c d^{2} c d a &\rightarrow d c^{2} a \\ + c^{2} d a c d a &\rightarrow a c d^{2} c d a c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/291.md b/docs/examples/1-relation/291.md new file mode 100644 index 0000000..32b1587 --- /dev/null +++ b/docs/examples/1-relation/291.md @@ -0,0 +1,36 @@ +#\#291 baabbabbaa=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a b^{2} a^{2} = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d c a^{2} = a d,\; c = b^{2},\; d = a c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a c a &\rightarrow d \\ + b a d &\rightarrow c a d c a^{2} \\ + b c &\rightarrow c b \\ + a c d &\rightarrow d c a \\ + c a d c a d^{2} &\rightarrow a d c a \\ + c a d c a d a &\rightarrow a d \\ + a^{2} d &\rightarrow d^{2} c a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/292.md b/docs/examples/1-relation/292.md new file mode 100644 index 0000000..9d8f99a --- /dev/null +++ b/docs/examples/1-relation/292.md @@ -0,0 +1,40 @@ +#\#292 baaaaabba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{4} d a &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + a b a^{2} &\rightarrow d a^{4} d a \\ + b a^{4} d^{2} &\rightarrow a d \\ + a b a d &\rightarrow d a^{4} d^{2} \\ + d a^{4} d a^{3} d^{2} &\rightarrow a^{2} d \\ + d a^{4} d a^{3} d a &\rightarrow a^{3} \\ + b a^{6} d &\rightarrow a^{5} d a^{3} d^{2} \\ + b a^{7} &\rightarrow a^{5} d a^{3} d a \\ + d a^{4} d a^{5} d &\rightarrow a^{6} d a^{3} d^{2} \\ + d a^{4} d a^{6} &\rightarrow a^{6} d a^{3} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/293.md b/docs/examples/1-relation/293.md new file mode 100644 index 0000000..d85b71d --- /dev/null +++ b/docs/examples/1-relation/293.md @@ -0,0 +1,60 @@ +#\#293 babaaaabba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{4} b^{2} a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c e c = a c^{2},\; c = b a,\; d = c a,\; e = d a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{2} &\rightarrow c e c \\ + d a^{2} b &\rightarrow e \\ + c a &\rightarrow d \\ + b c e c &\rightarrow c^{3} \\ + d a^{2} c &\rightarrow e a \\ + d c^{2} &\rightarrow c^{2} e c \\ + a c d &\rightarrow c e d \\ + b c e d &\rightarrow c^{2} d \\ + d a^{2} d &\rightarrow e a^{2} \\ + d c d &\rightarrow c^{2} e d \\ + a c e &\rightarrow c e^{2} \\ + b c e^{2} &\rightarrow c^{2} e \\ + d a^{2} e a^{2} &\rightarrow e a^{4} d \\ + d c e &\rightarrow c^{2} e^{2} \\ + e a^{4} b &\rightarrow d a^{2} e \\ + e a^{4} c &\rightarrow d a^{2} e a \\ + c^{2} e^{3} c^{2} &\rightarrow e c e c \\ + c^{2} e^{3} c e &\rightarrow e c e^{2} \\ + e a e &\rightarrow c^{2} e^{4} \\ + c^{2} e^{3} c d &\rightarrow e c e d \\ + e a d &\rightarrow c^{2} e^{3} d \\ + e a c &\rightarrow c^{2} e^{3} c \\ + a e c e d &\rightarrow c e c e^{3} c d \\ + a e c e^{2} &\rightarrow c e c e^{3} c e \\ + a e c e c &\rightarrow c e c e^{3} c^{2} \\ + d e c e d &\rightarrow c^{2} e c e^{3} c d \\ + d e c e^{2} &\rightarrow c^{2} e c e^{3} c e \\ + d e c e c &\rightarrow c^{2} e c e^{3} c^{2} \\ + c^{2} e^{4} c e d &\rightarrow e c e c e^{3} c d \\ + c^{2} e^{4} c e^{2} &\rightarrow e c e c e^{3} c e \\ + c^{2} e^{4} c e c &\rightarrow e c e c e^{3} c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/294.md b/docs/examples/1-relation/294.md new file mode 100644 index 0000000..e4cbe49 --- /dev/null +++ b/docs/examples/1-relation/294.md @@ -0,0 +1,41 @@ +#\#294 babbaaba=abba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a^{2} b a = a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d c = e c,\; c = b a,\; d = c b c a,\; e = a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow e \\ + e c &\rightarrow d c \\ + c b c a &\rightarrow d \\ + b e &\rightarrow c b \\ + a c &\rightarrow e a \\ + e d &\rightarrow d^{2} \\ + c b c e &\rightarrow d b \\ + b d^{2} &\rightarrow c b d \\ + b d c &\rightarrow c b c \\ + a d &\rightarrow d^{2} c a \\ + c b c d^{2} &\rightarrow d b d \\ + c b c d c &\rightarrow d b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/296.md b/docs/examples/1-relation/296.md new file mode 100644 index 0000000..9b40ebd --- /dev/null +++ b/docs/examples/1-relation/296.md @@ -0,0 +1,37 @@ +#\#296 ababbbabba=babbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{3} a b^{2} a = b a b^{2} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d c = c b c^{2},\; c = b a,\; d = b^{2} c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < b < a\): + +\( +\begin{aligned} + a c d c &\rightarrow c b c^{2} \\ + b a &\rightarrow c \\ + b^{2} c b &\rightarrow d \\ + b^{2} c d &\rightarrow d b c b \\ + b c b c^{2} &\rightarrow c^{2} d c \\ + d a &\rightarrow b^{2} c^{2} \\ + b^{2} c^{3} d c &\rightarrow d c b c^{2} \\ + b c^{2} d c &\rightarrow d c^{2} \\ + b c d c^{2} &\rightarrow c^{2} d c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/297.md b/docs/examples/1-relation/297.md new file mode 100644 index 0000000..04bb509 --- /dev/null +++ b/docs/examples/1-relation/297.md @@ -0,0 +1,40 @@ +#\#297 baabaabba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c b c b^{2} a = c,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{2} b a d a &\rightarrow a^{2} \\ + b a^{2} b a d^{2} &\rightarrow a d \\ + d a^{2} b a d a &\rightarrow a b a^{2} \\ + b a^{2} b a^{2} b a^{2} &\rightarrow a^{3} b a d a \\ + d a^{2} b a d^{2} &\rightarrow a b a d \\ + a b a^{3} b a d a &\rightarrow d a^{2} b a^{2} b a^{2} \\ + b a^{2} b a^{2} b a d &\rightarrow a^{3} b a d^{2} \\ + a b a^{3} b a d^{2} &\rightarrow d a^{2} b a^{2} b a d \\ + b a^{4} &\rightarrow a^{3} b a d^{2} a \\ + b a^{3} d &\rightarrow a^{3} b a d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/299.md b/docs/examples/1-relation/299.md new file mode 100644 index 0000000..fe9c5a5 --- /dev/null +++ b/docs/examples/1-relation/299.md @@ -0,0 +1,47 @@ +#\#299 babaababba=ababbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a b^{2} a = a b a b^{2} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a = a b a c a b a,\; c = b^{2},\; d = b a^{2} b a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a^{2} b a c &\rightarrow d \\ + a b a c a b a &\rightarrow b a d a \\ + b d &\rightarrow c a^{2} b a c \\ + b c &\rightarrow c b \\ + b a b a d a &\rightarrow d a b a \\ + b a d a^{2} b a c &\rightarrow a b a c a d \\ + a b a c b a d a &\rightarrow b a d a c a b a \\ + b a d b a d a &\rightarrow a b a c a c a d a \\ + b a b a c a d &\rightarrow c a d a^{2} b a c \\ + b a b a c a c a d a &\rightarrow c a d b a d a \\ + b a d a b a c a c a d a &\rightarrow a b a c a c a d b a d a \\ + b a d a b a d a &\rightarrow a b a c a d a b a \\ + b a d a b a c a d &\rightarrow a b a c a c a d a^{2} b a c \\ + a b a c a^{2} b a c a d &\rightarrow b a d a d a^{2} b a c \\ + a b a c a^{2} b a c a c a d a &\rightarrow b a d a d b a d a \\ + b a b a c a c a d b a d a &\rightarrow c a d a b a c a c a d a \\ + b a d a b a c a c a d b a d a &\rightarrow a b a c a c a d a b a c a c a d a \\ + a b a c a^{2} b a c a c a d b a d a &\rightarrow b a d a d a b a c a c a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/300.md b/docs/examples/1-relation/300.md new file mode 100644 index 0000000..5b6f9a8 --- /dev/null +++ b/docs/examples/1-relation/300.md @@ -0,0 +1,56 @@ +#\#300 aabbaabbba=baabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{2} a^{2} b^{3} a = b a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} b a = b c a,\; c = a^{2} b^{2},\; d = c b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b c a &\rightarrow c^{2} b a \\ + c b c &\rightarrow d \\ + a^{2} b^{2} &\rightarrow c \\ + b c^{2} &\rightarrow c d \\ + c^{3} b a &\rightarrow d a \\ + d b c &\rightarrow c b d \\ + a^{2} c d b a &\rightarrow c^{2} a \\ + b c d &\rightarrow c^{2} b d \\ + b d a &\rightarrow c d c b a \\ + c^{2} d &\rightarrow d c \\ + a^{2} c^{2} b d &\rightarrow c^{3} \\ + b d c &\rightarrow c d^{2} \\ + c^{3} b d &\rightarrow d^{2} \\ + c^{4} &\rightarrow a^{2} c d c d \\ + a^{2} c d b d &\rightarrow d c \\ + a^{2} c d c d b a &\rightarrow c d a \\ + a^{2} c d c^{2} b a &\rightarrow c^{3} a \\ + b d^{2} &\rightarrow c d c b d \\ + b a^{2} c d c d &\rightarrow c d c^{2} \\ + a^{2} c d c d c d &\rightarrow c d c^{2} \\ + a^{2} c d c d^{2} &\rightarrow d c^{2} \\ + a^{2} c d c^{2} b d &\rightarrow c d c \\ + c a^{2} c d c d &\rightarrow a^{2} c d c d c \\ + a^{2} c d c d c^{2} b a &\rightarrow c d c a \\ + a^{2} c d c d b d &\rightarrow c d^{2} \\ + a^{2} c d c d c b d &\rightarrow d c d \\ + a^{2} c d c d c b a &\rightarrow d c a \\ + a^{2} c d c d c^{2} b d &\rightarrow c d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/301.md b/docs/examples/1-relation/301.md new file mode 100644 index 0000000..bf06bf2 --- /dev/null +++ b/docs/examples/1-relation/301.md @@ -0,0 +1,45 @@ +#\#301 abaabababa=babaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b a b a = b a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a c^{3} = d c,\; c = b a,\; d = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c^{2} a &\rightarrow d \\ + a c a c^{3} &\rightarrow d c \\ + b d c &\rightarrow d c^{3} \\ + d c a c^{3} &\rightarrow c^{2} d c \\ + a c a c^{2} d &\rightarrow d^{2} \\ + a c a c d &\rightarrow d c a \\ + b c^{2} d c &\rightarrow d c d c^{3} \\ + b d^{2} &\rightarrow d c^{2} d \\ + d c a c^{2} d &\rightarrow c^{2} d^{2} \\ + c^{2} d c a &\rightarrow d c a c d \\ + b c^{2} d^{2} &\rightarrow d c d c^{2} d \\ + c^{4} d c &\rightarrow d c a c d c^{3} \\ + c^{4} d^{2} &\rightarrow d c a c d c^{2} d \\ + d c a c d c a c d &\rightarrow c^{2} d c d c a \\ + d c a d c a c d &\rightarrow c^{2} d^{2} c a \\ + a c a d c a c d &\rightarrow d^{2} c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/302.md b/docs/examples/1-relation/302.md new file mode 100644 index 0000000..4773c05 --- /dev/null +++ b/docs/examples/1-relation/302.md @@ -0,0 +1,76 @@ +#\#302 babaabbbba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b^{4} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} a b^{3} c = a c,\; c = b a,\; d = a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < a < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a^{2} b &\rightarrow d \\ + b a b d b^{3} a &\rightarrow a b a \\ + d a b d b^{3} a &\rightarrow a d a \\ + b a b d b^{3} d &\rightarrow a b d \\ + b a b d b^{2} a b a &\rightarrow d a \\ + d a b d b^{3} d &\rightarrow a d^{2} \\ + a^{2} d a &\rightarrow d a b d b^{2} a b a \\ + a d a b d b^{2} a b a &\rightarrow d a b d b^{2} d a \\ + a^{2} d^{2} &\rightarrow d a b d b^{2} a b d \\ + b a b d b^{2} d a &\rightarrow a d a \\ + b a b d b^{2} a b d &\rightarrow d^{2} \\ + b a b d b^{2} d^{2} &\rightarrow a d^{2} \\ + b a b d b^{2} a d a &\rightarrow d a b d b^{2} a b a \\ + a d a b d b^{2} a b d &\rightarrow d a b d b^{2} d^{2} \\ + a d a b d b^{2} d a &\rightarrow d a b d b^{2} a d a \\ + a d a b d b^{2} d^{2} &\rightarrow d a b d b^{2} a d^{2} \\ + a d a b d b^{2} a d a &\rightarrow d a b d b^{2} d a b d b^{2} a b a \\ + a d a b d b^{2} a d^{2} &\rightarrow d a b d b^{2} d a b d b^{2} a b d \\ + b a b d b^{2} a d^{2} &\rightarrow d a b d b^{2} a b d \\ + b a b d b a d a &\rightarrow d^{2} b^{2} d a \\ + b a b d b a d^{2} &\rightarrow d^{2} b^{2} d^{2} \\ + b a b d b a b a &\rightarrow d^{2} b^{3} a \\ + b a b d b a b d &\rightarrow d^{2} b^{3} d \\ + b a b d b d a &\rightarrow d^{2} b^{2} a b a \\ + b a b d b d^{2} &\rightarrow d^{2} b^{2} a b d \\ + a d a b d b a d a &\rightarrow d a b d b^{2} d^{2} b^{2} d a \\ + a d a b d b a d^{2} &\rightarrow d a b d b^{2} d^{2} b^{2} d^{2} \\ + a d a b d b a b a &\rightarrow d a b d b^{2} d^{2} b^{3} a \\ + a d a b d b a b d &\rightarrow d a b d b^{2} d^{2} b^{3} d \\ + a d a b d b d a &\rightarrow d a b d b^{2} d^{2} b^{2} a b a \\ + a d a b d b d^{2} &\rightarrow d a b d b^{2} d^{2} b^{2} a b d \\ + b a b d a d a &\rightarrow d^{2} b^{3} d b^{2} d a \\ + b a b d a d^{2} &\rightarrow d^{2} b^{3} d b^{2} d^{2} \\ + b a b d a b a &\rightarrow d^{2} b^{3} d b^{3} a \\ + b a b d a b d &\rightarrow d^{2} b^{3} d b^{3} d \\ + b a b d^{2} a &\rightarrow d^{2} b^{3} d b^{2} a b a \\ + b a b d^{3} &\rightarrow d^{2} b^{3} d b^{2} a b d \\ + a d a b d a d a &\rightarrow d a b d b^{2} d^{2} b^{3} d b^{2} d a \\ + a d a b d a d^{2} &\rightarrow d a b d b^{2} d^{2} b^{3} d b^{2} d^{2} \\ + a d a b d a b a &\rightarrow d a b d b^{2} d^{2} b^{3} d b^{3} a \\ + a d a b d a b d &\rightarrow d a b d b^{2} d^{2} b^{3} d b^{3} d \\ + a d a b d^{2} a &\rightarrow d a b d b^{2} d^{2} b^{3} d b^{2} a b a \\ + a d a b d^{3} &\rightarrow d a b d b^{2} d^{2} b^{3} d b^{2} a b d \\ + b a b a d a &\rightarrow d^{2} b^{3} d b^{3} d b^{3} a \\ + b a b a d^{2} &\rightarrow d^{2} b^{3} d b^{3} d b^{3} d \\ + a d a b a d a &\rightarrow d a b d b^{2} d^{2} b^{3} d b^{3} d b^{3} a \\ + a d a b a d^{2} &\rightarrow d a b d b^{2} d^{2} b^{3} d b^{3} d b^{3} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/303.md b/docs/examples/1-relation/303.md new file mode 100644 index 0000000..f717dd5 --- /dev/null +++ b/docs/examples/1-relation/303.md @@ -0,0 +1,38 @@ +#\#303 abbababaa=baabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a b a^{2} = b a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d e = c a d,\; c = b a,\; d = b c,\; e = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b c &\rightarrow d \\ + b a &\rightarrow c \\ + c^{2} a &\rightarrow e \\ + c a d &\rightarrow a d e \\ + b e &\rightarrow d c a \\ + c d e &\rightarrow d a d \\ + a d e^{2} &\rightarrow e d \\ + b d a d &\rightarrow d^{2} e \\ + c e d &\rightarrow e d e \\ + b d e d &\rightarrow d^{2} e^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/305.md b/docs/examples/1-relation/305.md new file mode 100644 index 0000000..7e7099d --- /dev/null +++ b/docs/examples/1-relation/305.md @@ -0,0 +1,36 @@ +#\#305 baaaabbaa=abaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a^{2} = a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d = a c,\; c = b a^{3},\; d = b^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{2} a^{2} &\rightarrow d \\ + b a^{3} &\rightarrow c \\ + a c &\rightarrow c a d \\ + b c &\rightarrow d a \\ + d a^{2} d a d a d &\rightarrow c^{2} \\ + b d a^{2} d a d &\rightarrow d c \\ + b d a^{2} d c a d c &\rightarrow d c a^{2} d a d a d \\ + d a^{2} d a d c a d c &\rightarrow c^{2} a^{2} d a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/306.md b/docs/examples/1-relation/306.md new file mode 100644 index 0000000..f3f7af3 --- /dev/null +++ b/docs/examples/1-relation/306.md @@ -0,0 +1,57 @@ +#\#306 aabababbaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b a b^{2} a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d a = b a c a,\; c = a b,\; d = c^{2} b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a c a &\rightarrow a c d a \\ + c^{2} b a &\rightarrow d \\ + a b &\rightarrow c \\ + b a c^{2} &\rightarrow a c d c \\ + c^{2} b c &\rightarrow d b \\ + c a c a &\rightarrow a^{2} c d a \\ + b a c d &\rightarrow a c d^{2} \\ + b a d &\rightarrow a c d c b a \\ + b a^{3} c d a &\rightarrow a c d a c a \\ + c a c^{2} &\rightarrow a^{2} c d c \\ + c a^{3} c d a &\rightarrow a^{2} c d a c a \\ + d b c b c &\rightarrow c^{2} b d b \\ + d b c b a &\rightarrow c^{2} b d \\ + b a^{3} c d c &\rightarrow a c d a c^{2} \\ + c a c d &\rightarrow a^{2} c d^{2} \\ + c a^{3} c d c &\rightarrow a^{2} c d a c^{2} \\ + a^{2} c d c b c &\rightarrow c a d b \\ + a^{2} c d c b a &\rightarrow c a d \\ + a^{2} c d a^{2} c d^{2} c &\rightarrow c a d c^{2} \\ + a^{2} c d a^{2} c d^{2} a &\rightarrow c a d c a \\ + d b a^{2} c d^{3} &\rightarrow c^{2} b d c d \\ + c a^{2} c d^{3} &\rightarrow d c d \\ + d b a^{2} c d^{2} c &\rightarrow c^{2} b d c^{2} \\ + d b a^{2} c d^{2} a &\rightarrow c^{2} b d c a \\ + c a^{2} c d^{2} c &\rightarrow d c^{2} \\ + c a^{2} c d^{2} a &\rightarrow d c a \\ + b a^{3} c d^{2} &\rightarrow a c d a c d \\ + a^{2} c d a^{2} c d^{3} &\rightarrow c a d c d \\ + c a^{3} c d^{2} &\rightarrow a^{2} c d a c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/307.md b/docs/examples/1-relation/307.md new file mode 100644 index 0000000..5e4604a --- /dev/null +++ b/docs/examples/1-relation/307.md @@ -0,0 +1,45 @@ +#\#307 baabbbbbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{6} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a = a^{2},\; c = b^{2},\; d = a c^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a c^{3} &\rightarrow d \\ + b c &\rightarrow c b \\ + b a^{2} &\rightarrow c a d a \\ + b a d &\rightarrow c a d^{2} \\ + c a d^{3} &\rightarrow a d \\ + c a d^{2} a &\rightarrow a^{2} \\ + a c^{2} a d &\rightarrow d a d^{3} \\ + a c^{2} a^{2} &\rightarrow d a d^{2} a \\ + a c a d &\rightarrow d a d^{5} \\ + a c a^{2} &\rightarrow d a d^{4} a \\ + d a d^{7} &\rightarrow a^{2} d \\ + d a d^{6} a &\rightarrow a^{3} \\ + c a d a^{2} d &\rightarrow a^{2} d^{7} \\ + c a d a^{3} &\rightarrow a^{2} d^{6} a \\ + d a d^{5} a^{2} d &\rightarrow a^{3} d^{7} \\ + d a d^{5} a^{3} &\rightarrow a^{3} d^{6} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/308.md b/docs/examples/1-relation/308.md new file mode 100644 index 0000000..9554008 --- /dev/null +++ b/docs/examples/1-relation/308.md @@ -0,0 +1,51 @@ +#\#308 baabbaa=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{2} = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} c a^{2} = a^{2} c a,\; c = b^{2},\; d = c a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + c a^{3} &\rightarrow d \\ + b c &\rightarrow c b \\ + a^{2} c a &\rightarrow c a^{2} d \\ + b a^{2} d &\rightarrow c a^{2} d a \\ + a^{2} c^{2} a^{2} d &\rightarrow c a^{2} d a c a \\ + c b a^{3} &\rightarrow b d \\ + c a^{2} d a^{2} &\rightarrow a^{2} d \\ + c^{2} a^{2} d a d &\rightarrow d a c a \\ + c a c a^{2} d &\rightarrow d c a \\ + b d a c^{2} a^{2} d &\rightarrow c^{2} a^{2} d^{2} a c a \\ + b d a c a &\rightarrow c^{2} a^{2} d^{2} \\ + c b a c a^{2} d &\rightarrow b d c a \\ + c a^{2} d a d a^{2} &\rightarrow a^{4} d \\ + c a^{2} d a c a^{2} d &\rightarrow a^{2} d a c a \\ + c a^{2} d c a^{2} d &\rightarrow a^{2} d c a \\ + b d a^{5} d &\rightarrow c^{2} a^{2} d^{2} a d a d a^{2} \\ + b d a^{3} d &\rightarrow c^{2} a^{2} d^{2} a d a^{2} \\ + b d a d &\rightarrow c^{2} a^{2} d^{2} a^{2} \\ + a^{6} d &\rightarrow c a^{2} d a d a d a^{2} \\ + b a^{4} d &\rightarrow c a^{2} d^{2} a^{2} \\ + c a^{2} d a d a c a^{2} d &\rightarrow a^{4} d a c a \\ + c a^{2} d a d c a^{2} d &\rightarrow a^{4} d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/309.md b/docs/examples/1-relation/309.md new file mode 100644 index 0000000..58487e5 --- /dev/null +++ b/docs/examples/1-relation/309.md @@ -0,0 +1,156 @@ +#\#309 baaaaababa=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b a b a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a b a b a = a b^{2} a^{2},\; c = a^{4},\; d = b a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a b c &\rightarrow d \\ + b c a b a b a &\rightarrow a b^{2} a^{2} \\ + a^{4} &\rightarrow c \\ + b a^{2} b^{2} a^{2} &\rightarrow d a b a b a \\ + a b^{2} c a &\rightarrow b c a d \\ + a b^{2} a^{2} b c &\rightarrow b c a b a d \\ + a c &\rightarrow c a \\ + b a^{2} b d a b a d &\rightarrow d b c a d^{2} b a d \\ + b a^{2} b d a d &\rightarrow d b c a d^{3} \\ + b a^{2} b^{2} c &\rightarrow d a b a b a^{3} \\ + b c b c a d^{2} b a d &\rightarrow a b d a b a d \\ + b c b c a d^{3} &\rightarrow a b d a d \\ + a b^{2} a^{3} b^{2} a^{2} &\rightarrow b c a b a d a b a b a \\ + a b^{2} c b c a b a d &\rightarrow b c a d b^{2} a^{2} b c \\ + a b^{2} c b c a d &\rightarrow b c a d b^{2} c a \\ + a b^{2} c^{2} &\rightarrow b c a d a^{3} \\ + a b a b^{2} a^{2} &\rightarrow b c a d b a b a \\ + b a^{2} b d a b a b a &\rightarrow d b c a d^{2} b a b a \\ + c b^{2} a^{2} b c &\rightarrow a^{3} b c a b a d \\ + c b^{2} c a &\rightarrow a^{3} b c a d \\ + b a b a^{3} b c a b a d &\rightarrow d b^{2} a^{2} b c \\ + b a b a^{3} b c a d &\rightarrow d b^{2} c a \\ + b a^{2} b d b c a d^{2} b a b a &\rightarrow d a b a b a b d a b a b a \\ + b a^{2} b d b c a d^{2} b a d &\rightarrow d a b a b a b d a b a d \\ + b a^{2} b d b c a d^{3} &\rightarrow d a b a b a b d a d \\ + b c b c a d^{2} b a b a &\rightarrow a b d a b a b a \\ + b c a b a d b c a d^{2} b a b a &\rightarrow a b^{2} a^{3} b d a b a b a \\ + b c a b a d b c a d^{2} b a d &\rightarrow a b^{2} a^{3} b d a b a d \\ + b c a b a d b c a d^{3} &\rightarrow a b^{2} a^{3} b d a d \\ + b c a b a d a b a b a b d a b a b a &\rightarrow a b^{2} a^{3} b d b c a d^{2} b a b a \\ + b c a b a d a b a b a b d a b a d &\rightarrow a b^{2} a^{3} b d b c a d^{2} b a d \\ + b c a b a d a b a b a b d a d &\rightarrow a b^{2} a^{3} b d b c a d^{3} \\ + b c a b a d a b a b a^{3} &\rightarrow a b^{2} a^{3} b^{2} c \\ + b c a d b^{2} c a d b a d &\rightarrow a b a b d a b a d \\ + b c a d b^{2} c a d^{2} &\rightarrow a b a b d a d \\ + b c a d b^{2} c a d b a b a &\rightarrow a b a b d a b a b a \\ + b c a d b a b a b^{2} c &\rightarrow a b a b d a b a b a^{3} \\ + b c a d b a b a b d a b a b a &\rightarrow a b a b d b c a d^{2} b a b a \\ + b c a d b a b a b d a b a d &\rightarrow a b a b d b c a d^{2} b a d \\ + b c a d b a b a b d a d &\rightarrow a b a b d b c a d^{3} \\ + b c a d b a b a^{3} &\rightarrow a b a b^{2} c \\ + a b^{2} a^{2} b a^{3} b c a b a d &\rightarrow b c a b a d b^{2} a^{2} b c \\ + a b^{2} a^{2} b a^{3} b c a d &\rightarrow b c a b a d b^{2} c a \\ + b a^{2} b a b d a b a d &\rightarrow d a d b^{2} c a d b a d \\ + b a^{2} b a b d a d &\rightarrow d a d b^{2} c a d^{2} \\ + c b^{2} a^{2} b a^{3} b c a b a d &\rightarrow a^{3} b c a b a d b^{2} a^{2} b c \\ + c b^{2} a^{2} b a^{3} b c a d &\rightarrow a^{3} b c a b a d b^{2} c a \\ + c b^{2} a^{3} b^{2} a^{2} &\rightarrow a^{3} b c a b a d a b a b a \\ + c b^{2} c b c a b a d &\rightarrow a^{3} b c a d b^{2} a^{2} b c \\ + c b^{2} c b c a d &\rightarrow a^{3} b c a d b^{2} c a \\ + c b^{2} c^{2} &\rightarrow a^{3} b c a d a^{3} \\ + c b a b^{2} a^{2} &\rightarrow a^{3} b c a d b a b a \\ + b c a d b a b a b d b c a d^{2} b a b a &\rightarrow a b a b d a b a b a b d a b a b a \\ + b c a d b a b a b d b c a d^{2} b a d &\rightarrow a b a b d a b a b a b d a b a d \\ + b c a d b a b a b d b c a d^{3} &\rightarrow a b a b d a b a b a b d a d \\ + b a^{2} b a b d a b a b a &\rightarrow d a d b^{2} c a d b a b a \\ + b c a b a d a d b^{2} c a d b a b a &\rightarrow a b^{2} a^{3} b a b d a b a b a \\ + b c a b a d a d b^{2} c a d b a d &\rightarrow a b^{2} a^{3} b a b d a b a d \\ + b c a b a d a d b^{2} c a d^{2} &\rightarrow a b^{2} a^{3} b a b d a d \\ + b c a d^{2} b^{2} c a d b a b a &\rightarrow a b a b a b d a b a b a \\ + b c a d^{2} b^{2} c a d b a d &\rightarrow a b a b a b d a b a d \\ + b c a d^{2} b^{2} c a d^{2} &\rightarrow a b a b a b d a d \\ + a b^{2} a^{3} b a b d a b a b a^{3} &\rightarrow b c a b a d a d b a b a b^{2} c \\ + a b a b a b d a b a b a^{3} &\rightarrow b c a d^{2} b a b a b^{2} c \\ + b c a d b a b a b a b d a b a b a &\rightarrow a b a b d a d b^{2} c a d b a b a \\ + b c a d b a b a b a b d a b a d &\rightarrow a b a b d a d b^{2} c a d b a d \\ + b c a d b a b a b a b d a d &\rightarrow a b a b d a d b^{2} c a d^{2} \\ + b a^{2} b a b a b d a b a b a &\rightarrow d a d^{2} b^{2} c a d b a b a \\ + b a^{2} b a b a b d a b a d &\rightarrow d a d^{2} b^{2} c a d b a d \\ + b a^{2} b a b a b d a d &\rightarrow d a d^{2} b^{2} c a d^{2} \\ + c b^{2} a^{3} b a b d b c a d^{2} b a b a &\rightarrow a^{3} b c a b a d a d b a b a b d a b a b a \\ + c b^{2} a^{3} b a b d a b a b a^{3} &\rightarrow a^{3} b c a b a d a d b a b a b^{2} c \\ + c b a b a b d b c a d^{2} b a b a &\rightarrow a^{3} b c a d^{2} b a b a b d a b a b a \\ + c b a b a b d a b a b a^{3} &\rightarrow a^{3} b c a d^{2} b a b a b^{2} c \\ + b a^{2} b a b^{2} c &\rightarrow d a d b a b a^{3} \\ + b a^{2} b a b d b c a d^{2} b a b a &\rightarrow d a d b a b a b d a b a b a \\ + b a^{2} b a b d b c a d^{2} b a d &\rightarrow d a d b a b a b d a b a d \\ + b a^{2} b a b d b c a d^{3} &\rightarrow d a d b a b a b d a d \\ + a b^{2} a^{3} b a b^{2} c &\rightarrow b c a b a d a d b a b a^{3} \\ + a b^{2} a^{3} b a b d b c a d^{2} b a b a &\rightarrow b c a b a d a d b a b a b d a b a b a \\ + a b^{2} a^{3} b a b d b c a d^{2} b a d &\rightarrow b c a b a d a d b a b a b d a b a d \\ + a b^{2} a^{3} b a b d b c a d^{3} &\rightarrow b c a b a d a d b a b a b d a d \\ + a b a b a b^{2} c &\rightarrow b c a d^{2} b a b a^{3} \\ + a b a b a b d b c a d^{2} b a b a &\rightarrow b c a d^{2} b a b a b d a b a b a \\ + a b a b a b d b c a d^{2} b a d &\rightarrow b c a d^{2} b a b a b d a b a d \\ + a b a b a b d b c a d^{3} &\rightarrow b c a d^{2} b a b a b d a d \\ + c b^{2} a^{3} b a b^{2} c &\rightarrow a^{3} b c a b a d a d b a b a^{3} \\ + c b^{2} a^{3} b a b d b c a d^{2} b a d &\rightarrow a^{3} b c a b a d a d b a b a b d a b a d \\ + c b^{2} a^{3} b a b d b c a d^{3} &\rightarrow a^{3} b c a b a d a d b a b a b d a d \\ + c b a b a b^{2} c &\rightarrow a^{3} b c a d^{2} b a b a^{3} \\ + c b a b a b d b c a d^{2} b a d &\rightarrow a^{3} b c a d^{2} b a b a b d a b a d \\ + c b a b a b d b c a d^{3} &\rightarrow a^{3} b c a d^{2} b a b a b d a d \\ + a b^{2} a^{3} b a b d a b a b a b d a b a b a &\rightarrow b c a b a d a d b a b a b d b c a d^{2} b a b a \\ + a b^{2} a^{3} b a b d a b a b a b d a b a d &\rightarrow b c a b a d a d b a b a b d b c a d^{2} b a d \\ + a b^{2} a^{3} b a b d a b a b a b d a d &\rightarrow b c a b a d a d b a b a b d b c a d^{3} \\ + a b a b a b d a b a b a b d a b a b a &\rightarrow b c a d^{2} b a b a b d b c a d^{2} b a b a \\ + a b a b a b d a b a b a b d a b a d &\rightarrow b c a d^{2} b a b a b d b c a d^{2} b a d \\ + a b a b a b d a b a b a b d a d &\rightarrow b c a d^{2} b a b a b d b c a d^{3} \\ + c b^{2} a^{3} b a b d a b a b a b d a b a b a &\rightarrow a^{3} b c a b a d a d b a b a b d b c a d^{2} b a b a \\ + c b^{2} a^{3} b a b d a b a b a b d a b a d &\rightarrow a^{3} b c a b a d a d b a b a b d b c a d^{2} b a d \\ + c b^{2} a^{3} b a b d a b a b a b d a d &\rightarrow a^{3} b c a b a d a d b a b a b d b c a d^{3} \\ + c b a b a b d a b a b a b d a b a b a &\rightarrow a^{3} b c a d^{2} b a b a b d b c a d^{2} b a b a \\ + c b a b a b d a b a b a b d a b a d &\rightarrow a^{3} b c a d^{2} b a b a b d b c a d^{2} b a d \\ + c b a b a b d a b a b a b d a d &\rightarrow a^{3} b c a d^{2} b a b a b d b c a d^{3} \\ + a b^{2} a^{3} b a b a b d a b a b a &\rightarrow b c a b a d a d^{2} b^{2} c a d b a b a \\ + a b^{2} a^{3} b a b d a d b^{2} c a d b a b a &\rightarrow b c a b a d a d b a b a b a b d a b a b a \\ + c b^{2} a^{3} b a b a b d a b a b a &\rightarrow a^{3} b c a b a d a d^{2} b^{2} c a d b a b a \\ + c b a b a b a b d a b a b a &\rightarrow a^{3} b c a d^{3} b^{2} c a d b a b a \\ + a b^{2} a^{3} b a b a b d a b a d &\rightarrow b c a b a d a d^{2} b^{2} c a d b a d \\ + a b^{2} a^{3} b a b a b d a d &\rightarrow b c a b a d a d^{2} b^{2} c a d^{2} \\ + a b^{2} a^{3} b a b d a d b^{2} c a d b a d &\rightarrow b c a b a d a d b a b a b a b d a b a d \\ + a b^{2} a^{3} b a b d a d b^{2} c a d^{2} &\rightarrow b c a b a d a d b a b a b a b d a d \\ + a b a b a b a b d a b a b a &\rightarrow b c a d^{3} b^{2} c a d b a b a \\ + a b a b a b a b d a b a d &\rightarrow b c a d^{3} b^{2} c a d b a d \\ + a b a b a b a b d a d &\rightarrow b c a d^{3} b^{2} c a d^{2} \\ + a b a b a b d a d b^{2} c a d b a b a &\rightarrow b c a d^{2} b a b a b a b d a b a b a \\ + a b a b a b d a d b^{2} c a d b a d &\rightarrow b c a d^{2} b a b a b a b d a b a d \\ + a b a b a b d a d b^{2} c a d^{2} &\rightarrow b c a d^{2} b a b a b a b d a d \\ + c b^{2} a^{3} b a b a b d a b a d &\rightarrow a^{3} b c a b a d a d^{2} b^{2} c a d b a d \\ + c b^{2} a^{3} b a b a b d a d &\rightarrow a^{3} b c a b a d a d^{2} b^{2} c a d^{2} \\ + c b a b a b a b d a b a d &\rightarrow a^{3} b c a d^{3} b^{2} c a d b a d \\ + c b a b a b a b d a d &\rightarrow a^{3} b c a d^{3} b^{2} c a d^{2} \\ + c b^{2} a^{3} b a b d a d b^{2} c a d b a b a &\rightarrow a^{3} b c a b a d a d b a b a b a b d a b a b a \\ + c b^{2} a^{3} b a b d a d b^{2} c a d b a d &\rightarrow a^{3} b c a b a d a d b a b a b a b d a b a d \\ + c b^{2} a^{3} b a b d a d b^{2} c a d^{2} &\rightarrow a^{3} b c a b a d a d b a b a b a b d a d \\ + c b a b a b d a d b^{2} c a d b a b a &\rightarrow a^{3} b c a d^{2} b a b a b a b d a b a b a \\ + c b a b a b d a d b^{2} c a d b a d &\rightarrow a^{3} b c a d^{2} b a b a b a b d a b a d \\ + c b a b a b d a d b^{2} c a d^{2} &\rightarrow a^{3} b c a d^{2} b a b a b a b d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/310.md b/docs/examples/1-relation/310.md new file mode 100644 index 0000000..fab0274 --- /dev/null +++ b/docs/examples/1-relation/310.md @@ -0,0 +1,43 @@ +#\#310 babaababa=abababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a b a = a b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c = a c^{3},\; c = b a,\; d = c^{2} a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{3} &\rightarrow d c \\ + c^{2} a c &\rightarrow d \\ + b d c &\rightarrow c^{4} \\ + a c^{2} d &\rightarrow d^{2} \\ + c^{2} d c &\rightarrow d c^{2} \\ + d c a c &\rightarrow c^{2} a d \\ + a c d &\rightarrow c^{2} a d \\ + b c^{2} a d &\rightarrow c^{2} d \\ + b d^{2} &\rightarrow c^{3} d \\ + a d c^{2} &\rightarrow d^{2} c \\ + c^{4} a d &\rightarrow d^{2} \\ + c^{2} d^{2} &\rightarrow d c d \\ + a d c d &\rightarrow d^{3} \\ + a d^{2} &\rightarrow d c^{2} a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/312.md b/docs/examples/1-relation/312.md new file mode 100644 index 0000000..ee4e8a8 --- /dev/null +++ b/docs/examples/1-relation/312.md @@ -0,0 +1,48 @@ +#\#312 baaaabbaba=abaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a b a = a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d a = a b a^{3},\; c = b^{2} a b,\; d = a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} c &\rightarrow d \\ + a b a^{3} &\rightarrow b a^{2} d a \\ + b^{2} a b &\rightarrow c \\ + a b a^{2} d &\rightarrow b a^{2} d^{2} \\ + b a^{2} d a c &\rightarrow a b a d \\ + b^{2} a c &\rightarrow c b a b \\ + b^{3} a^{2} d a &\rightarrow c a^{3} \\ + b a^{2} d b a^{2} d a &\rightarrow a b^{2} a^{2} d^{3} a \\ + a^{2} b a d &\rightarrow b a^{2} d^{2} a c \\ + a^{2} b^{2} a^{2} d^{3} a &\rightarrow b a^{2} d^{2} b a^{2} d a \\ + b a^{2} d b a^{2} d^{2} &\rightarrow a b^{2} a^{2} d^{4} \\ + b^{3} a^{2} d^{2} &\rightarrow c a^{2} d \\ + b a^{2} d a b a d &\rightarrow a b^{2} a^{2} d^{3} a c \\ + b a^{2} d a b^{2} a^{2} d^{3} a &\rightarrow a b^{2} a^{2} d^{3} b a^{2} d a \\ + b a^{2} d a b^{2} a^{2} d^{4} &\rightarrow a b^{2} a^{2} d^{3} b a^{2} d^{2} \\ + a^{2} b^{2} a^{2} d^{4} &\rightarrow b a^{2} d^{2} b a^{2} d^{2} \\ + a^{2} b^{2} a^{2} d^{3} b a^{2} d a &\rightarrow b a^{2} d^{2} a b^{2} a^{2} d^{3} a \\ + a^{2} b^{2} a^{2} d^{3} b a^{2} d^{2} &\rightarrow b a^{2} d^{2} a b^{2} a^{2} d^{4} \\ + b a^{2} d a b^{2} a^{2} d^{3} b a^{2} d a &\rightarrow a b^{2} a^{2} d^{3} a b^{2} a^{2} d^{3} a \\ + b a^{2} d a b^{2} a^{2} d^{3} b a^{2} d^{2} &\rightarrow a b^{2} a^{2} d^{3} a b^{2} a^{2} d^{4} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/313.md b/docs/examples/1-relation/313.md new file mode 100644 index 0000000..14fdae1 --- /dev/null +++ b/docs/examples/1-relation/313.md @@ -0,0 +1,68 @@ +#\#313 abaabbaaaa=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{2} a^{4} = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a b c e = b c a b a,\; c = a^{2},\; d = c b a,\; e = b^{2} c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < a < d < e < b\): + +\( +\begin{aligned} + b^{2} c^{2} &\rightarrow e \\ + a^{2} &\rightarrow c \\ + c b a &\rightarrow d \\ + b^{2} c d &\rightarrow e b a \\ + c b c &\rightarrow d a \\ + c a &\rightarrow a c \\ + a b c e &\rightarrow b a d \\ + b^{2} a c^{2} &\rightarrow e a \\ + d a b a &\rightarrow c b d \\ + d a b c &\rightarrow c b d a \\ + a b a d &\rightarrow d a e \\ + a d a e &\rightarrow d^{2} \\ + c b^{2} a d &\rightarrow d b c e \\ + b^{2} a c d &\rightarrow e a b a \\ + d b a d &\rightarrow c b d a e \\ + d a b^{2} a d &\rightarrow c b d b c e \\ + a b d^{2} &\rightarrow d a e a e \\ + a b a c b d &\rightarrow d a e a b a \\ + c d a e &\rightarrow a d^{2} \\ + c b^{2} d^{2} &\rightarrow d b c e a e \\ + c b^{2} a c b d &\rightarrow d b c e a b a \\ + c b d^{2} &\rightarrow d^{2} a e \\ + b^{2} a d^{2} &\rightarrow e b c e \\ + d b d^{2} &\rightarrow c b d a e a e \\ + d b a c b d &\rightarrow c b d a e a b a \\ + d a b^{2} d^{2} &\rightarrow c b d b c e a e \\ + d a b^{2} a c b d &\rightarrow c b d b c e a b a \\ + a b d c b d &\rightarrow d a e a e a b a \\ + c b^{2} d c b d &\rightarrow d b c e a e a b a \\ + c b d c b d &\rightarrow d^{2} a e a b a \\ + b^{2} a d c b d &\rightarrow e b c e a b a \\ + d b d c b d &\rightarrow c b d a e a e a b a \\ + d a b^{2} d c b d &\rightarrow c b d b c e a e a b a \\ + c b d b c e d &\rightarrow d a e b c e \\ + c e b c e &\rightarrow d b c e d \\ + a b d b c e d &\rightarrow b a d b c e \\ + c b d b c e c b d &\rightarrow d a e b c e a b a \\ + c e b d b c e d &\rightarrow d b c e d b c e \\ + a b d b c e c b d &\rightarrow b a d b c e a b a \\ + c e b d b c e c b d &\rightarrow d b c e d b c e a b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/315.md b/docs/examples/1-relation/315.md new file mode 100644 index 0000000..e1e35fe --- /dev/null +++ b/docs/examples/1-relation/315.md @@ -0,0 +1,40 @@ +#\#315 aaaabbaaba=baabaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{4} b^{2} a^{2} b a = b a^{2} b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d a = b c b c a,\; c = a^{2},\; d = b^{2} c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} c b &\rightarrow d \\ + b c b c a &\rightarrow c^{2} d a \\ + a^{2} &\rightarrow c \\ + b^{2} c d &\rightarrow d b c b \\ + b^{2} c^{3} d a &\rightarrow d c b c a \\ + b c b c^{2} &\rightarrow c^{2} d c \\ + b c^{2} d a &\rightarrow d c a \\ + a c &\rightarrow c a \\ + b^{2} c^{3} d c &\rightarrow d c b c^{2} \\ + b c d c a &\rightarrow c^{2} d c d a \\ + b c^{2} d c &\rightarrow d c^{2} \\ + b c d c^{2} &\rightarrow c^{2} d c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/317.md b/docs/examples/1-relation/317.md new file mode 100644 index 0000000..7735354 --- /dev/null +++ b/docs/examples/1-relation/317.md @@ -0,0 +1,38 @@ +#\#317 bababbaaba=ababba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b^{2} a^{2} b a = a b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c^{2} d c = e c,\; c = b a,\; d = b c a,\; e = a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow d \\ + a c b &\rightarrow e \\ + e c &\rightarrow c^{2} d c \\ + b e &\rightarrow c^{2} b \\ + b c e &\rightarrow d c b \\ + a c d &\rightarrow c^{2} d c a \\ + a c^{2} &\rightarrow e a \\ + b c^{2} d c &\rightarrow c^{2} b c \\ + b c^{3} d c &\rightarrow d c b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/318.md b/docs/examples/1-relation/318.md new file mode 100644 index 0000000..8fea268 --- /dev/null +++ b/docs/examples/1-relation/318.md @@ -0,0 +1,37 @@ +#\#318 baababba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{2} a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c b c = a,\; c = b a,\; d = a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + b d c^{2} &\rightarrow d c b c \\ + a &\rightarrow b d c b c \\ + c b d c b c &\rightarrow b d \\ + b^{2} d c b c &\rightarrow c \\ + b d c b^{2} d &\rightarrow d \\ + b d b d &\rightarrow d c^{2} \\ + b d c b d &\rightarrow d c b^{2} d \\ + c d &\rightarrow d c^{3} b c \\ + b d^{2} &\rightarrow d c^{3} b^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/321.md b/docs/examples/1-relation/321.md new file mode 100644 index 0000000..9420627 --- /dev/null +++ b/docs/examples/1-relation/321.md @@ -0,0 +1,48 @@ +#\#321 baabbaabaa=abaabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{2} b a^{2} = a b a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c b c^{2} = a c b c,\; c = b a^{2},\; d = c^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a c b c &\rightarrow c b c^{2} \\ + c^{2} b &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a c b d &\rightarrow c b c d \\ + d a^{2} &\rightarrow c^{3} \\ + b c b c^{3} &\rightarrow d c \\ + a c d c &\rightarrow c b d c^{3} \\ + d c b c^{3} &\rightarrow c^{2} d c \\ + d c b c^{2} d &\rightarrow c^{2} d^{2} \\ + b c b c^{2} d &\rightarrow d^{2} \\ + b c b c d &\rightarrow d c b \\ + a c^{3} d c &\rightarrow c b d c d c^{3} \\ + a c^{3} d^{2} &\rightarrow c b d c d c^{2} d \\ + a c d^{2} &\rightarrow c b d c^{2} d \\ + c^{2} d c b &\rightarrow d c b c d \\ + c^{4} d c &\rightarrow d c b c d c^{3} \\ + c^{4} d^{2} &\rightarrow d c b c d c^{2} d \\ + d c b c d c b c d &\rightarrow c^{2} d c d c b \\ + d c b d c b c d &\rightarrow c^{2} d^{2} c b \\ + b c b d c b c d &\rightarrow d^{2} c b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/322.md b/docs/examples/1-relation/322.md new file mode 100644 index 0000000..cd31f67 --- /dev/null +++ b/docs/examples/1-relation/322.md @@ -0,0 +1,34 @@ +#\#322 abaabaaba=baabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a^{2} b a = b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d^{2} c = d^{2},\; c = b a,\; d = c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a &\rightarrow d \\ + a d^{2} c &\rightarrow d^{2} \\ + c d^{2} &\rightarrow d^{3} c \\ + a d^{3} &\rightarrow d^{2} a \\ + b d^{2} &\rightarrow d^{3} c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/323.md b/docs/examples/1-relation/323.md new file mode 100644 index 0000000..c76349f --- /dev/null +++ b/docs/examples/1-relation/323.md @@ -0,0 +1,33 @@ +#\#323 aaaaababba=babaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{5} b a b^{2} a = b a b a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} a d a = b a b c^{2},\; c = a^{2},\; d = b a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a b^{2} &\rightarrow d \\ + b a b a^{4} &\rightarrow a^{5} d a \\ + b a b d &\rightarrow d a b^{2} \\ + d a b a^{4} &\rightarrow a^{5} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/324.md b/docs/examples/1-relation/324.md new file mode 100644 index 0000000..d6677b3 --- /dev/null +++ b/docs/examples/1-relation/324.md @@ -0,0 +1,39 @@ +#\#324 baabaaba=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b a = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} b a = d b a,\; c = b a^{2},\; d = a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a^{2} &\rightarrow c \\ + a^{2} b &\rightarrow d \\ + d b a &\rightarrow c^{2} b a \\ + b a d &\rightarrow c a b \\ + b d &\rightarrow c b \\ + a^{2} c &\rightarrow d a^{2} \\ + d c &\rightarrow c^{3} \\ + b a c^{2} b a &\rightarrow c a b^{2} a \\ + b a c^{3} &\rightarrow c a b c \\ + b c^{2} b a &\rightarrow c b^{2} a \\ + b c^{3} &\rightarrow c b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/325.md b/docs/examples/1-relation/325.md new file mode 100644 index 0000000..ac0b69b --- /dev/null +++ b/docs/examples/1-relation/325.md @@ -0,0 +1,38 @@ +#\#325 abaaaababa=baaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{4} b a b a = b a^{3} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c^{2} b d = b c a b c,\; c = a^{2},\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < b < a\): + +\( +\begin{aligned} + a^{2} &\rightarrow c \\ + a b a &\rightarrow d \\ + a b d &\rightarrow d b a \\ + a b c &\rightarrow d a \\ + a d &\rightarrow c b a \\ + a c &\rightarrow c a \\ + c b c^{2} b d &\rightarrow d c b c \\ + b c d a &\rightarrow d c d b a \\ + b c d c &\rightarrow d c d b c \\ + b c d^{2} &\rightarrow d c d b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/326.md b/docs/examples/1-relation/326.md new file mode 100644 index 0000000..b4dbc6d --- /dev/null +++ b/docs/examples/1-relation/326.md @@ -0,0 +1,47 @@ +#\#326 baaaabbaaa=aaaabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a^{3} = a^{4} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b d a^{3} = d a^{2},\; c = a^{3} b,\; d = a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b d a^{3} &\rightarrow d a^{2} \\ + a^{3} b &\rightarrow c \\ + a c b &\rightarrow d \\ + b d a^{2} d &\rightarrow d a d \\ + b d a^{2} c &\rightarrow d a c \\ + b d a c &\rightarrow d c \\ + b d c &\rightarrow d a^{2} b \\ + a^{3} d a^{2} &\rightarrow c d a^{3} \\ + a c d a^{2} &\rightarrow d^{2} a^{3} \\ + b d a d &\rightarrow d^{2} \\ + b d^{2} &\rightarrow d c b \\ + a^{3} d a d &\rightarrow c d a^{2} d \\ + a^{3} d a c &\rightarrow c d a^{2} c \\ + a^{3} d c &\rightarrow c d a c \\ + a c d a d &\rightarrow d^{2} a^{2} d \\ + a c d a c &\rightarrow d^{2} a^{2} c \\ + a c d c &\rightarrow d^{2} a c \\ + a^{3} d^{2} &\rightarrow c d a d \\ + a c d^{2} &\rightarrow d^{2} a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/327.md b/docs/examples/1-relation/327.md new file mode 100644 index 0000000..1829e3d --- /dev/null +++ b/docs/examples/1-relation/327.md @@ -0,0 +1,37 @@ +#\#327 abaabbaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{2} a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a b^{2} c = b c a,\; c = a b a,\; d = a b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a b a &\rightarrow c \\ + a b^{2} a &\rightarrow d \\ + b c a &\rightarrow c a b^{2} c \\ + c b a &\rightarrow a b c \\ + c b^{2} a &\rightarrow a b d \\ + d b a &\rightarrow a b^{2} c \\ + d b^{2} a &\rightarrow a b^{2} d \\ + b c d &\rightarrow c d b d \\ + b c^{2} &\rightarrow c d b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/328.md b/docs/examples/1-relation/328.md new file mode 100644 index 0000000..8ac29c9 --- /dev/null +++ b/docs/examples/1-relation/328.md @@ -0,0 +1,68 @@ +#\#328 abbabbbba=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{4} a = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d c a = b a c a,\; c = b^{2},\; d = c a c,\; e = a d^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < e < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a c a &\rightarrow a d c a \\ + c a c &\rightarrow d \\ + a d^{2} c &\rightarrow e \\ + b c &\rightarrow c b \\ + b a d &\rightarrow a d^{2} \\ + e a c &\rightarrow a d^{3} \\ + d a &\rightarrow e a \\ + c a d &\rightarrow a d^{3} \\ + b e &\rightarrow a d^{3} c \\ + d e &\rightarrow e^{2} \\ + b a e a &\rightarrow a e^{2} a \\ + b a^{2} d^{3} &\rightarrow a e a d^{3} \\ + c b a c &\rightarrow b d \\ + c e &\rightarrow a d^{4} c \\ + c a e a &\rightarrow a e^{3} a \\ + c a^{2} d^{3} &\rightarrow e a d \\ + a e^{2} a d^{3} &\rightarrow e a d \\ + e a^{2} d^{3} &\rightarrow a e^{3} a d \\ + b a e^{2} &\rightarrow a e^{3} \\ + b a^{2} e^{4} &\rightarrow a e a e^{4} \\ + b a^{2} e^{3} a &\rightarrow a e a e^{3} a \\ + c a e^{2} &\rightarrow a e^{4} \\ + c a^{2} e^{4} &\rightarrow e a e^{2} \\ + c a^{2} e^{3} a &\rightarrow e a e a \\ + d b a c &\rightarrow c a b d \\ + e b a c &\rightarrow a d^{2} b d \\ + a e^{3} a e^{2} &\rightarrow e a^{2} e^{4} \\ + a e^{3} a e a &\rightarrow e a^{2} e^{3} a \\ + a e^{2} a e^{3} a &\rightarrow e a e a \\ + a e^{2} a e^{4} &\rightarrow e a e^{2} \\ + a e^{3} a^{2} e^{4} &\rightarrow e a e a e^{2} \\ + a e^{3} a^{2} e^{3} a &\rightarrow e a e a e a \\ + e a^{2} e^{4} a e^{3} a &\rightarrow a e^{4} a e a \\ + e a^{2} e^{5} a e^{2} &\rightarrow a e^{4} a^{2} e^{4} \\ + e a^{2} e^{5} a e a &\rightarrow a e^{4} a^{2} e^{3} a \\ + e a^{2} e^{4} a d^{3} &\rightarrow a e^{4} a d \\ + e a^{2} e^{4} a e^{4} &\rightarrow a e^{4} a e^{2} \\ + e a^{2} e^{5} a^{2} e^{4} &\rightarrow a e^{4} a e a e^{2} \\ + e a^{2} e^{5} a^{2} e^{3} a &\rightarrow a e^{4} a e a e a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/329.md b/docs/examples/1-relation/329.md new file mode 100644 index 0000000..6d4aa1b --- /dev/null +++ b/docs/examples/1-relation/329.md @@ -0,0 +1,49 @@ +#\#329 babaaabbba=aabbbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{3} b^{3} a = a^{2} b^{3} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a b a^{2} c a = d a,\; c = a b^{3},\; d = a c a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a c a b &\rightarrow d \\ + a b^{3} &\rightarrow c \\ + b a b a^{2} c a &\rightarrow d a \\ + a c^{2} &\rightarrow d b^{2} \\ + d a b a^{2} c a &\rightarrow a c a d a \\ + c a b a^{2} c a &\rightarrow a b^{2} d a \\ + b a b a^{2} c d &\rightarrow d^{2} \\ + b a b a d &\rightarrow d a b \\ + a^{2} b^{2} d a &\rightarrow d a^{2} c a \\ + d a b a^{2} c d &\rightarrow a c a d^{2} \\ + a c a^{2} c a d a &\rightarrow d a b a d a^{2} c a \\ + a c a d a b &\rightarrow d a b a d \\ + c a b a^{2} c d &\rightarrow a b^{2} d^{2} \\ + c a b a d &\rightarrow a b^{2} d a b \\ + a^{2} b^{2} d^{2} &\rightarrow d a^{2} c d \\ + a c a^{2} c a d^{2} &\rightarrow d a b a d a^{2} c d \\ + a c a d c &\rightarrow d a b a d b^{2} \\ + a^{2} b^{2} a c a d a &\rightarrow d a d a^{2} c a \\ + a^{2} b^{2} a c a d^{2} &\rightarrow d a d a^{2} c d \\ + a^{2} b^{2} d c &\rightarrow d a d b^{2} \\ + d a b a d a b a d &\rightarrow a c a d a d a b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/330.md b/docs/examples/1-relation/330.md new file mode 100644 index 0000000..ddbd70e --- /dev/null +++ b/docs/examples/1-relation/330.md @@ -0,0 +1,52 @@ +#\#330 abaaababba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b a b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d a = b c a,\; c = a b a,\; d = a c b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b c a &\rightarrow c d a \\ + a b a &\rightarrow c \\ + a c b^{2} &\rightarrow d \\ + b c^{2} &\rightarrow c d c \\ + c^{2} b^{2} &\rightarrow a b d \\ + b c d &\rightarrow c d^{2} \\ + a b c &\rightarrow c b a \\ + a c^{2} d^{2} a &\rightarrow d c a \\ + b a b d &\rightarrow c d c b^{2} \\ + c b a c &\rightarrow a c d c \\ + c b a d &\rightarrow a c d^{2} \\ + c b a^{2} &\rightarrow a c d a \\ + a b d c^{2} &\rightarrow c^{3} d^{2} c \\ + a b d c d &\rightarrow c^{3} d^{3} \\ + a b d c a &\rightarrow c^{3} d^{2} a \\ + a c^{2} d^{2} c &\rightarrow d c^{2} \\ + a c^{2} d^{3} &\rightarrow d c d \\ + c b d c^{2} &\rightarrow a c d c^{2} d^{2} c \\ + c b d c d &\rightarrow a c d c^{2} d^{3} \\ + c b d c a &\rightarrow a c d c^{2} d^{2} a \\ + a b d a b d &\rightarrow c^{3} d^{2} c b^{2} \\ + a c d c b^{2} &\rightarrow c b d \\ + c b c b d &\rightarrow a c d c d c b^{2} \\ + c b d a b d &\rightarrow a c d c^{2} d^{2} c b^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/331.md b/docs/examples/1-relation/331.md new file mode 100644 index 0000000..dcdc9d1 --- /dev/null +++ b/docs/examples/1-relation/331.md @@ -0,0 +1,36 @@ +#\#331 abaabaaaba=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a^{3} b a = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a c a^{2} c = c a c,\; c = b a,\; d = a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a^{2} b &\rightarrow d \\ + b d^{2} &\rightarrow d^{2} a d a d^{2} \\ + b d a &\rightarrow d^{2} a d a d a \\ + a d^{2} a d a &\rightarrow d^{2} a \\ + a d^{2} a d^{2} &\rightarrow d^{3} \\ + a d^{4} a &\rightarrow d^{3} a d a \\ + a d^{5} &\rightarrow d^{3} a d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/332.md b/docs/examples/1-relation/332.md new file mode 100644 index 0000000..7a59816 --- /dev/null +++ b/docs/examples/1-relation/332.md @@ -0,0 +1,110 @@ +#\#332 abababaaaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b a^{4} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c b c a^{3} = b a c,\; c = a b a,\; d = c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + c b a &\rightarrow d \\ + a b a &\rightarrow c \\ + c b c &\rightarrow d b a \\ + a b c &\rightarrow d \\ + b a d &\rightarrow d b a^{3} c \\ + b a c &\rightarrow d b a^{4} \\ + c b d &\rightarrow d b c \\ + c d b a^{3} c &\rightarrow d^{2} \\ + c d b a^{4} &\rightarrow d c \\ + a b d &\rightarrow d b a \\ + a d b a^{3} c &\rightarrow c d \\ + a d b a^{4} &\rightarrow c^{2} \\ + d b a^{3} d a^{2} c &\rightarrow b c d \\ + d b a^{3} d a^{3} &\rightarrow b c^{2} \\ + c d b a^{3} d &\rightarrow d^{2} b a \\ + a d b a^{3} d &\rightarrow c d b a \\ + b c d b d &\rightarrow d b a^{3} d a^{2} d b c \\ + b c d b c &\rightarrow d b a^{3} d a^{2} d b a \\ + b c d b a &\rightarrow d b a^{3} d a^{2} d \\ + b c d^{2} b a^{3} d &\rightarrow d b a^{3} d a^{2} d^{2} b a \\ + b c d^{2} b a^{3} c &\rightarrow d b a^{3} d a^{2} d^{2} \\ + b c d^{2} b a^{4} &\rightarrow d b a^{3} d a^{2} d c \\ + c b^{2} c d &\rightarrow d b d a^{2} d a^{2} c \\ + c b^{2} c^{2} &\rightarrow d b d a^{2} d a^{3} \\ + a b^{2} c d &\rightarrow d b c a^{2} d a^{2} c \\ + a b^{2} c^{2} &\rightarrow d b c a^{2} d a^{3} \\ + d b a^{3} d a^{2} d a^{2} c &\rightarrow b d^{2} \\ + d b a^{3} d a^{2} d a^{3} &\rightarrow b d c \\ + c d b a^{2} d^{2} &\rightarrow d^{2} b c a^{2} d a^{2} c \\ + c d b a^{2} d c &\rightarrow d^{2} b c a^{2} d a^{3} \\ + c d b a^{2} c d &\rightarrow d^{2} b c a^{2} c \\ + c d b a^{2} c^{2} &\rightarrow d^{2} b c a^{3} \\ + a d b a^{2} d^{2} &\rightarrow c d b c a^{2} d a^{2} c \\ + a d b a^{2} d c &\rightarrow c d b c a^{2} d a^{3} \\ + a d b a^{2} c d &\rightarrow c d b c a^{2} c \\ + a d b a^{2} c^{2} &\rightarrow c d b c a^{3} \\ + b d^{2} b^{2} c d &\rightarrow d b a^{3} d a^{2} d a^{2} d b d a^{2} d a^{2} c \\ + b d^{2} b^{2} c^{2} &\rightarrow d b a^{3} d a^{2} d a^{2} d b d a^{2} d a^{3} \\ + b d^{2} b a &\rightarrow d b a^{3} d a^{2} d a^{2} d \\ + b c d b^{2} c d &\rightarrow d b a^{3} d a^{2} d b d a^{2} d a^{2} c \\ + b c d b^{2} c^{2} &\rightarrow d b a^{3} d a^{2} d b d a^{2} d a^{3} \\ + b c d^{2} b a^{2} d^{2} &\rightarrow d b a^{3} d a^{2} d^{2} b c a^{2} d a^{2} c \\ + b c d^{2} b a^{2} d c &\rightarrow d b a^{3} d a^{2} d^{2} b c a^{2} d a^{3} \\ + b c d^{2} b a^{2} c d &\rightarrow d b a^{3} d a^{2} d^{2} b c a^{2} c \\ + b c d^{2} b a^{2} c^{2} &\rightarrow d b a^{3} d a^{2} d^{2} b c a^{3} \\ + c b^{2} d^{2} &\rightarrow d b d a^{2} d a^{2} d a^{2} c \\ + c b^{2} d c &\rightarrow d b d a^{2} d a^{2} d a^{3} \\ + a b^{2} d^{2} &\rightarrow d b c a^{2} d a^{2} d a^{2} c \\ + a b^{2} d c &\rightarrow d b c a^{2} d a^{2} d a^{3} \\ + b d^{2} b d &\rightarrow d b a^{3} d a^{2} d a^{2} d b c \\ + b d^{2} b c &\rightarrow d b a^{3} d a^{2} d a^{2} d b a \\ + b d^{3} b a^{2} d^{2} &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b c a^{2} d a^{2} c \\ + b d^{3} b a^{2} d c &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b c a^{2} d a^{3} \\ + b d^{3} b a^{2} c d &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b c a^{2} c \\ + b d^{3} b a^{2} c^{2} &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b c a^{3} \\ + b d^{3} b a^{3} d &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b a \\ + b d^{3} b a^{3} c &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} \\ + b d^{3} b a^{4} &\rightarrow d b a^{3} d a^{2} d a^{2} d c \\ + b d b c d &\rightarrow d b a^{3} d a^{2} d a^{2} d a^{2} d a^{2} c \\ + b d b c^{2} &\rightarrow d b a^{3} d a^{2} d a^{2} d a^{2} d a^{3} \\ + b c d b^{2} d^{2} &\rightarrow d b a^{3} d a^{2} d b d a^{2} d a^{2} d a^{2} c \\ + b c d b^{2} d c &\rightarrow d b a^{3} d a^{2} d b d a^{2} d a^{2} d a^{3} \\ + c d b a^{2} d b d^{2} &\rightarrow d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} d a^{2} c \\ + c d b a^{2} d b d c &\rightarrow d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} d a^{3} \\ + c d b a^{2} d b c d &\rightarrow d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} c \\ + c d b a^{2} d b c^{2} &\rightarrow d^{2} b c a^{2} d a^{2} d a^{2} d a^{3} \\ + b d^{2} b^{2} d^{2} &\rightarrow d b a^{3} d a^{2} d a^{2} d b d a^{2} d a^{2} d a^{2} c \\ + b d^{2} b^{2} d c &\rightarrow d b a^{3} d a^{2} d a^{2} d b d a^{2} d a^{2} d a^{3} \\ + a d b a^{2} d b d^{2} &\rightarrow c d b c a^{2} d a^{2} d a^{2} d a^{2} d a^{2} c \\ + a d b a^{2} d b d c &\rightarrow c d b c a^{2} d a^{2} d a^{2} d a^{2} d a^{3} \\ + a d b a^{2} d b c d &\rightarrow c d b c a^{2} d a^{2} d a^{2} d a^{2} c \\ + a d b a^{2} d b c^{2} &\rightarrow c d b c a^{2} d a^{2} d a^{2} d a^{3} \\ + b d b d^{2} &\rightarrow d b a^{3} d a^{2} d a^{2} d a^{2} d a^{2} d a^{2} c \\ + b d b d c &\rightarrow d b a^{3} d a^{2} d a^{2} d a^{2} d a^{2} d a^{3} \\ + b c d^{2} b a^{2} d b d^{2} &\rightarrow d b a^{3} d a^{2} d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} d a^{2} c \\ + b c d^{2} b a^{2} d b d c &\rightarrow d b a^{3} d a^{2} d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} d a^{3} \\ + b c d^{2} b a^{2} d b c d &\rightarrow d b a^{3} d a^{2} d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} c \\ + b c d^{2} b a^{2} d b c^{2} &\rightarrow d b a^{3} d a^{2} d^{2} b c a^{2} d a^{2} d a^{2} d a^{3} \\ + b d^{3} b a^{2} d b d^{2} &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} d a^{2} c \\ + b d^{3} b a^{2} d b d c &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} d a^{3} \\ + b d^{3} b a^{2} d b c d &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b c a^{2} d a^{2} d a^{2} d a^{2} c \\ + b d^{3} b a^{2} d b c^{2} &\rightarrow d b a^{3} d a^{2} d a^{2} d^{2} b c a^{2} d a^{2} d a^{2} d a^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/333.md b/docs/examples/1-relation/333.md new file mode 100644 index 0000000..d65b9e0 --- /dev/null +++ b/docs/examples/1-relation/333.md @@ -0,0 +1,36 @@ +#\#333 aababababa=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b a b a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c^{4} a = d a,\; c = a b,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c a &\rightarrow d \\ + d a &\rightarrow a c^{4} a \\ + a b &\rightarrow c \\ + b c^{2} &\rightarrow d b \\ + a d &\rightarrow c^{2} a \\ + d c &\rightarrow a c^{5} \\ + a^{2} c^{5} &\rightarrow c^{2} a c \\ + a^{2} c^{4} a &\rightarrow c^{2} a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/334.md b/docs/examples/1-relation/334.md new file mode 100644 index 0000000..c9d8f67 --- /dev/null +++ b/docs/examples/1-relation/334.md @@ -0,0 +1,64 @@ +#\#334 abbabbbaaa=baaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{3} a^{3} = b a^{3} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a^{2} = b a^{2} c a,\; c = a b^{2},\; d = c^{2} b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a^{2} c a &\rightarrow d a^{2} \\ + c^{2} b a &\rightarrow d \\ + a b^{2} &\rightarrow c \\ + c^{2} d a^{2} &\rightarrow d a c a \\ + d b^{2} &\rightarrow c^{2} b c \\ + b a^{2} c^{2} &\rightarrow d a c \\ + a b d a^{2} &\rightarrow c a^{2} c a \\ + b a^{2} c d &\rightarrow d a d \\ + b a^{2} d &\rightarrow d a c b a \\ + c^{2} b c a^{2} c^{2} &\rightarrow d b d a c \\ + c^{2} b c a^{2} c a &\rightarrow d b d a^{2} \\ + c^{2} d a c &\rightarrow d a c^{2} \\ + a b d a c &\rightarrow c a^{2} c^{2} \\ + c^{2} b c a^{2} c d &\rightarrow d b d a d \\ + c^{2} b c a^{2} d a c^{2} &\rightarrow d b d a c d a c \\ + c^{2} b c a^{2} d a c a &\rightarrow d b d a c d a^{2} \\ + c^{2} d a d &\rightarrow d a c d \\ + d b d a c b c a^{2} c^{2} &\rightarrow c^{2} b c a^{2} d b d a c \\ + d b d a c b c a^{2} c a &\rightarrow c^{2} b c a^{2} d b d a^{2} \\ + d b d a c b a &\rightarrow c^{2} b c a^{2} d \\ + a b d a d &\rightarrow c a^{2} c d \\ + d b d a c b c a^{2} c d &\rightarrow c^{2} b c a^{2} d b d a d \\ + c^{2} b c a^{2} d a c d &\rightarrow d b d a c d a d \\ + d b d a c b c a^{2} d a c^{2} &\rightarrow c^{2} b c a^{2} d b d a c d a c \\ + d b d a c b c a^{2} d a c d &\rightarrow c^{2} b c a^{2} d b d a c d a d \\ + d b d a c b c a^{2} d a c a &\rightarrow c^{2} b c a^{2} d b d a c d a^{2} \\ + d b d a c d a c b c a^{2} c^{2} &\rightarrow c^{2} b c a^{2} d a d b d a c \\ + d b d a c d a c b c a^{2} c d &\rightarrow c^{2} b c a^{2} d a d b d a d \\ + d b d a c d a c b c a^{2} c a &\rightarrow c^{2} b c a^{2} d a d b d a^{2} \\ + d b d a c d a c b c a^{2} d a c^{2} &\rightarrow c^{2} b c a^{2} d a d b d a c d a c \\ + d b d a c d a c b c a^{2} d a c a &\rightarrow c^{2} b c a^{2} d a d b d a c d a^{2} \\ + d b d a c d a c b a &\rightarrow c^{2} b c a^{2} d a d \\ + d b d a c d a c d a c &\rightarrow c^{2} b c a^{2} d a d a c^{2} \\ + d b d a c d a c d a d &\rightarrow c^{2} b c a^{2} d a d a c d \\ + d b d a c d a c d a^{2} &\rightarrow c^{2} b c a^{2} d a d a c a \\ + d b d a c d a c b c a^{2} d a c d &\rightarrow c^{2} b c a^{2} d a d b d a c d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/335.md b/docs/examples/1-relation/335.md new file mode 100644 index 0000000..a7d4317 --- /dev/null +++ b/docs/examples/1-relation/335.md @@ -0,0 +1,45 @@ +#\#335 baaabbbaba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{3} a b a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d c = a c a,\; c = b a,\; d = a b^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a c a &\rightarrow c a d c \\ + a b^{2} c &\rightarrow d \\ + b a &\rightarrow c \\ + a c^{2} a d c &\rightarrow c a d c^{2} a \\ + c b^{2} c &\rightarrow b d \\ + b c a d c &\rightarrow c^{2} a \\ + a c d &\rightarrow c a d b d \\ + a c^{2} a d b d &\rightarrow c a d c^{2} d \\ + a b c^{2} a &\rightarrow d a d c \\ + a b^{3} d &\rightarrow d b^{2} c \\ + b c a d b d &\rightarrow c^{2} d \\ + c b^{3} d &\rightarrow b d b^{2} c \\ + b d a d c &\rightarrow c b c^{2} a \\ + a b c^{3} a d c &\rightarrow d a d c^{2} a \\ + a b c^{3} a d b d &\rightarrow d a d c^{2} d \\ + a b c^{2} d &\rightarrow d a d b d \\ + b d a d b d &\rightarrow c b c^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/336.md b/docs/examples/1-relation/336.md new file mode 100644 index 0000000..5f778ac --- /dev/null +++ b/docs/examples/1-relation/336.md @@ -0,0 +1,39 @@ +#\#336 baababbbaa=abba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{3} a^{2} = a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d a^{2} = a c a,\; c = b^{2},\; d = b a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a c b &\rightarrow d \\ + b a^{2} d a^{2} &\rightarrow a c a \\ + b c &\rightarrow c b \\ + b d &\rightarrow c a c b \\ + b a c^{2} &\rightarrow d b \\ + b a c d &\rightarrow d a c b \\ + b a c a &\rightarrow c a^{2} d a^{2} \\ + c a^{2} d a^{2} c a &\rightarrow d a^{2} d a^{2} \\ + b a d a^{2} d a^{2} &\rightarrow c a^{2} d a^{3} d a^{2} c a \\ + c a^{2} d a^{2} d a^{2} d a^{2} &\rightarrow d a^{2} d a^{3} d a^{2} c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/337.md b/docs/examples/1-relation/337.md new file mode 100644 index 0000000..0279444 --- /dev/null +++ b/docs/examples/1-relation/337.md @@ -0,0 +1,54 @@ +#\#337 baaaabbaa=abaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a^{2} = a b a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c d c = a b c^{2},\; c = a^{2},\; d = c b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a^{2} &\rightarrow c \\ + a b c^{2} &\rightarrow b c d c \\ + c b^{2} &\rightarrow d \\ + a c &\rightarrow c a \\ + a b c d &\rightarrow b c d^{2} \\ + c b c^{2} &\rightarrow b c d^{2} c \\ + a d &\rightarrow c a b^{2} \\ + c b c d &\rightarrow b c d^{3} \\ + a b^{2} c d^{4} c &\rightarrow b c d b c d^{2} c \\ + b c d^{2} b c d^{2} c &\rightarrow d c d^{4} c \\ + a b^{2} c d^{5} &\rightarrow b c d b c d^{3} \\ + a b^{2} c d^{4} b c d^{2} c &\rightarrow b c d^{2} c d^{4} c \\ + c d c d^{4} c &\rightarrow b c d^{4} b c d^{2} c \\ + b c d^{2} b c d^{3} &\rightarrow d c d^{5} \\ + c b d c d^{4} c &\rightarrow d c d^{2} b c d^{2} c \\ + b c d^{3} c d^{4} c &\rightarrow d c d^{4} b c d^{2} c \\ + b c d^{3} c d^{5} &\rightarrow d c d^{4} b c d^{3} \\ + a b^{2} c d^{4} b c d^{3} &\rightarrow b c d^{2} c d^{5} \\ + c d c d^{5} &\rightarrow b c d^{4} b c d^{3} \\ + c d c d^{4} b c d^{2} c &\rightarrow b c d^{5} c d^{4} c \\ + c d c d^{4} b c d^{3} &\rightarrow b c d^{5} c d^{5} \\ + c b d c d^{5} &\rightarrow d c d^{2} b c d^{3} \\ + c b d c d^{4} b c d^{2} c &\rightarrow d c d^{3} c d^{4} c \\ + c b d c d^{4} b c d^{3} &\rightarrow d c d^{3} c d^{5} \\ + b c d^{3} c d^{4} b c d^{2} c &\rightarrow d c d^{5} c d^{4} c \\ + b c d^{3} c d^{4} b c d^{3} &\rightarrow d c d^{5} c d^{5} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/338.md b/docs/examples/1-relation/338.md new file mode 100644 index 0000000..cfc3594 --- /dev/null +++ b/docs/examples/1-relation/338.md @@ -0,0 +1,35 @@ +#\#338 abaabaaaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a^{3} b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a c a^{2} c = c^{2} a,\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + b d a &\rightarrow d^{2} a d \\ + b d^{2} &\rightarrow d^{2} a d b a \\ + a b d &\rightarrow d b a \\ + d b a d &\rightarrow a d^{2} a d b a \\ + d b a^{2} &\rightarrow a d^{2} a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/339.md b/docs/examples/1-relation/339.md new file mode 100644 index 0000000..9486195 --- /dev/null +++ b/docs/examples/1-relation/339.md @@ -0,0 +1,67 @@ +#\#339 aaabaaabba=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a^{3} b^{2} a = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} b a = b c a,\; c = a^{3} b,\; d = c b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b c a &\rightarrow c^{2} b a \\ + c b c &\rightarrow d \\ + a^{3} b &\rightarrow c \\ + b c^{2} &\rightarrow c d \\ + a^{3} c^{2} b a &\rightarrow c^{2} a \\ + d b c &\rightarrow c b d \\ + b c d &\rightarrow c^{2} b d \\ + c^{2} d &\rightarrow d c \\ + c^{3} &\rightarrow a^{3} c d \\ + a^{3} c d b a &\rightarrow d a \\ + b d c &\rightarrow c d^{2} \\ + a^{3} c d b d &\rightarrow d^{2} \\ + c d c &\rightarrow a^{3} c d^{2} \\ + a^{3} c^{2} b d &\rightarrow d c \\ + a^{6} c d^{2} b a &\rightarrow c d a \\ + c a^{3} c d &\rightarrow a^{6} c d^{2} \\ + d c^{2} &\rightarrow a^{6} c d^{3} \\ + c a^{6} c d^{2} &\rightarrow a^{6} c d^{2} c \\ + b a^{3} c d &\rightarrow a^{3} c d^{2} \\ + b d a &\rightarrow a^{3} c d^{2} b a \\ + a^{3} c d^{2} c d &\rightarrow c d^{2} c \\ + a^{6} c d^{2} b d &\rightarrow c d^{2} \\ + b a^{6} c d^{2} &\rightarrow a^{3} c d^{2} c \\ + a^{3} c d^{2} a^{3} c d^{3} &\rightarrow c d^{3} c \\ + a^{3} c d^{2} a^{3} c d^{2} c &\rightarrow c d^{2} a^{3} c d \\ + b d^{2} &\rightarrow a^{3} c d^{2} b d \\ + a^{6} c d^{4} &\rightarrow d^{2} c \\ + a^{6} c d^{2} a^{6} c d^{3} &\rightarrow d a^{6} c d^{2} \\ + a^{6} c d^{3} c &\rightarrow d a^{3} c d \\ + c d a^{3} c d^{2} &\rightarrow a^{3} c d^{3} c \\ + c d a^{6} c d^{2} &\rightarrow a^{3} c d^{2} a^{3} c d \\ + a^{6} c d^{2} c b a &\rightarrow d c a \\ + a^{6} c d^{2} c b d &\rightarrow d c d \\ + a^{6} c d^{2} a^{3} c d^{2} &\rightarrow c d a^{3} c d \\ + a^{6} c d^{3} a^{6} c d^{2} &\rightarrow d a^{3} c d a^{3} c d \\ + a^{6} c d^{3} a^{3} c d &\rightarrow d a^{6} c d^{2} c \\ + a^{3} c d^{2} a^{3} c d^{2} a^{3} c d &\rightarrow c d^{2} a^{6} c d^{2} c \\ + a^{3} c d^{2} a^{3} c d^{2} a^{6} c d^{2} &\rightarrow c d^{2} a^{3} c d a^{3} c d \\ + c d a^{3} c d a^{3} c d &\rightarrow a^{3} c d^{2} a^{6} c d^{2} c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/340.md b/docs/examples/1-relation/340.md new file mode 100644 index 0000000..af99e9f --- /dev/null +++ b/docs/examples/1-relation/340.md @@ -0,0 +1,38 @@ +#\#340 baabababaa=aaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b a b a^{2} = a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c^{3} a = a d,\; c = b a,\; d = a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a^{2} c &\rightarrow d \\ + c a c^{3} a &\rightarrow a d \\ + b d &\rightarrow c a c \\ + a^{3} d &\rightarrow d a c^{3} a \\ + a d a c &\rightarrow c a c^{3} d \\ + c a c^{2} a d &\rightarrow a d c^{3} a \\ + b c a c^{3} d &\rightarrow c d a c \\ + a d a^{2} d &\rightarrow c a c^{3} d a c^{3} a \\ + c a c^{4} a c^{3} d &\rightarrow a d^{2} a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/342.md b/docs/examples/1-relation/342.md new file mode 100644 index 0000000..048f6db --- /dev/null +++ b/docs/examples/1-relation/342.md @@ -0,0 +1,43 @@ +#\#342 babaababba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a b^{2} a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c = a c^{2},\; c = b a,\; d = c a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{2} &\rightarrow c d c \\ + c a c b &\rightarrow d \\ + b c d c &\rightarrow c^{3} \\ + d a &\rightarrow c^{2} d c \\ + a c d &\rightarrow c d^{2} \\ + b c d^{2} &\rightarrow c^{2} d \\ + c^{2} d c^{3} &\rightarrow d c d c \\ + c^{2} d c^{2} d &\rightarrow d c d^{2} \\ + a d c d c &\rightarrow c d c d c^{3} \\ + a d c d^{2} &\rightarrow c d c d c^{2} d \\ + c^{2} d c d c d c &\rightarrow d c d c d c^{3} \\ + c^{2} d c d c d^{2} &\rightarrow d c d c d c^{2} d \\ + c^{2} d^{2} c d c &\rightarrow d c d^{2} c^{3} \\ + c^{2} d^{2} c d^{2} &\rightarrow d c d^{2} c^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/344.md b/docs/examples/1-relation/344.md new file mode 100644 index 0000000..cfb7a20 --- /dev/null +++ b/docs/examples/1-relation/344.md @@ -0,0 +1,66 @@ +#\#344 babbbabba=abbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{3} a b^{2} a = a b^{3} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b d c a = d,\; c = b^{2},\; d = a c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a c b a &\rightarrow d \\ + b d &\rightarrow c d c a \\ + b c &\rightarrow c b \\ + c d c a c a &\rightarrow d \\ + a c^{2} d c d &\rightarrow d c^{2} d c a \\ + a c^{2} d c^{2} d c a d &\rightarrow d c^{2} d^{2} \\ + c d c d c^{2} d c d c d &\rightarrow d c^{2} d c d \\ + a c^{2} d^{2} &\rightarrow d c d c a \\ + a c d &\rightarrow c d c^{2} d c a d \\ + c d c d c^{2} d c d c a &\rightarrow d c^{2} d c a \\ + c d c d c^{2} d c d c^{2} d c^{2} d c^{2} d c a d &\rightarrow d c^{2} d c^{2} d c^{2} d c^{2} d c a d \\ + a c^{2} d c^{2} d c^{2} d c^{2} d c a d &\rightarrow d c^{2} d c d \\ + a c^{2} d c a &\rightarrow c d c^{2} d c^{2} d c a d \\ + d c b a &\rightarrow c d c^{2} d c^{2} d c a d \\ + a c^{2} d c^{2} d c^{2} d c a d &\rightarrow d c^{2} d c d c a \\ + c d c d c^{2} d^{2} &\rightarrow d c d \\ + c d c d c^{2} d c d c^{2} d c^{2} d c a d &\rightarrow d c^{2} d c^{2} d c^{2} d c a d \\ + c d c d c^{2} d c d^{2} &\rightarrow d c^{2} d^{2} \\ + c d c a d c a &\rightarrow a d \\ + a d c^{2} d c a &\rightarrow d c^{2} d c d c d c d c d \\ + a d c^{2} d c d &\rightarrow d c^{2} d c d c d c d c^{2} d c^{2} d c^{2} d c a d \\ + a d c^{2} d c^{2} d c^{2} d c^{2} d c a d &\rightarrow d c^{2} d c d c d c d c d c^{2} d c a \\ + c d c d c^{2} d c d c^{2} d c a d &\rightarrow d c^{2} d c^{2} d c a d \\ + a d c^{2} d c^{2} d c^{2} d c a d &\rightarrow d c^{2} d c d c d c d c d c d \\ + a c a d &\rightarrow d c^{2} d^{2} c a c a \\ + a d c d &\rightarrow d c^{2} d c d c d c d c d c a \\ + a d c^{2} d^{2} &\rightarrow d c^{2} d c d c d c d c^{2} d c a d \\ + a d c a d &\rightarrow d c^{2} d c d c d c d c d^{2} c a \\ + a d c^{2} d c^{2} d c a d &\rightarrow d c^{2} d c d c d c d c d c d c a \\ + c d c d c^{2} d a d &\rightarrow d c a d \\ + a c^{2} d a d &\rightarrow d c d^{2} c a \\ + c d c d c^{2} d c d a d &\rightarrow d c^{2} d a d \\ + a d c^{2} d a d &\rightarrow d c^{2} d c d c d^{2} c^{2} d^{2} c a c a \\ + b a d &\rightarrow c d^{2} c a \\ + a^{2} d &\rightarrow d c^{2} d c d c d^{2} c a c a \\ + a d a d &\rightarrow d c^{2} d c d c d c d^{3} c a \\ + a d^{2} &\rightarrow d c^{2} d c d c d c d^{2} c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/346.md b/docs/examples/1-relation/346.md new file mode 100644 index 0000000..ab9023b --- /dev/null +++ b/docs/examples/1-relation/346.md @@ -0,0 +1,41 @@ +#\#346 baababbaa=abaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{2} a^{2} = a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c d e = d^{2},\; c = b a,\; d = a c,\; e = b c a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < e < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow e \\ + a c &\rightarrow d \\ + c d e &\rightarrow d^{2} \\ + b c d &\rightarrow e c \\ + b d &\rightarrow c^{2} \\ + a d^{2} &\rightarrow d^{2} e \\ + c^{2} d &\rightarrow e c e \\ + a e c e &\rightarrow d c d \\ + e c e^{2} &\rightarrow c d^{2} \\ + a e^{2} c e d &\rightarrow d c d c e^{2} \\ + c d c d^{2} &\rightarrow d^{2} c e^{2} \\ + e c e c d^{2} &\rightarrow c d^{2} c e^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/347.md b/docs/examples/1-relation/347.md new file mode 100644 index 0000000..ee0a0e0 --- /dev/null +++ b/docs/examples/1-relation/347.md @@ -0,0 +1,53 @@ +#\#347 ababbbabba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{3} a b^{2} a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d b c = c^{2},\; c = b a,\; d = c b^{2} c,\; e = c d b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a d b c &\rightarrow c^{2} \\ + c b^{2} c &\rightarrow d \\ + c d b &\rightarrow e \\ + b c^{2} &\rightarrow e c \\ + e a &\rightarrow c d c \\ + c b^{2} d &\rightarrow d b^{2} c \\ + a d b d &\rightarrow c d \\ + b c d &\rightarrow e d \\ + b c e &\rightarrow e^{2} \\ + a d e c &\rightarrow c^{3} \\ + c d e c &\rightarrow e c^{2} \\ + c d e d &\rightarrow e c d \\ + c d e^{2} &\rightarrow e c e \\ + b e &\rightarrow e d b \\ + a d e d &\rightarrow c^{2} d \\ + a d e^{2} &\rightarrow c^{2} e \\ + c e d b d &\rightarrow d^{2} \\ + c e d b c &\rightarrow d c \\ + b d^{2} &\rightarrow e^{2} d b d \\ + b d e &\rightarrow e^{2} d e d b \\ + b d c &\rightarrow e^{2} d b c \\ + c e d e d &\rightarrow d c d \\ + c e d e^{2} &\rightarrow d c e \\ + c e d e c &\rightarrow d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/348.md b/docs/examples/1-relation/348.md new file mode 100644 index 0000000..808fedb --- /dev/null +++ b/docs/examples/1-relation/348.md @@ -0,0 +1,33 @@ +#\#348 baaaabbaba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a b a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{3} d = a c a,\; c = b a,\; d = b c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b c^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + a c a &\rightarrow c a^{3} d \\ + b c a^{3} d &\rightarrow c^{2} a \\ + a c^{2} a^{3} d &\rightarrow c a^{3} d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/350.md b/docs/examples/1-relation/350.md new file mode 100644 index 0000000..908dd95 --- /dev/null +++ b/docs/examples/1-relation/350.md @@ -0,0 +1,36 @@ +#\#350 aaaabaabba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{4} b a^{2} b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{4} c a b c = c^{2} a,\; c = b a,\; d = a b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a^{2} &\rightarrow d \\ + a^{3} d b^{2} a &\rightarrow b d \\ + a b^{2} d &\rightarrow d a d b^{2} a \\ + a b a b d &\rightarrow d a^{2} d b^{2} a \\ + a b a d &\rightarrow d b a^{2} \\ + a^{3} d b^{3} d &\rightarrow b d a^{2} d b^{2} a \\ + a^{3} d b^{2} d &\rightarrow b d b a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/352.md b/docs/examples/1-relation/352.md new file mode 100644 index 0000000..7696e1d --- /dev/null +++ b/docs/examples/1-relation/352.md @@ -0,0 +1,45 @@ +#\#352 baabaaaba=abaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{3} b a = a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a c a = a b c a,\; c = a^{2} b,\; d = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b c a c a &\rightarrow a b c a \\ + a^{2} b &\rightarrow c \\ + c^{2} a &\rightarrow d \\ + b c a c^{2} &\rightarrow a b c^{2} \\ + a d &\rightarrow d c a \\ + d a b &\rightarrow c^{3} \\ + b c d c a &\rightarrow a b d \\ + a c^{3} &\rightarrow d c^{2} \\ + c^{2} d c a &\rightarrow d^{2} \\ + c^{2} d c^{2} &\rightarrow d c^{3} \\ + b c d c^{2} &\rightarrow a b c^{3} \\ + a c^{2} d &\rightarrow d c d \\ + a c d &\rightarrow d^{2} \\ + c^{2} d c d &\rightarrow d c^{2} d \\ + c^{2} d^{2} &\rightarrow d c d \\ + b c d^{2} &\rightarrow a b c d \\ + b c d c d &\rightarrow a b c^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/353.md b/docs/examples/1-relation/353.md new file mode 100644 index 0000000..4524d3e --- /dev/null +++ b/docs/examples/1-relation/353.md @@ -0,0 +1,55 @@ +#\#353 abaabbabaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{2} a b a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a c e = c a c,\; c = b a,\; d = b c,\; e = a d c a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(e < a < c < d < b\): + +\( +\begin{aligned} + b c &\rightarrow d \\ + b a &\rightarrow c \\ + c a c &\rightarrow a c e \\ + a d c a &\rightarrow e \\ + d a c &\rightarrow c^{2} e \\ + b e &\rightarrow c d c a \\ + a d c e &\rightarrow e d c a \\ + a c^{2} e^{2} &\rightarrow e c \\ + a c e a c &\rightarrow c a^{2} c e \\ + d a^{2} c e &\rightarrow c^{2} e a c \\ + d c a^{2} c e &\rightarrow c^{2} e^{2} a c \\ + c^{2} a^{2} c e &\rightarrow a c e^{2} a c \\ + a c e c a^{2} c e &\rightarrow c a^{2} c e^{2} a c \\ + a c e c e^{2} &\rightarrow c e c \\ + d c e c &\rightarrow c^{2} e^{2} c e^{2} \\ + c^{2} e c &\rightarrow a c e^{2} c e^{2} \\ + a c e c e c &\rightarrow c a^{2} c e^{2} c e^{2} \\ + d e a c e &\rightarrow a c e^{2} c e^{4} a c \\ + d e c &\rightarrow a c e^{2} c e^{4} \\ + d c e a c e &\rightarrow c^{2} e^{2} c e^{2} a c \\ + a c e^{2} c e^{2} c a^{2} c e &\rightarrow c^{2} e a c e^{2} a c \\ + a c e^{2} c e^{2} c e c &\rightarrow c^{2} e a c e^{2} c e^{2} \\ + a c e^{2} c e^{2} a c &\rightarrow c^{2} e a c e \\ + d c^{2} e a c e &\rightarrow c^{2} e^{3} c e^{2} a c \\ + c^{3} e a c e &\rightarrow a c e^{3} c e^{2} a c \\ + a c e c^{2} e a c e &\rightarrow c a^{2} c e^{3} c e^{2} a c \\ + a c e^{2} c e^{2} c^{2} e a c e &\rightarrow c^{2} e a c e^{3} c e^{2} a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/354.md b/docs/examples/1-relation/354.md new file mode 100644 index 0000000..a773e17 --- /dev/null +++ b/docs/examples/1-relation/354.md @@ -0,0 +1,56 @@ +#\#354 baabbaaaba=aabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a^{3} b a = a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a c d a = a c b a^{2},\; c = a b,\; d = b a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a c b a^{2} &\rightarrow b a c d a \\ + a b &\rightarrow c \\ + b a^{2} c &\rightarrow d \\ + a c b a c &\rightarrow b a c d c \\ + c a^{2} c &\rightarrow a d \\ + b a c d a c &\rightarrow a c d \\ + b c a c d a &\rightarrow d b a^{2} \\ + b a c d b a c d a &\rightarrow a c d b a^{2} \\ + a^{2} c d &\rightarrow c a c d a c \\ + a d b a^{2} &\rightarrow c^{2} a c d a \\ + a c a c d &\rightarrow b a c d c d a c \\ + b a c d b a c d c &\rightarrow a c d b a c \\ + a c b^{2} a c d a &\rightarrow b a c d c b a^{2} \\ + a c b^{2} a c d c &\rightarrow b a c d c b a c \\ + b c a c d b a c d a &\rightarrow d^{2} b a^{2} \\ + b c a c d b a c d c &\rightarrow d^{2} b a c \\ + b c a c d c &\rightarrow d b a c \\ + c a^{3} d &\rightarrow a d a^{2} c \\ + a d b a c &\rightarrow c^{2} a c d c \\ + b a^{3} d &\rightarrow d a^{2} c \\ + b a c d a^{2} d &\rightarrow a c d a^{2} c \\ + a d a c d &\rightarrow c^{2} a c d c d a c \\ + c^{2} a c d b a c d a &\rightarrow a d^{2} b a^{2} \\ + c^{2} a c d b a c d c &\rightarrow a d^{2} b a c \\ + a d b^{2} a c d a &\rightarrow c^{2} a c d c b a^{2} \\ + a d b^{2} a c d c &\rightarrow c^{2} a c d c b a c \\ + c^{2} a c d a^{2} d &\rightarrow a d^{2} a^{2} c \\ + c^{2} a c d a c &\rightarrow a d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/355.md b/docs/examples/1-relation/355.md new file mode 100644 index 0000000..c5e49e7 --- /dev/null +++ b/docs/examples/1-relation/355.md @@ -0,0 +1,53 @@ +#\#355 baaabbaba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a b a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b d b a = a,\; c = a^{2} b,\; d = a c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a^{2} b &\rightarrow c \\ + d b a &\rightarrow b d a \\ + d b c &\rightarrow b d c \\ + d b d &\rightarrow b d^{2} \\ + a c &\rightarrow c b d a b \\ + c b d a b d &\rightarrow d c b a b d \\ + c b d a b c &\rightarrow d c b a b c \\ + c b d a b a &\rightarrow d c b a b a \\ + c b d a b^{2} d &\rightarrow d c b a \\ + c b d a b^{2} c &\rightarrow d a b \\ + b^{2} d^{2} &\rightarrow d \\ + b^{2} d c &\rightarrow c \\ + a^{3} &\rightarrow c b d a \\ + b^{2} d a &\rightarrow a \\ + c b d a b^{2} a &\rightarrow d \\ + c b d c &\rightarrow d c b c \\ + c b d a^{2} &\rightarrow d c b a^{2} \\ + a d &\rightarrow d c b c b a \\ + c b d^{2} &\rightarrow d c b d \\ + a b d a &\rightarrow d c b c b a b a \\ + a b d c &\rightarrow d c b c b a b c \\ + a b d^{2} &\rightarrow d c b c b a b d \\ + c d a &\rightarrow d c a \\ + c d c &\rightarrow d c^{2} \\ + c d^{2} &\rightarrow d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/356.md b/docs/examples/1-relation/356.md new file mode 100644 index 0000000..d9ea146 --- /dev/null +++ b/docs/examples/1-relation/356.md @@ -0,0 +1,38 @@ +#\#356 baabaabaa=ababaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b a^{2} = a b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a c^{2} a = c d,\; c = a b a,\; d = b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a b a &\rightarrow c \\ + b a^{2} &\rightarrow d \\ + a d &\rightarrow c a \\ + a b c &\rightarrow c b a \\ + b a c &\rightarrow d b a \\ + c d &\rightarrow d^{3} \\ + a b d^{3} &\rightarrow c b c a \\ + b c^{3} a &\rightarrow d b c a \\ + b c^{4} &\rightarrow d b c^{2} \\ + b d^{9} &\rightarrow d b d^{5} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/357.md b/docs/examples/1-relation/357.md new file mode 100644 index 0000000..a85fadb --- /dev/null +++ b/docs/examples/1-relation/357.md @@ -0,0 +1,58 @@ +#\#357 abbabbaaa=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a^{3} = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d a = b a c a,\; c = b^{2},\; d = a c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a c a &\rightarrow a c d a \\ + a c a^{2} &\rightarrow d \\ + b c &\rightarrow c b \\ + b d &\rightarrow a c d a^{2} \\ + b a c d &\rightarrow a c d^{2} \\ + a c a d &\rightarrow d c a^{2} \\ + c a c a &\rightarrow a c d^{2} a \\ + c a c d &\rightarrow a c d^{3} \\ + a c d^{2} a^{2} &\rightarrow c d \\ + b a^{2} c d^{2} a &\rightarrow a c d a c a \\ + c d c a^{2} &\rightarrow a c d^{2} a d \\ + c a^{2} c d^{2} a &\rightarrow a c d^{2} a c a \\ + c^{2} d &\rightarrow a c d^{4} a^{2} \\ + c d c d^{2} a^{2} &\rightarrow a c d^{2} a c d \\ + c d a c d^{3} &\rightarrow a c d^{2} a d c d^{2} a^{2} \\ + c d c a d &\rightarrow a c d^{2} a d c a^{2} \\ + c d a c d^{2} a c a &\rightarrow a c d^{2} a d c d^{2} a \\ + a^{2} c d^{3} &\rightarrow d c d^{2} a^{2} \\ + a^{2} c d^{2} a c a &\rightarrow d c d^{2} a \\ + c d a c d^{2} a c d &\rightarrow a c d^{2} a d c d^{3} \\ + c d c d c d^{2} a &\rightarrow a c d^{2} a d c d^{2} a c a \\ + c d c d^{3} &\rightarrow a c d^{3} c d^{2} a^{2} \\ + c d c d^{2} a c a &\rightarrow a c d^{3} c d^{2} a \\ + c d c d^{2} c d &\rightarrow a c d^{3} c d^{4} a^{2} \\ + a^{2} c d^{2} a c d &\rightarrow d c d^{3} \\ + a c d c d^{2} a &\rightarrow d c d^{2} a c a \\ + c d c d^{2} a c d &\rightarrow a c d^{3} c d^{3} \\ + c d a c d^{2} c d &\rightarrow a c d^{2} a d c d^{4} a^{2} \\ + a^{2} c d^{2} c d &\rightarrow d c d^{4} a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/358.md b/docs/examples/1-relation/358.md new file mode 100644 index 0000000..f33eec6 --- /dev/null +++ b/docs/examples/1-relation/358.md @@ -0,0 +1,35 @@ +#\#358 baaabbabaa=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a b a^{2} = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c d b c = c b^{2} a,\; c = a^{2},\; d = a b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} a &\rightarrow d \\ + b a^{2} d b a^{2} &\rightarrow a d \\ + b a^{2} d b a d &\rightarrow a d b^{2} a \\ + b a^{2} d a d &\rightarrow a d^{2} b a^{2} \\ + a b^{2} d &\rightarrow d b^{2} a \\ + a b a d &\rightarrow d a d b a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/359.md b/docs/examples/1-relation/359.md new file mode 100644 index 0000000..367a9e4 --- /dev/null +++ b/docs/examples/1-relation/359.md @@ -0,0 +1,54 @@ +#\#359 ababaaabaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{3} b a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c^{2} a^{2} c a = c a c,\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + d b a^{2} d a &\rightarrow b a d \\ + b a d b a &\rightarrow d b a^{2} d^{2} \\ + a b d &\rightarrow d b a \\ + d b a^{2} d^{2} a d a &\rightarrow b d^{2} \\ + b a d b d &\rightarrow d b a^{2} d^{2} b a \\ + b a d^{2} b a^{2} d^{2} &\rightarrow d b a^{2} d^{3} b a \\ + a b^{2} a d &\rightarrow d b d a d a \\ + a d b a^{2} d^{2} &\rightarrow d^{2} b a \\ + b d^{2} b^{2} a d &\rightarrow d b a^{2} d^{2} a d^{2} b d a d a \\ + b d^{2} b d &\rightarrow d b a^{2} d^{2} a d^{2} b a \\ + b d^{2} b a &\rightarrow d b a^{2} d^{2} a d^{2} \\ + b d^{3} b a^{2} d^{2} &\rightarrow d b a^{2} d^{2} a d^{3} b a \\ + b d b a d &\rightarrow d b a^{2} d^{2} a d^{2} a d a \\ + b a d b^{2} a d &\rightarrow d b a^{2} d^{2} b d a d a \\ + a b^{2} d^{2} &\rightarrow d b d a d^{2} a d a \\ + b a d b^{2} d^{2} &\rightarrow d b a^{2} d^{2} b d a d^{2} a d a \\ + b a d^{2} b a^{2} d b d^{2} &\rightarrow d b a^{2} d^{3} b d a d^{2} a d a \\ + b a d^{2} b a^{2} d b a d &\rightarrow d b a^{2} d^{3} b d a d a \\ + a d b a^{2} d b d^{2} &\rightarrow d^{2} b d a d^{2} a d a \\ + a d b a^{2} d b a d &\rightarrow d^{2} b d a d a \\ + b d b d^{2} &\rightarrow d b a^{2} d^{2} a d^{2} a d^{2} a d a \\ + b d^{2} b^{2} d^{2} &\rightarrow d b a^{2} d^{2} a d^{2} b d a d^{2} a d a \\ + b d^{3} b a^{2} d b d^{2} &\rightarrow d b a^{2} d^{2} a d^{3} b d a d^{2} a d a \\ + b d^{3} b a^{2} d b a d &\rightarrow d b a^{2} d^{2} a d^{3} b d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/360.md b/docs/examples/1-relation/360.md new file mode 100644 index 0000000..31c34fb --- /dev/null +++ b/docs/examples/1-relation/360.md @@ -0,0 +1,34 @@ +#\#360 baabaabaa=abaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b a^{2} = a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d^{3} = a d c,\; c = b a,\; d = c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a &\rightarrow d \\ + a d c &\rightarrow d^{3} \\ + b d^{3} &\rightarrow c d c \\ + c d^{3} &\rightarrow d^{2} c \\ + a d^{2} &\rightarrow d^{3} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/361.md b/docs/examples/1-relation/361.md new file mode 100644 index 0000000..6e8ddbe --- /dev/null +++ b/docs/examples/1-relation/361.md @@ -0,0 +1,36 @@ +#\#361 baaaabaaba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a^{2} b a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d c d e = a,\; c = a^{2},\; d = b c,\; e = b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < e < c < a < b\): + +\( +\begin{aligned} + b c &\rightarrow d \\ + a &\rightarrow d c d e \\ + b d c d e &\rightarrow e \\ + e d c d e &\rightarrow d \\ + d c d^{2} &\rightarrow c \\ + e c &\rightarrow d^{2} c d e \\ + d c d c &\rightarrow c^{2} d^{2} \\ + d c^{2} &\rightarrow c^{2} d^{4} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/362.md b/docs/examples/1-relation/362.md new file mode 100644 index 0000000..2d0f882 --- /dev/null +++ b/docs/examples/1-relation/362.md @@ -0,0 +1,42 @@ +#\#362 abbaabbaba=bababbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b^{2} a b a = b a b a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d b c^{2} = e,\; c = b a,\; d = b c a,\; e = c^{2} d +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b c a &\rightarrow d \\ + b a &\rightarrow c \\ + c^{2} d &\rightarrow e \\ + a d b c^{2} &\rightarrow e \\ + c d b c^{2} &\rightarrow b e \\ + d^{2} b c^{2} &\rightarrow b c e \\ + a d b c e &\rightarrow e c d \\ + a d b e &\rightarrow e d \\ + c b e &\rightarrow e b c^{2} \\ + c^{2} b c e &\rightarrow e d b c^{2} \\ + c d b c e &\rightarrow b e c d \\ + c d b e &\rightarrow b e d \\ + d^{2} b c e &\rightarrow b c e c d \\ + d^{2} b e &\rightarrow b c e d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/363.md b/docs/examples/1-relation/363.md new file mode 100644 index 0000000..3a946b4 --- /dev/null +++ b/docs/examples/1-relation/363.md @@ -0,0 +1,37 @@ +#\#363 aabaabbaba=babaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b^{2} a b a = b a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d c^{2} = c^{2} a c,\; c = b a,\; d = a c a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a d c^{2} &\rightarrow c^{2} a c \\ + a c a b &\rightarrow d \\ + b d &\rightarrow c^{2} a b \\ + b c^{2} a c &\rightarrow c d c^{2} \\ + a c a c &\rightarrow d a \\ + b c^{2} d &\rightarrow c d c^{2} a b \\ + d a^{2} b &\rightarrow a c d \\ + d a^{2} c &\rightarrow a c d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/364.md b/docs/examples/1-relation/364.md new file mode 100644 index 0000000..0f0eb5e --- /dev/null +++ b/docs/examples/1-relation/364.md @@ -0,0 +1,37 @@ +#\#364 aabaababaa=baabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b a b a^{2} = b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d c^{2} a = c d,\; c = b a,\; d = a c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c a &\rightarrow d \\ + a d c^{2} a &\rightarrow c d \\ + b c d &\rightarrow c d c^{2} a \\ + b d &\rightarrow c^{2} a \\ + a c^{2} d &\rightarrow d^{2} c^{2} a \\ + a c d &\rightarrow d c a \\ + a d c^{3} d &\rightarrow c d^{2} c^{2} a \\ + a d c^{2} d &\rightarrow c d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/365.md b/docs/examples/1-relation/365.md new file mode 100644 index 0000000..a4f2a5b --- /dev/null +++ b/docs/examples/1-relation/365.md @@ -0,0 +1,49 @@ +#\#365 abaababbaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b^{2} a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a c d a = c a c,\; c = b a,\; d = a c b c,\; e = a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < e < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow e \\ + b e &\rightarrow c b \\ + c^{2} b c &\rightarrow b d \\ + a c &\rightarrow e a \\ + c d &\rightarrow e a d e c \\ + e^{3} a d a &\rightarrow d e a \\ + e^{2} c &\rightarrow d \\ + c e a &\rightarrow e a d a \\ + b d e a &\rightarrow e a d a d a d a d e c a \\ + b d^{2} &\rightarrow e a d a d a d a d e c e c \\ + a e a d a &\rightarrow e a e a \\ + c e^{2} &\rightarrow e a d e \\ + d c b c &\rightarrow e^{2} b d \\ + e^{3} a d e &\rightarrow d e^{2} \\ + b d e^{2} &\rightarrow e a d a d a d a d e c e \\ + c e d &\rightarrow e a d^{2} \\ + a e a d e &\rightarrow e a e^{2} \\ + e^{3} a d^{2} &\rightarrow d e d \\ + b d e d &\rightarrow e a d a d a d a d e^{2} a d e c \\ + a e a d^{2} &\rightarrow e a e d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/366.md b/docs/examples/1-relation/366.md new file mode 100644 index 0000000..3935df6 --- /dev/null +++ b/docs/examples/1-relation/366.md @@ -0,0 +1,36 @@ +#\#366 aaababaaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a b a^{2} b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{3} c^{2} a c = c^{2} a,\; c = b a,\; d = a b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a^{2} &\rightarrow d \\ + a^{3} b d b a &\rightarrow b d \\ + a b^{2} d &\rightarrow d a b d b a \\ + a b a b d &\rightarrow d a^{2} b d b a \\ + a b a d &\rightarrow d b a^{2} \\ + a^{3} b d b^{2} d &\rightarrow b d a^{2} b d b a \\ + a^{3} b d b d &\rightarrow b d b a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/367.md b/docs/examples/1-relation/367.md new file mode 100644 index 0000000..e42ea75 --- /dev/null +++ b/docs/examples/1-relation/367.md @@ -0,0 +1,58 @@ +#\#367 baaababaaa=abaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b a^{3} = a b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; b c d e a = e d a,\; c = a^{2},\; d = a b,\; e = d c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < e < c < a < b\): + +\( +\begin{aligned} + b c d e a &\rightarrow e d a \\ + a b &\rightarrow d \\ + a^{2} &\rightarrow c \\ + d c &\rightarrow e \\ + b c d e c &\rightarrow e^{2} \\ + b c d e d &\rightarrow e d^{2} \\ + a c &\rightarrow c a \\ + a e d a &\rightarrow e d e a \\ + c b &\rightarrow a d \\ + b c d e^{2} &\rightarrow e d e \\ + a e^{2} &\rightarrow e d e c \\ + a e d^{2} &\rightarrow e d e d \\ + e b &\rightarrow d a d \\ + a e d e &\rightarrow e d e^{2} \\ + c e d a &\rightarrow e d e^{2} a \\ + c e^{2} &\rightarrow e d e^{2} c \\ + c e d e &\rightarrow e d e^{3} \\ + c e d^{2} &\rightarrow e d e^{2} d \\ + d e d e^{2} a &\rightarrow e^{2} d a \\ + d e d e^{2} c &\rightarrow e^{3} \\ + d e d e^{3} &\rightarrow e^{2} d e \\ + d e d e^{2} d &\rightarrow e^{2} d^{2} \\ + b e d e^{2} c e &\rightarrow e d^{2} e^{2} c \\ + b e d e^{2} c d a &\rightarrow e d^{2} e^{2} a \\ + b e d e^{2} c d e &\rightarrow e d^{2} e^{3} \\ + b e d e^{2} c d^{2} &\rightarrow e d^{2} e^{2} d \\ + b e d e^{3} d e^{2} a &\rightarrow e d^{2} e^{2} c d a \\ + b e d e^{3} d e^{2} c &\rightarrow e d^{2} e^{2} c e \\ + b e d e^{3} d e^{3} &\rightarrow e d^{2} e^{2} c d e \\ + b e d e^{3} d e^{2} d &\rightarrow e d^{2} e^{2} c d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/368.md b/docs/examples/1-relation/368.md new file mode 100644 index 0000000..10be075 --- /dev/null +++ b/docs/examples/1-relation/368.md @@ -0,0 +1,33 @@ +#\#368 aababbbaba=babbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b^{3} a b a = b a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c b^{2} c^{2} = c b c a,\; c = b a,\; d = b a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b a b &\rightarrow d \\ + d b a^{2} &\rightarrow a^{2} d b d a \\ + b a d &\rightarrow d a b \\ + d a b^{2} a^{2} &\rightarrow b a^{3} d b d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/369.md b/docs/examples/1-relation/369.md new file mode 100644 index 0000000..fe33e78 --- /dev/null +++ b/docs/examples/1-relation/369.md @@ -0,0 +1,68 @@ +#\#369 ababbabbaa=babbaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b^{2} a^{2} = b a b^{2} a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b a b^{2} a c = b a c b a,\; c = b^{2} a^{2},\; d = c b a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b^{2} a^{2} &\rightarrow c \\ + c b a b &\rightarrow d \\ + a b a b^{2} a c &\rightarrow b a c b a \\ + b^{2} a b a c b a &\rightarrow d b a c \\ + a b a b^{2} a d &\rightarrow b a d a b \\ + c b^{2} a c b a &\rightarrow d a b^{2} a c \\ + c b a c &\rightarrow d b a^{2} \\ + b^{2} a b a d &\rightarrow d b a c b \\ + d b a b a c b a &\rightarrow c b a d b a c \\ + d b a^{2} b^{2} a c b a &\rightarrow c b a d a b^{2} a c \\ + d b a^{2} b a b &\rightarrow c b a d \\ + d b a^{2} b a c &\rightarrow c b a d b a^{2} \\ + d a b^{2} a c b &\rightarrow c b^{2} a d \\ + c b^{2} a d b a^{2} &\rightarrow d a b^{2} a c^{2} \\ + c b^{2} a d a b &\rightarrow d a b^{2} a d \\ + b^{2} a b a c b^{2} a d &\rightarrow d b a d b a c b \\ + d b a b a d b a^{2} &\rightarrow c b a d b a c^{2} \\ + d b a b a d a b &\rightarrow c b a d b a d \\ + d b a^{2} b^{2} a d b a^{2} &\rightarrow c b a d a b^{2} a c^{2} \\ + d b a^{2} b^{2} a d a b &\rightarrow c b a d a b^{2} a d \\ + d b a^{2} b a d b a c b &\rightarrow c b a d b a b a d \\ + d a b^{2} a d b a b a d &\rightarrow c b^{2} a d a d b a c b \\ + d a b^{2} a d b a^{2} &\rightarrow c b^{2} a d a c \\ + d a b^{2} a d a b^{2} a d &\rightarrow c b^{2} a d b a d a b \\ + d a b^{2} a d a b^{2} a c &\rightarrow c b^{2} a d b a c b a \\ + c b^{2} a d b a b a d &\rightarrow d a b^{2} a c d b a c b \\ + c b^{2} a c b^{2} a d &\rightarrow d a b^{2} a d b a c b \\ + c b a d b a c b &\rightarrow d b a b a d \\ + d b a b a c b^{2} a d &\rightarrow c b a d b a d b a c b \\ + d b a^{2} b^{2} a c b^{2} a d &\rightarrow c b a d a b^{2} a d b a c b \\ + d b a^{2} b a d b a d b a^{2} &\rightarrow c b a d b a b a d a c \\ + d b a^{2} b a d b a d a b^{2} a d &\rightarrow c b a d b a b a d b a d a b \\ + d b a^{2} b a d b a d a b^{2} a c &\rightarrow c b a d b a b a d b a c b a \\ + c b a d b a d b a b a d &\rightarrow d b a b a d a d b a c b \\ + c b a d b a d b a^{2} &\rightarrow d b a b a d a c \\ + c b a d b a d a b^{2} a d &\rightarrow d b a b a d b a d a b \\ + c b a d b a d a b^{2} a c &\rightarrow d b a b a d b a c b a \\ + d b a b a d b a b a d &\rightarrow c b a d b a c d b a c b \\ + d b a^{2} b^{2} a d b a b a d &\rightarrow c b a d a b^{2} a c d b a c b \\ + d b a^{2} b a d b a d b a b a d &\rightarrow c b a d b a b a d a d b a c b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/370.md b/docs/examples/1-relation/370.md new file mode 100644 index 0000000..3be36b3 --- /dev/null +++ b/docs/examples/1-relation/370.md @@ -0,0 +1,86 @@ +#\#370 baaababaaa=aababaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b a^{3} = a^{2} b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d = a c a^{2},\; c = a b a b,\; d = c a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a^{2} d &\rightarrow a c a^{2} \\ + a b a b &\rightarrow c \\ + c a^{3} &\rightarrow d \\ + a b c &\rightarrow c a b \\ + d b a b &\rightarrow c a^{2} c \\ + a^{2} c a^{2} &\rightarrow c a^{2} d a \\ + b c a^{2} d a c &\rightarrow a c a c \\ + d b c &\rightarrow c a^{2} c a b \\ + c a b a^{3} &\rightarrow a b d \\ + b c a^{2} d a d &\rightarrow a c a d \\ + a^{2} c^{2} a^{2} d a &\rightarrow c a^{2} d a c a^{2} \\ + a^{2} c a c &\rightarrow c a^{2} d c \\ + a b d a c a^{2} &\rightarrow c^{2} a^{2} d^{2} a \\ + c a b a c a^{2} d a &\rightarrow a b d c a^{2} \\ + c a^{2} d a^{2} &\rightarrow a^{2} d \\ + c^{2} a^{2} d a d c &\rightarrow d a c a c \\ + c^{2} a^{2} d a d a &\rightarrow d a c a^{2} \\ + c a c a^{2} d a &\rightarrow d c a^{2} \\ + a b d a^{3} d &\rightarrow c^{2} a^{2} d^{2} a d a^{2} \\ + a b d a d &\rightarrow c^{2} a^{2} d^{2} a^{2} \\ + a b d a c^{2} a^{2} d a &\rightarrow c^{2} a^{2} d^{2} a c a^{2} \\ + a b d a c^{2} a^{2} d c &\rightarrow c^{2} a^{2} d^{2} a c a c \\ + c a b a c a^{2} d c &\rightarrow a b d c a c \\ + c a^{2} d c a^{2} d c &\rightarrow a^{2} d c a c \\ + a^{2} c^{2} a^{2} d c &\rightarrow c a^{2} d a c a c \\ + c^{2} a^{2} d a d^{2} &\rightarrow d a c a d \\ + a^{2} c a d &\rightarrow c a^{2} d^{2} \\ + a b d a c a c &\rightarrow c^{2} a^{2} d^{2} c \\ + c a^{2} d a d a^{2} &\rightarrow a^{4} d \\ + c a^{2} d a c a^{2} d a &\rightarrow a^{2} d a c a^{2} \\ + c a^{2} d a c a^{2} d c &\rightarrow a^{2} d a c a c \\ + c a^{2} d c a^{2} d a &\rightarrow a^{2} d c a^{2} \\ + d b d a c a c &\rightarrow c^{2} a^{2} d a c a d c \\ + c a c a^{2} d c &\rightarrow d c a c \\ + d b d a c a^{2} &\rightarrow c^{2} a^{2} d a c a d a \\ + a b d a c^{2} a^{2} d^{2} &\rightarrow c^{2} a^{2} d^{2} a c a d \\ + c a b a c a^{2} d^{2} &\rightarrow a b d c a d \\ + c a^{2} d c a^{2} d^{2} &\rightarrow a^{2} d c a d \\ + d b d a^{3} d &\rightarrow c^{2} a^{2} d a c a d a d a^{2} \\ + d b d a d &\rightarrow c^{2} a^{2} d a c a d a^{2} \\ + d b d a c^{2} a^{2} d a &\rightarrow c^{2} a^{2} d a c a d a c a^{2} \\ + d b d a c^{2} a^{2} d c &\rightarrow c^{2} a^{2} d a c a d a c a c \\ + c a^{2} d a d c a^{2} d c &\rightarrow a^{4} d c a c \\ + b a^{4} d &\rightarrow a c a d a^{2} \\ + a b d a^{5} d &\rightarrow c^{2} a^{2} d^{2} a d a d a^{2} \\ + a b d a c a d &\rightarrow c^{2} a^{2} d^{3} \\ + a^{6} d &\rightarrow c a^{2} d a d a d a^{2} \\ + a^{2} c^{2} a^{2} d^{2} &\rightarrow c a^{2} d a c a d \\ + c a^{2} d a d a c a^{2} d a &\rightarrow a^{4} d a c a^{2} \\ + c a^{2} d a d a c a^{2} d^{2} &\rightarrow a^{4} d a c a d \\ + c a^{2} d a d a c a^{2} d c &\rightarrow a^{4} d a c a c \\ + c a^{2} d a d c a^{2} d a &\rightarrow a^{4} d c a^{2} \\ + c a^{2} d a d c a^{2} d^{2} &\rightarrow a^{4} d c a d \\ + c a^{2} d a c a^{2} d^{2} &\rightarrow a^{2} d a c a d \\ + d b d a c a d &\rightarrow c^{2} a^{2} d a c a d^{2} \\ + c a c a^{2} d^{2} &\rightarrow d c a d \\ + d b d a^{5} d &\rightarrow c^{2} a^{2} d a c a d a d a d a^{2} \\ + d b d a c^{2} a^{2} d^{2} &\rightarrow c^{2} a^{2} d a c a d a c a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/372.md b/docs/examples/1-relation/372.md new file mode 100644 index 0000000..7e6195f --- /dev/null +++ b/docs/examples/1-relation/372.md @@ -0,0 +1,52 @@ +#\#372 abbabbaaba=babba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a^{2} b a = b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d a = b c a,\; c = a b^{2},\; d = a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b c a &\rightarrow c^{2} d a \\ + a b^{2} &\rightarrow c \\ + a^{2} b &\rightarrow d \\ + b c d &\rightarrow c^{2} d^{2} \\ + d b &\rightarrow a c \\ + b c^{2} &\rightarrow c^{2} d c \\ + a c^{2} a &\rightarrow d c^{2} d a \\ + a c^{2} d &\rightarrow d c^{2} d^{2} \\ + a c^{3} &\rightarrow d c^{2} d c \\ + d c^{2} d^{2} c d c &\rightarrow c^{3} \\ + d c^{2} d^{2} c d^{2} &\rightarrow c^{2} d \\ + a d c^{2} d^{2} c &\rightarrow d c^{2} \\ + a d c^{2} d^{3} &\rightarrow d c d \\ + d c^{2} d^{2} c d a &\rightarrow c^{2} a \\ + a d c^{2} d^{2} a &\rightarrow d c a \\ + a d c^{2} d c^{2} a &\rightarrow d c^{3} d^{2} c d a \\ + a d c^{2} d c^{2} d &\rightarrow d c^{3} d^{2} c d^{2} \\ + a d c^{2} d c^{3} &\rightarrow d c^{3} d^{2} c d c \\ + d c^{2} d^{2} c^{3} a &\rightarrow c^{4} d^{2} c d a \\ + d c^{2} d^{2} c^{3} d &\rightarrow c^{4} d^{2} c d^{2} \\ + d c^{2} d^{2} c^{4} &\rightarrow c^{4} d^{2} c d c \\ + a d c^{4} a &\rightarrow d c^{3} d^{2} c d^{3} c d a \\ + a d c^{4} d &\rightarrow d c^{3} d^{2} c d^{3} c d^{2} \\ + a d c^{5} &\rightarrow d c^{3} d^{2} c d^{3} c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/374.md b/docs/examples/1-relation/374.md new file mode 100644 index 0000000..b2c14cf --- /dev/null +++ b/docs/examples/1-relation/374.md @@ -0,0 +1,31 @@ +#\#374 abaabababa=babaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b a b a = b a b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a c^{3} = c^{2} a c a,\; c = d,\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow d \\ + b a &\rightarrow d \\ + d^{2} a d a &\rightarrow a d a d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/375.md b/docs/examples/1-relation/375.md new file mode 100644 index 0000000..0385344 --- /dev/null +++ b/docs/examples/1-relation/375.md @@ -0,0 +1,37 @@ +#\#375 baabaabba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{2} a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d = a c,\; c = b a^{2},\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} a &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a c &\rightarrow c^{2} d \\ + b c &\rightarrow d a \\ + d c^{2} d^{2} c d &\rightarrow c^{2} \\ + b d c^{2} d^{2} &\rightarrow d c \\ + b d c^{2} d c^{2} &\rightarrow d c^{3} d^{2} c d \\ + d c^{2} d^{2} c^{3} &\rightarrow c^{4} d^{2} c d \\ + b d c^{4} &\rightarrow d c^{3} d^{2} c d^{3} c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/376.md b/docs/examples/1-relation/376.md new file mode 100644 index 0000000..b48f38d --- /dev/null +++ b/docs/examples/1-relation/376.md @@ -0,0 +1,36 @@ +#\#376 abaabababa=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c^{3} a = d a,\; c = a b,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c a &\rightarrow d \\ + d a &\rightarrow c a c^{3} a \\ + a b &\rightarrow c \\ + b c^{2} &\rightarrow d b \\ + d c &\rightarrow c a c^{4} \\ + a d &\rightarrow c^{2} a \\ + a c a c^{4} &\rightarrow c^{2} a c \\ + a c a c^{3} a &\rightarrow c^{2} a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/377.md b/docs/examples/1-relation/377.md new file mode 100644 index 0000000..83a0257 --- /dev/null +++ b/docs/examples/1-relation/377.md @@ -0,0 +1,35 @@ +#\#377 babaababaa=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a b a^{2} = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d^{2} = a c^{2},\; c = b a,\; d = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{2} &\rightarrow d^{2} \\ + c^{2} a &\rightarrow d \\ + b d^{2} &\rightarrow c^{3} \\ + a d &\rightarrow d^{2} a \\ + a c d &\rightarrow d^{2} c a \\ + c^{2} d^{2} &\rightarrow d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/378.md b/docs/examples/1-relation/378.md new file mode 100644 index 0000000..8dc29b8 --- /dev/null +++ b/docs/examples/1-relation/378.md @@ -0,0 +1,61 @@ +#\#378 baabaaaba=aaabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{3} b a = a^{3} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c a^{2} c = a^{3} b c a,\; c = b a,\; d = a c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < b < a\): + +\( +\begin{aligned} + a^{3} b c a &\rightarrow c a c a^{2} c \\ + a c^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + a^{3} b c d &\rightarrow c a c a d c \\ + b d &\rightarrow c^{3} \\ + b c a c a^{2} c &\rightarrow c a^{2} b c a \\ + a^{3} c a^{2} b c a &\rightarrow c a c a d a^{2} c \\ + c a^{2} b c a c &\rightarrow b c a c a d \\ + c a^{2} b c d &\rightarrow b c a c a d c \\ + a^{3} b c^{2} a c a d &\rightarrow c a b c a c a d a d \\ + a^{3} b^{2} c a c a d &\rightarrow c a c a^{2} c a b c a c \\ + a^{3} c a^{2} b^{2} c a c a d &\rightarrow c a c a d a^{2} c a b c a c \\ + a^{3} c a^{2} b c^{2} a b c a c a d a d &\rightarrow c a c a d a^{2} c a^{2} b c^{2} a c a d \\ + a^{3} c a^{2} b c^{2} a c a^{2} c &\rightarrow c a c a d a^{2} c a^{2} b c a \\ + a c b c a c a d &\rightarrow d a^{2} b c a c \\ + b c a b c a c a d a d &\rightarrow c a^{2} b c^{2} a c a d \\ + c a^{2} b c a b c a c a d &\rightarrow b c a c a d a^{2} b c a c \\ + c a^{2} c a^{2} b c a &\rightarrow b c a c a d a^{2} c \\ + a^{3} b c^{2} a b c a c a d a d &\rightarrow c a b c a c a d a^{2} b c a c a d \\ + a^{3} b c^{2} a c a^{2} c &\rightarrow c a b c a c a d a^{2} c \\ + a^{3} b c^{2} a b c a c a d a^{2} c &\rightarrow c a b c a c a d a^{2} c a^{2} b c a \\ + a^{3} c a^{2} b c^{2} a b c a c a d a^{2} b c a c a d &\rightarrow c a c a d a^{2} c a^{2} b c^{2} a b c a c a d a d \\ + a^{3} c a^{2} b c^{2} a b c a c a d a^{2} c &\rightarrow c a c a d a^{2} c a^{2} b c^{2} a c a^{2} c \\ + a^{3} c a^{2} b c^{2} a c a d &\rightarrow c a c a d a^{2} b c a c a d \\ + b c a b c a c a d a^{2} b c a c a d &\rightarrow c a^{2} b c^{2} a b c a c a d a d \\ + b c a b c a c a d a^{2} c &\rightarrow c a^{2} b c^{2} a c a^{2} c \\ + c a^{2} c a^{2} b^{2} c a c a d &\rightarrow b c a c a d a^{2} c a b c a c \\ + c a^{2} c a^{2} b c^{2} a b c a c a d a^{2} b c a c a d &\rightarrow b c a c a d a^{2} c a^{2} b c^{2} a b c a c a d a d \\ + c a^{2} c a^{2} b c^{2} a b c a c a d a^{2} c &\rightarrow b c a c a d a^{2} c a^{2} b c^{2} a c a^{2} c \\ + c a^{2} c a^{2} b c^{2} a b c a c a d a d &\rightarrow b c a c a d a^{2} c a^{2} b c^{2} a c a d \\ + c a^{2} c a^{2} b c^{2} a c a^{2} c &\rightarrow b c a c a d a^{2} c a^{2} b c a \\ + c a^{2} c a^{2} b c^{2} a c a d &\rightarrow b c a c a d a^{2} b c a c a d \\ + a^{3} b c^{2} a b c a c a d a^{2} b c a c a d &\rightarrow c a b c a c a d a^{2} c a^{2} b c^{2} a c a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/379.md b/docs/examples/1-relation/379.md new file mode 100644 index 0000000..069d081 --- /dev/null +++ b/docs/examples/1-relation/379.md @@ -0,0 +1,37 @@ +#\#379 baaaabaaba=aabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a^{2} b a = a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{2} d c = a d,\; c = b a,\; d = a c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c a &\rightarrow d \\ + b d &\rightarrow c^{2} a \\ + a^{2} d &\rightarrow d a d c \\ + a c d &\rightarrow d c a \\ + a d^{2} a d c^{2} &\rightarrow c d a d c a d \\ + c d a d c^{2} &\rightarrow a d \\ + b c d a d c a d &\rightarrow c d^{2} a d c^{2} \\ + a c^{2} d a d c a d &\rightarrow d^{3} a d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/381.md b/docs/examples/1-relation/381.md new file mode 100644 index 0000000..c97fea2 --- /dev/null +++ b/docs/examples/1-relation/381.md @@ -0,0 +1,41 @@ +#\#381 babbaaaba=ababba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a^{3} b a = a b a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d b a = a c b a,\; c = b a b,\; d = b a^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + b a^{3} &\rightarrow d \\ + a c b a &\rightarrow c d b a \\ + b a d &\rightarrow c a^{3} \\ + b a c &\rightarrow c a b \\ + a c^{2} &\rightarrow c d c \\ + a c d &\rightarrow c d^{2} \\ + c a b d^{2} b a &\rightarrow d c b a \\ + c a b d^{3} &\rightarrow d c d \\ + c a b d^{2} c &\rightarrow d c^{2} \\ + b c d b a &\rightarrow c a b^{2} a \\ + b c d^{2} &\rightarrow c a b d \\ + b c d c &\rightarrow c a b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/382.md b/docs/examples/1-relation/382.md new file mode 100644 index 0000000..8addce3 --- /dev/null +++ b/docs/examples/1-relation/382.md @@ -0,0 +1,37 @@ +#\#382 aaabbababa=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b^{2} a b a b a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{3} d a = c a^{3},\; c = b a b,\; d = b c a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b c a b &\rightarrow d \\ + b a b &\rightarrow c \\ + c a^{3} &\rightarrow a^{3} d a \\ + b c a c &\rightarrow d a b \\ + b c a d &\rightarrow d c a b \\ + b a c &\rightarrow c a b \\ + b a d &\rightarrow c^{2} a b \\ + b a^{3} d a^{2} d a &\rightarrow d a b a^{3} \\ + b a^{4} d a &\rightarrow c a b a^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/383.md b/docs/examples/1-relation/383.md new file mode 100644 index 0000000..eadc12f --- /dev/null +++ b/docs/examples/1-relation/383.md @@ -0,0 +1,36 @@ +#\#383 bababbbbba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b^{5} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d c = a c,\; c = b a,\; d = b^{4} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{4} &\rightarrow d \\ + b a &\rightarrow c \\ + a c &\rightarrow c^{2} d c \\ + b^{3} c &\rightarrow d a \\ + b d &\rightarrow d b \\ + b c^{2} &\rightarrow d c^{2} d c d c d c \\ + d c^{2} d c d c d c d c &\rightarrow c^{2} \\ + d c^{2} d c d c d c^{3} &\rightarrow c^{3} d c d c d c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/384.md b/docs/examples/1-relation/384.md new file mode 100644 index 0000000..d8ba2be --- /dev/null +++ b/docs/examples/1-relation/384.md @@ -0,0 +1,36 @@ +#\#384 abaabaaaba=baabaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a^{3} b a = b a^{2} b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d c = c d,\; c = b a,\; d = a c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c a^{2} &\rightarrow d \\ + a c d c &\rightarrow c d \\ + b d &\rightarrow c^{2} a^{2} \\ + b c d &\rightarrow c^{2} d c \\ + a c a d &\rightarrow d c a^{2} \\ + a c a c d &\rightarrow d c d c \\ + a c^{2} d &\rightarrow d c d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/385.md b/docs/examples/1-relation/385.md new file mode 100644 index 0000000..5289c8b --- /dev/null +++ b/docs/examples/1-relation/385.md @@ -0,0 +1,42 @@ +#\#385 bbaaabbbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{3} b^{4} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b^{2} c d b a = c,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b^{2} a^{2} d b a &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + b^{2} a^{2} d b d &\rightarrow a d \\ + a b^{2} a^{2} &\rightarrow d b a^{2} d b a \\ + a b a^{2} &\rightarrow d a^{2} d b a \\ + a b^{2} a d &\rightarrow d b a^{2} d b d \\ + d b a^{2} d b a d b d &\rightarrow a^{2} d \\ + d b a^{2} d b a d b a &\rightarrow a^{3} \\ + a b a d &\rightarrow d a^{2} d b d \\ + b^{2} a^{4} d &\rightarrow a^{3} d b a d b d \\ + b^{2} a^{5} &\rightarrow a^{3} d b a d b a \\ + d b a^{2} d b a^{3} d &\rightarrow a^{4} d b a d b d \\ + d b a^{2} d b a^{4} &\rightarrow a^{4} d b a d b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/386.md b/docs/examples/1-relation/386.md new file mode 100644 index 0000000..626f6da --- /dev/null +++ b/docs/examples/1-relation/386.md @@ -0,0 +1,46 @@ +#\#386 baabaaabaa=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{3} b a^{2} = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d c a = a c,\; c = b a,\; d = c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a^{2} &\rightarrow d \\ + c a d c a &\rightarrow a c \\ + a c a &\rightarrow c a d^{2} \\ + c a d a c &\rightarrow a c d c a \\ + b c a d^{2} &\rightarrow c^{2} a \\ + a^{2} c &\rightarrow c a d^{3} c a \\ + c a d a d &\rightarrow a c d^{2} a \\ + a c^{2} a d^{2} &\rightarrow c a d^{2} c a \\ + c a d^{2} a &\rightarrow a d \\ + c a d c^{2} a d^{2} &\rightarrow a c^{2} a \\ + c^{2} a d^{4} &\rightarrow d c a \\ + a^{2} d &\rightarrow c a d^{4} a \\ + c a d^{2} c a d^{2} &\rightarrow a d c a \\ + c^{2} a d^{3} c a &\rightarrow d c \\ + c^{2} a d^{3} a d &\rightarrow d c d^{2} a \\ + c^{2} a d^{3} a c &\rightarrow d c d c a \\ + c^{2} a d^{3} c^{2} a d^{2} &\rightarrow d c^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/388.md b/docs/examples/1-relation/388.md new file mode 100644 index 0000000..a3ac17c --- /dev/null +++ b/docs/examples/1-relation/388.md @@ -0,0 +1,35 @@ +#\#388 babaabaaba=aabbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a^{2} b a = a^{2} b^{2} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a b c^{2} a = c b d,\; c = a^{2} b,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a^{2} b &\rightarrow c \\ + a b a &\rightarrow d \\ + a d &\rightarrow c a \\ + a b c &\rightarrow d a b \\ + a b d &\rightarrow d b a \\ + c b d &\rightarrow b d^{3} \\ + a b^{2} d^{3} &\rightarrow d a b^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/389.md b/docs/examples/1-relation/389.md new file mode 100644 index 0000000..17fb63b --- /dev/null +++ b/docs/examples/1-relation/389.md @@ -0,0 +1,37 @@ +#\#389 baaababaa=aababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b a^{2} = a^{2} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a = d,\; c = b a b,\; d = a^{2} c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + a^{2} c a &\rightarrow d \\ + b a d &\rightarrow c a d a \\ + b a c &\rightarrow c a b \\ + a^{2} c d &\rightarrow d a c a \\ + c a d a d &\rightarrow d a c a \\ + c a d a^{2} &\rightarrow d \\ + c a d^{2} &\rightarrow d c a \\ + a^{2} d &\rightarrow d^{2} a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/390.md b/docs/examples/1-relation/390.md new file mode 100644 index 0000000..35b8905 --- /dev/null +++ b/docs/examples/1-relation/390.md @@ -0,0 +1,80 @@ +#\#390 abaaabbaba=babaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b^{2} a b a = b a b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c b^{2} a b a = b a c b a,\; c = b a^{3},\; d = a b a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a c b^{2} a b a &\rightarrow b a c b a \\ + a b a c &\rightarrow d \\ + b a^{3} &\rightarrow c \\ + a b^{2} a c b a &\rightarrow d b^{2} a b a \\ + b a^{2} d &\rightarrow c b a c \\ + b a d b a &\rightarrow c^{2} b^{2} a b a \\ + a c b^{2} a c &\rightarrow b a c^{2} \\ + b a c b a c &\rightarrow a c b^{2} d \\ + a c b^{2} a b d &\rightarrow b a c b d \\ + a c b^{2} d b^{2} a b a &\rightarrow b^{2} a c^{2} b a \\ + a^{2} c b^{2} d &\rightarrow d b a c \\ + a c b a c b^{2} d &\rightarrow b a c^{2} b a c \\ + b^{2} a c^{2} a c b^{2} d &\rightarrow a c b^{2} d b^{2} d b a c \\ + b^{2} a c^{2} b^{2} a c b a &\rightarrow a c b^{2} d b^{2} d b^{2} a b a \\ + a c b^{3} a c^{2} &\rightarrow b a c^{2} b^{2} a c \\ + a c b^{3} a c b a &\rightarrow b a c^{2} b^{2} a b a \\ + a c b^{3} a c b d &\rightarrow b a c^{2} b^{2} a b d \\ + a c b^{4} a c^{2} b a &\rightarrow b a c^{2} b^{2} d b^{2} a b a \\ + a b^{2} a c^{2} &\rightarrow d b^{2} a c \\ + a b^{2} a c b d &\rightarrow d b^{2} a b d \\ + a b^{3} a c^{3} &\rightarrow d b^{2} d b^{2} a c \\ + a b^{3} a c^{2} b a &\rightarrow d b^{2} d b^{2} a b a \\ + a b^{3} a c^{2} b d &\rightarrow d b^{2} d b^{2} a b d \\ + b a d c &\rightarrow c^{2} b^{2} a c \\ + b a d a c b^{2} d &\rightarrow c^{2} b^{2} d b a c \\ + b a d b^{2} a c^{2} &\rightarrow c^{2} b^{2} d b^{2} a c \\ + b a d b^{2} a c b a &\rightarrow c^{2} b^{2} d b^{2} a b a \\ + b a d b^{2} a c b d &\rightarrow c^{2} b^{2} d b^{2} a b d \\ + b a d b^{3} a c^{2} b a &\rightarrow c^{2} b^{2} d b^{2} d b^{2} a b a \\ + b a d b d &\rightarrow c^{2} b^{2} a b d \\ + b a c a c b^{2} d &\rightarrow a c b^{2} d b a c \\ + a c b^{2} d b^{2} a c &\rightarrow b^{2} a c^{3} \\ + a c b^{2} d b^{2} a b d &\rightarrow b^{2} a c^{2} b d \\ + b^{2} a c^{2} b a c &\rightarrow a c b^{2} d b^{2} d \\ + b^{2} a c^{2} b^{3} a c^{2} b a &\rightarrow a c b^{2} d b^{2} d b^{2} d b^{2} a b a \\ + b^{2} a c^{2} b^{2} a c b d &\rightarrow a c b^{2} d b^{2} d b^{2} a b d \\ + b^{2} a c^{2} b^{2} a c^{2} &\rightarrow a c b^{2} d b^{2} d b^{2} a c \\ + a c b^{2} d b a c b^{2} d &\rightarrow b^{2} a c^{3} b a c \\ + a c b^{2} d b^{3} a c^{2} &\rightarrow b^{2} a c^{3} b^{2} a c \\ + a c b^{2} d b^{3} a c b a &\rightarrow b^{2} a c^{3} b^{2} a b a \\ + a c b^{2} d b^{3} a c b d &\rightarrow b^{2} a c^{3} b^{2} a b d \\ + a c b^{2} d b^{4} a c^{2} b a &\rightarrow b^{2} a c^{3} b^{2} d b^{2} a b a \\ + b^{2} a c^{2} b^{3} a c^{3} &\rightarrow a c b^{2} d b^{2} d b^{2} d b^{2} a c \\ + b^{2} a c^{2} b^{3} a c^{2} b d &\rightarrow a c b^{2} d b^{2} d b^{2} d b^{2} a b d \\ + a c b^{4} a c^{3} &\rightarrow b a c^{2} b^{2} d b^{2} a c \\ + a c b^{4} a c^{2} b d &\rightarrow b a c^{2} b^{2} d b^{2} a b d \\ + b a d b^{3} a c^{3} &\rightarrow c^{2} b^{2} d b^{2} d b^{2} a c \\ + b a d b^{3} a c^{2} b d &\rightarrow c^{2} b^{2} d b^{2} d b^{2} a b d \\ + a c a c b^{2} d b^{2} d &\rightarrow b a c^{3} b a c \\ + a c b^{2} d a c b^{2} d b^{2} d &\rightarrow b^{2} a c^{4} b a c \\ + a c b^{2} d b^{4} a c^{3} &\rightarrow b^{2} a c^{3} b^{2} d b^{2} a c \\ + a c b^{2} d b^{4} a c^{2} b d &\rightarrow b^{2} a c^{3} b^{2} d b^{2} a b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/391.md b/docs/examples/1-relation/391.md new file mode 100644 index 0000000..f33e63b --- /dev/null +++ b/docs/examples/1-relation/391.md @@ -0,0 +1,37 @@ +#\#391 bababaaba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a^{2} b a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c = a c^{2},\; c = b a,\; d = c^{3} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{2} &\rightarrow d c \\ + c^{3} a &\rightarrow d \\ + b d c &\rightarrow c^{3} \\ + a c d &\rightarrow d^{2} \\ + a d &\rightarrow d c^{2} a \\ + c^{3} d c &\rightarrow d c^{2} \\ + b d^{2} &\rightarrow c^{2} d \\ + c^{3} d^{2} &\rightarrow d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/392.md b/docs/examples/1-relation/392.md new file mode 100644 index 0000000..663c4a4 --- /dev/null +++ b/docs/examples/1-relation/392.md @@ -0,0 +1,38 @@ +#\#392 baababbaba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{2} a b a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c b c^{2} = a,\; c = b a,\; d = a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + b d c b^{2} d &\rightarrow d c b c^{2} \\ + a &\rightarrow b d c b c^{2} \\ + c b d c b c^{2} &\rightarrow b d \\ + b^{2} d c b c^{2} &\rightarrow c \\ + b d c b c b d &\rightarrow d \\ + b d c^{2} &\rightarrow d c b c^{3} b c^{2} \\ + b d b d &\rightarrow d c b c^{3} b^{2} d \\ + b d c b d &\rightarrow d c b c^{3} b c b d \\ + c d &\rightarrow d c b c^{4} \\ + b d^{2} &\rightarrow d c b c^{3} b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/393.md b/docs/examples/1-relation/393.md new file mode 100644 index 0000000..a16852d --- /dev/null +++ b/docs/examples/1-relation/393.md @@ -0,0 +1,56 @@ +#\#393 babaaaabba=ababaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{4} b^{2} a = a b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d c = a c^{2} a,\; c = b a,\; d = a^{3} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{3} b &\rightarrow d \\ + a c^{2} a &\rightarrow c^{2} d c \\ + b a &\rightarrow c \\ + a^{3} c &\rightarrow d a \\ + c a^{2} b &\rightarrow b d \\ + a c^{2} d &\rightarrow c^{2} d b d \\ + c^{3} a &\rightarrow b c^{2} d c \\ + a c^{4} d c &\rightarrow c^{2} d b c^{2} d c \\ + d a c a &\rightarrow c^{2} d b d b d c \\ + d a c d &\rightarrow c^{2} d b d b d b d \\ + d a c^{3} d c &\rightarrow c^{2} d b d b d b c^{2} d c \\ + a c^{4} d b d &\rightarrow c^{2} d c^{3} d \\ + a c^{4} d b c^{2} d c &\rightarrow c^{2} d c^{5} d c \\ + a c b d &\rightarrow c^{2} d c a b \\ + c a^{2} c &\rightarrow b d a \\ + b c^{2} d b d &\rightarrow c^{3} d \\ + b c^{2} d c a b &\rightarrow c^{2} b d \\ + b c^{2} d b c^{2} d c &\rightarrow c^{5} d c \\ + b c^{2} d c^{3} d &\rightarrow c^{5} d b d \\ + b c^{2} d c^{5} d c &\rightarrow c^{5} d b c^{2} d c \\ + d a b d &\rightarrow c^{2} d b d b d c a b \\ + d a c^{3} d b d &\rightarrow c^{2} d b d b d c^{3} d \\ + d a c^{3} d b c^{2} d c &\rightarrow c^{2} d b d b d c^{5} d c \\ + c^{5} d c a b &\rightarrow b c^{2} d c^{2} b d \\ + b c^{2} d c^{5} d b d &\rightarrow c^{5} d c^{3} d \\ + b c^{2} d c^{5} d b c^{2} d c &\rightarrow c^{5} d c^{5} d c \\ + b c^{2} d c a c &\rightarrow c^{2} b d a \\ + c^{5} d c a c &\rightarrow b c^{2} d c^{2} b d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/394.md b/docs/examples/1-relation/394.md new file mode 100644 index 0000000..1a21a88 --- /dev/null +++ b/docs/examples/1-relation/394.md @@ -0,0 +1,115 @@ +#\#394 baaaababa=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a b a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a = a b^{2} a^{2},\; c = b a b,\; d = a^{3} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{3} c &\rightarrow d \\ + a b^{2} a^{2} &\rightarrow b a d a \\ + b a b &\rightarrow c \\ + b a d b a d a &\rightarrow a b c a d a \\ + a b^{2} a d &\rightarrow b a d^{2} \\ + b a d a^{2} c &\rightarrow a b^{2} d \\ + b a c &\rightarrow c a b \\ + b^{2} a d a &\rightarrow c b a^{2} \\ + a c b a^{2} &\rightarrow b a d^{2} a \\ + b a d^{2} b a d a &\rightarrow a c^{2} a d a \\ + b a d b a d^{2} &\rightarrow a b c a d^{2} \\ + a b c a d b a d a &\rightarrow b a d a b c a d a \\ + a b c a d b a d^{2} &\rightarrow b a d a b c a d^{2} \\ + c a d a^{2} c &\rightarrow b a^{2} b^{2} d \\ + b^{2} a d^{2} &\rightarrow c b a d \\ + b a^{2} b c a d a &\rightarrow c a d b a d a \\ + b a d^{2} a^{2} c &\rightarrow a c b d \\ + a^{3} b a^{2} b^{2} d &\rightarrow d a d a^{2} c \\ + a^{2} b a d^{2} a &\rightarrow d b a^{2} \\ + b a d^{2} b a d^{2} &\rightarrow a c^{2} a d^{2} \\ + a c^{2} a d b a d a &\rightarrow b a d^{2} a b c a d a \\ + a c^{2} a d b a d^{2} &\rightarrow b a d^{2} a b c a d^{2} \\ + a c b a d &\rightarrow b a d^{3} \\ + a c a b c a d a &\rightarrow b a d^{3} b a d a \\ + a b c a d^{2} b a d a &\rightarrow b a d a c^{2} a d a \\ + c a d a^{2} b a^{2} b^{2} d &\rightarrow b a^{2} b^{2} d a d a^{2} c \\ + c a d a b a d^{2} a &\rightarrow b a^{2} b^{2} d b a^{2} \\ + c a d^{2} a^{2} c &\rightarrow b a^{2} c b d \\ + b a^{2} b c a d^{2} &\rightarrow c a d b a d^{2} \\ + b a^{2} c^{2} a d a &\rightarrow c a d^{2} b a d a \\ + b a d a^{2} b a^{2} b^{2} d &\rightarrow a b^{2} d a d a^{2} c \\ + b a d a b a d^{2} a &\rightarrow a b^{2} d b a^{2} \\ + b a d^{2} a^{2} b a^{2} b^{2} d &\rightarrow a c b d a d a^{2} c \\ + b a d^{2} a b a d^{2} a &\rightarrow a c b d b a^{2} \\ + b a d^{3} a^{2} c &\rightarrow a c a b^{2} d \\ + a^{2} b a d^{3} &\rightarrow d b a d \\ + a c a c^{2} a d a &\rightarrow b a d^{4} b a d a \\ + a c a c^{2} a d^{2} &\rightarrow b a d^{4} b a d^{2} \\ + a c a b c a d^{2} &\rightarrow b a d^{3} b a d^{2} \\ + a c^{2} a d^{2} b a d a &\rightarrow b a d^{2} a c^{2} a d a \\ + a c^{2} a d^{2} b a d^{2} &\rightarrow b a d^{2} a c^{2} a d^{2} \\ + a b c a d^{2} b a d^{2} &\rightarrow b a d a c^{2} a d^{2} \\ + c a d a b a d^{3} &\rightarrow b a^{2} b^{2} d b a d \\ + b a^{2} c^{2} a d^{2} &\rightarrow c a d^{2} b a d^{2} \\ + b a d a b a d^{3} &\rightarrow a b^{2} d b a d \\ + b a d^{2} a b a d^{3} &\rightarrow a c b d b a d \\ + b a d^{4} a^{2} c &\rightarrow a c a c b d \\ + a^{3} b a^{2} c b d &\rightarrow d a d^{2} a^{2} c \\ + c a d a^{2} b a^{2} c b d &\rightarrow b a^{2} b^{2} d a d^{2} a^{2} c \\ + c a d^{2} a^{2} b a^{2} c b d &\rightarrow b a^{2} c b d a d^{2} a^{2} c \\ + c a d^{2} a^{2} b a^{2} b^{2} d &\rightarrow b a^{2} c b d a d a^{2} c \\ + c a d^{2} a b a d^{2} a &\rightarrow b a^{2} c b d b a^{2} \\ + c a d^{2} a b a d^{3} &\rightarrow b a^{2} c b d b a d \\ + b a d a^{2} b a^{2} c b d &\rightarrow a b^{2} d a d^{2} a^{2} c \\ + b a d^{2} a^{2} b a^{2} c b d &\rightarrow a c b d a d^{2} a^{2} c \\ + b a d^{3} a^{2} b a^{2} b^{2} d &\rightarrow a c a b^{2} d a d a^{2} c \\ + b a d^{3} a b a d^{2} a &\rightarrow a c a b^{2} d b a^{2} \\ + b a d^{4} a^{2} b a^{2} b^{2} d &\rightarrow a c a c b d a d a^{2} c \\ + a c a c a b^{2} d &\rightarrow b a d^{5} a^{2} c \\ + a c^{2} a d^{3} a^{2} c &\rightarrow b a d^{2} a c a b^{2} d \\ + a b c a d^{3} a^{2} c &\rightarrow b a d a c a b^{2} d \\ + b a^{2} c a b^{2} d &\rightarrow c a d^{3} a^{2} c \\ + b a d^{3} a^{2} b a^{2} c b d &\rightarrow a c a b^{2} d a d^{2} a^{2} c \\ + b a d^{3} a b a d^{3} &\rightarrow a c a b^{2} d b a d \\ + b a d^{4} a b a d^{2} a &\rightarrow a c a c b d b a^{2} \\ + a c a c a c b d &\rightarrow b a d^{6} a^{2} c \\ + a c^{2} a d^{4} a^{2} c &\rightarrow b a d^{2} a c a c b d \\ + a b c a d^{4} a^{2} c &\rightarrow b a d a c a c b d \\ + b a^{2} c a c b d &\rightarrow c a d^{4} a^{2} c \\ + b a d^{4} a^{2} b a^{2} c b d &\rightarrow a c a c b d a d^{2} a^{2} c \\ + b a d^{4} a b a d^{3} &\rightarrow a c a c b d b a d \\ + a c^{2} a d^{3} a^{2} b a^{2} b^{2} d &\rightarrow b a d^{2} a c a b^{2} d a d a^{2} c \\ + a c^{2} a d^{3} a b a d^{2} a &\rightarrow b a d^{2} a c a b^{2} d b a^{2} \\ + a b c a d^{3} a^{2} b a^{2} b^{2} d &\rightarrow b a d a c a b^{2} d a d a^{2} c \\ + a b c a d^{3} a b a d^{2} a &\rightarrow b a d a c a b^{2} d b a^{2} \\ + a c^{2} a d^{3} a^{2} b a^{2} c b d &\rightarrow b a d^{2} a c a b^{2} d a d^{2} a^{2} c \\ + a c^{2} a d^{3} a b a d^{3} &\rightarrow b a d^{2} a c a b^{2} d b a d \\ + a c^{2} a d^{4} a^{2} b a^{2} b^{2} d &\rightarrow b a d^{2} a c a c b d a d a^{2} c \\ + a b c a d^{3} a^{2} b a^{2} c b d &\rightarrow b a d a c a b^{2} d a d^{2} a^{2} c \\ + a b c a d^{3} a b a d^{3} &\rightarrow b a d a c a b^{2} d b a d \\ + a b c a d^{4} a^{2} b a^{2} b^{2} d &\rightarrow b a d a c a c b d a d a^{2} c \\ + a c^{2} a d^{4} a b a d^{2} a &\rightarrow b a d^{2} a c a c b d b a^{2} \\ + a b c a d^{4} a b a d^{2} a &\rightarrow b a d a c a c b d b a^{2} \\ + a c^{2} a d^{4} a^{2} b a^{2} c b d &\rightarrow b a d^{2} a c a c b d a d^{2} a^{2} c \\ + a c^{2} a d^{4} a b a d^{3} &\rightarrow b a d^{2} a c a c b d b a d \\ + a b c a d^{4} a^{2} b a^{2} c b d &\rightarrow b a d a c a c b d a d^{2} a^{2} c \\ + a b c a d^{4} a b a d^{3} &\rightarrow b a d a c a c b d b a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/395.md b/docs/examples/1-relation/395.md new file mode 100644 index 0000000..c96728b --- /dev/null +++ b/docs/examples/1-relation/395.md @@ -0,0 +1,42 @@ +#\#395 baaababba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c a^{2} d = a^{2},\; c = b a,\; d = c b c,\; e = d a d^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(e < d < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c b c &\rightarrow d \\ + d a d^{2} &\rightarrow e \\ + c^{2} a &\rightarrow d a^{2} d \\ + c b d &\rightarrow d b c \\ + d a d e &\rightarrow e a d^{2} \\ + c b e &\rightarrow d b c a d^{2} \\ + c a^{2} &\rightarrow d a^{2} d a d \\ + d a^{2} e &\rightarrow a^{2} \\ + c d a^{2} d a d &\rightarrow d a^{2} d a \\ + d a d a^{2} &\rightarrow e a^{2} e \\ + c d a^{2} d a e &\rightarrow d a^{2} d a^{2} d^{2} \\ + d a^{3} &\rightarrow e a^{2} e^{2} \\ + d a e a^{2} e^{2} &\rightarrow e a^{2} e a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/396.md b/docs/examples/1-relation/396.md new file mode 100644 index 0000000..b43c1be --- /dev/null +++ b/docs/examples/1-relation/396.md @@ -0,0 +1,33 @@ +#\#396 abaaaababa=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{4} b a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a^{3} c^{2} = c^{2} a,\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + d b a &\rightarrow a b d \\ + b d^{2} &\rightarrow d a^{2} d b d \\ + b d a &\rightarrow d a^{3} b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/397.md b/docs/examples/1-relation/397.md new file mode 100644 index 0000000..c454d47 --- /dev/null +++ b/docs/examples/1-relation/397.md @@ -0,0 +1,52 @@ +#\#397 abbabbbbaa=bbaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{4} a^{2} = b^{2} a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e = d c a,\; c = b^{2},\; d = c a^{2},\; e = c a c d +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < e < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + c a c d &\rightarrow e \\ + c a^{2} &\rightarrow d \\ + d c a &\rightarrow a e \\ + b c &\rightarrow c b \\ + c a c a e &\rightarrow e c a \\ + a e c d &\rightarrow d e \\ + a e a &\rightarrow d^{2} \\ + c b a c a e &\rightarrow b e c a \\ + c b a c d &\rightarrow b e \\ + c b a^{2} &\rightarrow b d \\ + c a d e &\rightarrow d e c d \\ + c a d^{2} &\rightarrow d e a \\ + a e c a e &\rightarrow d e c a \\ + d c d e &\rightarrow a e^{2} c d \\ + d c d^{2} &\rightarrow a e^{2} a \\ + d^{2} e c d &\rightarrow a e d e \\ + d^{2} e a &\rightarrow a e d^{2} \\ + b d e c a e &\rightarrow c b a d e c a \\ + b d e c d &\rightarrow c b a d e \\ + b d e a &\rightarrow c b a d^{2} \\ + c a d a e &\rightarrow d e a c a \\ + d c d a e &\rightarrow a e^{2} a c a \\ + d^{2} e c a e &\rightarrow a e d e c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/399.md b/docs/examples/1-relation/399.md new file mode 100644 index 0000000..9bb0ea4 --- /dev/null +++ b/docs/examples/1-relation/399.md @@ -0,0 +1,46 @@ +#\#399 ababaaabba=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{3} b^{2} a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a c^{2} a^{2} b c = c^{2} a^{2},\; c = b a,\; d = a b^{2},\; e = c^{2} a d +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c^{2} a d &\rightarrow e \\ + a b^{2} &\rightarrow d \\ + c b^{2} &\rightarrow b d \\ + a b c &\rightarrow d a \\ + c^{2} a^{2} &\rightarrow a e a \\ + c b c &\rightarrow b d a \\ + d a c a^{2} &\rightarrow a c e a \\ + a e d &\rightarrow e \\ + b e &\rightarrow c e d \\ + c^{2} a e &\rightarrow a e^{2} \\ + a e a c e a &\rightarrow e a c a^{2} \\ + d a c a e &\rightarrow a c e^{2} \\ + d a c a d &\rightarrow a c e d \\ + c^{2} e &\rightarrow a e^{2} d \\ + a e a c e^{2} &\rightarrow e a c a e \\ + a e a c e d &\rightarrow e a c a d \\ + d a c e &\rightarrow a c e^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/400.md b/docs/examples/1-relation/400.md new file mode 100644 index 0000000..7ec6c6b --- /dev/null +++ b/docs/examples/1-relation/400.md @@ -0,0 +1,40 @@ +#\#400 baaaabbaaa=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a^{3} = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{2} c a = a,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{2} &\rightarrow d \\ + b a^{3} d a^{3} &\rightarrow a \\ + b a^{3} d a^{2} d &\rightarrow d \\ + a b a &\rightarrow d a^{3} d a^{3} \\ + a b d &\rightarrow d a^{3} d a^{2} d \\ + d a^{3} d a^{5} d a^{2} d &\rightarrow a d \\ + d a^{3} d a^{5} d a^{3} &\rightarrow a^{2} \\ + b a^{4} d &\rightarrow a d a^{5} d a^{2} d \\ + b a^{5} &\rightarrow a d a^{5} d a^{3} \\ + d a^{3} d a^{6} d &\rightarrow a^{2} d a^{5} d a^{2} d \\ + d a^{3} d a^{7} &\rightarrow a^{2} d a^{5} d a^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/403.md b/docs/examples/1-relation/403.md new file mode 100644 index 0000000..f815f84 --- /dev/null +++ b/docs/examples/1-relation/403.md @@ -0,0 +1,36 @@ +#\#403 babaaaaaba=aaaababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{5} b a = a^{4} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a^{3} c a = a^{3} c^{2} a,\; c = a b,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b c a &\rightarrow d \\ + d a^{3} c a &\rightarrow a^{3} c^{2} a \\ + a b &\rightarrow c \\ + b c^{2} &\rightarrow d b \\ + a d &\rightarrow c^{2} a \\ + d a^{3} c^{2} &\rightarrow a^{3} c^{3} \\ + a^{4} c^{2} a &\rightarrow c^{2} a^{4} c a \\ + a^{4} c^{3} &\rightarrow c^{2} a^{4} c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/404.md b/docs/examples/1-relation/404.md new file mode 100644 index 0000000..fb43195 --- /dev/null +++ b/docs/examples/1-relation/404.md @@ -0,0 +1,35 @@ +#\#404 abaaabaaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b a^{2} b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a^{2} c a c = c^{2} a,\; c = b a,\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b a &\rightarrow d \\ + b d a &\rightarrow d a d^{2} \\ + b d^{2} &\rightarrow d a d^{2} b a \\ + a b d &\rightarrow d b a \\ + d b a d &\rightarrow a d a d^{2} b a \\ + d b a^{2} &\rightarrow a d a d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/405.md b/docs/examples/1-relation/405.md new file mode 100644 index 0000000..1bba6de --- /dev/null +++ b/docs/examples/1-relation/405.md @@ -0,0 +1,38 @@ +#\#405 aabaabbaaa=baaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b^{2} a^{3} = b a^{3} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} d = c a c,\; c = b a^{2},\; d = c b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a^{2} &\rightarrow c \\ + a^{2} d &\rightarrow c a c \\ + c b c a &\rightarrow d \\ + b a c a c &\rightarrow c a d \\ + b c a c &\rightarrow c d \\ + c b c^{2} a c &\rightarrow d a d \\ + b a c a d &\rightarrow c a d b c a \\ + b c a d &\rightarrow c d b c a \\ + c b c^{2} a d &\rightarrow d a d b c a \\ + c^{2} d &\rightarrow d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/407.md b/docs/examples/1-relation/407.md new file mode 100644 index 0000000..87bf911 --- /dev/null +++ b/docs/examples/1-relation/407.md @@ -0,0 +1,42 @@ +#\#407 abbabbabaa=baabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{2} a b a^{2} = b a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a c d a = b a^{2} c a,\; c = b^{2},\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a^{2} c a &\rightarrow a c a c d a \\ + a b a &\rightarrow d \\ + b c &\rightarrow c b \\ + b a c a c d a &\rightarrow c a^{2} c a \\ + b a^{2} c d &\rightarrow a c a c d^{2} \\ + a b d &\rightarrow d b a \\ + a^{2} c a c d a &\rightarrow d a c a \\ + b d a c a &\rightarrow a c a c d a c d a \\ + b a c a c d^{2} &\rightarrow c a^{2} c d \\ + d c a c d a &\rightarrow a c a^{2} c a \\ + a^{2} c a c d^{2} &\rightarrow d a c d \\ + b d a c d &\rightarrow a c a c d a c d^{2} \\ + d c a c d^{2} &\rightarrow a c a^{2} c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/408.md b/docs/examples/1-relation/408.md new file mode 100644 index 0000000..f010526 --- /dev/null +++ b/docs/examples/1-relation/408.md @@ -0,0 +1,34 @@ +#\#408 babaababaa=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a b a^{2} = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c d = a c,\; c = b a,\; d = c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c &\rightarrow c d c d \\ + c a &\rightarrow d \\ + b c d c d &\rightarrow c^{2} \\ + a d &\rightarrow c d c d a \\ + c^{2} d c d &\rightarrow d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/409.md b/docs/examples/1-relation/409.md new file mode 100644 index 0000000..587d000 --- /dev/null +++ b/docs/examples/1-relation/409.md @@ -0,0 +1,54 @@ +#\#409 aaababbaaa=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a b^{2} a^{3} = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} d a = b a^{2} c a,\; c = a b,\; d = c^{2} b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a^{2} c a &\rightarrow a^{2} d a \\ + a b &\rightarrow c \\ + c^{2} b a^{2} &\rightarrow d \\ + b a^{2} c^{2} &\rightarrow a^{2} d c \\ + a^{3} d a &\rightarrow c a^{2} c a \\ + c^{2} b a c &\rightarrow d b \\ + c^{2} a^{2} d a &\rightarrow d c a \\ + b a^{2} d &\rightarrow a^{2} d c b a^{2} \\ + b a^{2} c d &\rightarrow a^{2} d^{2} \\ + a^{3} d c &\rightarrow c a^{2} c^{2} \\ + d b c b a^{2} &\rightarrow c^{2} b a d \\ + d b c b a c &\rightarrow c^{2} b a d b \\ + d b c a^{2} d a &\rightarrow c^{2} b a d c a \\ + c^{2} a^{2} d c &\rightarrow d c^{2} \\ + c^{2} b c a^{2} c a &\rightarrow d a d a \\ + a^{3} d^{2} &\rightarrow c a^{2} c d \\ + d b c b c a^{2} c a &\rightarrow c^{2} b a d a d a \\ + d b c b c a^{2} c^{2} &\rightarrow c^{2} b a d a d c \\ + d b c a^{2} d^{2} &\rightarrow c^{2} b a d c d \\ + d b c a^{2} d c &\rightarrow c^{2} b a d c^{2} \\ + c^{2} a^{2} d^{2} &\rightarrow d c d \\ + c^{2} b c a^{2} c^{2} &\rightarrow d a d c \\ + d b c b c a^{2} d &\rightarrow c^{2} b a d a d c b a^{2} \\ + d b c b c a^{2} c d &\rightarrow c^{2} b a d a d^{2} \\ + c^{2} b c a^{2} d &\rightarrow d a d c b a^{2} \\ + c^{2} b c a^{2} c d &\rightarrow d a d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/410.md b/docs/examples/1-relation/410.md new file mode 100644 index 0000000..79ea441 --- /dev/null +++ b/docs/examples/1-relation/410.md @@ -0,0 +1,36 @@ +#\#410 baaaaaabba=abaaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{6} b^{2} a = a b a^{5} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d = a c,\; c = b a^{5},\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{2} a &\rightarrow d \\ + b a^{5} &\rightarrow c \\ + a c &\rightarrow c a d \\ + b c &\rightarrow d a^{4} \\ + d a^{5} d a d a d a d a d &\rightarrow c^{2} \\ + b d a^{5} d &\rightarrow d c \\ + b d c a d a d a d a d a d c &\rightarrow d c a^{5} d a d a d a d a d \\ + d a^{5} d a d a d a d c a d c &\rightarrow c^{2} a^{5} d a d a d a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/411.md b/docs/examples/1-relation/411.md new file mode 100644 index 0000000..e46d65e --- /dev/null +++ b/docs/examples/1-relation/411.md @@ -0,0 +1,39 @@ +#\#411 aabaabbaba=babaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b^{2} a b a = b a b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d e = c^{2} d,\; c = b a,\; d = a c a,\; e = b c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c^{2} &\rightarrow e \\ + a d e &\rightarrow c^{2} d \\ + a c a &\rightarrow d \\ + b d &\rightarrow c^{2} a \\ + c d e &\rightarrow e d \\ + a c d &\rightarrow d c a \\ + a c^{3} d &\rightarrow d^{2} e \\ + b c e d &\rightarrow e d e \\ + a e d &\rightarrow d c a e \\ + a c^{2} e d &\rightarrow d^{2} e^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/412.md b/docs/examples/1-relation/412.md new file mode 100644 index 0000000..255c16f --- /dev/null +++ b/docs/examples/1-relation/412.md @@ -0,0 +1,46 @@ +#\#412 abaababbaa=baabaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b^{2} a^{2} = b a^{2} b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d = c^{2} b a,\; c = b a^{2},\; d = b a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b a b c &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + c^{2} b a &\rightarrow a c d \\ + b a b a c d &\rightarrow d c b a \\ + a c d b c &\rightarrow c^{2} d \\ + a c d a &\rightarrow c^{3} \\ + b a c^{2} d &\rightarrow c^{2} d b c \\ + b a c^{3} &\rightarrow c^{2} d a \\ + a c d b a c d &\rightarrow c^{2} d c b a \\ + c^{2} b c^{2} d &\rightarrow a c d c d b c \\ + c^{2} b c^{3} &\rightarrow a c d c d a \\ + c^{4} d b c &\rightarrow a c d c^{2} d \\ + c^{4} d a &\rightarrow a c d c^{3} \\ + b a c a c d &\rightarrow c^{2} d a b a \\ + b a c^{2} a c d &\rightarrow c^{2} d a c b a \\ + c^{2} b c a c d &\rightarrow a c d c d a b a \\ + c^{2} b c^{2} a c d &\rightarrow a c d c d a c b a \\ + c^{4} d b a c d &\rightarrow a c d c^{2} d c b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/413.md b/docs/examples/1-relation/413.md new file mode 100644 index 0000000..647e7d8 --- /dev/null +++ b/docs/examples/1-relation/413.md @@ -0,0 +1,44 @@ +#\#413 ababaabbaa=baababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{2} b^{2} a^{2} = b a^{2} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a c^{2} d c a = c a c^{2},\; c = b a,\; d = a b,\; e = d a d c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < e < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow d \\ + d a d c &\rightarrow e \\ + b d &\rightarrow c b \\ + a c &\rightarrow d a \\ + c d^{3} &\rightarrow d e d \\ + c d^{2} a &\rightarrow d e a \\ + b e &\rightarrow c^{2} d c \\ + c d e &\rightarrow d e a d c \\ + c d^{2} e &\rightarrow d e^{2} \\ + a d e a &\rightarrow d a d^{2} a \\ + a d e d &\rightarrow d a d^{3} \\ + d a d^{2} e a &\rightarrow e d^{2} a \\ + d a d^{2} e d &\rightarrow e d^{3} \\ + a d e^{2} &\rightarrow d a d^{2} e \\ + d a d^{2} e^{2} &\rightarrow e d^{2} e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/415.md b/docs/examples/1-relation/415.md new file mode 100644 index 0000000..9b1b156 --- /dev/null +++ b/docs/examples/1-relation/415.md @@ -0,0 +1,39 @@ +#\#415 bbabaaabba=abbabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a b a^{3} b^{2} a = a b^{2} a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} a^{2} b c = a b c^{2} a,\; c = b a,\; d = b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < a < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b a^{2} &\rightarrow d \\ + b^{2} a d a b^{2} a &\rightarrow a b^{2} a d \\ + a b^{2} a d a &\rightarrow b^{2} a d a b d \\ + a^{2} b^{2} a d &\rightarrow b^{2} a d a b d b^{2} a \\ + b^{3} a d a b d b d &\rightarrow d b^{2} a d a \\ + b^{2} a d a b^{4} a d a b d &\rightarrow a b^{2} a d b^{2} a d a \\ + b^{2} a d b^{2} a d a b d &\rightarrow a b^{2} a d^{2} a \\ + b^{3} a d a b d b^{2} a &\rightarrow d b^{2} a d \\ + b^{3} a d a b d a b^{2} a d &\rightarrow d b^{2} a d^{2} a b^{2} a \\ + b^{3} a d a b d b^{4} a d a b d &\rightarrow d b^{2} a d b^{2} a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/416.md b/docs/examples/1-relation/416.md new file mode 100644 index 0000000..659817a --- /dev/null +++ b/docs/examples/1-relation/416.md @@ -0,0 +1,41 @@ +#\#416 baaaababaa=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a b a^{2} = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b a b c = c,\; c = a^{2},\; d = a b a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b a b &\rightarrow d \\ + b a^{3} d a^{2} &\rightarrow a^{2} \\ + a b d &\rightarrow d a b \\ + a b a^{3} &\rightarrow d a^{3} d a^{2} \\ + b a^{3} d a d &\rightarrow a d \\ + a b a^{2} d &\rightarrow d a^{3} d a d \\ + d a^{3} d a^{2} d a d &\rightarrow a^{2} d \\ + d a^{3} d a^{2} d a^{2} &\rightarrow a^{3} \\ + b a^{5} d &\rightarrow a^{3} d a^{2} d a d \\ + b a^{6} &\rightarrow a^{3} d a^{2} d a^{2} \\ + d a^{3} d a^{4} d &\rightarrow a^{4} d a^{2} d a d \\ + d a^{3} d a^{5} &\rightarrow a^{4} d a^{2} d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/417.md b/docs/examples/1-relation/417.md new file mode 100644 index 0000000..07aec7c --- /dev/null +++ b/docs/examples/1-relation/417.md @@ -0,0 +1,36 @@ +#\#417 baaababbaa=aaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b^{2} a^{2} = a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{2} c d = a^{3} c,\; c = b a,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow d \\ + c a^{2} c &\rightarrow d a c d \\ + b c d a c d^{2} &\rightarrow d a^{2} c \\ + a^{3} c &\rightarrow d a c d^{2} \\ + b d a c d &\rightarrow d a c \\ + a^{3} d a c d &\rightarrow d a c d^{2} a^{2} c \\ + c a^{2} d a c d &\rightarrow d a c d a^{2} c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/418.md b/docs/examples/1-relation/418.md new file mode 100644 index 0000000..e4f35a1 --- /dev/null +++ b/docs/examples/1-relation/418.md @@ -0,0 +1,32 @@ +#\#418 abaababbaa=babaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b^{2} a^{2} = b a b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c a c b c a = c^{2} a c,\; c = b a,\; d = b^{2} a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + b^{2} a^{2} &\rightarrow d \\ + b a b a^{2} b a &\rightarrow a b a^{2} b a d \\ + b a b a^{3} b a^{2} b a d &\rightarrow a b a^{2} b a d b a^{2} b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/419.md b/docs/examples/1-relation/419.md new file mode 100644 index 0000000..93df609 --- /dev/null +++ b/docs/examples/1-relation/419.md @@ -0,0 +1,31 @@ +#\#419 aababababa=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b a b a b a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c^{4} = c^{2} a^{2},\; c = d,\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow d \\ + b a &\rightarrow d \\ + d^{2} a^{2} &\rightarrow a^{2} d^{4} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/421.md b/docs/examples/1-relation/421.md new file mode 100644 index 0000000..c53807f --- /dev/null +++ b/docs/examples/1-relation/421.md @@ -0,0 +1,43 @@ +#\#421 abaababa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a b a = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d c = d,\; c = b a,\; d = c a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a c &\rightarrow d \\ + a d c &\rightarrow d \\ + d a c &\rightarrow a d^{2} \\ + c a d &\rightarrow a d^{2} \\ + c d &\rightarrow a d^{2} c \\ + c a^{2} d^{2} &\rightarrow d a d \\ + a d a d^{2} &\rightarrow d a d \\ + a d^{2} a d &\rightarrow d a^{2} d^{2} \\ + b d &\rightarrow a d^{2} c^{2} \\ + a d^{2} a^{2} d^{2} &\rightarrow d a d a d \\ + d a^{2} d^{2} c &\rightarrow a d^{3} \\ + d a^{2} d^{3} a d &\rightarrow a d^{3} a^{2} d^{2} \\ + d a d a^{2} d^{3} &\rightarrow a d^{3} a d \\ + d a^{2} d^{3} a^{2} d^{2} &\rightarrow a d^{3} a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/422.md b/docs/examples/1-relation/422.md new file mode 100644 index 0000000..25a88e4 --- /dev/null +++ b/docs/examples/1-relation/422.md @@ -0,0 +1,67 @@ +#\#422 aabbbaaba=baabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{3} a^{2} b a = b a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} b d = c b c,\; c = b a^{2},\; d = b c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c b a &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a^{2} b d &\rightarrow c b c \\ + b c b c &\rightarrow c b d \\ + b c^{2} &\rightarrow d a \\ + b a c b c &\rightarrow c a b d \\ + c b d b c &\rightarrow d a b d \\ + b c b d a b d &\rightarrow c b d b d b c \\ + b c d a b d &\rightarrow d a b d b c \\ + b a c b d a b d &\rightarrow c a b d b d b c \\ + b a c^{2} b d &\rightarrow c a b d b c \\ + d a b d b a &\rightarrow c b d^{2} \\ + c b d b d a b d &\rightarrow d a b d b d b c \\ + c b d b a &\rightarrow b c d \\ + c b d^{2} a &\rightarrow d a b d c \\ + c b d c &\rightarrow b c d a \\ + c a b d b a &\rightarrow b a c d \\ + c a b d c &\rightarrow b a c d a \\ + b c b^{2} c d &\rightarrow c b d b d b a \\ + b c b^{2} a c d &\rightarrow c b c b d^{2} \\ + b a c b^{2} c d &\rightarrow c a b d b d b a \\ + b a c b^{2} a c d &\rightarrow c a b c b d^{2} \\ + b a c d a b d b d a b d &\rightarrow c a b d^{2} a b d b d b c \\ + b a c d a b d b c &\rightarrow c a b d^{2} a b d \\ + b a c d a b d^{2} a &\rightarrow c a b d^{2} a b d c \\ + b a c d a b d c &\rightarrow c a b d b c d a \\ + c b d b d a b c b d^{2} &\rightarrow d a b d b d b^{2} a c d \\ + b c d a b c b d^{2} &\rightarrow d a b d b^{2} a c d \\ + d a b d c a b d &\rightarrow c b d^{2} c b c \\ + c b d b^{2} c d &\rightarrow d a b d b d b a \\ + c b d b^{2} a c d &\rightarrow d a b c b d^{2} \\ + b c b d a b c b d^{2} &\rightarrow c b d b d b^{2} a c d \\ + a^{2} b c b d^{2} &\rightarrow c b^{2} a c d \\ + b a c b d a b c b d^{2} &\rightarrow c a b d b d b^{2} a c d \\ + b a c^{2} b c b d^{2} &\rightarrow c a b d b^{2} a c d \\ + b a c d a b d b^{2} c d &\rightarrow c a b d^{2} a b d b d b a \\ + b a c d a b d b^{2} a c d &\rightarrow c a b d^{2} a b c b d^{2} \\ + b a c d a b d b d a b c b d^{2} &\rightarrow c a b d^{2} a b d b d b^{2} a c d \\ + d a b d c a b c b d^{2} &\rightarrow c b d^{2} c b^{2} a c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/423.md b/docs/examples/1-relation/423.md new file mode 100644 index 0000000..cc4ee58 --- /dev/null +++ b/docs/examples/1-relation/423.md @@ -0,0 +1,71 @@ +#\#423 baaaaabbaa=abaaaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a^{2} = a b a^{5} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c d a^{2} = a b c a^{2},\; c = a^{3},\; d = a^{2} b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a^{3} &\rightarrow c \\ + a^{2} b^{2} &\rightarrow d \\ + a b c a^{2} &\rightarrow b c d a^{2} \\ + a c &\rightarrow c a \\ + a d &\rightarrow c b^{2} \\ + a b c^{2} &\rightarrow b c d c \\ + a b c d &\rightarrow b c d^{2} \\ + c b c a^{2} &\rightarrow b c d^{3} a^{2} \\ + c b c^{2} &\rightarrow b c d^{3} c \\ + c b c d &\rightarrow b c d^{4} \\ + c b^{2} c d^{5} a^{2} &\rightarrow b c d^{3} b c d^{2} a^{2} \\ + c b^{2} c d^{5} c &\rightarrow b c d^{3} b c d^{2} c \\ + c b^{2} c d^{6} &\rightarrow b c d^{3} b c d^{3} \\ + a b^{2} c d^{6} &\rightarrow b c d b c d^{3} \\ + a b^{2} c d^{5} c &\rightarrow b c d b c d^{2} c \\ + a b^{2} c d^{5} a^{2} &\rightarrow b c d b c d^{2} a^{2} \\ + b c d^{2} b c d^{2} c &\rightarrow d c d^{5} c \\ + b c d^{2} b c d^{3} &\rightarrow d c d^{6} \\ + b c d^{3} c d^{5} c &\rightarrow d c d^{5} b c d^{2} c \\ + b c d^{3} c d^{6} &\rightarrow d c d^{5} b c d^{3} \\ + c b^{2} c d^{5} b c d^{3} &\rightarrow b c d^{4} c d^{6} \\ + c b^{2} c d^{5} b c d^{2} c &\rightarrow b c d^{4} c d^{5} c \\ + a b^{2} c d^{5} b c d^{2} c &\rightarrow b c d^{2} c d^{5} c \\ + a b^{2} c d^{5} b c d^{3} &\rightarrow b c d^{2} c d^{6} \\ + a^{2} b d c d^{5} c &\rightarrow d c d^{2} b c d^{2} c \\ + a^{2} b d c d^{6} &\rightarrow d c d^{2} b c d^{3} \\ + a^{2} b d c d^{5} b c d^{2} c &\rightarrow d c d^{3} c d^{5} c \\ + a^{2} b d c d^{5} b c d^{3} &\rightarrow d c d^{3} c d^{6} \\ + b c d^{2} b c d^{2} a^{2} &\rightarrow d c d^{5} a^{2} \\ + c d c d^{5} c &\rightarrow b c d^{5} b c d^{2} c \\ + c d c d^{6} &\rightarrow b c d^{5} b c d^{3} \\ + c d c d^{5} b c d^{2} a^{2} &\rightarrow b c d^{6} c d^{5} a^{2} \\ + c d c d^{5} b c d^{2} c &\rightarrow b c d^{6} c d^{5} c \\ + c d c d^{5} b c d^{3} &\rightarrow b c d^{6} c d^{6} \\ + b c d^{3} c d^{5} b c d^{2} c &\rightarrow d c d^{6} c d^{5} c \\ + b c d^{3} c d^{5} b c d^{3} &\rightarrow d c d^{6} c d^{6} \\ + a^{2} b d c d^{5} b c d^{2} a^{2} &\rightarrow d c d^{3} c d^{5} a^{2} \\ + b c d^{3} c d^{5} b c d^{2} a^{2} &\rightarrow d c d^{6} c d^{5} a^{2} \\ + b c d^{3} c d^{5} a^{2} &\rightarrow d c d^{5} b c d^{2} a^{2} \\ + a b^{2} c d^{5} b c d^{2} a^{2} &\rightarrow b c d^{2} c d^{5} a^{2} \\ + c b^{2} c d^{5} b c d^{2} a^{2} &\rightarrow b c d^{4} c d^{5} a^{2} \\ + a^{2} b d c d^{5} a^{2} &\rightarrow d c d^{2} b c d^{2} a^{2} \\ + c d c d^{5} a^{2} &\rightarrow b c d^{5} b c d^{2} a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/424.md b/docs/examples/1-relation/424.md new file mode 100644 index 0000000..272c7a1 --- /dev/null +++ b/docs/examples/1-relation/424.md @@ -0,0 +1,31 @@ +#\#424 aababababa=bababaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b a b a b a = b a b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c^{4} = c^{3} a^{2},\; c = d,\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow d \\ + b a &\rightarrow d \\ + d^{3} a^{2} &\rightarrow a^{2} d^{4} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/426.md b/docs/examples/1-relation/426.md new file mode 100644 index 0000000..daa93b6 --- /dev/null +++ b/docs/examples/1-relation/426.md @@ -0,0 +1,43 @@ +#\#426 bababaaaba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a b a^{3} b a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c e c = a c^{2},\; c = b a,\; d = a^{2},\; e = c^{2} d +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a^{2} &\rightarrow d \\ + a c^{2} &\rightarrow c e c \\ + c^{2} d &\rightarrow e \\ + b d &\rightarrow c a \\ + b c e c &\rightarrow c^{3} \\ + a d &\rightarrow d a \\ + a e &\rightarrow c e c d \\ + a c e &\rightarrow c e^{2} \\ + d c^{2} &\rightarrow c e^{2} c \\ + b c e^{2} &\rightarrow c^{2} e \\ + d c e &\rightarrow c e^{3} \\ + d e &\rightarrow c e^{2} c d \\ + c^{3} e^{2} c &\rightarrow e c^{2} \\ + c^{3} e^{3} &\rightarrow e c e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/427.md b/docs/examples/1-relation/427.md new file mode 100644 index 0000000..4d56836 --- /dev/null +++ b/docs/examples/1-relation/427.md @@ -0,0 +1,37 @@ +#\#427 aaabababba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{3} b a b a b^{2} a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{3} d a = c a^{2},\; c = b a b,\; d = c a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + c a b^{2} &\rightarrow d \\ + c a^{2} &\rightarrow a^{3} d a \\ + b a c &\rightarrow c a b \\ + c a b c &\rightarrow d a b \\ + b a^{4} d a &\rightarrow c a b a^{2} \\ + b a d &\rightarrow c a c b \\ + c a b a^{3} d a &\rightarrow d a b a^{2} \\ + c a b d &\rightarrow d a c b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/428.md b/docs/examples/1-relation/428.md new file mode 100644 index 0000000..be9e645 --- /dev/null +++ b/docs/examples/1-relation/428.md @@ -0,0 +1,56 @@ +#\#428 aabbaabbaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{2} a^{2} b^{2} a^{2} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} d a d a = c a c,\; c = b a,\; d = b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c &\rightarrow d \\ + b a &\rightarrow c \\ + a^{2} d a d a &\rightarrow c a c \\ + c a d a d a &\rightarrow d a c \\ + a^{2} d a d c a c &\rightarrow c a d a c \\ + b d a c &\rightarrow d a d a d a \\ + c a d a d c a c &\rightarrow d a d a c \\ + c a d a d c a d a c &\rightarrow d a d a d a c \\ + a^{2} d a d c a d a c &\rightarrow d a c^{2} \\ + b d a d a d a c &\rightarrow d a d a d c a d a c \\ + b d a d a c &\rightarrow d a d a d c a c \\ + d a d a d a d a c &\rightarrow c a d a d^{2} a c^{2} \\ + a^{2} d a d^{2} a c^{2} &\rightarrow d a c d a c \\ + a^{2} d a d^{2} a c d a d a c &\rightarrow d a c d a d a d a c \\ + a^{2} d a d^{2} a c d a c &\rightarrow d a c d a d a c \\ + d a c a d a d^{2} a c^{2} &\rightarrow c a d a d^{2} a c d a c \\ + d a c a d a d^{2} a c d a c &\rightarrow c a d a d^{2} a c d a d a c \\ + d a c a d a d^{2} a c d a d a c &\rightarrow c a d a d^{2} a c d a d a d a c \\ + d a c a d a d^{2} a c d a d a d a c &\rightarrow c a d a d^{2} a c^{2} a d a d^{2} a c^{2} \\ + c a c a d a d^{2} a c d a d a d a c &\rightarrow d a c^{2} a d a d^{2} a c d a c \\ + c a c a d a d^{2} a c d a d a c &\rightarrow d a c^{2} a d a d^{2} a c^{2} \\ + c a c a d a d^{2} a c d a c &\rightarrow d a c d a d a d a c \\ + c a c a d a d^{2} a c^{2} &\rightarrow d a c d a d a c \\ + a^{2} d a d^{2} a c d a d a d a c &\rightarrow d a c^{2} a d a d^{2} a c^{2} \\ + a^{2} c a d a d^{2} a c d a d a d a c &\rightarrow c a c^{2} a d a d^{2} a c d a c \\ + a^{2} c a d a d^{2} a c d a d a c &\rightarrow c a c^{2} a d a d^{2} a c^{2} \\ + a^{2} c a d a d^{2} a c d a c &\rightarrow c a c d a d a d a c \\ + a^{2} c a d a d^{2} a c^{2} &\rightarrow c a c d a d a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/429.md b/docs/examples/1-relation/429.md new file mode 100644 index 0000000..59610fb --- /dev/null +++ b/docs/examples/1-relation/429.md @@ -0,0 +1,40 @@ +#\#429 baaaaabba=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c a,\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{4} d a &\rightarrow a^{3} \\ + a b^{2} &\rightarrow d \\ + a b a^{3} &\rightarrow d a^{4} d a \\ + b a^{4} d^{2} &\rightarrow a^{2} d \\ + a b a^{2} d &\rightarrow d a^{4} d^{2} \\ + d a^{4} d a^{2} d^{2} &\rightarrow a^{3} d \\ + d a^{4} d a^{2} d a &\rightarrow a^{4} \\ + b a^{7} d &\rightarrow a^{6} d a^{2} d^{2} \\ + b a^{8} &\rightarrow a^{6} d a^{2} d a \\ + d a^{4} d a^{5} d &\rightarrow a^{7} d a^{2} d^{2} \\ + d a^{4} d a^{6} &\rightarrow a^{7} d a^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/430.md b/docs/examples/1-relation/430.md new file mode 100644 index 0000000..c0658c0 --- /dev/null +++ b/docs/examples/1-relation/430.md @@ -0,0 +1,45 @@ +#\#430 baabaabba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{2} a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; e c = d c,\; c = b a,\; d = a c,\; e = c d a b +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c &\rightarrow d \\ + d c &\rightarrow e c \\ + c d a b &\rightarrow e \\ + b d &\rightarrow c^{2} \\ + d^{2} a b &\rightarrow a e \\ + d e &\rightarrow e^{2} \\ + e a &\rightarrow c d^{2} \\ + b e^{2} &\rightarrow c^{2} e \\ + b e c &\rightarrow c^{3} \\ + e d &\rightarrow c e^{2} c \\ + a e^{3} &\rightarrow e^{3} c e \\ + a e^{2} c &\rightarrow e^{3} c^{2} \\ + c e^{2} c e &\rightarrow e^{3} \\ + c e^{2} c^{2} &\rightarrow e^{2} c \\ + c e^{5} &\rightarrow e^{4} c e \\ + c e^{4} c &\rightarrow e^{4} c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/431.md b/docs/examples/1-relation/431.md new file mode 100644 index 0000000..a417ebb --- /dev/null +++ b/docs/examples/1-relation/431.md @@ -0,0 +1,34 @@ +#\#431 abababbaba=babababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b^{2} a b a = b a b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c^{2} = c^{4},\; c = b a,\; d = a c^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c^{2} b &\rightarrow d \\ + d c^{2} &\rightarrow c^{4} \\ + b d &\rightarrow c^{3} b \\ + a c^{3} &\rightarrow d a \\ + b c^{4} &\rightarrow c^{3} b c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/432.md b/docs/examples/1-relation/432.md new file mode 100644 index 0000000..0796500 --- /dev/null +++ b/docs/examples/1-relation/432.md @@ -0,0 +1,39 @@ +#\#432 baaabbbaaa=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{3} a^{3} = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d = a b a,\; c = b^{3} a,\; d = a c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b^{3} a &\rightarrow c \\ + a b a &\rightarrow b a^{2} d \\ + a c a^{2} &\rightarrow d \\ + c b a &\rightarrow b c a d \\ + b^{3} d &\rightarrow c^{2} a^{2} \\ + a b^{2} a^{2} d &\rightarrow b a^{2} d b a \\ + a b d &\rightarrow b a^{2} d c a^{2} \\ + a b c a d a d a d &\rightarrow d b a \\ + a c a d &\rightarrow d c a^{2} \\ + c b^{2} a^{2} d &\rightarrow b c a d b a \\ + c b d &\rightarrow b c a d c a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/433.md b/docs/examples/1-relation/433.md new file mode 100644 index 0000000..29500ef --- /dev/null +++ b/docs/examples/1-relation/433.md @@ -0,0 +1,43 @@ +#\#433 baaaabaaba=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a^{2} b a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b c b a = a b^{2} c,\; c = a^{2},\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b a &\rightarrow d \\ + a b^{2} a^{2} &\rightarrow b a^{3} d^{2} \\ + a b^{2} a d &\rightarrow b a^{3} d^{2} b a \\ + d b^{2} a^{2} &\rightarrow b a^{3} d^{2} a d^{2} \\ + a b d &\rightarrow d b a \\ + b a^{3} d b a^{3} d^{2} a d^{2} &\rightarrow a b^{2} d a^{2} d^{2} \\ + b a^{3} d^{2} a d b a^{3} d^{2} a d^{2} &\rightarrow d b^{2} d a^{2} d^{2} \\ + d b^{2} a d &\rightarrow b a^{3} d^{2} a d^{2} b a \\ + d b a b^{2} d a^{2} d^{2} &\rightarrow b a^{3} d^{2} a d^{2} a d b a^{3} d^{2} a d^{2} \\ + a b^{2} d a^{2} d b a^{3} d^{2} a d^{2} &\rightarrow b a^{3} d^{2} b^{2} d a^{2} d^{2} \\ + a^{2} b^{2} d a^{2} d^{2} &\rightarrow d a^{2} d b a^{3} d^{2} a d^{2} \\ + d b^{2} d a^{2} d b a^{3} d^{2} a d^{2} &\rightarrow b a^{3} d^{2} a d^{2} b^{2} d a^{2} d^{2} \\ + d b d b^{2} d a^{2} d^{2} &\rightarrow b a^{3} d^{2} a d^{2} a d^{2} a d b a^{3} d^{2} a d^{2} \\ + a d b^{2} d a^{2} d^{2} &\rightarrow d a^{2} d^{2} a d b a^{3} d^{2} a d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/434.md b/docs/examples/1-relation/434.md new file mode 100644 index 0000000..5335157 --- /dev/null +++ b/docs/examples/1-relation/434.md @@ -0,0 +1,45 @@ +#\#434 ababaaaba=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{3} b a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d c = b c a^{2},\; c = a b a,\; d = c b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < c < d < b\): + +\( +\begin{aligned} + b c a^{2} &\rightarrow d c \\ + c b a^{2} &\rightarrow d \\ + a b a &\rightarrow c \\ + b c a c &\rightarrow d c b a \\ + c b a c &\rightarrow d b a \\ + a b c &\rightarrow c b a \\ + c b a d &\rightarrow d b c a \\ + b c a d &\rightarrow d c b c a \\ + a b d &\rightarrow c b c a \\ + a d c &\rightarrow d a \\ + b c d a &\rightarrow d c d c b a \\ + a d^{2} &\rightarrow d c a \\ + c b d a &\rightarrow d^{2} c b a \\ + c b d c &\rightarrow d^{2} c b c \\ + b c d c &\rightarrow d c d c b c \\ + c b d^{2} &\rightarrow d^{2} c b d \\ + b c d^{2} &\rightarrow d c d c b d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/436.md b/docs/examples/1-relation/436.md new file mode 100644 index 0000000..52ca123 --- /dev/null +++ b/docs/examples/1-relation/436.md @@ -0,0 +1,46 @@ +#\#436 baabaaaaba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{4} b a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d = a c a,\; c = b a,\; d = c a^{3} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a c a &\rightarrow c a d \\ + c a^{3} c &\rightarrow d \\ + b c a d &\rightarrow c^{2} a \\ + a c^{2} a d &\rightarrow c a d c a \\ + c^{2} a d^{3} &\rightarrow d a \\ + c a d a^{2} c &\rightarrow a d \\ + a d a^{3} c &\rightarrow c a d a^{2} d \\ + c a d c a d^{2} &\rightarrow a d a \\ + c a d^{2} a^{2} c &\rightarrow a^{2} d \\ + b c^{2} a d a^{2} d &\rightarrow c^{2} a^{4} c \\ + a^{3} d &\rightarrow c a d^{3} a^{2} c \\ + a^{2} d a &\rightarrow c a d^{2} c a d^{2} \\ + a^{2} d c a d &\rightarrow c a d^{2} c a d^{2} c a \\ + c a d c a^{4} c &\rightarrow a c^{3} a d a^{2} d \\ + a^{2} c^{3} a d a^{2} d &\rightarrow c a d^{2} c a^{4} c \\ + c a d^{2} c a d^{2} c a^{4} c &\rightarrow a^{2} d c^{2} a d a^{2} d \\ + c a d a^{2} d c^{2} a d a^{2} d &\rightarrow a d c a d^{3} c a^{4} c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/437.md b/docs/examples/1-relation/437.md new file mode 100644 index 0000000..4480a02 --- /dev/null +++ b/docs/examples/1-relation/437.md @@ -0,0 +1,42 @@ +#\#437 baaaaabbba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{3} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{4} d a &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + a b^{2} a^{2} &\rightarrow d a^{4} d a \\ + b a^{4} d^{2} &\rightarrow a d \\ + a b^{2} a d &\rightarrow d a^{4} d^{2} \\ + a b a d &\rightarrow d a^{4} d a^{3} d^{2} \\ + a b a^{2} &\rightarrow d a^{4} d a^{3} d a \\ + d a^{4} d a^{3} d a^{3} d^{2} &\rightarrow a^{2} d \\ + d a^{4} d a^{3} d a^{3} d a &\rightarrow a^{3} \\ + b a^{6} d &\rightarrow a^{5} d a^{3} d a^{3} d^{2} \\ + b a^{7} &\rightarrow a^{5} d a^{3} d a^{3} d a \\ + d a^{4} d a^{3} d a^{5} d &\rightarrow a^{6} d a^{3} d a^{3} d^{2} \\ + d a^{4} d a^{3} d a^{6} &\rightarrow a^{6} d a^{3} d a^{3} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/438.md b/docs/examples/1-relation/438.md new file mode 100644 index 0000000..2a37cfd --- /dev/null +++ b/docs/examples/1-relation/438.md @@ -0,0 +1,35 @@ +#\#438 baabbababa=abbaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{2} a b a b a = a b^{2} a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d c^{3} = d c d c,\; c = b a,\; d = a b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a b &\rightarrow d \\ + d c d c &\rightarrow c d c^{3} \\ + b d &\rightarrow c b \\ + a c &\rightarrow d a \\ + d c^{2} d c^{3} &\rightarrow c d c^{3} d c \\ + b c d c^{3} &\rightarrow c b c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/439.md b/docs/examples/1-relation/439.md new file mode 100644 index 0000000..61ef88c --- /dev/null +++ b/docs/examples/1-relation/439.md @@ -0,0 +1,62 @@ +#\#439 baaaabbaaa=aaaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{2} a^{3} = a^{4} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; b c^{2} d c a = c^{2} d a,\; c = a^{2},\; d = b^{2},\; e = d c^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < e < c < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow d \\ + a^{2} &\rightarrow c \\ + d c^{3} &\rightarrow e \\ + b d &\rightarrow d b \\ + a c &\rightarrow c a \\ + d b c^{3} &\rightarrow b e \\ + c^{2} d c &\rightarrow d c^{2} e \\ + d^{2} c^{2} e c a &\rightarrow c^{2} d a \\ + b c^{2} e &\rightarrow d c^{2} e c \\ + b c^{2} d a &\rightarrow d^{2} c^{2} e a \\ + a d c^{2} e &\rightarrow c^{2} a d c \\ + c^{2} d^{2} c^{2} e &\rightarrow d c^{2} e c d c \\ + b e c d c &\rightarrow d^{2} c^{2} e^{2} \\ + d b c d c^{2} e &\rightarrow b e d c \\ + d c^{2} e c^{2} &\rightarrow c^{2} e \\ + d^{2} c^{2} e c e &\rightarrow e c d c \\ + d c d c^{2} e &\rightarrow e d c \\ + b e c^{3} e &\rightarrow d^{2} c^{2} e^{2} c e c^{2} \\ + b e c e &\rightarrow d^{2} c^{2} e^{2} c^{2} \\ + b e c d^{2} c^{2} e &\rightarrow d^{2} c^{2} e^{2} c d c \\ + d c^{2} e c e c^{2} &\rightarrow c^{4} e \\ + d c^{2} e c d c^{2} a &\rightarrow c^{4} d a \\ + d c^{2} e c d c^{2} e &\rightarrow c^{2} e c d c \\ + d c^{2} e d c^{2} e &\rightarrow c^{2} e d c \\ + b e c^{5} e &\rightarrow d^{2} c^{2} e^{2} c e c e c^{2} \\ + b e c^{5} d a &\rightarrow d^{2} c^{2} e^{2} c e c d c^{2} a \\ + b e c^{3} d a &\rightarrow d^{2} c^{2} e^{2} c d c^{2} a \\ + c^{6} e &\rightarrow d c^{2} e c e c e c^{2} \\ + c^{6} d a &\rightarrow d c^{2} e c e c d c^{2} a \\ + b c^{4} e &\rightarrow d c^{2} e^{2} c^{2} \\ + b c^{4} d a &\rightarrow d c^{2} e d c^{2} a \\ + d c^{2} e c e c d c^{2} e &\rightarrow c^{4} e c d c \\ + d c^{2} e c e d c^{2} e &\rightarrow c^{4} e d c \\ + b e c d a &\rightarrow d^{2} c^{2} e d c^{2} a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/440.md b/docs/examples/1-relation/440.md new file mode 100644 index 0000000..08ebd92 --- /dev/null +++ b/docs/examples/1-relation/440.md @@ -0,0 +1,55 @@ +#\#440 ababbabba=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b^{2} a = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a = c a,\; c = b a b,\; d = a c^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a b &\rightarrow c \\ + c a &\rightarrow d a \\ + a c^{2} b &\rightarrow d \\ + d a b &\rightarrow a c^{3} \\ + c d &\rightarrow d^{2} \\ + b a c &\rightarrow a c^{3} \\ + b d &\rightarrow a c^{4} b \\ + b a d^{2} &\rightarrow a d^{4} \\ + b a d a &\rightarrow a d^{3} a \\ + b a^{2} c^{3} &\rightarrow d a c \\ + a d^{2} a c^{3} &\rightarrow d a c \\ + d a^{2} c^{3} &\rightarrow a d^{3} a c \\ + b a^{2} d^{4} &\rightarrow d a d^{2} \\ + b a^{2} d^{3} a &\rightarrow d a d a \\ + a d^{3} a d^{2} &\rightarrow d a^{2} d^{4} \\ + a d^{3} a d a &\rightarrow d a^{2} d^{3} a \\ + a d^{2} a d^{4} &\rightarrow d a d^{2} \\ + a d^{2} a d^{3} a &\rightarrow d a d a \\ + d a^{2} d^{4} a d^{4} &\rightarrow a d^{4} a d^{2} \\ + d a^{2} d^{4} a d^{3} a &\rightarrow a d^{4} a d a \\ + d a^{2} d^{5} a d^{2} &\rightarrow a d^{4} a^{2} d^{4} \\ + d a^{2} d^{5} a d a &\rightarrow a d^{4} a^{2} d^{3} a \\ + a d^{3} a^{2} d^{4} &\rightarrow d a d a d^{2} \\ + a d^{3} a^{2} d^{3} a &\rightarrow d a d a d a \\ + d a^{2} d^{4} a c^{3} &\rightarrow a d^{4} a c \\ + d a^{2} d^{5} a^{2} d^{4} &\rightarrow a d^{4} a d a d^{2} \\ + d a^{2} d^{5} a^{2} d^{3} a &\rightarrow a d^{4} a d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/441.md b/docs/examples/1-relation/441.md new file mode 100644 index 0000000..2e7c588 --- /dev/null +++ b/docs/examples/1-relation/441.md @@ -0,0 +1,34 @@ +#\#441 abaabbaaaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b^{2} a^{4} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c d = b c b a,\; c = a^{2},\; d = b^{2} c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < a < d < b\): + +\( +\begin{aligned} + b^{2} c^{2} &\rightarrow d \\ + b c b a &\rightarrow a b c d \\ + a^{2} &\rightarrow c \\ + b c b c &\rightarrow a b c d a \\ + a c &\rightarrow c a \\ + b c a b c d &\rightarrow a b c d a b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/444.md b/docs/examples/1-relation/444.md new file mode 100644 index 0000000..b4ee042 --- /dev/null +++ b/docs/examples/1-relation/444.md @@ -0,0 +1,55 @@ +#\#444 abbababaa=bababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b a b a^{2} = b a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d a = c^{3},\; c = b a,\; d = b c^{3},\; e = d a d +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < c < e < d < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a e a &\rightarrow a e \\ + b c^{2} a e &\rightarrow e c^{3} \\ + b c a e &\rightarrow e c^{3} a \\ + d &\rightarrow a e a^{3} \\ + a e a^{3} c^{2} a e &\rightarrow e a^{2} e a \\ + c e &\rightarrow e c^{3} a^{2} \\ + a e a^{4} e &\rightarrow e a^{2} e a^{3} \\ + a e a^{3} c^{3} &\rightarrow e a \\ + a^{2} e a^{4} &\rightarrow c^{3} \\ + a e a^{5} e a^{3} &\rightarrow e \\ + b c^{3} &\rightarrow a e a^{3} \\ + a e a c^{3} &\rightarrow e c^{3} a^{2} c^{3} a^{3} \\ + b e &\rightarrow e c^{3} a^{7} e a^{3} \\ + a e a^{3} e &\rightarrow e c^{3} a^{4} e a^{3} \\ + a e a^{2} c^{3} &\rightarrow e c^{3} a^{3} \\ + a e c^{3} &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{3} \\ + a e a^{2} c^{2} a e &\rightarrow e c^{3} a^{4} e a \\ + a e^{2} &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{2} c^{3} a^{4} e a^{3} \\ + a e a c^{2} a e &\rightarrow e c^{3} a^{2} c^{3} a^{4} e a \\ + a e a^{2} e &\rightarrow e c^{3} a^{2} c^{3} a^{4} e a^{3} \\ + a e a^{3} c a e &\rightarrow e a^{2} e a^{2} \\ + a e a e &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{4} e a^{3} \\ + a e c^{2} a e &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{4} e a \\ + a e a c a e &\rightarrow e c^{3} a^{2} c^{3} a^{4} e a^{2} \\ + a e a^{2} c a e &\rightarrow e c^{3} a^{4} e a^{2} \\ + a e c a e &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{4} e a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/445.md b/docs/examples/1-relation/445.md new file mode 100644 index 0000000..12154be --- /dev/null +++ b/docs/examples/1-relation/445.md @@ -0,0 +1,33 @@ +#\#445 abaaaababa=baabaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{4} b a b a = b a^{2} b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c^{2} d a = b c b c a,\; c = a^{2},\; d = b a b +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a b &\rightarrow d \\ + b a^{2} b a^{3} &\rightarrow a b a^{4} d a \\ + d a^{2} b a^{3} &\rightarrow a b a^{4} d a^{2} d a \\ + d a b &\rightarrow b a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/446.md b/docs/examples/1-relation/446.md new file mode 100644 index 0000000..fd40b8f --- /dev/null +++ b/docs/examples/1-relation/446.md @@ -0,0 +1,40 @@ +#\#446 baaaaabba=aaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{5} b^{2} a = a^{4} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} d a = c^{2},\; c = a^{2},\; d = a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{4} d a &\rightarrow a^{4} \\ + a b^{2} &\rightarrow d \\ + a b a^{4} &\rightarrow d a^{4} d a \\ + b a^{4} d^{2} &\rightarrow a^{3} d \\ + a b a^{3} d &\rightarrow d a^{4} d^{2} \\ + d a^{4} d a d^{2} &\rightarrow a^{4} d \\ + d a^{4} d a d a &\rightarrow a^{5} \\ + b a^{8} d &\rightarrow a^{7} d a d^{2} \\ + b a^{9} &\rightarrow a^{7} d a d a \\ + d a^{4} d a^{5} d &\rightarrow a^{8} d a d^{2} \\ + d a^{4} d a^{6} &\rightarrow a^{8} d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/447.md b/docs/examples/1-relation/447.md new file mode 100644 index 0000000..65b9d3c --- /dev/null +++ b/docs/examples/1-relation/447.md @@ -0,0 +1,50 @@ +#\#447 baaababa=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a b a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d a c^{2} = a,\; c = b a,\; d = c a,\; e = a c d a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + c e c^{2} &\rightarrow d^{2} \\ + c e c d c^{3} d &\rightarrow e c d c^{2} \\ + d c^{2} e c d c^{2} &\rightarrow c e \\ + b e &\rightarrow c^{3} e c d c^{2} \\ + a &\rightarrow c e c d c^{4} \\ + c e c d c^{6} e c d c^{2} &\rightarrow e \\ + c^{2} e c d c^{4} &\rightarrow d \\ + c e c d c^{5} e &\rightarrow e c^{4} e c d c^{2} \\ + c e c d c^{4} e &\rightarrow e c d c^{4} e c d c^{2} \\ + c e c d c^{4} d &\rightarrow e c^{2} \\ + b c e c d c^{4} &\rightarrow c \\ + c e c d c^{3} e &\rightarrow e c d c d c^{4} e c d c^{2} \\ + c e c d c^{2} d &\rightarrow e c d c d c^{2} \\ + d c e c d c^{2} &\rightarrow c e c d \\ + c e c d c d &\rightarrow e c d c d c d c^{2} \\ + d e &\rightarrow e c d c d \\ + c e c d^{2} &\rightarrow e c d c d c d c d c^{2} \\ + c e d &\rightarrow e c d c d c d c d c d c^{4} \\ + c e c e &\rightarrow e c d c d c d c d c d c^{6} e c d c^{2} \\ + c e c d c^{2} e &\rightarrow e c d c d c d c^{4} e c d c^{2} \\ + c e c d c e &\rightarrow e c d c d c d c d c^{4} e c d c^{2} \\ + c e^{2} &\rightarrow e c d c d c d c d c d c^{3} d c^{4} e c d c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/448.md b/docs/examples/1-relation/448.md new file mode 100644 index 0000000..426f7c0 --- /dev/null +++ b/docs/examples/1-relation/448.md @@ -0,0 +1,36 @@ +#\#448 baabaaabba=abaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{3} b^{2} a = a b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} a d = a c^{2},\; c = b a^{2},\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b^{2} a &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a c^{2} &\rightarrow c^{2} a d \\ + b c &\rightarrow d a \\ + d a c a d a d &\rightarrow c^{3} \\ + b d a c a d &\rightarrow d c^{2} \\ + b d c^{2} a d c a d c &\rightarrow d c^{2} a c a d a d \\ + d a c a d c^{2} a d c &\rightarrow c^{3} a c a d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/450.md b/docs/examples/1-relation/450.md new file mode 100644 index 0000000..b1e3552 --- /dev/null +++ b/docs/examples/1-relation/450.md @@ -0,0 +1,48 @@ +#\#450 baabababaa=aabbaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b a b a^{2} = a^{2} b^{2} a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c a c d = a^{2} e,\; c = b a,\; d = c^{2} a,\; e = b c a c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b c a c &\rightarrow e \\ + b a &\rightarrow c \\ + c^{2} a &\rightarrow d \\ + c a c d &\rightarrow a^{2} e \\ + b c a d &\rightarrow e c a \\ + b c a^{3} e &\rightarrow e a c d \\ + c a e &\rightarrow e d \\ + d c d &\rightarrow c a^{2} e \\ + b e d^{2} &\rightarrow e a e \\ + c e d &\rightarrow d e \\ + a^{2} e c d &\rightarrow c a d a e \\ + b e d c a^{2} e &\rightarrow e a e c d \\ + d a e c d &\rightarrow c d^{2} a e \\ + c e c a^{2} e &\rightarrow d e c d \\ + b e c a^{2} e d a e &\rightarrow e a e a e c d \\ + c e c d^{2} a e &\rightarrow d e a e c d \\ + c a c^{2} d^{2} a e &\rightarrow a^{2} e a e c d \\ + d c^{2} d^{2} a e &\rightarrow c a^{2} e a e c d \\ + d a e c^{2} d^{2} a e &\rightarrow c d^{2} a e a e c d \\ + a^{2} e c^{2} d^{2} a e &\rightarrow c a d a e a e c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/451.md b/docs/examples/1-relation/451.md new file mode 100644 index 0000000..188fb8e --- /dev/null +++ b/docs/examples/1-relation/451.md @@ -0,0 +1,45 @@ +#\#451 baaabaaba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a^{2} b a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d a d c = a,\; c = b a,\; d = c a,\; e = a d^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + c e d c d c d^{2} &\rightarrow e d c d c \\ + d c e d c &\rightarrow c e \\ + b e &\rightarrow c^{2} e d c \\ + a &\rightarrow c e d c d c d c \\ + c e d c d c d c^{2} e d c &\rightarrow e \\ + c^{2} e d c d c d c &\rightarrow d \\ + c e d c d c d c e &\rightarrow e d c d c^{2} e d c \\ + b c e d c d c d c &\rightarrow c \\ + c e d c d c^{2} e &\rightarrow e d c d c^{2} e d c d c \\ + d e &\rightarrow e d c d^{2} \\ + c e d c d c e &\rightarrow e d c d^{3} c d c^{2} e d c \\ + c e d c d^{2} &\rightarrow e d c d^{3} c d c \\ + c e d^{2} &\rightarrow e d c d^{3} c d^{3} c d c \\ + c e d c^{2} e &\rightarrow e d c d^{3} c d c^{2} e d c d c \\ + c e^{2} &\rightarrow e d c d^{3} c d^{3} c d^{3} c d c^{2} e d c \\ + c e d c e &\rightarrow e d c d^{3} c d^{3} c d c^{2} e d c \\ + c e c e &\rightarrow e d c d^{3} c d^{3} c d c^{2} e d c d c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/452.md b/docs/examples/1-relation/452.md new file mode 100644 index 0000000..3ebc472 --- /dev/null +++ b/docs/examples/1-relation/452.md @@ -0,0 +1,36 @@ +#\#452 baabaabbaa=aaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{2} a^{2} = a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c a d = a^{3} c,\; c = b a,\; d = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow d \\ + a^{3} c &\rightarrow c a c a d \\ + c a^{2} c &\rightarrow d c a d \\ + b c^{2} a c a d &\rightarrow d a^{2} c \\ + b d c a d &\rightarrow d a c \\ + a^{3} d c a d &\rightarrow c a c a d a^{2} c \\ + c a^{2} d c a d &\rightarrow d c a d a^{2} c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/453.md b/docs/examples/1-relation/453.md new file mode 100644 index 0000000..374c94a --- /dev/null +++ b/docs/examples/1-relation/453.md @@ -0,0 +1,40 @@ +#\#453 aabbababaa=baaabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{2} a b a b a^{2} = b a^{3} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a^{2} d e = c a^{2} d,\; c = b a,\; d = b c,\; e = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b c &\rightarrow d \\ + b a &\rightarrow c \\ + c^{2} a &\rightarrow e \\ + c a^{2} d &\rightarrow a^{2} d e \\ + b e &\rightarrow d c a \\ + c a d e &\rightarrow d a^{2} d \\ + a^{2} d e^{2} &\rightarrow e a d \\ + b d a^{2} d &\rightarrow d a d e \\ + c e a d &\rightarrow e a d e \\ + c d a^{2} d &\rightarrow e d e \\ + b d e a d &\rightarrow d a d e^{3} \\ + c d e a d &\rightarrow e d e^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/454.md b/docs/examples/1-relation/454.md new file mode 100644 index 0000000..531912c --- /dev/null +++ b/docs/examples/1-relation/454.md @@ -0,0 +1,34 @@ +#\#454 ababababaa=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b a b a^{2} = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d = c^{2} a,\; c = b a,\; d = c^{3} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c^{2} a &\rightarrow a c d \\ + c a c d &\rightarrow d \\ + a c d c d &\rightarrow c d \\ + c^{2} d &\rightarrow d c d \\ + b c d &\rightarrow d c d c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/457.md b/docs/examples/1-relation/457.md new file mode 100644 index 0000000..2eb76ce --- /dev/null +++ b/docs/examples/1-relation/457.md @@ -0,0 +1,39 @@ +#\#457 ababaabbaa=baaababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a^{2} b^{2} a^{2} = b a^{3} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a c e = c a^{2} c^{2},\; c = b a,\; d = b c a,\; e = c a d +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < e < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow d \\ + c a^{2} c^{2} &\rightarrow a c e \\ + c a d &\rightarrow e \\ + d a c^{2} &\rightarrow c^{2} e \\ + b e &\rightarrow d^{2} \\ + c a^{2} c a c e &\rightarrow a c e a^{2} c^{2} \\ + c a^{2} c e &\rightarrow a c e a d \\ + d a c a c e &\rightarrow c^{2} e a^{2} c^{2} \\ + d a c e &\rightarrow c^{2} e a d \\ + c a c^{2} e &\rightarrow e a c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/458.md b/docs/examples/1-relation/458.md new file mode 100644 index 0000000..4141f80 --- /dev/null +++ b/docs/examples/1-relation/458.md @@ -0,0 +1,65 @@ +#\#458 baaabbbaa=aaabbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{3} a^{2} = a^{3} b^{3} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b d a = d,\; c = b^{3} a,\; d = a^{3} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{3} a &\rightarrow c \\ + a^{3} c &\rightarrow d \\ + c a^{2} c d &\rightarrow d c \\ + b c a^{2} c a &\rightarrow c a^{2} c \\ + c a^{2} c a d &\rightarrow d a c \\ + b d &\rightarrow c a^{2} c a^{2} \\ + a^{3} d c &\rightarrow d a^{2} c d \\ + c a^{2} c a^{2} d &\rightarrow d a^{2} c \\ + c a^{2} c a^{3} &\rightarrow d \\ + b c a^{2} d &\rightarrow c a^{2} c a c a^{3} \\ + a^{3} d a^{2} c &\rightarrow d a^{2} c a^{2} d \\ + a^{3} d a c &\rightarrow d a^{2} c a d \\ + d a^{2} c a^{3} &\rightarrow a^{3} d \\ + d a^{2} c d a^{2} c a^{2} d &\rightarrow a^{3} d^{2} a^{2} c \\ + d a^{2} c d a^{2} c a d &\rightarrow a^{3} d^{2} a c \\ + d a^{2} c d a^{2} c d &\rightarrow a^{3} d^{2} c \\ + b a^{3} d &\rightarrow d a c a^{3} \\ + b c a^{5} d &\rightarrow c a^{2} c a c a^{2} d a^{3} \\ + a^{6} d^{2} c &\rightarrow d a^{2} c a^{2} d^{2} a^{2} c d \\ + d a^{2} c a^{2} d a^{3} &\rightarrow a^{6} d \\ + d a^{2} c a^{2} d a^{2} c a^{2} d &\rightarrow a^{3} d a^{2} d a^{2} c \\ + d a^{2} c a^{2} d a^{2} c a d &\rightarrow a^{3} d a^{2} d a c \\ + d a^{2} c a^{2} d a^{2} c d &\rightarrow a^{3} d a^{2} d c \\ + d a^{2} c a^{2} d^{2} a^{2} c a^{2} d &\rightarrow a^{6} d^{2} a^{2} c \\ + d a^{2} c a^{2} d^{2} a^{2} c a d &\rightarrow a^{6} d^{2} a c \\ + d a^{2} c a d a^{2} c a^{2} d &\rightarrow a^{3} d a d a^{2} c \\ + d a^{2} c a d a^{2} c a d &\rightarrow a^{3} d a d a c \\ + d a^{2} c a d a^{2} c d &\rightarrow a^{3} d a d c \\ + b a^{6} d &\rightarrow d a c a^{2} d a^{3} \\ + b c a^{8} d &\rightarrow c a^{2} c a c a^{2} d a^{2} d a^{3} \\ + a^{9} d &\rightarrow d a^{2} c a^{2} d a^{2} d a^{3} \\ + a^{6} d a^{2} d c &\rightarrow d a^{2} c a^{2} d a^{2} d a^{2} c d \\ + a^{6} d a d c &\rightarrow d a^{2} c a^{2} d a d a^{2} c d \\ + d a^{2} c a^{2} d a^{2} d a^{2} c a^{2} d &\rightarrow a^{6} d a^{2} d a^{2} c \\ + d a^{2} c a^{2} d a^{2} d a^{2} c a d &\rightarrow a^{6} d a^{2} d a c \\ + d a^{2} c a^{2} d a d a^{2} c a^{2} d &\rightarrow a^{6} d a d a^{2} c \\ + d a^{2} c a^{2} d a d a^{2} c a d &\rightarrow a^{6} d a d a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/459.md b/docs/examples/1-relation/459.md new file mode 100644 index 0000000..af28a8f --- /dev/null +++ b/docs/examples/1-relation/459.md @@ -0,0 +1,35 @@ +#\#459 baaabaaaba=aabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b a^{3} b a = a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a d a c = a^{2} d,\; c = b a,\; d = c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + a^{2} d &\rightarrow d a d a c \\ + c a &\rightarrow d \\ + b d a d a c &\rightarrow d^{2} \\ + c d a d a c &\rightarrow d a d \\ + b d a d a d &\rightarrow d^{2} a \\ + c d a d a d &\rightarrow d a d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/460.md b/docs/examples/1-relation/460.md new file mode 100644 index 0000000..be8e7be --- /dev/null +++ b/docs/examples/1-relation/460.md @@ -0,0 +1,39 @@ +#\#460 aabbaaabba=baabbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b^{2} a^{3} b^{2} a = b a^{2} b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d a d a = b d a^{2},\; c = b^{2},\; d = a^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b d a^{2} &\rightarrow d a d a \\ + a^{2} c &\rightarrow d \\ + b c &\rightarrow c b \\ + b d^{2} &\rightarrow d a d a c \\ + b d a d &\rightarrow d a d^{2} \\ + c d a^{2} &\rightarrow d a d^{2} a \\ + c d a d &\rightarrow d a d^{3} \\ + c d^{2} &\rightarrow d a d^{2} a c \\ + a^{2} d a d^{2} a &\rightarrow d^{2} a^{2} \\ + a^{2} d a d^{3} &\rightarrow d^{2} a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/461.md b/docs/examples/1-relation/461.md new file mode 100644 index 0000000..83b3393 --- /dev/null +++ b/docs/examples/1-relation/461.md @@ -0,0 +1,105 @@ +#\#461 baaaababaa=abaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a b a^{2} = a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a d a^{2} = a b a^{3},\; c = a^{2} b a,\; d = a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a^{2} b a &\rightarrow c \\ + a c b &\rightarrow d \\ + a b a^{3} &\rightarrow b a d a^{2} \\ + a^{2} b c &\rightarrow c a b a \\ + a^{2} b d &\rightarrow c^{2} b \\ + a^{2} b^{2} a d a^{2} &\rightarrow c b a^{3} \\ + a b c d a^{2} &\rightarrow b a d c a^{2} \\ + a b a^{2} c &\rightarrow b a d a c \\ + a b a^{2} d &\rightarrow b a d a d \\ + a b a c &\rightarrow b a d c \\ + a^{2} b^{2} a d a c &\rightarrow c b a^{2} c \\ + a^{2} b^{2} a d a d &\rightarrow c b a^{2} d \\ + a^{2} b^{2} a d c &\rightarrow c b a c \\ + a c a^{2} &\rightarrow c d a^{2} \\ + a b c d a c &\rightarrow b a d c a c \\ + a b c d a d &\rightarrow b a d c a d \\ + a b c d c &\rightarrow b a d c^{2} \\ + a b a d &\rightarrow b a d^{2} \\ + b a d c b &\rightarrow a b d \\ + b a d^{2} c &\rightarrow c^{2} \\ + b a d^{2} a d &\rightarrow c a d \\ + b a d^{2} a c &\rightarrow c a c \\ + b a d^{2} a^{2} &\rightarrow c a^{2} \\ + a^{2} b^{2} a d^{2} &\rightarrow c b a d \\ + d a d^{2} a^{2} &\rightarrow c d c a^{2} \\ + d a d^{2} a c &\rightarrow c d c a c \\ + d a d^{2} a d &\rightarrow c d c a d \\ + d a d^{2} c &\rightarrow c d c^{2} \\ + d a d^{3} &\rightarrow c d c d \\ + a c a b d &\rightarrow d a d c b \\ + a b c d^{2} &\rightarrow b a d c d \\ + b a d^{3} &\rightarrow c d \\ + a b d a d c b &\rightarrow b a d c a b d \\ + a c^{2} &\rightarrow c d c \\ + a c a d &\rightarrow c d a d \\ + a c a c &\rightarrow c d a c \\ + a^{2} b^{2} c d^{2} c a^{2} &\rightarrow c b a^{2} d^{2} a^{2} \\ + a^{2} b^{2} c d^{2} c a c &\rightarrow c b a^{2} d^{2} a c \\ + a^{2} b^{2} c d^{2} c a d &\rightarrow c b a^{2} d^{2} a d \\ + a^{2} b^{2} c d^{2} c^{2} &\rightarrow c b a^{2} d^{2} c \\ + a^{2} b^{2} c d^{2} c d &\rightarrow c b a^{2} d^{3} \\ + b a d c d c a^{2} &\rightarrow c a d^{2} a^{2} \\ + b a d c d c a c &\rightarrow c a d^{2} a c \\ + b a d c d c a d &\rightarrow c a d^{2} a d \\ + b a d c d c^{2} &\rightarrow c a d^{2} c \\ + b a d c d c d &\rightarrow c a d^{3} \\ + d a d c d c a^{2} &\rightarrow c d c a d^{2} a^{2} \\ + d a d c d c a c &\rightarrow c d c a d^{2} a c \\ + d a d c d c a d &\rightarrow c d c a d^{2} a d \\ + d a d c d c^{2} &\rightarrow c d c a d^{2} c \\ + d a d c d c d &\rightarrow c d c a d^{3} \\ + b a d c a d^{2} a^{2} &\rightarrow a b c^{2} d c a^{2} \\ + b a d c a d^{2} a c &\rightarrow a b c^{2} d c a c \\ + b a d c a d^{2} a d &\rightarrow a b c^{2} d c a d \\ + b a d c a d^{2} c &\rightarrow a b c^{2} d c^{2} \\ + b a d c a d^{3} &\rightarrow a b c^{2} d c d \\ + a c d &\rightarrow c d^{2} \\ + b a d c a d c d c a^{2} &\rightarrow a b c^{2} d c a d^{2} a^{2} \\ + b a d c a d c d c a c &\rightarrow a b c^{2} d c a d^{2} a c \\ + b a d c a d c d c a d &\rightarrow a b c^{2} d c a d^{2} a d \\ + b a d c a d c d c^{2} &\rightarrow a b c^{2} d c a d^{2} c \\ + b a d c a d c d c d &\rightarrow a b c^{2} d c a d^{3} \\ + a c a b c^{2} d c a^{2} &\rightarrow d a d c a d^{2} a^{2} \\ + a c a b c^{2} d c a c &\rightarrow d a d c a d^{2} a c \\ + a c a b c^{2} d c a d &\rightarrow d a d c a d^{2} a d \\ + a c a b c^{2} d c^{2} &\rightarrow d a d c a d^{2} c \\ + a c a b c^{2} d c d &\rightarrow d a d c a d^{3} \\ + a b d a d c a d^{2} a^{2} &\rightarrow b a d c a b c^{2} d c a^{2} \\ + a b d a d c a d^{2} a c &\rightarrow b a d c a b c^{2} d c a c \\ + a b d a d c a d^{2} a d &\rightarrow b a d c a b c^{2} d c a d \\ + a b d a d c a d^{2} c &\rightarrow b a d c a b c^{2} d c^{2} \\ + a b d a d c a d^{3} &\rightarrow b a d c a b c^{2} d c d \\ + a b d a d c a d c d c a^{2} &\rightarrow b a d c a b c^{2} d c a d^{2} a^{2} \\ + a b d a d c a d c d c a c &\rightarrow b a d c a b c^{2} d c a d^{2} a c \\ + a b d a d c a d c d c a d &\rightarrow b a d c a b c^{2} d c a d^{2} a d \\ + a b d a d c a d c d c^{2} &\rightarrow b a d c a b c^{2} d c a d^{2} c \\ + a b d a d c a d c d c d &\rightarrow b a d c a b c^{2} d c a d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/462.md b/docs/examples/1-relation/462.md new file mode 100644 index 0000000..431aed9 --- /dev/null +++ b/docs/examples/1-relation/462.md @@ -0,0 +1,31 @@ +#\#462 abbaaababa=baabbaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{3} b a b a = b a^{2} b^{2} a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b^{2} c a d^{2} = b c b^{2} c a,\; c = a^{2},\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a &\rightarrow d \\ + d a b d a^{2} &\rightarrow a b d a^{2} d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/463.md b/docs/examples/1-relation/463.md new file mode 100644 index 0000000..ad5dee9 --- /dev/null +++ b/docs/examples/1-relation/463.md @@ -0,0 +1,47 @@ +#\#463 baabababba=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b a b^{2} a = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c^{2} b c = a c,\; c = b a,\; d = a c a c +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b d c b c b d &\rightarrow d c^{3} \\ + b d c b c^{3} b d &\rightarrow d c b^{2} d \\ + b d c b c^{2} b d &\rightarrow d c b^{2} d c b c \\ + a b d &\rightarrow b d c b c^{2} b^{2} d \\ + c b d c b c &\rightarrow b d \\ + b d c b c^{4} &\rightarrow d c b c \\ + a c &\rightarrow b d c b c^{2} b c \\ + b d c b c^{2} b^{2} d c b c &\rightarrow d \\ + a d &\rightarrow d c b^{2} d c b c \\ + b d c b^{2} d &\rightarrow d c^{4} b c \\ + b^{2} d c b c^{2} b c &\rightarrow c^{2} \\ + b^{2} d c b c^{2} b^{2} d &\rightarrow c b d \\ + c d &\rightarrow d c^{4} b c^{2} b c \\ + b d c b d &\rightarrow d c^{4} b c^{2} b c^{2} b^{2} d c b c \\ + b d b d &\rightarrow d c^{4} b c^{2} b c^{4} \\ + b d c^{2} b d &\rightarrow d c^{4} b c^{2} b c^{2} b^{2} d \\ + b d c^{3} &\rightarrow d c^{4} b c^{2} b c^{2} b c \\ + b d^{2} &\rightarrow d c^{4} b c^{2} b c^{5} b c^{2} b^{2} d c b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/465.md b/docs/examples/1-relation/465.md new file mode 100644 index 0000000..047a824 --- /dev/null +++ b/docs/examples/1-relation/465.md @@ -0,0 +1,52 @@ +#\#465 abbaababaa=baababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b a b a^{2} = b a^{2} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d a^{2} = b a c a,\; c = a b a b,\; d = b^{2} a c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} a c &\rightarrow d \\ + b a c a &\rightarrow a d a^{2} \\ + a b a b &\rightarrow c \\ + b a d a^{2} &\rightarrow d a \\ + c b a c &\rightarrow a b a d \\ + a b c &\rightarrow c a b \\ + b a c^{2} &\rightarrow a d a c \\ + b^{2} a^{2} b a d &\rightarrow d b a c \\ + b a d a c &\rightarrow d c \\ + a b a d b a c &\rightarrow c b a^{2} b a d \\ + a b a d c &\rightarrow c a d a c \\ + a b a d a &\rightarrow c a d a^{2} \\ + c a d a^{2} c &\rightarrow a d c \\ + c a d a^{3} &\rightarrow a d a \\ + b a^{2} d c &\rightarrow a d a^{2} d a^{2} c \\ + b a^{2} d a &\rightarrow a d a^{2} d a^{3} \\ + c a c^{2} &\rightarrow a^{2} d a^{2} d a^{3} c \\ + c a c a &\rightarrow a^{2} d a^{2} d a^{4} \\ + b a^{3} d a^{2} d a^{3} c &\rightarrow a d a^{2} c^{2} \\ + b a^{3} d a^{2} d a^{4} &\rightarrow a d a^{2} c a \\ + c a^{2} d c &\rightarrow a^{2} d a^{2} d a^{4} d a^{2} c \\ + c a^{2} d a &\rightarrow a^{2} d a^{2} d a^{4} d a^{3} \\ + c a^{3} d a^{2} d a^{3} c &\rightarrow a^{2} d a^{2} d a^{4} c^{2} \\ + c a^{3} d a^{2} d a^{4} &\rightarrow a^{2} d a^{2} d a^{4} c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/468.md b/docs/examples/1-relation/468.md new file mode 100644 index 0000000..0ec1b20 --- /dev/null +++ b/docs/examples/1-relation/468.md @@ -0,0 +1,36 @@ +#\#468 baabbbbbaa=aba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b^{5} a^{2} = a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a b^{2} d = a c,\; c = b a,\; d = b^{2} c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a c &\rightarrow c a b^{2} d \\ + b a &\rightarrow c \\ + b^{2} c a &\rightarrow d \\ + b c a b^{2} d &\rightarrow c^{2} \\ + b c^{2} &\rightarrow d b^{2} d \\ + b d b^{2} d a b^{2} d &\rightarrow d c \\ + b c a b d c &\rightarrow c^{2} b^{2} d a b^{2} d \\ + b d b^{2} d a b d c &\rightarrow d c b^{2} d a b^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/470.md b/docs/examples/1-relation/470.md new file mode 100644 index 0000000..26fd927 --- /dev/null +++ b/docs/examples/1-relation/470.md @@ -0,0 +1,41 @@ +#\#470 ababbabaa=baba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b a^{2} = b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c d a = c^{2},\; c = b a,\; d = b c^{2} +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c^{2} &\rightarrow a c d a \\ + a c d a d a &\rightarrow d \\ + b d &\rightarrow d^{2} a \\ + a c d a c &\rightarrow c a c d a \\ + a c d a d^{2} &\rightarrow d c d a d a \\ + b c a c d a &\rightarrow d c \\ + a c d^{2} &\rightarrow c d^{2} a \\ + a c d c a c d a &\rightarrow c d c \\ + a c d a d c a c d a &\rightarrow d c d a c \\ + b c d &\rightarrow d c d a \\ + a c d c d &\rightarrow c d c d a \\ + a c d a d c d &\rightarrow d c d a c d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/471.md b/docs/examples/1-relation/471.md new file mode 100644 index 0000000..48071aa --- /dev/null +++ b/docs/examples/1-relation/471.md @@ -0,0 +1,42 @@ +#\#471 abaaababa=baabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{3} b a b a = b a^{2} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a c b a = b a c a,\; c = a b a,\; d = b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < b < a\): + +\( +\begin{aligned} + a b a &\rightarrow c \\ + b a^{2} &\rightarrow d \\ + a b c &\rightarrow c b a \\ + a d &\rightarrow c a \\ + b a c &\rightarrow d b a \\ + c a c b c &\rightarrow d^{2} b a \\ + c a c d &\rightarrow d^{2} a \\ + c a c b a &\rightarrow d^{2} \\ + b c^{2} a &\rightarrow d^{2} c b a \\ + c a c b d^{2} &\rightarrow d^{3} c b a \\ + a b d^{2} &\rightarrow c d c b a \\ + b c^{3} &\rightarrow d^{2} c b c \\ + b c d^{2} &\rightarrow d^{2} c d b c \\ + b c^{2} d^{2} &\rightarrow d^{2} c b d^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/472.md b/docs/examples/1-relation/472.md new file mode 100644 index 0000000..6874d68 --- /dev/null +++ b/docs/examples/1-relation/472.md @@ -0,0 +1,39 @@ +#\#472 baababbbaa=abbabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b^{3} a^{2} = a b^{2} a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b a^{2} d a^{2} = a c a c a,\; c = b^{2},\; d = b a c b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a c b &\rightarrow d \\ + b a^{2} d a^{2} &\rightarrow a c a c a \\ + b c &\rightarrow c b \\ + b d &\rightarrow c a c b \\ + b a c^{2} &\rightarrow d b \\ + b a c d &\rightarrow d a c b \\ + b a c a c a &\rightarrow c a^{2} d a^{2} \\ + c a^{2} d a^{2} c a &\rightarrow d a^{2} d a^{2} \\ + b a c a d a^{2} d a^{2} &\rightarrow c a^{2} d a^{3} d a^{2} c a \\ + c a^{2} d a^{2} d a^{2} d a^{2} &\rightarrow d a^{2} d a^{3} d a^{2} c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/473.md b/docs/examples/1-relation/473.md new file mode 100644 index 0000000..46ce714 --- /dev/null +++ b/docs/examples/1-relation/473.md @@ -0,0 +1,36 @@ +#\#473 abaabaaba=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{2} b a^{2} b a = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d^{3} = e d,\; c = b a,\; d = a c,\; e = c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a &\rightarrow e \\ + e d &\rightarrow d^{3} \\ + a c &\rightarrow d \\ + b d &\rightarrow c^{2} \\ + c d &\rightarrow e c \\ + a e &\rightarrow d a \\ + a d^{3} &\rightarrow d a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/474.md b/docs/examples/1-relation/474.md new file mode 100644 index 0000000..a987cd3 --- /dev/null +++ b/docs/examples/1-relation/474.md @@ -0,0 +1,45 @@ +#\#474 bbaaabbaa=abba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{3} b^{2} a^{2} = a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d a = a c a,\; c = b^{2},\; d = a^{2} c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b c &\rightarrow c b \\ + a d &\rightarrow d^{2} a \\ + a c d &\rightarrow c d^{4} a \\ + a c a &\rightarrow c d^{2} a^{2} \\ + c d^{6} a^{3} &\rightarrow d \\ + c b d^{6} a^{3} &\rightarrow b d \\ + a c^{2} d^{2} a^{2} &\rightarrow c d^{3} \\ + c d^{8} a &\rightarrow d c a \\ + c d^{9} &\rightarrow d c d \\ + a c^{2} d^{3} &\rightarrow c d^{3} c a \\ + c d^{8} c d^{2} a^{2} &\rightarrow d c^{2} d^{2} a^{2} \\ + c d^{8} c d^{3} &\rightarrow d c^{2} d^{3} \\ + c b d^{8} a &\rightarrow b d c a \\ + c b d^{9} &\rightarrow b d c d \\ + c b d^{8} c d^{2} a^{2} &\rightarrow b d c^{2} d^{2} a^{2} \\ + c b d^{8} c d^{3} &\rightarrow b d c^{2} d^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/475.md b/docs/examples/1-relation/475.md new file mode 100644 index 0000000..3f2375d --- /dev/null +++ b/docs/examples/1-relation/475.md @@ -0,0 +1,61 @@ +#\#475 baaabbaaba=abbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{3} b^{2} a^{2} b a = a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c a b^{2} c b a = a b^{2} c,\; c = a^{2},\; d = b a^{2} b +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + b a^{2} b &\rightarrow d \\ + b a^{2} d &\rightarrow d a^{2} b \\ + b a^{3} b d &\rightarrow d a^{3} b d a b \\ + d a^{3} b d a b a d &\rightarrow a b d a^{2} b \\ + a b d a^{3} b d a b &\rightarrow d a^{3} b d a d^{2} \\ + a b a b d &\rightarrow d a^{3} b d a d^{2} a b \\ + a b d a^{3} b d^{2} a^{3} b d a b a &\rightarrow d a^{3} b d a d^{2} b a^{2} \\ + d a^{3} b d a b a b &\rightarrow a b d \\ + d a^{3} b d^{2} a^{3} b d a d^{2} a &\rightarrow a b d b a^{2} \\ + b a d a^{3} b d a b a &\rightarrow d b a^{2} \\ + a b^{2} a^{2} &\rightarrow d a^{3} b d a b a \\ + b a d a^{3} b d a d &\rightarrow d b a^{3} b \\ + d a^{3} b d^{2} a^{3} b d a d^{3} a^{3} b d a b a &\rightarrow a b d^{2} b a^{2} \\ + a b d b a^{4} b d a b a b &\rightarrow d a^{3} b d^{2} a^{3} b d a d a b d \\ + a b d b a^{4} b d a b a d &\rightarrow d a^{3} b d^{2} a^{3} b d a d a b d a^{2} b \\ + a b d b a^{4} b d^{2} a^{3} b d a d^{2} a &\rightarrow d a^{3} b d^{2} a^{3} b d a d a b d b a^{2} \\ + a b d a^{3} b d a d &\rightarrow d a^{3} b d a d^{2} a^{2} b \\ + d a^{3} b d a d b a^{2} &\rightarrow a b d a^{3} b d a \\ + a b d a^{3} b d^{2} a^{3} b d a d &\rightarrow d a^{3} b d a d^{2} b a^{3} b \\ + b a d a^{3} b d^{2} a^{3} b d a d a b d &\rightarrow d^{2} b a^{4} b d a b a b \\ + d a^{3} b d^{2} a^{3} b d a d^{3} a^{3} b d^{2} a^{3} b d a d a b d &\rightarrow a b d^{3} b a^{4} b d a b a b \\ + a b d b a^{4} b d^{2} a^{3} b d a d^{3} a^{3} b d a b a &\rightarrow d a^{3} b d^{2} a^{3} b d a d a b d^{2} b a^{2} \\ + a b d a^{3} b d^{2} a^{3} b d^{2} a^{3} b d a d a b d &\rightarrow d a^{3} b d a d^{3} b a^{4} b d a b a b \\ + b a d a^{3} b d a^{2} b d &\rightarrow d b a^{4} b d a b \\ + a b d a^{3} b d a^{2} b d &\rightarrow d a^{3} b d a d^{2} a^{3} b d a b \\ + a b d b a^{4} b d^{2} a^{3} b d a d^{3} a^{3} b d^{2} a^{3} b d a d a b d &\rightarrow d a^{3} b d^{2} a^{3} b d a d a b d^{3} b a^{4} b d a b a b \\ + a b d b a^{4} b d a d b a^{2} &\rightarrow d a^{3} b d^{2} a^{3} b d a d a b d a^{3} b d a \\ + d a^{3} b d^{2} a^{3} b d a d^{3} a^{3} b d a d &\rightarrow a b d^{2} b a^{3} b \\ + a b d a^{3} b d^{2} a^{3} b d a^{2} b d &\rightarrow d a^{3} b d a d^{2} b a^{4} b d a b \\ + a b d b a^{4} b d^{2} a^{3} b d a d^{3} a^{3} b d a d &\rightarrow d a^{3} b d^{2} a^{3} b d a d a b d^{2} b a^{3} b \\ + d a^{3} b d^{2} a^{3} b d a d^{3} a^{3} b d a^{2} b d &\rightarrow a b d^{2} b a^{4} b d a b \\ + a b d b a^{4} b d^{2} a^{3} b d a d^{3} a^{3} b d a^{2} b d &\rightarrow d a^{3} b d^{2} a^{3} b d a d a b d^{2} b a^{4} b d a b +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/476.md b/docs/examples/1-relation/476.md new file mode 100644 index 0000000..4a61d5d --- /dev/null +++ b/docs/examples/1-relation/476.md @@ -0,0 +1,40 @@ +#\#476 abababbaaa=baaababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b a b^{2} a^{3} = b a^{3} b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a e a = c a^{2} c^{2},\; c = b a,\; d = b c a,\; e = c^{2} d +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c a &\rightarrow d \\ + c a^{2} c^{2} &\rightarrow a e a \\ + c^{2} d &\rightarrow e \\ + d a c^{2} &\rightarrow c e a \\ + c a^{2} e &\rightarrow a e a d \\ + c a^{2} c a e a &\rightarrow a e a^{3} c^{2} \\ + c a^{2} c e &\rightarrow a e a c d \\ + d a c a e a &\rightarrow c e a^{3} c^{2} \\ + d a e &\rightarrow c e a d \\ + d a c e &\rightarrow c e a c d \\ + c^{3} e a &\rightarrow e a c^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/477.md b/docs/examples/1-relation/477.md new file mode 100644 index 0000000..e9940fb --- /dev/null +++ b/docs/examples/1-relation/477.md @@ -0,0 +1,115 @@ +#\#477 abbabbbaba=baabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a b^{3} a b a = b a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a c^{2} a = b a^{2} b^{2} a,\; c = b^{2} a b,\; d = c a b c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + c a b c &\rightarrow d \\ + b a^{2} b^{2} a &\rightarrow a c^{2} a \\ + b^{2} a b &\rightarrow c \\ + c a b d &\rightarrow d a b c \\ + b a^{2} c &\rightarrow a c^{2} a b \\ + c a^{2} b^{2} a &\rightarrow b a c d a \\ + b^{2} a c &\rightarrow c b a b \\ + b a^{2} b a c^{2} a &\rightarrow a c b a c d a \\ + a c^{2} a b a b d &\rightarrow b a^{2} d a b c \\ + a c^{2} a b a b c &\rightarrow b a^{2} d \\ + c a^{2} c &\rightarrow b a c d a b \\ + c a^{2} b a c^{2} a &\rightarrow b a c d a^{2} b^{2} a \\ + c a^{2} b a c b a c d a &\rightarrow b a c d a^{2} b a c^{2} a \\ + c a c b a b d a &\rightarrow d a^{2} b^{2} a \\ + c b a b a b d &\rightarrow b^{2} a d a b c \\ + c b a b a b c &\rightarrow b^{2} a d \\ + b a^{2} b a c d &\rightarrow a c b a c d a b c \\ + b a^{2} b a c b a c d a &\rightarrow a c b a c d a^{2} b^{2} a \\ + a c^{2} a b a c b a b d a &\rightarrow b a^{2} d a^{2} b^{2} a \\ + a c^{2} a b a b^{3} a d &\rightarrow b a^{2} d b a b a b c \\ + b a b a^{2} d &\rightarrow a c d a b a b c \\ + c a^{2} b^{3} a^{2} d &\rightarrow b a c d b a^{2} d \\ + c a b^{3} a d &\rightarrow d b a b a b c \\ + c b a b a c b a b d a &\rightarrow b^{2} a d a^{2} b^{2} a \\ + c b a b a b^{3} a d &\rightarrow b^{2} a d b a b a b c \\ + c b a b c a b a b d &\rightarrow b^{3} a^{2} d a b c \\ + c b a b c a b a b c &\rightarrow b^{3} a^{2} d \\ + b a^{2} b a c^{2} b a^{2} d &\rightarrow a c b a c d b a^{2} d \\ + b a^{2} b^{3} a^{2} d &\rightarrow a c^{2} b a^{2} d \\ + c a^{2} b a c b a c d b a^{2} d &\rightarrow b a c d a^{2} b a c^{2} b a^{2} d \\ + c a^{2} b a c d &\rightarrow b a c d a^{2} c^{2} \\ + b a c d a b a c b a b d a &\rightarrow c a^{2} d a^{2} b^{2} a \\ + b a c d a b a b d &\rightarrow c a^{2} d a b c \\ + b a c d a b a b c &\rightarrow c a^{2} d \\ + b a c d a b c a b a b d &\rightarrow c a b a^{2} d a b c \\ + b a c d a b c a b a b c &\rightarrow c a b a^{2} d \\ + c a c b a b d b a^{2} d &\rightarrow d a^{2} b^{3} a^{2} d \\ + b a^{2} b a c b a c d b a^{2} d &\rightarrow a c b a c d a^{2} b^{3} a^{2} d \\ + c a^{2} b a c^{2} b a^{2} d &\rightarrow b a c d a^{2} b^{3} a^{2} d \\ + a c^{2} a b a c b a b d b a^{2} d &\rightarrow b a^{2} d a^{2} b^{3} a^{2} d \\ + a c^{2} a b a b^{4} a^{2} d &\rightarrow b a^{2} d b a b c a b a b c \\ + b a c d a b a c b a b d b a^{2} d &\rightarrow c a^{2} d a^{2} b^{3} a^{2} d \\ + c a b^{4} a^{2} d &\rightarrow d b a b c a b a b c \\ + c b a b a c b a b d b a^{2} d &\rightarrow b^{2} a d a^{2} b^{3} a^{2} d \\ + c b a b a b^{4} a^{2} d &\rightarrow b^{2} a d b a b c a b a b c \\ + c b a b d a b a c b a b d a &\rightarrow b c a^{2} d a^{2} b^{2} a \\ + c b a b c a b a c b a b d a &\rightarrow b^{3} a^{2} d a^{2} b^{2} a \\ + c b a b c a b a c b a b d b a^{2} d &\rightarrow b^{3} a^{2} d a^{2} b^{3} a^{2} d \\ + c b a b c a b a b^{3} a d &\rightarrow b^{3} a^{2} d b a b a b c \\ + c b a b c a b a b^{4} a^{2} d &\rightarrow b^{3} a^{2} d b a b c a b a b c \\ + b a c d a b a b^{3} a d &\rightarrow c a^{2} d b a b a b c \\ + b a c d a b c a b a c b a b d a &\rightarrow c a b a^{2} d a^{2} b^{2} a \\ + b a c d a b c a b a b^{3} a d &\rightarrow c a b a^{2} d b a b a b c \\ + c a^{2} b c a^{2} d &\rightarrow b a c d a c d a b a b c \\ + c a^{2} b c a b a^{2} d &\rightarrow b a c d a c d a b c a b a b c \\ + c b a b d a b a c b a b d b a^{2} d &\rightarrow b c a^{2} d a^{2} b^{3} a^{2} d \\ + c b a b d a b a b d &\rightarrow b c a^{2} d a b c \\ + c b a b d a b a b c &\rightarrow b c a^{2} d \\ + c b a b d a b c a b a b d &\rightarrow b c a b a^{2} d a b c \\ + c b a b d a b c a b a b c &\rightarrow b c a b a^{2} d \\ + b a^{2} b c a^{2} d &\rightarrow a c^{2} a c d a b a b c \\ + b a^{2} b c a b a^{2} d &\rightarrow a c^{2} a c d a b c a b a b c \\ + b a c d a b a b^{4} a^{2} d &\rightarrow c a^{2} d b a b c a b a b c \\ + b a c d a b c a b a c b a b d b a^{2} d &\rightarrow c a b a^{2} d a^{2} b^{3} a^{2} d \\ + b a c d a b c a b a b^{4} a^{2} d &\rightarrow c a b a^{2} d b a b c a b a b c \\ + c b a b d a b a b^{3} a d &\rightarrow b c a^{2} d b a b a b c \\ + c b a b d a b c a b a c b a b d a &\rightarrow b c a b a^{2} d a^{2} b^{2} a \\ + c b a b d a b c a b a b^{3} a d &\rightarrow b c a b a^{2} d b a b a b c \\ + a c^{2} a b a b^{2} c a^{2} d &\rightarrow b a^{2} d b a b d a b a b c \\ + c a b^{2} c a^{2} d &\rightarrow d b a b d a b a b c \\ + c b a b a b^{2} c a^{2} d &\rightarrow b^{2} a d b a b d a b a b c \\ + c b a b d a b a b^{2} c a^{2} d &\rightarrow b c a^{2} d b a b d a b a b c \\ + c b a b d a b a b^{2} c a b a^{2} d &\rightarrow b c a^{2} d b a b d a b c a b a b c \\ + c b a b d a b a b^{4} a^{2} d &\rightarrow b c a^{2} d b a b c a b a b c \\ + c b a b d a b c a b a b^{2} c a^{2} d &\rightarrow b c a b a^{2} d b a b d a b a b c \\ + c b a b d a b c a b a b^{2} c a b a^{2} d &\rightarrow b c a b a^{2} d b a b d a b c a b a b c \\ + c b a b c a b a b^{2} c a^{2} d &\rightarrow b^{3} a^{2} d b a b d a b a b c \\ + b a c d a b a b^{2} c a^{2} d &\rightarrow c a^{2} d b a b d a b a b c \\ + b a c d a b c a b a b^{2} c a^{2} d &\rightarrow c a b a^{2} d b a b d a b a b c \\ + a c^{2} a b a b^{2} c a b a^{2} d &\rightarrow b a^{2} d b a b d a b c a b a b c \\ + c a b^{2} c a b a^{2} d &\rightarrow d b a b d a b c a b a b c \\ + c b a b a b^{2} c a b a^{2} d &\rightarrow b^{2} a d b a b d a b c a b a b c \\ + c b a b d a b c a b a c b a b d b a^{2} d &\rightarrow b c a b a^{2} d a^{2} b^{3} a^{2} d \\ + c b a b d a b c a b a b^{4} a^{2} d &\rightarrow b c a b a^{2} d b a b c a b a b c \\ + c b a b c a b a b^{2} c a b a^{2} d &\rightarrow b^{3} a^{2} d b a b d a b c a b a b c \\ + b a c d a b a b^{2} c a b a^{2} d &\rightarrow c a^{2} d b a b d a b c a b a b c \\ + b a c d a b c a b a b^{2} c a b a^{2} d &\rightarrow c a b a^{2} d b a b d a b c a b a b c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/478.md b/docs/examples/1-relation/478.md new file mode 100644 index 0000000..8aa8c95 --- /dev/null +++ b/docs/examples/1-relation/478.md @@ -0,0 +1,39 @@ +#\#478 babbabbaba=abbabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a b^{2} a b a = a b^{2} a b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c e = a d^{2},\; c = b a,\; d = b c,\; e = d^{2} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(e < d < c < a < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + b c &\rightarrow d \\ + d^{2} c &\rightarrow e \\ + c d^{2} &\rightarrow d e \\ + c e &\rightarrow d e c \\ + c d e &\rightarrow d e d c \\ + b d e &\rightarrow d^{3} \\ + d^{3} e &\rightarrow e d^{2} \\ + a e &\rightarrow d e c^{2} \\ + a d e &\rightarrow d e c d c \\ + a d^{2} &\rightarrow d e c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/479.md b/docs/examples/1-relation/479.md new file mode 100644 index 0000000..44c55eb --- /dev/null +++ b/docs/examples/1-relation/479.md @@ -0,0 +1,33 @@ +#\#479 bbaabbabba=a + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{2} b^{2} a b^{2} a = a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{2} c a c a = a,\; c = b^{2},\; d = b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(d < a < b < c\): + +\( +\begin{aligned} + c &\rightarrow b^{2} \\ + b^{2} a &\rightarrow d \\ + d a d^{2} &\rightarrow a \\ + d a d a &\rightarrow a^{2} d^{2} \\ + d a^{2} &\rightarrow a^{2} d^{4} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/481.md b/docs/examples/1-relation/481.md new file mode 100644 index 0000000..c5c1a5c --- /dev/null +++ b/docs/examples/1-relation/481.md @@ -0,0 +1,74 @@ +#\#481 babbabbbba=abbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b^{2} a b^{4} a = a b^{3} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; d b^{2} a = a b^{3} a,\; c = b a b^{2},\; d = c a b^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < c < a < b\): + +\( +\begin{aligned} + b a b^{2} &\rightarrow c \\ + a b^{3} a &\rightarrow d b^{2} a \\ + c a b^{2} &\rightarrow d \\ + b a b c &\rightarrow d \\ + b d b^{2} a &\rightarrow c b a \\ + c d b^{2} a &\rightarrow d b a \\ + a b^{2} c &\rightarrow d b c \\ + d a b^{2} &\rightarrow c a b c \\ + b a b d &\rightarrow c a b c \\ + b d b c &\rightarrow c^{2} \\ + b d b d &\rightarrow c d \\ + d a b c &\rightarrow c a b d \\ + c a b c a b c &\rightarrow d a b d \\ + c d b c &\rightarrow d c \\ + a b^{2} d &\rightarrow d b d \\ + c d b d &\rightarrow d^{2} \\ + b d c b a &\rightarrow d b a \\ + b d c^{2} &\rightarrow d c \\ + b d c d &\rightarrow d^{2} \\ + c a b c b^{2} a &\rightarrow b a c b a \\ + c a b c b c &\rightarrow b a c^{2} \\ + c a b c b d &\rightarrow b a c d \\ + a b c b a &\rightarrow d c b a \\ + a b c^{2} &\rightarrow d c^{2} \\ + a b c d &\rightarrow d c d \\ + d^{2} b^{2} a &\rightarrow c d c b a \\ + d^{2} b c &\rightarrow c d c^{2} \\ + d^{2} b d &\rightarrow c d c d \\ + c d c d c b a &\rightarrow d^{3} b a \\ + c d c d c^{2} &\rightarrow d^{3} c \\ + b d^{2} b a &\rightarrow c d c b a \\ + b d^{2} c &\rightarrow c d c^{2} \\ + b d^{3} &\rightarrow c d c d \\ + c d c d c d &\rightarrow d^{4} \\ + c d^{2} b a &\rightarrow d^{2} c b a \\ + c d^{2} c &\rightarrow d^{2} c^{2} \\ + c d^{3} &\rightarrow d^{2} c d \\ + a b d b a &\rightarrow d^{2} b a \\ + a b d c &\rightarrow d^{2} c \\ + a b d^{2} &\rightarrow d^{3} \\ + a d b a &\rightarrow d^{2} c b a \\ + a d c &\rightarrow d^{2} c^{2} \\ + a d^{2} &\rightarrow d^{2} c d \\ + a c d c b a &\rightarrow d^{3} b a \\ + a c d c^{2} &\rightarrow d^{3} c \\ + a c d c d &\rightarrow d^{4} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/482.md b/docs/examples/1-relation/482.md new file mode 100644 index 0000000..8e4810e --- /dev/null +++ b/docs/examples/1-relation/482.md @@ -0,0 +1,56 @@ +#\#482 aabaaabbaa=baaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{3} b^{2} a^{2} = b a^{3} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c d a = b a c a,\; c = a^{2} b,\; d = a c b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b a c a &\rightarrow c d a \\ + a c b a &\rightarrow d \\ + a^{2} b &\rightarrow c \\ + b a c^{2} &\rightarrow c d c \\ + b a c d &\rightarrow c d^{2} \\ + c a c a &\rightarrow a^{2} c d a \\ + a c b c &\rightarrow d a b \\ + a c b d &\rightarrow d c b a \\ + a c^{2} d a &\rightarrow d c a \\ + b d c a &\rightarrow c d c d a \\ + b a^{3} c d a &\rightarrow c d a c a \\ + c a c^{2} &\rightarrow a^{2} c d c \\ + c a c d &\rightarrow a^{2} c d^{2} \\ + c a^{3} c d a &\rightarrow a^{2} c d a c a \\ + a c^{2} d^{2} &\rightarrow d c d \\ + a c^{2} d c &\rightarrow d c^{2} \\ + b d c^{2} &\rightarrow c d c d c \\ + b d c d &\rightarrow c d c d^{2} \\ + b d a^{2} c d c &\rightarrow c d c d a c^{2} \\ + b d a^{2} c d^{2} &\rightarrow c d c d a c d \\ + b d a^{2} c d a &\rightarrow c d c d a c a \\ + b a^{3} c d c &\rightarrow c d a c^{2} \\ + b a^{3} c d^{2} &\rightarrow c d a c d \\ + c a^{3} c d c &\rightarrow a^{2} c d a c^{2} \\ + c a^{3} c d^{2} &\rightarrow a^{2} c d a c d \\ + a^{2} c d c d c &\rightarrow c d c^{2} \\ + a^{2} c d c d^{2} &\rightarrow c d c d \\ + a^{2} c d c d a &\rightarrow c d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/483.md b/docs/examples/1-relation/483.md new file mode 100644 index 0000000..104adc3 --- /dev/null +++ b/docs/examples/1-relation/483.md @@ -0,0 +1,45 @@ +#\#483 aabaabbaaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b^{2} a^{3} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a^{2} c a d = c a c,\; c = b a,\; d = b c a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b c a^{2} &\rightarrow d \\ + b a &\rightarrow c \\ + c a c &\rightarrow a^{2} c a d \\ + c a^{3} c a d &\rightarrow a^{2} c a d a c \\ + a^{2} c a d a d a c &\rightarrow d a c a d \\ + b c d a c a d &\rightarrow d c a d a d a c \\ + b c a d a c a d &\rightarrow d a c a d a d a c \\ + b d a c a d &\rightarrow a^{2} c a d a d a d a c \\ + a^{2} c a d a c a d a c &\rightarrow c a d a c a d \\ + a^{2} c a d a d a^{3} c a d &\rightarrow d a c a d a c \\ + b c^{2} a d a c a d &\rightarrow d c a d a c a d a c \\ + d a c a d a c a d a c &\rightarrow a^{2} c a d a d a d a c a d \\ + d a c a d a c a d a^{3} c a d &\rightarrow a^{2} c a d a d a d a c a d a c \\ + a^{2} c a d a c a d a^{3} c a d &\rightarrow c a d a c a d a c \\ + d a^{3} c a d a c a d a d a c a d &\rightarrow a^{2} c a d a d a d a c a d a d a c \\ + a^{2} c a d a^{3} c a d a d a d a c a d &\rightarrow d a c a d a d a c a d a c \\ + a^{4} c a d a c a d a d a c a d &\rightarrow c a d a c a d a d a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/484.md b/docs/examples/1-relation/484.md new file mode 100644 index 0000000..39ef861 --- /dev/null +++ b/docs/examples/1-relation/484.md @@ -0,0 +1,47 @@ +#\#484 ababbabaaa=baaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{2} a b a^{3} = b a^{2} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b a b^{2} d a = b c,\; c = a^{2} b a,\; d = a b a^{2} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < c < d < a\): + +\( +\begin{aligned} + a^{2} b a &\rightarrow c \\ + a b a^{2} &\rightarrow d \\ + a b a b^{2} d a &\rightarrow b c \\ + a^{2} b d &\rightarrow c b a^{2} \\ + a^{2} b^{2} c &\rightarrow c b a b^{2} d a \\ + a d &\rightarrow c a \\ + a b a c &\rightarrow d a b a \\ + a b a b^{2} d^{2} &\rightarrow b c b a^{2} \\ + a b a b^{2} d c &\rightarrow b c a b a \\ + a b c &\rightarrow c b^{2} d a \\ + d b a &\rightarrow c b^{2} d a \\ + a c b^{2} d a &\rightarrow c a b a \\ + d b d &\rightarrow c b^{2} d^{2} \\ + d b c &\rightarrow c b^{2} d c \\ + a c b^{2} d c &\rightarrow c^{2} b^{2} d a \\ + a c b^{2} d^{2} &\rightarrow c a b d \\ + a c b^{2} c b^{2} d c &\rightarrow c a b^{2} c \\ + d b^{2} c &\rightarrow c b^{2} c b^{2} d c \\ + a b a b^{2} c b^{2} d c &\rightarrow b c b a b^{2} d a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/485.md b/docs/examples/1-relation/485.md new file mode 100644 index 0000000..932d463 --- /dev/null +++ b/docs/examples/1-relation/485.md @@ -0,0 +1,41 @@ +#\#485 baaaababba=aa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b a b^{2} a = a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c a^{3} d = a^{2},\; c = b a,\; d = c b c,\; e = d a^{2} d +\right\rangle +\) + +## Complete rewriting system + +Using reversed recursive path order (`RevRPOCmp`) with \(e < d < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c b c &\rightarrow d \\ + d a^{2} d &\rightarrow e \\ + c^{2} a &\rightarrow d a^{3} d \\ + c b d &\rightarrow d b c \\ + d a^{2} e &\rightarrow e a^{2} d \\ + c a^{2} &\rightarrow d a^{3} e \\ + c b e &\rightarrow d b d a^{3} e d \\ + d a^{3} e a e &\rightarrow a^{4} d \\ + d a^{3} e a d &\rightarrow a^{2} \\ + c d a^{3} e &\rightarrow d a^{3} d a \\ + d a^{4} &\rightarrow e a^{3} e a d \\ + d a^{3} e a^{3} &\rightarrow a^{5} e a d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/487.md b/docs/examples/1-relation/487.md new file mode 100644 index 0000000..18ab903 --- /dev/null +++ b/docs/examples/1-relation/487.md @@ -0,0 +1,35 @@ +#\#487 abbaabbaba=babbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{2} a^{2} b^{2} a b a = b a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c a b c^{2} = c b c a,\; c = b a,\; d = a b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow b a \\ + a b^{2} a &\rightarrow d \\ + b d a &\rightarrow d^{2} b a \\ + b d^{2} &\rightarrow d^{2} b d \\ + a b^{2} d &\rightarrow d b^{2} a \\ + d b^{2} a d &\rightarrow a d^{2} b d b d \\ + d b^{2} a^{2} &\rightarrow a d^{2} b d b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/488.md b/docs/examples/1-relation/488.md new file mode 100644 index 0000000..bb46d4b --- /dev/null +++ b/docs/examples/1-relation/488.md @@ -0,0 +1,54 @@ +#\#488 baababaaba=aababbaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a b a^{2} b a = a^{2} b a b^{2} a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c a d = a^{2} c e,\; c = b a,\; d = c^{2} a c,\; e = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b c a &\rightarrow e \\ + b a &\rightarrow c \\ + c^{2} a c &\rightarrow d \\ + a^{2} c e &\rightarrow c a d \\ + c a c e &\rightarrow e d \\ + c^{2} a d &\rightarrow d c a c \\ + b d c a c &\rightarrow e a c e \\ + b e d &\rightarrow e c e \\ + b d c a e d &\rightarrow e a c e a c e \\ + b d c a d &\rightarrow e a c e c a c \\ + b d e d &\rightarrow e a c e^{2} \\ + c^{2} a e d &\rightarrow d a c e \\ + c e d &\rightarrow d e \\ + c a d e &\rightarrow e d^{2} \\ + c a d^{2} &\rightarrow a^{2} d e \\ + b d c a^{3} d e &\rightarrow e a c e a d^{2} \\ + b d a^{2} d e &\rightarrow e a c e c a c d \\ + c^{2} a^{3} d e &\rightarrow d a d^{2} \\ + d c a c d &\rightarrow c a^{2} d e \\ + b e c a^{2} d e &\rightarrow e c e c a c d \\ + b d c a e c a^{2} d e &\rightarrow e a c e a c e c a c d \\ + b d e c a^{2} d e &\rightarrow e a c e^{2} c a c d \\ + c^{2} a e c a^{2} d e &\rightarrow d a c e c a c d \\ + c e c a^{2} d e &\rightarrow d e c a c d \\ + c a d c a^{2} d e &\rightarrow a^{2} d e c a c d \\ + d c a c^{2} a^{2} d e &\rightarrow c a^{2} d e c a c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/490.md b/docs/examples/1-relation/490.md new file mode 100644 index 0000000..06f33ab --- /dev/null +++ b/docs/examples/1-relation/490.md @@ -0,0 +1,48 @@ +#\#490 babaabaaba=ababa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a b a^{2} b a^{2} b a = a b a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; e c = a c^{2},\; c = b a,\; d = c a c,\; e = c d a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < e < a < c < b\): + +\( +\begin{aligned} + b a &\rightarrow c \\ + c a c &\rightarrow d \\ + c d a &\rightarrow e \\ + a c^{2} &\rightarrow e c \\ + b e c &\rightarrow c^{3} \\ + c a d &\rightarrow d a c \\ + c e c &\rightarrow d c \\ + c d e c &\rightarrow e c^{2} \\ + a c d &\rightarrow e d \\ + a c e &\rightarrow e^{2} \\ + b e^{2} &\rightarrow c^{2} e \\ + b e d &\rightarrow c^{2} d \\ + c e^{2} &\rightarrow d e \\ + c e d &\rightarrow d^{2} \\ + c d e^{2} &\rightarrow e c e \\ + c d e d &\rightarrow e c d \\ + a e &\rightarrow e d a \\ + a d c &\rightarrow e^{2} c \\ + a d e &\rightarrow e^{3} \\ + a d^{2} &\rightarrow e^{2} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/491.md b/docs/examples/1-relation/491.md new file mode 100644 index 0000000..349343f --- /dev/null +++ b/docs/examples/1-relation/491.md @@ -0,0 +1,38 @@ +#\#491 baabaabbba=abaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{2} b a^{2} b^{3} a = a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c^{2} d = a c,\; c = b a^{2},\; d = b^{3} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{3} a &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a c &\rightarrow c^{2} d \\ + b^{2} c &\rightarrow d a \\ + b d c^{2} d^{2} &\rightarrow d c \\ + b c^{2} &\rightarrow d c^{2} d^{2} c d \\ + d c^{2} d^{2} c d^{2} c d &\rightarrow c^{2} \\ + b d c^{2} d c^{2} &\rightarrow d c^{3} d^{2} c d^{2} c d \\ + d c^{2} d^{2} c d^{2} c^{3} &\rightarrow c^{4} d^{2} c d^{2} c d \\ + b d c^{4} &\rightarrow d c^{3} d^{2} c d^{2} c d^{3} c d^{2} c d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/493.md b/docs/examples/1-relation/493.md new file mode 100644 index 0000000..e1941ca --- /dev/null +++ b/docs/examples/1-relation/493.md @@ -0,0 +1,34 @@ +#\#493 ababbbabba=bbabbaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a b^{3} a b^{2} a = b^{2} a b^{2} a b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a d c d c a = c a c a d,\; c = b^{2},\; d = b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < c < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + b a &\rightarrow d \\ + c a c a d &\rightarrow a d c d c a \\ + b c &\rightarrow c b \\ + b d &\rightarrow c a \\ + c d c a d &\rightarrow d^{2} c d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/494.md b/docs/examples/1-relation/494.md new file mode 100644 index 0000000..b74842f --- /dev/null +++ b/docs/examples/1-relation/494.md @@ -0,0 +1,42 @@ +#\#494 baaaabbbaa=aaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a^{2} = a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b c^{2} b^{3} c = c a,\; c = a^{2},\; d = a b^{3} +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b^{3} &\rightarrow d \\ + b a^{3} d a^{2} &\rightarrow a^{3} \\ + a b^{2} a^{3} &\rightarrow d a^{3} d a^{2} \\ + b a^{3} d a d &\rightarrow a^{2} d \\ + a b^{2} a^{2} d &\rightarrow d a^{3} d a d \\ + a b a^{2} d &\rightarrow d a^{3} d a^{2} d a d \\ + a b a^{3} &\rightarrow d a^{3} d a^{2} d a^{2} \\ + d a^{3} d a^{2} d a^{2} d a d &\rightarrow a^{3} d \\ + d a^{3} d a^{2} d a^{2} d a^{2} &\rightarrow a^{4} \\ + b a^{6} d &\rightarrow a^{4} d a^{2} d a^{2} d a d \\ + b a^{7} &\rightarrow a^{4} d a^{2} d a^{2} d a^{2} \\ + d a^{3} d a^{2} d a^{5} d &\rightarrow a^{5} d a^{2} d a^{2} d a d \\ + d a^{3} d a^{2} d a^{6} &\rightarrow a^{5} d a^{2} d a^{2} d a^{2} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/495.md b/docs/examples/1-relation/495.md new file mode 100644 index 0000000..b5786dd --- /dev/null +++ b/docs/examples/1-relation/495.md @@ -0,0 +1,44 @@ +#\#495 aababbaba=babaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a b^{2} a b a = b a b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; d c a = e a,\; c = a b,\; d = a c^{2} b,\; e = b c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < e < b\): + +\( +\begin{aligned} + b c a &\rightarrow e \\ + e a &\rightarrow d c a \\ + a b &\rightarrow c \\ + a c^{2} b &\rightarrow d \\ + b c^{2} &\rightarrow e b \\ + a e &\rightarrow c^{2} a \\ + a c^{2} e &\rightarrow d c a \\ + e c &\rightarrow d c^{2} \\ + e d &\rightarrow d c d \\ + b c d &\rightarrow d c^{3} b \\ + a d c a &\rightarrow c^{2} a^{2} \\ + a d c d &\rightarrow c^{2} a d \\ + a d c^{2} &\rightarrow c^{2} a c \\ + a c^{2} d c a &\rightarrow d c a^{2} \\ + a c^{2} d c d &\rightarrow d c a d \\ + a c^{2} d c^{2} &\rightarrow d c a c +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/496.md b/docs/examples/1-relation/496.md new file mode 100644 index 0000000..c8c4d3c --- /dev/null +++ b/docs/examples/1-relation/496.md @@ -0,0 +1,35 @@ +#\#496 abaaaababa=babaaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b a^{4} b a b a = b a b a^{3} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; a b c^{2} b d = b a b c a,\; c = a^{2},\; d = a b a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < b < c\): + +\( +\begin{aligned} + c &\rightarrow a^{2} \\ + a b a &\rightarrow d \\ + a b d &\rightarrow d b a \\ + b d a d &\rightarrow d a^{2} d b d \\ + b d a^{2} &\rightarrow d a^{2} d b a \\ + d b a^{2} d &\rightarrow a d a^{2} d b d \\ + d b a^{3} &\rightarrow a d a^{2} d b a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/497.md b/docs/examples/1-relation/497.md new file mode 100644 index 0000000..a1eb656 --- /dev/null +++ b/docs/examples/1-relation/497.md @@ -0,0 +1,66 @@ +#\#497 aabaabbaaa=baaaaba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a^{2} b a^{2} b^{2} a^{3} = b a^{4} b a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; c e = b c^{2} b a,\; c = a^{2},\; d = b c a,\; e = b c b d +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < e < d < a < b\): + +\( +\begin{aligned} + b c b d &\rightarrow e \\ + b c a &\rightarrow d \\ + a^{2} &\rightarrow c \\ + b c^{2} &\rightarrow d a \\ + a c &\rightarrow c a \\ + d a b c &\rightarrow c e a \\ + d a b a &\rightarrow c e \\ + b c b c e &\rightarrow e a b a \\ + c e a b d &\rightarrow d a e \\ + d a d &\rightarrow c e c \\ + b c d a e &\rightarrow d a e a b d \\ + d a b d a e &\rightarrow c e a e a b d \\ + d a e a b a &\rightarrow c e a b c e \\ + c a e a b d &\rightarrow a d a e \\ + c e a b c e a &\rightarrow d a e a b c \\ + c e a b c e c &\rightarrow d a e a d \\ + c e c a b a &\rightarrow d c a e \\ + c e c a b c &\rightarrow d c a e a \\ + c e c a d &\rightarrow d c a e c \\ + b a d a e &\rightarrow d e a b d \\ + b c d c a e &\rightarrow d a e c a b a \\ + d a b d c a e &\rightarrow c e a e c a b a \\ + c a e a b c e a &\rightarrow a d a e a b c \\ + c a e a b c e c &\rightarrow a d a e a d \\ + c a e c a b a &\rightarrow a d c a e \\ + c a e c a b c &\rightarrow a d c a e a \\ + c a e c a d &\rightarrow a d c a e c \\ + c e a b c e d a e &\rightarrow d a e a d e a b d \\ + c e a b c e d c a e &\rightarrow d a e a d e c a b a \\ + c e c a b d a e &\rightarrow d c a e a e a b d \\ + c e c a b d c a e &\rightarrow d c a e a e c a b a \\ + c e c a e a b a &\rightarrow d c a e a b c e \\ + b a d c a e &\rightarrow d e c a b a \\ + c a e a b c e d a e &\rightarrow a d a e a d e a b d \\ + c a e a b c e d c a e &\rightarrow a d a e a d e c a b a \\ + c a e c a b d a e &\rightarrow a d c a e a e a b d \\ + c a e c a b d c a e &\rightarrow a d c a e a e c a b a \\ + c a e c a e a b a &\rightarrow a d c a e a b c e +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/498.md b/docs/examples/1-relation/498.md new file mode 100644 index 0000000..bf884ca --- /dev/null +++ b/docs/examples/1-relation/498.md @@ -0,0 +1,39 @@ +#\#498 bbaaabbaaa=aabba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b^{2} a^{3} b^{2} a^{3} = a^{2} b^{2} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a d a^{2} = d,\; c = b^{2},\; d = a^{2} c a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(c < d < a < b\): + +\( +\begin{aligned} + b^{2} &\rightarrow c \\ + a^{2} c a &\rightarrow d \\ + c a d a^{2} &\rightarrow d \\ + b c &\rightarrow c b \\ + a^{2} d &\rightarrow d^{2} a^{2} \\ + a^{2} c d &\rightarrow d a c a \\ + c a d a d &\rightarrow d a c a \\ + c a d^{2} &\rightarrow d c a \\ + c b a d a^{2} &\rightarrow b d \\ + c b a d a d &\rightarrow b d a c a \\ + c b a d^{2} &\rightarrow b d c a +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/499.md b/docs/examples/1-relation/499.md new file mode 100644 index 0000000..ff548a8 --- /dev/null +++ b/docs/examples/1-relation/499.md @@ -0,0 +1,40 @@ +#\#499 baaaaaabba=aaabaa + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{6} b^{2} a = a^{3} b a^{2} +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; c a^{3} d = a^{3} c,\; c = b a^{2},\; d = a b^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(b < d < c < a\): + +\( +\begin{aligned} + a^{3} c &\rightarrow c a^{3} d \\ + a b^{2} a &\rightarrow d \\ + b a^{2} &\rightarrow c \\ + a b c &\rightarrow d a \\ + a b^{2} d &\rightarrow d b^{2} a \\ + b a c a^{3} d &\rightarrow c a^{2} c \\ + b a d &\rightarrow c b^{2} a \\ + b c a^{3} d &\rightarrow c a c \\ + d a^{4} d &\rightarrow a c a c \\ + d a^{2} c a^{3} d a c &\rightarrow a c a c a^{4} d \\ + b a c a c a^{3} d a c &\rightarrow c a^{2} c a^{4} d \\ + b c a c a^{3} d a c &\rightarrow c a c a^{4} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/500.md b/docs/examples/1-relation/500.md new file mode 100644 index 0000000..162eacc --- /dev/null +++ b/docs/examples/1-relation/500.md @@ -0,0 +1,39 @@ +#\#500 abbbababaa=baabbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; a b^{3} a b a b a^{2} = b a^{2} b^{3} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d, e \;\middle|\; a d e = c a d,\; c = b a,\; d = b^{2} c,\; e = c^{2} a +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(a < d < e < c < b\): + +\( +\begin{aligned} + b^{2} c &\rightarrow d \\ + b a &\rightarrow c \\ + c^{2} a &\rightarrow e \\ + c a d &\rightarrow a d e \\ + b^{2} e &\rightarrow d c a \\ + b c d e &\rightarrow d a d \\ + a d e^{2} &\rightarrow e d \\ + b e d &\rightarrow c d e^{2} \\ + b d a d &\rightarrow d^{2} e \\ + c e d &\rightarrow e d e \\ + b d e d &\rightarrow d^{2} e^{3} +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/501.md b/docs/examples/1-relation/501.md new file mode 100644 index 0000000..3018cbf --- /dev/null +++ b/docs/examples/1-relation/501.md @@ -0,0 +1,46 @@ +#\#501 baaaabbbaa=aaaabbba + +## Original presentation + +\( +\left\langle + a, b \;\middle|\; b a^{4} b^{3} a^{2} = a^{4} b^{3} a +\right\rangle +\) + +## Modified presentation + +\( +\left\langle + a, b, c, d \;\middle|\; b d a^{2} = d a,\; c = a b^{3},\; d = a^{3} c +\right\rangle +\) + +## Complete rewriting system + +Using recursive path order (`RPOCmp`) with \(d < a < c < b\): + +\( +\begin{aligned} + b d a^{2} &\rightarrow d a \\ + a b^{3} &\rightarrow c \\ + a^{3} c &\rightarrow d \\ + b d a d &\rightarrow d^{2} \\ + b d^{2} &\rightarrow d a^{2} c \\ + a b^{2} d a &\rightarrow c d a^{2} \\ + b d a c &\rightarrow d c \\ + a b^{2} d c &\rightarrow c d a c \\ + c d a d &\rightarrow a d a c \\ + a b d c &\rightarrow c d a^{2} c \\ + a b d a &\rightarrow c d a^{3} \\ + c d a^{2} d &\rightarrow a d a^{2} c \\ + c d^{2} &\rightarrow a d c \\ + c d a^{4} &\rightarrow a d a \\ + c d a^{3} d &\rightarrow a d^{2} \\ + a^{4} d c &\rightarrow d^{3} \\ + a^{4} d a &\rightarrow d^{2} a^{4} \\ + a^{4} d^{2} &\rightarrow d^{2} a^{3} d +\end{aligned} +\) + + diff --git a/docs/examples/1-relation/index.md b/docs/examples/1-relation/index.md new file mode 100644 index 0000000..f41ed27 --- /dev/null +++ b/docs/examples/1-relation/index.md @@ -0,0 +1,249 @@ +# Finding complete rewriting systems for 1-relation monoids + +In +[Cirpons, Hivert, Mahboubi, Melquiond, Mitchell and Smith (2026)](https://doi.org/10.1145/3779031.3779101), +the authors present a toolbox for the +[Rocq proof assistant](https://rocq-prover.org/), that can be used to verify the +decidability of the word problem for a given monoid and, in some cases, to +produce the corresponding decision procedure. In particular, the techniques +described in this paper were applied to a large number of 1-relation monoids to +determine their decidability. Of these, 501 required a modified version of the +Knuth-Bendix procedure along with the termination solver `Matchbox` to find a +complete reqriting system, thus proving decidability. + +On the following pages, you will find complete rewriting systems for 412 of +those 501 presentations. The complete rewriting systems described here were +found using the Knuth-Bendix procedure implemented in +[`libsemigroups`](https://libsemigroups.github.io/libsemigroups/), using the +morphocompletion technique as described in +[Pedersen (2005)](https://doi.org/10.1007/3-540-51081-8_141) and by Pestov in +[Slava's Monoid Zoo](https://monoids.net/). + +## Hall of fame + +The following information may be of particular interest. + +### The presentations whose complete rewriting systems contains the most rules (128) + +- [#309 baaaaababa=abbaa](309.md) (128 rules) +- [#394 baaaababa=abbaa](394.md) (87 rules) +- [#010 baaaaabbba=abaa](010.md) (87 rules) +- [#477 abbabbbaba=baabba](477.md) (87 rules) +- [#214 ababbaaaba=baaababa](214.md) (85 rules) +- [#332 abababaaaa=baaba](332.md) (82 rules) + +### The presentations whose complete rewriting systems contains the fewest rules + +- [#064 aaaabababa=babaaaaa](064.md) (3 rules) +- [#273 aaabababa=babaaaa](273.md) (3 rules) +- [#374 abaabababa=babaabaa](374.md) (3 rules) +- [#419 aababababa=babaaa](419.md) (3 rules) +- [#424 aababababa=bababaaa](424.md) (3 rules) +- [#462 abbaaababa=baabbaaa](462.md) (3 rules) + +### The most common rules + +Across all of the complete rewriting systems, there are $7,781$ rules in total, +and $6,572$ unique rules. There are $6,148$ rules that appear exactly once. The +following are the most common rules: + +| Rule | Occurrences | +| :---------: | :---------: | +| $ba \to c$ | 131 | +| $c \to aa$ | 55 | +| $c \to ba$ | 29 | +| $bc \to cb$ | 26 | +| $abb \to d$ | 25 | +| $bb \to c$ | 25 | + +## All presentations + +### 000–049 + +| | | | +| :--------------------------------- | :--------------------------------- | :--------------------------------- | +| [#000 ababbaaaaa=baaaaaba](000.md) | [#001 aabaababba=baaba](001.md) | [#002 abbabbaaaa=babba](002.md) | +| [#003 babbabaaba=abba](003.md) | [#004 baabbaaaba=aabba](004.md) | [#005 baaaaabbba=aaaaa](005.md) | +| [#007 abaaabaaba=babaaa](007.md) | [#008 aabaaaabba=baaabaa](008.md) | [#010 baaaaabbba=abaa](010.md) | +| [#011 babaaababa=aabababa](011.md) | [#012 bababbba=aba](012.md) | [#013 ababaaabba=baaababa](013.md) | +| [#014 babbaababa=abba](014.md) | [#015 baabaabbaa=abbaaba](015.md) | [#016 baaaabbbaa=aaaa](016.md) | +| [#017 aababbabba=babaa](017.md) | [#018 aaabaaabba=baaabaa](018.md) | [#020 aaabaabbaa=baaaba](020.md) | +| [#021 aaaabaabba=baabaaa](021.md) | [#023 baaabbbbba=aa](023.md) | [#024 baaabaaaba=abbaa](024.md) | +| [#025 abbababaaa=baabba](025.md) | [#026 baaabaabaa=ababa](026.md) | [#028 aaabababba=babaaa](028.md) | +| [#029 abbabbaa=baba](029.md) | [#030 baabbaabba=abababa](030.md) | [#031 aaaabababa=babaa](031.md) | +| [#032 baaabababa=abbaa](032.md) | [#034 baaaabbbaa=abaaa](034.md) | [#036 aababbaaaa=baaaaaba](036.md) | +| [#037 baaaabbbaa=aba](037.md) | [#038 babaabbbba=abbbbaba](038.md) | [#040 baabbaabba=aa](040.md) | +| [#042 baabababba=a](042.md) | [#043 baaabbabba=aa](043.md) | [#045 baabbaabaa=aabbaa](045.md) | +| [#047 ababbababa=babababa](047.md) | [#048 abbababbba=baba](048.md) | [#049 baabaabba=a](049.md) | + +### 050–099 + +| | | | +| :--------------------------------- | :--------------------------------- | :--------------------------------- | +| [#050 babaabaaba=aabbabaa](050.md) | [#051 baaaaaabba=aaaaa](051.md) | [#052 abaabbabba=babbaaba](052.md) | +| [#053 baaababbaa=abababa](053.md) | [#054 bbaaabbaaa=abbaa](054.md) | [#055 baabaabbba=aa](055.md) | +| [#056 abbabbabaa=baba](056.md) | [#057 abaabbbaba=babaa](057.md) | [#058 baabbaaaba=abbaa](058.md) | +| [#059 abbabbbaba=babba](059.md) | [#060 baabbbabba=a](060.md) | [#061 baaababbba=abaa](061.md) | +| [#062 baabbbbba=aa](062.md) | [#064 aaaabababa=babaaaaa](064.md) | [#066 abababbba=baba](066.md) | +| [#067 baaaaabba=aabaa](067.md) | [#068 baaaabbba=aa](068.md) | [#069 abaabaabba=babaa](069.md) | +| [#070 abababbaaa=baababa](070.md) | [#071 baababbba=aba](071.md) | [#072 baaaaabbba=aaa](072.md) | +| [#074 abaababaa=baaba](074.md) | [#075 baaababba=aba](075.md) | [#076 baaaabbbba=aa](076.md) | +| [#077 aabbabbaba=babaa](077.md) | [#078 baaaabbbba=aaa](078.md) | [#080 aaaababbaa=babaaa](080.md) | +| [#081 aabaababba=baabaa](081.md) | [#082 abbaaabbaa=babba](082.md) | [#083 baabaabbaa=aabbaaba](083.md) | +| [#086 aaabbbaaba=baabbaa](086.md) | [#088 bbaaabbbba=abbaa](088.md) | [#089 baabbaabba=aba](089.md) | +| [#091 abaaabbaba=babaa](091.md) | [#092 abbaababba=bababbaa](092.md) | [#093 aabaaaabba=babaa](093.md) | +| [#094 baaaabba=aa](094.md) | [#095 babbabbbba=abba](095.md) | [#096 baabaabbaa=abababa](096.md) | +| [#097 baababbbaa=aaba](097.md) | [#098 baaaabba=aaa](098.md) | | + +### 100–149 + +| | | | +| :--------------------------------- | :--------------------------------- | :-------------------------------- | +| [#100 aaabababaa=baaaaba](100.md) | [#101 baaaaabbba=aaaa](101.md) | [#102 baaabbaba=aa](102.md) | +| [#103 abbabbaa=babba](103.md) | [#104 babaaabbba=aba](104.md) | [#107 baaaabbbba=aaaa](107.md) | +| [#108 baabbabba=a](108.md) | [#110 abbaababa=baabbaa](110.md) | [#111 baabbabaa=aabba](111.md) | +| [#112 baababba=aba](112.md) | [#113 abbababbaa=babba](113.md) | [#114 aababbabaa=babaaba](114.md) | +| [#115 bbaabbbbaa=aabba](115.md) | [#116 abababaaba=bababaa](116.md) | [#117 baaaaabbba=abaaa](117.md) | +| [#118 ababaaaaba=baba](118.md) | [#119 bbaabbaa=abba](119.md) | [#121 baaaaaabba=aa](121.md) | +| [#122 babaababaa=abaabba](122.md) | [#123 baabbaaba=abbaa](123.md) | [#124 bbaabbbbaa=aabbbba](124.md) | +| [#126 baaaababba=aaa](126.md) | [#127 baaabbabaa=abaaabba](127.md) | [#128 aaaabbabba=babaa](128.md) | +| [#129 baaaaabba=abaaaa](129.md) | [#130 baaaaabbaa=aaa](130.md) | [#131 babbaaabba=abbaa](131.md) | +| [#132 baabaababa=abaabbaa](132.md) | [#133 bbbaabbbaa=abbba](133.md) | [#134 baabbbba=aa](134.md) | +| [#135 baabbba=aa](135.md) | [#136 abaaababa=baaaba](136.md) | [#137 abaabbbbba=babaa](137.md) | +| [#138 aababbbaa=baaba](138.md) | [#139 abababbaaa=baaba](139.md) | [#140 abaaabaaba=baaaba](140.md) | +| [#141 ababaaabba=baba](141.md) | [#142 baabaaaaba=aabaaba](142.md) | [#143 baaaabbbba=abaa](143.md) | +| [#144 baaaaababa=abaaa](144.md) | [#145 baababaaba=a](145.md) | [#147 baaaaabbaa=aaaa](147.md) | +| [#148 ababaabbba=baba](148.md) | [#149 abaababaa=baabaa](149.md) | | + +### 150–199 + +| | | | +| :--------------------------------- | :--------------------------------- | :--------------------------------- | +| [#151 ababaababa=bababaa](151.md) | [#152 babaaaabba=aaabbaba](152.md) | [#153 baaaababaa=abaaba](153.md) | +| [#154 aababaabba=babaa](154.md) | [#156 ababaabaaa=baba](156.md) | [#157 bababaabba=ababa](157.md) | +| [#160 baaabbabba=abbbaa](160.md) | [#161 baaaababba=abaaa](161.md) | [#163 aabbaababa=baaabbaa](163.md) | +| [#166 abbabaaaba=baba](166.md) | [#168 baaabbbba=aa](168.md) | [#169 abbaababaa=baaba](169.md) | +| [#170 baaabbbaa=aaa](170.md) | [#171 baaabbabaa=aaabba](171.md) | [#172 baaabbbba=aaa](172.md) | +| [#173 baabbabbba=a](173.md) | [#174 ababaaaba=baba](174.md) | [#175 baaabaabaa=aabba](175.md) | +| [#176 bababbbba=aba](176.md) | [#177 abbababbaa=bababa](177.md) | [#179 babababbba=abbaba](179.md) | +| [#180 babababbba=ababa](180.md) | [#181 aabbaabaa=baaba](181.md) | [#182 baaabbba=aa](182.md) | +| [#183 abbabbabba=bbabbaa](183.md) | [#184 abababbaba=bababa](184.md) | [#185 baaabbba=aaa](185.md) | +| [#186 baabaaaba=abaa](186.md) | [#187 babbababa=abba](187.md) | [#188 abbaaababa=baabbaa](188.md) | +| [#189 ababaabbaa=baabaaba](189.md) | [#190 abaaabbaba=babaaa](190.md) | [#191 baaaaaabba=aaaaaa](191.md) | +| [#192 baaabbbbaa=aaa](192.md) | [#194 baaabbaaba=aabbbaa](194.md) | [#195 baabbababa=abbaba](195.md) | +| [#196 babbababba=abbabba](196.md) | [#198 baaaaabbaa=abaaaa](198.md) | [#199 ababbababa=baaba](199.md) | + +### 200–249 + +| | | | +| :--------------------------------- | :--------------------------------- | :--------------------------------- | +| [#200 ababaaabaa=baba](200.md) | [#203 baabbabbaa=aba](203.md) | [#204 aabaaababa=baabaa](204.md) | +| [#205 baababbaa=aaba](205.md) | [#206 abababbaa=baba](206.md) | [#207 aababbaaaa=baaaaba](207.md) | +| [#208 abbbabbbaa=babbba](208.md) | [#209 abbaabbbaa=baabba](209.md) | [#210 baaabbaaba=abbabaa](210.md) | +| [#211 baaaabbba=aaa](211.md) | [#213 abbaabbbba=babbaa](213.md) | [#214 ababbaaaba=baaababa](214.md) | +| [#215 bababababa=ababbaba](215.md) | [#216 abbababbaa=bababba](216.md) | [#217 babaaabba=ababa](217.md) | +| [#218 abaababba=babaa](218.md) | [#219 bababaaaba=aababa](219.md) | [#220 baaabbaaba=abaa](220.md) | +| [#221 aaababbaba=babaa](221.md) | [#222 aabaaababa=babaaa](222.md) | [#224 babababbba=aba](224.md) | +| [#225 abbaabbaa=baba](225.md) | [#226 aabaabbaaa=baaaba](226.md) | [#227 baaaabbba=aaaa](227.md) | +| [#229 abbabbabaa=bababba](229.md) | [#231 abbaababba=babbaa](231.md) | [#232 baaabbbbaa=aba](232.md) | +| [#233 abaababaaa=baaabaa](233.md) | [#234 baaababbaa=abbaaaba](234.md) | [#235 abaabababa=baaba](235.md) | +| [#236 babaaaaaba=ababaa](236.md) | [#237 ababbaba=bababa](237.md) | [#238 baabaabaa=aabbaa](238.md) | +| [#239 abaabaabba=baababaa](239.md) | [#242 abbabbbaaa=bababba](242.md) | [#243 baaaabbbaa=a](243.md) | +| [#244 baaaabba=aaaa](244.md) | [#245 abaabaaba=babaa](245.md) | [#247 abaaabbaaa=baaaaba](247.md) | +| [#248 baaaaabbaa=a](248.md) | [#249 baaaabaaaa=aaaaaba](249.md) | | + +### 250–299 + +| | | | +| :--------------------------------- | :--------------------------------- | :--------------------------------- | +| [#250 ababbbbaaa=baba](250.md) | [#251 baabaabaa=ababa](251.md) | [#252 baaaaaabba=aaa](252.md) | +| [#255 baabababaa=abbaaba](255.md) | [#256 ababbaabaa=baabaaba](256.md) | [#257 baaabbbaba=aa](257.md) | +| [#258 aabbaabbaa=baabba](258.md) | [#259 bababbbaba=ababbba](259.md) | [#260 baaabbaa=abaaa](260.md) | +| [#262 baabaaabba=aabaa](262.md) | [#263 aabaabaaba=baabaa](263.md) | [#264 abaabbaaba=baaba](264.md) | +| [#265 baaaabbaba=aa](265.md) | [#266 ababbabbaa=baba](266.md) | [#267 ababbabbba=babbbaba](267.md) | +| [#268 aababaabaa=baaba](268.md) | [#269 baaaaaabba=aaaa](269.md) | [#270 baaaabbaba=aaa](270.md) | +| [#271 baaaaabbba=abbaa](271.md) | [#272 ababababa=babaaba](272.md) | [#273 aaabababa=babaaaa](273.md) | +| [#275 baaabba=aa](275.md) | [#276 aabaababaa=baaba](276.md) | [#277 baaaababaa=aaaba](277.md) | +| [#278 baababbba=a](278.md) | [#279 abbabaa=baba](279.md) | [#280 abaabbabaa=babaaaba](280.md) | +| [#282 abababaaba=babaa](282.md) | [#283 baabbbaaba=aabbbaa](283.md) | [#285 baaabaaaba=aaabba](285.md) | +| [#286 bababaaaba=aabababa](286.md) | [#288 abbaabbaa=babba](288.md) | [#289 abbabbbaba=bababba](289.md) | +| [#290 abaabbaa=baaba](290.md) | [#291 baabbabbaa=aabba](291.md) | [#292 baaaaabba=aa](292.md) | +| [#293 babaaaabba=ababa](293.md) | [#294 babbaaba=abba](294.md) | [#296 ababbbabba=babbaba](296.md) | +| [#297 baabaabba=aa](297.md) | [#299 babaababba=ababbaba](299.md) | | + +### 300–349 + +| | | | +| :--------------------------------- | :-------------------------------- | :--------------------------------- | +| [#300 aabbaabbba=baabba](300.md) | [#301 abaabababa=babaaba](301.md) | [#302 babaabbbba=aba](302.md) | +| [#303 abbababaa=baabba](303.md) | [#305 baaaabbaa=abaaa](305.md) | [#306 aabababbaa=baaba](306.md) | +| [#307 baabbbbbba=aa](307.md) | [#308 baabbaa=aabba](308.md) | [#309 baaaaababa=abbaa](309.md) | +| [#310 babaababa=abababa](310.md) | [#312 baaaabbaba=abaaa](312.md) | [#313 abaabbaaaa=baaaba](313.md) | +| [#315 aaaabbaaba=baabaaa](315.md) | [#317 bababbaaba=ababba](317.md) | [#318 baababba=a](318.md) | +| [#321 baabbaabaa=abaabbaa](321.md) | [#322 abaabaaba=baabaa](322.md) | [#323 aaaaababba=babaaaa](323.md) | +| [#324 baabaaba=aabba](324.md) | [#325 abaaaababa=baaabaa](325.md) | [#326 baaaabbaaa=aaaabbaa](326.md) | +| [#327 abaabbaba=babaa](327.md) | [#328 abbabbbba=babba](328.md) | [#329 babaaabbba=aabbbaba](329.md) | +| [#330 abaaababba=babaa](330.md) | [#331 abaabaaaba=baaba](331.md) | [#332 abababaaaa=baaba](332.md) | +| [#333 aababababa=babaa](333.md) | [#334 abbabbbaaa=baaabba](334.md) | [#335 baaabbbaba=abaa](335.md) | +| [#336 baababbbaa=abba](336.md) | [#337 baaaabbaa=abaaaa](337.md) | [#338 abaabaaaba=babaa](338.md) | +| [#339 aaabaaabba=baaaba](339.md) | [#340 baabababaa=aaaba](340.md) | [#342 babaababba=ababa](342.md) | +| [#344 babbbabba=abbba](344.md) | [#346 baababbaa=abaaba](346.md) | [#347 ababbbabba=baba](347.md) | +| [#348 baaaabbaba=abaa](348.md) | | | + +### 350–399 + +| | | | +| :--------------------------------- | :-------------------------------- | :--------------------------------- | +| [#350 aaaabaabba=babaa](350.md) | [#352 baabaaaba=abaaba](352.md) | [#353 abaabbabaa=baaba](353.md) | +| [#354 baabbaaaba=aabbaa](354.md) | [#355 baaabbaba=a](355.md) | [#356 baabaabaa=ababaa](356.md) | +| [#357 abbabbaaa=babba](357.md) | [#358 baaabbabaa=aabba](358.md) | [#359 ababaaabaa=baaba](359.md) | +| [#360 baabaabaa=abaaba](360.md) | [#361 baaaabaaba=a](361.md) | [#362 abbaabbaba=bababbaa](362.md) | +| [#363 aabaabbaba=babaaba](363.md) | [#364 aabaababaa=baabaa](364.md) | [#365 abaababbaa=baaba](365.md) | +| [#366 aaababaaba=babaa](366.md) | [#367 baaababaaa=abaaaba](367.md) | [#368 aababbbaba=babbaa](368.md) | +| [#369 ababbabbaa=babbaaba](369.md) | [#370 baaababaaa=aababaa](370.md) | [#372 abbabbaaba=babba](372.md) | +| [#374 abaabababa=babaabaa](374.md) | [#375 baabaabba=abaa](375.md) | [#376 abaabababa=babaa](376.md) | +| [#377 babaababaa=ababa](377.md) | [#378 baabaaaba=aaabbaa](378.md) | [#379 baaaabaaba=aabaa](379.md) | +| [#381 babbaaaba=ababba](381.md) | [#382 aaabbababa=babaaa](382.md) | [#383 bababbbbba=aba](383.md) | +| [#384 abaabaaaba=baabaaa](384.md) | [#385 bbaaabbbba=aa](385.md) | [#386 baabaaabaa=aba](386.md) | +| [#388 babaabaaba=aabbaba](388.md) | [#389 baaababaa=aababa](389.md) | [#390 abaaabbaba=babaaaba](390.md) | +| [#391 bababaaba=ababa](391.md) | [#392 baababbaba=a](392.md) | [#393 babaaaabba=ababaa](393.md) | +| [#394 baaaababa=abbaa](394.md) | [#395 baaababba=aa](395.md) | [#396 abaaaababa=babaa](396.md) | +| [#397 abbabbbbaa=bbaabba](397.md) | [#399 ababaaabba=babaaa](399.md) | | + +### 400–449 + +| | | | +| :--------------------------------- | :--------------------------------- | :--------------------------------- | +| [#400 baaaabbaaa=a](400.md) | [#403 babaaaaaba=aaaababa](403.md) | [#404 abaaabaaba=babaa](404.md) | +| [#405 aabaabbaaa=baaabaa](405.md) | [#407 abbabbabaa=baabba](407.md) | [#408 babaababaa=aba](408.md) | +| [#409 aaababbaaa=baaaba](409.md) | [#410 baaaaaabba=abaaaaa](410.md) | [#411 aabaabbaba=babaabaa](411.md) | +| [#412 abaababbaa=baabaaba](412.md) | [#413 ababaabbaa=baababa](413.md) | [#415 bbabaaabba=abbabaa](415.md) | +| [#416 baaaababaa=aa](416.md) | [#417 baaababbaa=aaaba](417.md) | [#418 abaababbaa=babaaba](418.md) | +| [#419 aababababa=babaaa](419.md) | [#421 abaababa=baaba](421.md) | [#422 aabbbaaba=baabbaa](422.md) | +| [#423 baaaaabbaa=abaaaaa](423.md) | [#424 aababababa=bababaaa](424.md) | [#426 bababaaaba=ababa](426.md) | +| [#427 aaabababba=babaa](427.md) | [#428 aabbaabbaa=baaba](428.md) | [#429 baaaaabba=aaa](429.md) | +| [#430 baabaabba=ababa](430.md) | [#431 abababbaba=babababa](431.md) | [#432 baaabbbaaa=aba](432.md) | +| [#433 baaaabaaba=abbaa](433.md) | [#434 ababaaaba=babaaa](434.md) | [#436 baabaaaaba=abaa](436.md) | +| [#437 baaaaabbba=aa](437.md) | [#438 baabbababa=abbaabba](438.md) | [#439 baaaabbaaa=aaaabba](439.md) | +| [#440 ababbabba=baba](440.md) | [#441 abaabbaaaa=baaba](441.md) | [#444 abbababaa=bababa](444.md) | +| [#445 abaaaababa=baabaaa](445.md) | [#446 baaaaabba=aaaa](446.md) | [#447 baaababa=a](447.md) | +| [#448 baabaaabba=abaabaa](448.md) | | | + +### 450–501 + +| | | | +| :--------------------------------- | :--------------------------------- | :--------------------------------- | +| [#450 baabababaa=aabbaaba](450.md) | [#451 baaabaaba=a](451.md) | [#452 baabaabbaa=aaaba](452.md) | +| [#453 aabbababaa=baaabba](453.md) | [#454 ababababaa=babaa](454.md) | [#457 ababaabbaa=baaababa](457.md) | +| [#458 baaabbbaa=aaabbba](458.md) | [#459 baaabaaaba=aabaa](459.md) | [#460 aabbaaabba=baabbaa](460.md) | +| [#461 baaaababaa=abaaa](461.md) | [#462 abbaaababa=baabbaaa](462.md) | [#463 baabababba=aba](463.md) | +| [#465 abbaababaa=baababa](465.md) | [#468 baabbbbbaa=aba](468.md) | [#470 ababbabaa=baba](470.md) | +| [#471 abaaababa=baabaa](471.md) | [#472 baababbbaa=abbabba](472.md) | [#473 abaabaaba=baaaba](473.md) | +| [#474 bbaaabbaa=abba](474.md) | [#475 baaabbaaba=abbaa](475.md) | [#476 abababbaaa=baaababa](476.md) | +| [#477 abbabbbaba=baabba](477.md) | [#478 babbabbaba=abbabba](478.md) | [#479 bbaabbabba=a](479.md) | +| [#481 babbabbbba=abbba](481.md) | [#482 aabaaabbaa=baaaba](482.md) | [#483 aabaabbaaa=baaba](483.md) | +| [#484 ababbabaaa=baaba](484.md) | [#485 baaaababba=aa](485.md) | [#487 abbaabbaba=babbaa](487.md) | +| [#488 baababaaba=aababbaa](488.md) | [#490 babaabaaba=ababa](490.md) | [#491 baabaabbba=abaa](491.md) | +| [#493 ababbbabba=bbabbaba](493.md) | [#494 baaaabbbaa=aaa](494.md) | [#495 aababbaba=babaa](495.md) | +| [#496 abaaaababa=babaaa](496.md) | [#497 aabaabbaaa=baaaaba](497.md) | [#498 bbaaabbaaa=aabba](498.md) | +| [#499 baaaaaabba=aaabaa](499.md) | [#500 abbbababaa=baabbba](500.md) | [#501 baaaabbbaa=aaaabbba](501.md) | diff --git a/docs/examples/atlas/index.md b/docs/examples/atlas/index.md index 5c19066..5150c6e 100644 --- a/docs/examples/atlas/index.md +++ b/docs/examples/atlas/index.md @@ -1,17 +1,27 @@ -# Verifying presentations in the ATLAS of Finite Groups +# Verifying presentations for sporadic simple groups In this example, we will verify the validity of the presentations of the -sporadic groups as given in [this ATLAS][atlas]. -In particular, where possible, we will use the Todd-Coxeter algorithm to check -that the size of group defined by the presentations is equal to the claimed -size. Where the claimed size of the group is very large, we will instead -calculate the size of small-index subgroups. +sporadic groups as given in +[this ATLAS](https://brauer.maths.qmul.ac.uk/Atlas/v3/) we will use the +Todd-Coxeter algorithm to check that the size of group defined by the +presentations is equal to the claimed size. Where the claimed size of the group +is very large, we will instead calculate the index of some low-index subgroups. +This is a work-in-progress, so some of the pages may contain incomplete +information. When this is the case, the pages will be clearly marked as such. -??? info "libsemigroups_pybind11 version" +!!! warning "Disclaimer" + + The purpose of the following pages is not to prove the validity of the + presentations of the sporadic groups; there are better approaches than those + taken here. Instead, the purpose of these pages is to highlight some of the + ways you might use [libsemigroups_pybind11][] and its implementation of the + Todd-Coxeter algorithm to perform difficult enumerations. + +??? info "[libsemigroups_pybind11][] version" All examples provided on the subsequent subpages were run using - libsemigroups_pybind11 version 1.4.4 on a laptop with a 13th Gen Intel(R) - Core(TM) i7-13700H processor and 64 GB RAM. + [libsemigroups_pybind11][] version 1.4.4 on a either laptop with a 13th Gen Intel(R) + Core(TM) i7-13700H processor and 64 GB RAM or a 2024 MacBook Pro M4 with 48GB of RAM. ??? info "Defining groups with monoid presentations" @@ -36,7 +46,7 @@ calculate the size of small-index subgroups. In the subsequent subpages, we will use lowercase letters for the generators that are given in the presentations in the ATLAS, and their uppercase counterparts to represent their inverses. Therefore, many of our - examples will in a similar way to: + examples will begin in a similar way to: ```py from libsemigroups_pybind11 import presentation, Presentation @@ -48,8 +58,8 @@ calculate the size of small-index subgroups. The algorithms in [libsemigroups_pybind11][] were written for semigroups and monoids. This means that there are no group-specific optimisations. -The following tables summarise the results of this project. Click on a group -to see more information. +The following tables summarise the results of this project. Click on a group to +see more information.
@@ -68,10 +78,10 @@ to see more information. M22 - + M23 - + M24 @@ -83,16 +93,16 @@ to see more information. Leech lattice groups - HS - + HS + J2 Co1 - + Co2 Co3 - McL + McL Suz @@ -103,10 +113,10 @@ to see more information. Monster sections - He + He HN Th - + Fi22 Fi23 @@ -122,15 +132,15 @@ to see more information. Pariahs - + J1 O'N - + J3 - Ru - + Ru + J4 Ly @@ -143,7 +153,7 @@ to see more information. Miscellaneous - T + T
@@ -157,24 +167,37 @@ to see more information. - The presentation defines a group of the correct size + The presentation defines a group of the correct size. - The presentation has a subgroup of the correct size + The presentation has at least 1 subgroup of the correct index. + + The source of the presentation is not the ATLAS and the + presentation has at least 1 subgroup of the correct index. + + + + + + The presentation has at least 1 subgroup of the correct index, and + at least 1 subgroup whose index could not be verified. + + + - Todd-Coxeter did not terminate + Work in progress. - No presentation is provided + No presentation is provided.
@@ -182,4 +205,3 @@ to see more information. [atlas]: https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ -[libsemigroups_pybind11]: https://libsemigroups.github.io/libsemigroups_pybind11/index.html diff --git a/docs/examples/atlas/leech-lattice/co2/index.md b/docs/examples/atlas/leech-lattice/co2/index.md index 0c900db..8f49820 100644 --- a/docs/examples/atlas/leech-lattice/co2/index.md +++ b/docs/examples/atlas/leech-lattice/co2/index.md @@ -2,11 +2,9 @@ > Order: $42,305,421,312,000$ -> Presentation: TODO +> Presentation: +> $\langle a, b \mid a^2 = b^5 = (abb)^9 = [a, b]^4 = [a, b^2]^4 = [a, bab]^3 = [a, bab^2ab]^2 = [a, bab^{−2}]^3 = [a, b^{−2}abab^{−2}]^2 = (abab^2ab^{−1}ab^{−2})^7 = 1 \rangle$ -On this page, we provide links to verifications that the maximal subgroups of -the Conway group Co~2~ define groups of the correct order. +!!! info "Work in Progress" -## Maximal subgroups - -The following are maximal subgroups that have been verified: + This page is a work in progress. diff --git a/docs/examples/atlas/leech-lattice/hs.md b/docs/examples/atlas/leech-lattice/hs.md index 01d3061..7d8b6c4 100644 --- a/docs/examples/atlas/leech-lattice/hs.md +++ b/docs/examples/atlas/leech-lattice/hs.md @@ -1,28 +1,166 @@ # Higman-Sims group HS -> Order: $44,352,000$ +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/HS/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/HS/). -> Presentation: TODO +## Claim -On this page, we verify that the above claimed presentation of the Higman-Sims group HS -defines a group of order $44,352,000$. +$$ + \mathrm{HS} = \langle a,b \mid a^{2} = b^{5} = (ab)^{11} = (ab^{2})^{10} = [a,b]^{5} = [a,b^{2}]^{6} = [a,bab]^{3} = ababab^{2}ab^{-1}ab^{-2}ab^{-1}ab^{2}abab(ab^{-2})^{4} = ab (ab^{2}(ab^{-2})^{2})^{2} ab^{2}abab^{2} (ab^{-1}ab^{2})^{2} = abab(ab^{2})^{2}ab (ab^{-1})^{2} ab(ab^{2})^{2} ababab^{-2}ab^{-1}ab^{-2} = 1 +\rangle +$$ + +with + +$$ + |\mathrm{HS}| = 44,352,000. +$$ + +On this page, we verify that the above claimed presentation of the Higman-Sims +group HS defines a group of order $44,352,000$. ## The code -=== "Python" +In [libsemigroups_pybind11][], the following script constructs the presentation +for HS and runs the Todd-Coxeter algorithm, whilst taking advantage of +techniques such as `lookahead` and `lookbehind`. + +The code below took about 1 hour 20 minutes to run on a 2024 MacBook Pro M4 with +48GB of RAM: + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + KnuthBendix, + Order, + Presentation, + ToddCoxeter, + congruence_kind, + presentation, + to, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^5"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^10"), "") +presentation.add_rule(p, parse_relations("(a,b)^5"), "") +presentation.add_rule(p, parse_relations("(a,bab)^3"), "") +presentation.add_rule(p, parse_relations("(a,b^2)^6"), "") +presentation.add_rule(p, parse_relations("ababab^2aBaB^2aBab^2abab(aB^2)^4"), "") +presentation.add_rule(p, parse_relations("ab(ab^2(aB^2)^2)^2ab^2abab^2(aBab^2)^2"), "") +presentation.add_rule(p, parse_relations("abab(ab^2)^2ab(aB)^2ab(ab^2)^2ababaB^2aBaB^2"), "") +presentation.add_rule(p, parse_relations("(ababab^2aBaB^2ababaB)^2"), "") +presentation.add_rule(p, parse_relations("(ababab^2)^2ababaBabab(ab^2)^3ababaB"), "") +presentation.add_rule(p, parse_relations("ab(abab^2)^3ababab^2aBabaB^2abaBab^2"), "") + +# The "coset enumerator" in libsemigroups is for semigroups and monoids, and +# has no optimizations for groups. To avoid longer relation words, we try to +# balance the length of the left- and right-hand sides. +presentation.balance(p, "abAB", "ABab") + +# Remove the inverses to limit memory usage later. +presentation.replace_subword(p, "A", "a") +presentation.replace_subword(p, "B", "bbbb") +p.alphabet("ab") + +# Reduce the rules in the presentation to avoid unnecessarily long relations +kb = KnuthBendix(congruence_kind.twosided, p) +kb.run_for(timedelta(microseconds=1)) - Coming soon! +# Create the ToddCoxeter instance using the reduced rules from the KnuthBendix +tc = ToddCoxeter(congruence_kind.twosided, to(kb, rtype=(Presentation,))) +tc.lookahead_extent(ToddCoxeter.options.lookahead_extent.full) +limit = 100_000_000 -=== "ACE" +# If we just run Todd-Coxeter, this will very quickly use all the memory +# available, and seemingly only produces a graph with ~73% of the edges +# defined. So, we limit the number of nodes in the graph to "limit", +# and then repeatedly try to shrink the graph using lookaheads and lookbehinds. +# Lookbehinds are relatively slow, and so we only do this when lookahead isn't +# any longer shrinking the graph. +while not tc.finished(): + tc.run_until(lambda: tc.number_of_nodes_active() >= limit) + num_lookaheads = 0 + while True: + num_lookaheads += 1 + num_nodes = tc.number_of_nodes_active() + tc.perform_lookahead() + diff = num_nodes - tc.number_of_nodes_active() + if diff <= 1_000_000: + break + if num_lookaheads == 1: + tc.standardize(Order.shortlex) + tc.perform_lookbehind() + limit *= 1.1 - Coming soon! +print(f"The size of the group is {tc.number_of_classes()}") +``` ## The output -=== "Python" +The truncated output of the enumeration is below: - Coming soon! +??? info "Truncated output from the Python script" -=== "ACE" + ``` + #0: KnuthBendix: running for approx. 1µs + ----------------------------------------------------------------------------------------------- + #0: KnuthBendix: locally confluent for: 1 / 13 ( 7.7%) rules (0s) + +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + #0: KnuthBendix: STARTING . . . + +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + #0: KnuthBendix: |A| = 2, |R| = 13, |u| + |v| ∈ [2, 57], ∑(|u| + |v|) = 519 + #0: KnuthBendix: rules 13 (active) | 16 (defined) + #0: KnuthBendix: diff -3 (active) | +0 (defined) + #0: KnuthBendix: time 122µs (total) | 130,390/s (defined) + ----------------------------------------------------------------------------------------------- + #0: KnuthBendix: |A| = 2, |R| = 13, |u| + |v| ∈ [2, 57], ∑(|u| + |v|) = 519 + +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + #0: KnuthBendix: STOPPING -- timed out! + +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: running until predicate returns true or finished + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 2, |R| = 13, |u| + |v| ∈ [2, 57], ∑(|u| + |v|) = 519 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 2 | 0.0% + #0: ToddCoxeter: time | run 0 = 18µs | all runs = 18µs | elapsed = 49µs + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 STOP + #0: ToddCoxeter: HLT 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 5,000,150 | 85,626 | 5,085,776 + #0: ToddCoxeter: diff 0.0.0 | +5,000,149 | +85,626 | +5,085,775 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,277,936 | 4,722,364 | 52.8% + #0: ToddCoxeter: diff 0.0.0 | +5,277,936 | +4,722,362 | +52.8% + #0: ToddCoxeter: phase 0.0 = 87ms | run 0 = 87ms | all runs = 87ms | elapsed = 87ms + [... lines omitted ...] + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 26 STOP (finished) + #0: ToddCoxeter: run 26 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 8min50s (99%) | - (0%) + #0: ToddCoxeter: all runs | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 26 | 1 | 10 | 0 + #0: ToddCoxeter: time spent in | 40min59s (52%) | 28min20s (36%) | 9min26s (12%) | - (0%) + #0: ToddCoxeter: phase 26.1 = 8min53s | run 26 = 8min53s | all runs = 1h18min48s | elapsed = 1h19min23s + #0: ToddCoxeter: lenlex standardizing the word graph, this might take a few moments! + #0: ToddCoxeter: the word graph was lenlex standardized in 7.676s + The size of the group is 44352000 + ``` - Coming soon! +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $44,352,000$ diff --git a/docs/examples/atlas/leech-lattice/j2.md b/docs/examples/atlas/leech-lattice/j2.md deleted file mode 100644 index ad93c5e..0000000 --- a/docs/examples/atlas/leech-lattice/j2.md +++ /dev/null @@ -1,28 +0,0 @@ -# Janko group J~2~ - -> Order: $604,800$ - -> Presentation: TODO - -On this page, we verify that the above claimed presentation of the Janko group J~2~ -defines a group of order $604,800$. - -## The code - -=== "Python" - - Coming soon! - -=== "ACE" - - Coming soon! - -## The output - -=== "Python" - - Coming soon! - -=== "ACE" - - Coming soon! diff --git a/docs/examples/atlas/leech-lattice/j2/index.md b/docs/examples/atlas/leech-lattice/j2/index.md new file mode 100644 index 0000000..d78869f --- /dev/null +++ b/docs/examples/atlas/leech-lattice/j2/index.md @@ -0,0 +1,8 @@ +# Janko group J2, Hall–Janko group HJ + +On this page, we provide links to verifications that the claimed presentations +for the Janko group J~2~ defines groups of the correct order. Two presentations +are given for J~2~: + +- [Presentation 1](j2-p1.md) +- [Presentation 2](j2-p2.md) diff --git a/docs/examples/atlas/leech-lattice/j2/j2-p1.md b/docs/examples/atlas/leech-lattice/j2/j2-p1.md new file mode 100644 index 0000000..333e2f8 --- /dev/null +++ b/docs/examples/atlas/leech-lattice/j2/j2-p1.md @@ -0,0 +1,160 @@ +# Janko group J2, Hall–Janko group HJ (Presentation 1) + +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/pres/J2G1-P1](https://brauer.maths.qmul.ac.uk/Atlas/v3/pres/J2G1-P1). + +## Claim + +$$ + \mathrm{J}_2 = \langle a,b \mid a^{2} = b^{3} = (ab)^{7} = [a,b]^{12} = (ababab^{-1}abab^{-1}ab^{-1} ababab^{-1}ab^{-1}abab^{-1})^{3} = 1 \rangle +$$ + +with + +$$ + |\mathrm{J}_2| = 604,800. +$$ + +On this page, we verify that the above claimed presentation of the Janko group +J~2~ defines a group of order $604,800$. + +## The code + +In [libsemigroups_pybind11][], the following script constructs the presentation +for J~2~ and runs the Todd-Coxeter algorithm. + +The code below took about 3.25 seconds to run on a 2024 MacBook Pro M4 with 48GB +of RAM: + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^7"), "") +presentation.add_rule(p, parse_relations("(a,b)^12"), "") +presentation.add_rule(p, parse_relations("(ababaBabaBaBababaBaBabaB)^3"), "") + +presentation.balance(p, "abAB", "ABab") +presentation.replace_subword(p, "A", "a") +presentation.replace_subword(p, "B", "bb") +p.alphabet("ab") + +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.strategy(ToddCoxeter.options.strategy.felsch) + +print(f"The size of the group is {tc.number_of_classes()}") +``` + +## The output + +The output of the enumeration is below: + +??? info "Output from the Python script" + + ``` + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 2, |R| = 9, |u| + |v| ∈ [2, 90], ∑(|u| + |v|) = 184 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 2 | 0.0% + #0: ToddCoxeter: time | run 0 = 28µs | all runs = 28µs | elapsed = 70µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #1: ToddCoxeter: nodes | 733,508 | 29,013 | 762,536 + #1: ToddCoxeter: diff 0.0.0 | +733,507 | +29,013 | +762,535 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,337,701 | 129,315 | 91.2% + #1: ToddCoxeter: diff 0.0.0 | +1,337,701 | +129,313 | +91.2% + #1: ToddCoxeter: phase 0.0 = 1.001s | run 0 = 1.001s | all runs = 1.001s | elapsed = 1.001s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined + #1: ToddCoxeter: nodes | 534,945 | 533,309 | 1,068,260 + #1: ToddCoxeter: diff 0.0.1 | -198,563 | +504,296 | +305,724 + #1: ToddCoxeter: diff 0.0.0 | +534,944 | +533,309 | +1,068,259 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,044,193 | 25,697 | 97.6% + #1: ToddCoxeter: diff 0.0.1 | -293,508 | -103,618 | +6.4% + #1: ToddCoxeter: diff 0.0.0 | +1,044,193 | +25,695 | +97.6% + #1: ToddCoxeter: phase 0.0 = 2.003s | run 0 = 2.003s | all runs = 2.003s | elapsed = 2.003s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined + #0: ToddCoxeter: nodes | 604,800 | 538,867 | 1,143,667 + #0: ToddCoxeter: diff 0.0.2 | +69,855 | +5,558 | +75,407 + #0: ToddCoxeter: diff 0.0.0 | +604,799 | +538,867 | +1,143,666 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 1,209,600 | 0 | 100.0% + #0: ToddCoxeter: diff 0.0.2 | +165,407 | -25,697 | +2.4% + #0: ToddCoxeter: diff 0.0.0 | +1,209,600 | -2 | +100.0% + #0: ToddCoxeter: phase 0.0 = 3.007s | run 0 = 3.007s | all runs = 3.007s | elapsed = 3.007s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 604,800 | 538,867 | 1,143,667 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 1,209,600 | 0 | 100.0% + #0: ToddCoxeter: time | run 0 = 3.007s | all runs = 3.007s | elapsed = 3.007s + #0: ToddCoxeter: triggered because there are skipped definitions (604,800 active nodes)! + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #1: ToddCoxeter: nodes | 604,800 | 538,867 | 1,143,667 + #1: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,209,600 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.1 = 549µs | run 0 = 3.008s | all runs = 3.008s | elapsed = 3.008s + #1: ToddCoxeter: lookahead progress | ~0.1% + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.2 | active | killed | defined + #0: ToddCoxeter: nodes | 604,800 | 538,867 | 1,143,667 + #0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 1,209,600 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.1 = 296ms | run 0 = 3.303s | all runs = 3.303s | elapsed = 3.303s + #0: ToddCoxeter: lookahead_next() is now max(f x a = 1,209,600, m = 10,000) (-3,790,400) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 604,800 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.3 | active | killed | defined + #0: ToddCoxeter: nodes | 604,800 | 538,867 | 1,143,667 + #0: ToddCoxeter: diff 0.2.2 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 1,209,600 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.2 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 296ms | run 0 = 3.303s | all runs = 3.303s | elapsed = 3.303s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 296ms (9%) | - (0%) | - (0%) | 3.007s (91%) + #0: ToddCoxeter: phase 0.2 = 298ms | run 0 = 3.305s | all runs = 3.305s | elapsed = 3.305s + The size of the group is 604800 + ``` + +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $604,800$. diff --git a/docs/examples/atlas/leech-lattice/j2/j2-p2.md b/docs/examples/atlas/leech-lattice/j2/j2-p2.md new file mode 100644 index 0000000..4d209fd --- /dev/null +++ b/docs/examples/atlas/leech-lattice/j2/j2-p2.md @@ -0,0 +1,666 @@ +# Janko group J2, Hall–Janko group HJ (Presentation 2) + +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/pres/J2G1-P2](https://brauer.maths.qmul.ac.uk/Atlas/v3/pres/J2G1-P2). + +## Claim + +$$ + \mathrm{J}_2 = \langle a,b \mid a^{2} = b^{3} = (ab)^{7} = [a,b]^{12} = (ababab^{-1}abab^{-1})^{6} = 1 \rangle +$$ + +with + +$$ + |\mathrm{J}_2| = 604,800. +$$ + +On this page, we verify that the above claimed presentation of the Janko group +J~2~ defines a group of order $604,800$. + +## The code + +In [libsemigroups_pybind11][], the following script constructs the presentation +for J~2~ and runs the Todd-Coxeter algorithm. + +The code below took about 27 seconds to run on a 2024 MacBook Pro M4 with 48GB + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^7"), "") +presentation.add_rule(p, parse_relations("(a,b)^12"), "") +presentation.add_rule(p, parse_relations("(ababaBabaB)^6"), "") + +presentation.balance(p, "abAB", "ABab") +presentation.replace_subword(p, "A", "a") +presentation.replace_subword(p, "B", "bb") +p.alphabet("ab") + +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.lookahead_extent(ToddCoxeter.options.lookahead_extent.full) +tc.strategy(ToddCoxeter.options.strategy.felsch) + +print(f"The size of the group is {tc.number_of_classes()}") +``` + +## The output + +The output of the enumeration is below: + +??? info "Output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 2, |R| = 9, |u| + |v| ∈ [2, 75], ∑(|u| + |v|) = 169 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | + defined + #0: ToddCoxeter: nodes | 1 | 0 | + 1 + #0: ToddCoxeter: | active | missing | % + complete + #0: ToddCoxeter: edges | 0 | 2 | + 0.0% + #0: ToddCoxeter: time | run 0 = 29µs | all runs = 29µs | elapsed = + 76µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.1 | active | killed | + defined + #1: ToddCoxeter: nodes | 1,054,757 | 11,665 | + 1,066,442 + #1: ToddCoxeter: diff 0.0.0 | +1,054,756 | +11,665 | + +1,066,441 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 1,918,733 | 190,781 | + 91.0% + #1: ToddCoxeter: diff 0.0.0 | +1,918,733 | +190,779 | + +91.0% + #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | + elapsed = 1.005s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.2 | active | killed | + defined + #1: ToddCoxeter: nodes | 2,068,730 | 24,927 | + 2,093,682 + #1: ToddCoxeter: diff 0.0.1 | +1,013,973 | +13,262 | + +1,027,240 + #1: ToddCoxeter: diff 0.0.0 | +2,068,729 | +24,927 | + +2,093,681 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 3,773,664 | 363,796 | + 91.2% + #1: ToddCoxeter: diff 0.0.1 | +1,854,931 | +173,015 | + +0.3% + #1: ToddCoxeter: diff 0.0.0 | +3,773,664 | +363,794 | + +91.2% + #1: ToddCoxeter: phase 0.0 = 2.010s | run 0 = 2.010s | all runs = 2.010s | + elapsed = 2.010s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.3 | active | killed | + defined + #1: ToddCoxeter: nodes | 3,048,312 | 38,150 | + 3,086,472 + #1: ToddCoxeter: diff 0.0.2 | +979,582 | +13,223 | + +992,790 + #1: ToddCoxeter: diff 0.0.0 | +3,048,311 | +38,150 | + +3,086,471 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 5,546,882 | 549,742 | + 91.0% + #1: ToddCoxeter: diff 0.0.2 | +1,773,218 | +185,946 | + -0.2% + #1: ToddCoxeter: diff 0.0.0 | +5,546,882 | +549,740 | + +91.0% + #1: ToddCoxeter: phase 0.0 = 3.015s | run 0 = 3.015s | all runs = 3.015s | + elapsed = 3.015s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.4 | active | killed | + defined + #1: ToddCoxeter: nodes | 4,041,704 | 52,159 | + 4,093,872 + #1: ToddCoxeter: diff 0.0.3 | +993,392 | +14,009 | + +1,007,400 + #1: ToddCoxeter: diff 0.0.0 | +4,041,703 | +52,159 | + +4,093,871 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 7,357,321 | 726,087 | + 91.0% + #1: ToddCoxeter: diff 0.0.3 | +1,810,439 | +176,345 | + +0.0% + #1: ToddCoxeter: diff 0.0.0 | +7,357,321 | +726,085 | + +91.0% + #1: ToddCoxeter: phase 0.0 = 4.020s | run 0 = 4.020s | all runs = 4.020s | + elapsed = 4.020s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.5 | active | killed | + defined + #1: ToddCoxeter: nodes | 5,007,440 | 65,309 | + 5,072,763 + #1: ToddCoxeter: diff 0.0.4 | +965,736 | +13,150 | + +978,891 + #1: ToddCoxeter: diff 0.0.0 | +5,007,439 | +65,309 | + +5,072,762 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 9,119,814 | 895,066 | + 91.1% + #1: ToddCoxeter: diff 0.0.4 | +1,762,493 | +168,979 | + +0.0% + #1: ToddCoxeter: diff 0.0.0 | +9,119,814 | +895,064 | + +91.1% + #1: ToddCoxeter: phase 0.0 = 5.024s | run 0 = 5.024s | all runs = 5.024s | + elapsed = 5.024s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.6 | active | killed | + defined + #1: ToddCoxeter: nodes | 5,986,526 | 79,692 | + 6,066,241 + #1: ToddCoxeter: diff 0.0.5 | +979,086 | +14,383 | + +993,478 + #1: ToddCoxeter: diff 0.0.0 | +5,986,525 | +79,692 | + +6,066,240 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 10,912,342 | 1,060,710 | + 91.1% + #1: ToddCoxeter: diff 0.0.5 | +1,792,528 | +165,644 | + +0.1% + #1: ToddCoxeter: diff 0.0.0 | +10,912,342 | +1,060,708 | + +91.1% + #1: ToddCoxeter: phase 0.0 = 6.029s | run 0 = 6.029s | all runs = 6.029s | + elapsed = 6.029s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.7 | active | killed | + defined + #1: ToddCoxeter: nodes | 6,950,902 | 94,545 | + 7,045,467 + #1: ToddCoxeter: diff 0.0.6 | +964,376 | +14,853 | + +979,226 + #1: ToddCoxeter: diff 0.0.0 | +6,950,901 | +94,545 | + +7,045,466 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 12,679,959 | 1,221,845 | + 91.2% + #1: ToddCoxeter: diff 0.0.6 | +1,767,617 | +161,135 | + +0.1% + #1: ToddCoxeter: diff 0.0.0 | +12,679,959 | +1,221,843 | + +91.2% + #1: ToddCoxeter: phase 0.0 = 7.034s | run 0 = 7.034s | all runs = 7.034s | + elapsed = 7.034s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.8 | active | killed | + defined + #1: ToddCoxeter: nodes | 7,908,946 | 106,977 | + 8,015,937 + #1: ToddCoxeter: diff 0.0.7 | +958,044 | +12,432 | + +970,470 + #1: ToddCoxeter: diff 0.0.0 | +7,908,945 | +106,977 | + +8,015,936 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 14,433,768 | 1,384,124 | + 91.2% + #1: ToddCoxeter: diff 0.0.7 | +1,753,809 | +162,279 | + +0.0% + #1: ToddCoxeter: diff 0.0.0 | +14,433,768 | +1,384,122 | + +91.2% + #1: ToddCoxeter: phase 0.0 = 8.039s | run 0 = 8.039s | all runs = 8.039s | + elapsed = 8.039s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.9 | active | killed | + defined + #1: ToddCoxeter: nodes | 8,826,087 | 120,204 | + 8,946,309 + #1: ToddCoxeter: diff 0.0.8 | +917,141 | +13,227 | + +930,372 + #1: ToddCoxeter: diff 0.0.0 | +8,826,086 | +120,204 | + +8,946,308 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 16,072,677 | 1,579,497 | + 91.1% + #1: ToddCoxeter: diff 0.0.8 | +1,638,909 | +195,373 | + -0.2% + #1: ToddCoxeter: diff 0.0.0 | +16,072,677 | +1,579,495 | + +91.1% + #1: ToddCoxeter: phase 0.0 = 9.044s | run 0 = 9.044s | all runs = 9.044s | + elapsed = 9.044s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.10 | active | killed | + defined + #1: ToddCoxeter: nodes | 9,775,831 | 135,310 | + 9,911,156 + #1: ToddCoxeter: diff 0.0.9 | +949,744 | +15,106 | + +964,847 + #1: ToddCoxeter: diff 0.0.0 | +9,775,830 | +135,310 | + +9,911,155 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 17,827,462 | 1,724,200 | + 91.2% + #1: ToddCoxeter: diff 0.0.9 | +1,754,785 | +144,703 | + +0.1% + #1: ToddCoxeter: diff 0.0.0 | +17,827,462 | +1,724,198 | + +91.2% + #1: ToddCoxeter: phase 0.0 = 10.049s | run 0 = 10.049s | all runs = 10.049s | + elapsed = 10.049s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.11 | active | killed | + defined + #1: ToddCoxeter: nodes | 10,718,880 | 146,360 | + 10,865,256 + #1: ToddCoxeter: diff 0.0.10 | +943,049 | +11,050 | + +954,100 + #1: ToddCoxeter: diff 0.0.0 | +10,718,879 | +146,360 | + +10,865,255 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 19,534,800 | 1,902,960 | + 91.1% + #1: ToddCoxeter: diff 0.0.10 | +1,707,338 | +178,760 | + -0.1% + #1: ToddCoxeter: diff 0.0.0 | +19,534,800 | +1,902,958 | + +91.1% + #1: ToddCoxeter: phase 0.0 = 11.050s | run 0 = 11.050s | all runs = 11.050s | + elapsed = 11.050s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.12 | active | killed | + defined + #1: ToddCoxeter: nodes | 11,650,984 | 163,136 | + 11,814,139 + #1: ToddCoxeter: diff 0.0.11 | +932,104 | +16,776 | + +948,883 + #1: ToddCoxeter: diff 0.0.0 | +11,650,983 | +163,136 | + +11,814,138 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 21,236,938 | 2,065,030 | + 91.1% + #1: ToddCoxeter: diff 0.0.11 | +1,702,138 | +162,070 | + +0.0% + #1: ToddCoxeter: diff 0.0.0 | +21,236,938 | +2,065,028 | + +91.1% + #1: ToddCoxeter: phase 0.0 = 12.055s | run 0 = 12.055s | all runs = 12.055s | + elapsed = 12.055s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.13 | active | killed | + defined + #1: ToddCoxeter: nodes | 12,585,643 | 183,529 | + 12,769,191 + #1: ToddCoxeter: diff 0.0.12 | +934,659 | +20,393 | + +955,052 + #1: ToddCoxeter: diff 0.0.0 | +12,585,642 | +183,529 | + +12,769,190 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 23,004,143 | 2,167,143 | + 91.4% + #1: ToddCoxeter: diff 0.0.12 | +1,767,205 | +102,113 | + +0.3% + #1: ToddCoxeter: diff 0.0.0 | +23,004,143 | +2,167,141 | + +91.4% + #1: ToddCoxeter: phase 0.0 = 13.060s | run 0 = 13.060s | all runs = 13.060s | + elapsed = 13.060s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.14 | active | killed | + defined + #1: ToddCoxeter: nodes | 13,520,513 | 196,671 | + 13,717,195 + #1: ToddCoxeter: diff 0.0.13 | +934,870 | +13,142 | + +948,004 + #1: ToddCoxeter: diff 0.0.0 | +13,520,512 | +196,671 | + +13,717,194 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 24,688,003 | 2,353,023 | + 91.3% + #1: ToddCoxeter: diff 0.0.13 | +1,683,860 | +185,880 | + -0.1% + #1: ToddCoxeter: diff 0.0.0 | +24,688,003 | +2,353,021 | + +91.3% + #1: ToddCoxeter: phase 0.0 = 14.065s | run 0 = 14.065s | all runs = 14.065s | + elapsed = 14.065s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.15 | active | killed | + defined + #1: ToddCoxeter: nodes | 14,451,287 | 208,571 | + 14,659,875 + #1: ToddCoxeter: diff 0.0.14 | +930,774 | +11,900 | + +942,680 + #1: ToddCoxeter: diff 0.0.0 | +14,451,286 | +208,571 | + +14,659,874 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 26,322,838 | 2,579,736 | + 91.1% + #1: ToddCoxeter: diff 0.0.14 | +1,634,835 | +226,713 | + -0.2% + #1: ToddCoxeter: diff 0.0.0 | +26,322,838 | +2,579,734 | + +91.1% + #1: ToddCoxeter: phase 0.0 = 15.070s | run 0 = 15.070s | all runs = 15.070s | + elapsed = 15.070s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.16 | active | killed | + defined + #1: ToddCoxeter: nodes | 15,364,601 | 230,647 | + 15,595,262 + #1: ToddCoxeter: diff 0.0.15 | +913,314 | +22,076 | + +935,387 + #1: ToddCoxeter: diff 0.0.0 | +15,364,600 | +230,647 | + +15,595,261 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 28,048,246 | 2,680,956 | + 91.3% + #1: ToddCoxeter: diff 0.0.15 | +1,725,408 | +101,220 | + +0.2% + #1: ToddCoxeter: diff 0.0.0 | +28,048,246 | +2,680,954 | + +91.3% + #1: ToddCoxeter: phase 0.0 = 16.075s | run 0 = 16.075s | all runs = 16.075s | + elapsed = 16.075s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.17 | active | killed | + defined + #1: ToddCoxeter: nodes | 16,278,171 | 251,128 | + 16,529,313 + #1: ToddCoxeter: diff 0.0.16 | +913,570 | +20,481 | + +934,051 + #1: ToddCoxeter: diff 0.0.0 | +16,278,170 | +251,128 | + +16,529,312 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 29,733,273 | 2,823,069 | + 91.3% + #1: ToddCoxeter: diff 0.0.16 | +1,685,027 | +142,113 | + +0.1% + #1: ToddCoxeter: diff 0.0.0 | +29,733,273 | +2,823,067 | + +91.3% + #1: ToddCoxeter: phase 0.0 = 17.080s | run 0 = 17.080s | all runs = 17.080s | + elapsed = 17.080s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.18 | active | killed | + defined + #1: ToddCoxeter: nodes | 17,136,800 | 264,123 | + 17,400,945 + #1: ToddCoxeter: diff 0.0.17 | +858,629 | +12,995 | + +871,632 + #1: ToddCoxeter: diff 0.0.0 | +17,136,799 | +264,123 | + +17,400,944 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 31,328,380 | 2,945,220 | + 91.4% + #1: ToddCoxeter: diff 0.0.17 | +1,595,107 | +122,151 | + +0.1% + #1: ToddCoxeter: diff 0.0.0 | +31,328,380 | +2,945,218 | + +91.4% + #1: ToddCoxeter: phase 0.0 = 18.085s | run 0 = 18.085s | all runs = 18.085s | + elapsed = 18.085s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.19 | active | killed | + defined + #1: ToddCoxeter: nodes | 18,042,421 | 284,551 | + 18,326,991 + #1: ToddCoxeter: diff 0.0.18 | +905,621 | +20,428 | + +926,046 + #1: ToddCoxeter: diff 0.0.0 | +18,042,420 | +284,551 | + +18,326,990 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 32,928,600 | 3,156,242 | + 91.3% + #1: ToddCoxeter: diff 0.0.18 | +1,600,220 | +211,022 | + -0.2% + #1: ToddCoxeter: diff 0.0.0 | +32,928,600 | +3,156,240 | + +91.3% + #1: ToddCoxeter: phase 0.0 = 19.090s | run 0 = 19.090s | all runs = 19.090s | + elapsed = 19.090s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.20 | active | killed | + defined + #1: ToddCoxeter: nodes | 18,949,003 | 299,208 | + 19,248,204 + #1: ToddCoxeter: diff 0.0.19 | +906,582 | +14,657 | + +921,213 + #1: ToddCoxeter: diff 0.0.0 | +18,949,002 | +299,208 | + +19,248,203 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 34,554,079 | 3,343,927 | + 91.2% + #1: ToddCoxeter: diff 0.0.19 | +1,625,479 | +187,685 | + -0.1% + #1: ToddCoxeter: diff 0.0.0 | +34,554,079 | +3,343,925 | + +91.2% + #1: ToddCoxeter: phase 0.0 = 20.095s | run 0 = 20.095s | all runs = 20.095s | + elapsed = 20.095s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.21 | active | killed | + defined + #1: ToddCoxeter: nodes | 19,836,819 | 326,032 | + 20,162,865 + #1: ToddCoxeter: diff 0.0.20 | +887,816 | +26,824 | + +914,661 + #1: ToddCoxeter: diff 0.0.0 | +19,836,818 | +326,032 | + +20,162,864 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 36,181,798 | 3,491,840 | + 91.2% + #1: ToddCoxeter: diff 0.0.20 | +1,627,719 | +147,913 | + +0.0% + #1: ToddCoxeter: diff 0.0.0 | +36,181,798 | +3,491,838 | + +91.2% + #1: ToddCoxeter: phase 0.0 = 21.100s | run 0 = 21.100s | all runs = 21.100s | + elapsed = 21.100s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.22 | active | killed | + defined + #1: ToddCoxeter: nodes | 20,746,679 | 346,083 | + 21,092,769 + #1: ToddCoxeter: diff 0.0.21 | +909,860 | +20,051 | + +929,904 + #1: ToddCoxeter: diff 0.0.0 | +20,746,678 | +346,083 | + +21,092,768 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 37,905,561 | 3,587,797 | + 91.4% + #1: ToddCoxeter: diff 0.0.21 | +1,723,763 | +95,957 | + +0.2% + #1: ToddCoxeter: diff 0.0.0 | +37,905,561 | +3,587,795 | + +91.4% + #1: ToddCoxeter: phase 0.0 = 22.105s | run 0 = 22.105s | all runs = 22.105s | + elapsed = 22.105s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.23 | active | killed | + defined + #1: ToddCoxeter: nodes | 21,642,109 | 365,426 | + 22,007,543 + #1: ToddCoxeter: diff 0.0.22 | +895,430 | +19,343 | + +914,774 + #1: ToddCoxeter: diff 0.0.0 | +21,642,108 | +365,426 | + +22,007,542 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 39,527,455 | 3,756,763 | + 91.3% + #1: ToddCoxeter: diff 0.0.22 | +1,621,894 | +168,966 | + -0.0% + #1: ToddCoxeter: diff 0.0.0 | +39,527,455 | +3,756,761 | + +91.3% + #1: ToddCoxeter: phase 0.0 = 23.110s | run 0 = 23.110s | all runs = 23.110s | + elapsed = 23.110s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.24 | active | killed | + defined + #1: ToddCoxeter: nodes | 22,545,900 | 383,811 | + 22,929,721 + #1: ToddCoxeter: diff 0.0.23 | +903,791 | +18,385 | + +922,178 + #1: ToddCoxeter: diff 0.0.0 | +22,545,899 | +383,811 | + +22,929,720 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 41,219,413 | 3,872,387 | + 91.4% + #1: ToddCoxeter: diff 0.0.23 | +1,691,958 | +115,624 | + +0.1% + #1: ToddCoxeter: diff 0.0.0 | +41,219,413 | +3,872,385 | + +91.4% + #1: ToddCoxeter: phase 0.0 = 24.115s | run 0 = 24.115s | all runs = 24.115s | + elapsed = 24.115s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.25 | active | killed | + defined + #1: ToddCoxeter: nodes | 21,779,189 | 1,989,573 | + 23,768,083 + #1: ToddCoxeter: diff 0.0.24 | -766,711 | +1,605,762 | + +838,362 + #1: ToddCoxeter: diff 0.0.0 | +21,779,188 | +1,989,573 | + +23,768,082 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 39,817,438 | 3,740,940 | + 91.4% + #1: ToddCoxeter: diff 0.0.24 | -1,401,975 | -131,447 | + -0.0% + #1: ToddCoxeter: diff 0.0.0 | +39,817,438 | +3,740,938 | + +91.4% + #1: ToddCoxeter: phase 0.0 = 25.120s | run 0 = 25.120s | all runs = 25.120s | + elapsed = 25.120s + #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = + large_collapse()! + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.26 | active | killed | + defined + #1: ToddCoxeter: nodes | 2,806,820 | 20,961,781 | + 23,768,083 + #1: ToddCoxeter: diff 0.0.25 | -18,972,369 | +18,972,208 | + +0 + #1: ToddCoxeter: diff 0.0.0 | +2,806,819 | +20,961,781 | + +23,768,082 + #1: ToddCoxeter: | active | missing | + % complete + #1: ToddCoxeter: edges | 0 | 5,613,640 | + 0.0% + #1: ToddCoxeter: diff 0.0.25 | -39,817,438 | +1,872,700 | + -91.4% + #1: ToddCoxeter: diff 0.0.0 | +0 | +5,613,638 | + +0.0% + #1: ToddCoxeter: phase 0.0 = 26.121s | run 0 = 26.121s | all runs = 26.121s | + elapsed = 26.121s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.27 | active | killed | + defined + #0: ToddCoxeter: nodes | 604,800 | 23,163,283 | + 23,768,083 + #0: ToddCoxeter: diff 0.0.26 | -2,202,020 | +2,201,502 | + +0 + #0: ToddCoxeter: diff 0.0.0 | +604,799 | +23,163,283 | + +23,768,082 + #0: ToddCoxeter: | active | missing | + % complete + #0: ToddCoxeter: edges | 1,209,600 | 0 | + 100.0% + #0: ToddCoxeter: diff 0.0.26 | +1,209,600 | -5,613,640 | + +100.0% + #0: ToddCoxeter: diff 0.0.0 | +1,209,600 | -2 | + +100.0% + #0: ToddCoxeter: phase 0.0 = 26.308s | run 0 = 26.308s | all runs = 26.308s | + elapsed = 26.308s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, + lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | + defined + #0: ToddCoxeter: nodes | 604,800 | 23,163,283 | + 23,768,083 + #0: ToddCoxeter: | active | missing | + % complete + #0: ToddCoxeter: edges | 1,209,600 | 0 | + 100.0% + #0: ToddCoxeter: time | run 0 = 26.308s | all runs = 26.308s | + elapsed = 26.308s + #0: ToddCoxeter: triggered because there are skipped definitions (604,800 + active nodes)! + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | + defined + #0: ToddCoxeter: nodes | 604,800 | 23,163,283 | + 23,768,083 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | + +0 + #0: ToddCoxeter: | active | missing | + % complete + #0: ToddCoxeter: edges | 1,209,600 | 0 | + 100.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | + +0.0% + #0: ToddCoxeter: phase 0.1 = 254ms | run 0 = 26.563s | all runs = 26.563s | + elapsed = 26.563s + #0: ToddCoxeter: lookahead_next() is now max(f x a = 1,209,600, m = 10,000) + (-3,790,400) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 604,800 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.2 | active | killed | + defined + #0: ToddCoxeter: nodes | 604,800 | 23,163,283 | + 23,768,083 + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | + +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | + +0 + #0: ToddCoxeter: | active | missing | + % complete + #0: ToddCoxeter: edges | 1,209,600 | 0 | + 100.0% + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | + +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | + +0.0% + #0: ToddCoxeter: phase 0.2 = 254ms | run 0 = 26.563s | all runs = 26.563s | + elapsed = 26.563s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | + hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | + 0 | 1 + #0: ToddCoxeter: time spent in phases | 254ms (1%) | - (0%) | + - (0%) | 26.308s (99%) + #0: ToddCoxeter: phase 0.2 = 256ms | run 0 = 26.565s | all runs = 26.565s | + elapsed = 26.565s + The size of the group is 604800 + ``` + +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $604,800$. diff --git a/docs/examples/atlas/leech-lattice/mcl/index.md b/docs/examples/atlas/leech-lattice/mcl/index.md index 5db3fb3..60a2e75 100644 --- a/docs/examples/atlas/leech-lattice/mcl/index.md +++ b/docs/examples/atlas/leech-lattice/mcl/index.md @@ -2,11 +2,9 @@ > Order: $898,128,000$ -> Presentation: TODO +> Presentation: +> $\langle a, b \mid a^2 = b^5 = (ab)^{11} = (ab^2)^{12} = [a, b]^5 = [a, b^2]^6 = (abab^{−2})^7 = [a, b^{−2}ab^2ab^{−1}ab(ab^2)^2abab^{−1}] = [a, b^2ab(ab^2)^2]^2 = abab^2ab^{−2}abab^{−1}ab^2(ab^{−2}ab)^2(ab^2ab^{−2}ab^2)^2 = [a, b^2ab^2ab^{−1}ab^2]^2 = [a, b^2ab]^4 = ^1 \rangle$ -On this page, we provide links to verifications that the maximal subgroups of -the McLaughlin group McL define groups of the correct order. +!!! info "Work in Progress" -## Maximal subgroups - -The following are maximal subgroups that have been verified: + This page is a work in progress. diff --git a/docs/examples/atlas/mathieu/m11.md b/docs/examples/atlas/mathieu/m11.md index cfd7a84..94bf24b 100644 --- a/docs/examples/atlas/mathieu/m11.md +++ b/docs/examples/atlas/mathieu/m11.md @@ -1,97 +1,103 @@ # Mathieu group M~11~ -> Order: $7,920$ +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). -> Presentation: -> $\langle{ a, b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^6 = ababab^{−1}abab^2ab^{−1}abab^{−1}ab^{−1} = 1 }\rangle$ +## Claim -On this page, we verify that the above claimed presentation of the Mathieu group -M~11~ defines a group of order $7,920$. - -## The code - -=== "Python" - - In [libsemigroups_pybind11][], the following script constructs the - presentation for M~11~ and runs the Todd-Coxeter algorithm. +$$ + \mathrm{M}_{11} = \langle a, b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^6 = ababab^{−1}abab^2ab^{−1}abab^{−1}ab^{−1} = 1 \rangle +$$ - ```python - from libsemigroups_pybind11 import Presentation, ToddCoxeter, congruence_kind, presentation - from libsemigroups_pybind11.words import parse_relations +with - # Setup the presentation object with the empty and inverses, so it can represent a group - p = Presentation("abAB") - p.contains_empty_word(True) - presentation.add_inverse_rules(p, "ABab") +$$ + |\mathrm{M}_{11}| = 7,920. +$$ - # Add the defining relations - presentation.add_rule(p, parse_relations("a^2"), "") - presentation.add_rule(p, parse_relations("b^4"), "") - presentation.add_rule(p, parse_relations("(ab)^11"), "") - presentation.add_rule(p, parse_relations("(ab^2)^6"), "") - presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") - - # Run the Todd-Coxeter algorithm - tc = ToddCoxeter(congruence_kind.twosided, p) - tc.run() - - assert tc.number_of_classes() == 7920 - ``` +On this page, we verify that the above claimed presentation of the Mathieu group +M~11~ defines a group of order $7,920$. -=== "ACE" +## The code - Coming soon! +In [libsemigroups_pybind11][], the following script constructs the presentation +for M~11~ and runs the Todd-Coxeter algorithm. + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") + +# Run the Todd-Coxeter algorithm +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.run() + +print(f"The size of the group is {tc.number_of_classes()}") +``` ## The output -=== "Python" - - Running the above Python script produces the following output: - - ``` - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 START (strategy() = hlt) - #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 START - #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined - #0: ToddCoxeter: nodes | 1 | 0 | 1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 0 | 4 | 0.0% - #0: ToddCoxeter: time | run 0 = 28µs | all runs = 28µs | elapsed = 92µs - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 STOP - #0: ToddCoxeter: HLT 0.0.1 | active | killed | defined - #0: ToddCoxeter: nodes | 7,920 | 1,992,427 | 2,000,347 - #0: ToddCoxeter: diff 0.0.0 | +7,919 | +1,992,427 | +2,000,346 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 31,680 | 0 | 100.0% - #0: ToddCoxeter: diff 0.0.0 | +31,680 | -4 | +100.0% - #0: ToddCoxeter: phase 0.0 = 176ms | run 0 = 176ms | all runs = 176ms | elapsed = 176ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.1.2 | active | killed | defined - #0: ToddCoxeter: nodes | 7,920 | 1,992,427 | 2,000,347 - #0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0 - #0: ToddCoxeter: diff 0.1.0 | +7,919 | +1,992,427 | +2,000,346 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 31,680 | 0 | 100.0% - #0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0.0% - #0: ToddCoxeter: diff 0.1.0 | +31,680 | -4 | +100.0% - #0: ToddCoxeter: phase 0.1 = 176ms | run 0 = 176ms | all runs = 176ms | elapsed = 176ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 STOP (finished) - #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch - #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 - #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 176ms (100%) | - (0%) - #0: ToddCoxeter: phase 0.1 = 176ms | run 0 = 176ms | all runs = 176ms | elapsed = 176ms - ``` - -=== "ACE" - - Coming soon! - -:simple-ticktick: The computed size of the group matches the size of the group -provided on [the ATLAS][atlas]: $7,920$. +Running the above script produces the following output: + +``` +++++++++++++++++++++++++++++++++ +#0: ToddCoxeter: running for approx. 10.000s +++++++++++++++++++++++++++++++++ +#0: ToddCoxeter: RUN 0 START (strategy() = hlt) +#0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 +++++++++++++++++++++++++++++++++ +#0: ToddCoxeter: HLT 0.0 START +#0: ToddCoxeter: HLT 0.0.0 | active | killed | defined +#0: ToddCoxeter: nodes | 1 | 0 | 1 +#0: ToddCoxeter: | active | missing | % complete +#0: ToddCoxeter: edges | 0 | 4 | 0.0% +#0: ToddCoxeter: time | run 0 = 19µs | all runs = 19µs | elapsed = 641µs +++++++++++++++++++++++++++++++++ +#0: ToddCoxeter: HLT 0.0 STOP +#0: ToddCoxeter: HLT 0.0.1 | active | killed | defined +#0: ToddCoxeter: nodes | 7,920 | 1,992,427 | 2,000,347 +#0: ToddCoxeter: diff 0.0.0 | +7,919 | +1,992,427 | +2,000,346 +#0: ToddCoxeter: | active | missing | % complete +#0: ToddCoxeter: edges | 31,680 | 0 | 100.0% +#0: ToddCoxeter: diff 0.0.0 | +31,680 | -4 | +100.0% +#0: ToddCoxeter: phase 0.0 = 209ms | run 0 = 209ms | all runs = 209ms | elapsed = 210ms +++++++++++++++++++++++++++++++++ +#0: ToddCoxeter: HLT 0.1.2 | active | killed | defined +#0: ToddCoxeter: nodes | 7,920 | 1,992,427 | 2,000,347 +#0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0 +#0: ToddCoxeter: diff 0.1.0 | +7,919 | +1,992,427 | +2,000,346 +#0: ToddCoxeter: | active | missing | % complete +#0: ToddCoxeter: edges | 31,680 | 0 | 100.0% +#0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0.0% +#0: ToddCoxeter: diff 0.1.0 | +31,680 | -4 | +100.0% +#0: ToddCoxeter: phase 0.1 = 209ms | run 0 = 209ms | all runs = 209ms | elapsed = 210ms +++++++++++++++++++++++++++++++++ +#0: ToddCoxeter: RUN 0 STOP (finished) +#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch +#0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 +#0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 209ms (100%) | - (0%) +#0: ToddCoxeter: phase 0.1 = 209ms | run 0 = 209ms | all runs = 209ms | elapsed = 210ms +The size of the group is 7920 +``` + +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $7,920$. !!! note @@ -99,7 +105,3 @@ provided on [the ATLAS][atlas]: $7,920$. the above; the numbers produced and time taken will depend on the machine you are using. However, the size of the group that is reported should be the same as above. - -[atlas]: https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ -[libsemigroups_pybind11]: - https://libsemigroups.github.io/libsemigroups_pybind11/index.html diff --git a/docs/examples/atlas/mathieu/m12.md b/docs/examples/atlas/mathieu/m12.md index 999046f..3254fac 100644 --- a/docs/examples/atlas/mathieu/m12.md +++ b/docs/examples/atlas/mathieu/m12.md @@ -1,143 +1,155 @@ # Mathieu group M~12~ -> Order: $95,040$ +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). -> Presentation: -> $\langle{ a, b \mid a^2 = b^3 = (ab)^{11} = [a, b]^6 = (ababab^{−1})^6 = 1 }\rangle$ +## Claim + +$$ + \mathrm{M}_{12} = \langle a, b \mid a^2 = b^3 = (ab)^{11} = [a, b]^6 = (ababab^{−1})^6 = 1 \rangle +$$ + +with + +$$ + |\mathrm{M}_{12}| = 95,040. +$$ On this page, we verify that the above claimed presentation of the Mathieu group M~12~ defines a group of order $95,040$. ## The code -=== "Python" +In [libsemigroups_pybind11][], the following script constructs the presentation +for M~12~ and runs the Todd-Coxeter algorithm. - In [libsemigroups_pybind11][], the following script constructs the - presentation for M~12~ and runs the Todd-Coxeter algorithm. +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations - ```python - from libsemigroups_pybind11 import Presentation, ToddCoxeter, congruence_kind, presentation - from libsemigroups_pybind11.words import parse_relations +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") - # Setup the presentation object with the empty and inverses, so it can represent a group - p = Presentation("abAB") - p.contains_empty_word(True) - presentation.add_inverse_rules(p, "ABab") +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(a,b)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") - # Add the defining relations - presentation.add_rule(p, parse_relations("a^2"), "") - presentation.add_rule(p, parse_relations("b^3"), "") - presentation.add_rule(p, parse_relations("(ab)^11"), "") - presentation.add_rule(p, parse_relations("(a,b)^6"), "") - presentation.add_rule(p, parse_relations("(ababaB)^6"), "") +# Run the Todd-Coxeter algorithm +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.run() - # Run the Todd-Coxeter algorithm - tc = ToddCoxeter(congruence_kind.twosided, p) - tc.run() - - assert tc.number_of_classes() == 95040 - ``` +print(f"The size of the group is {tc.number_of_classes()}") +``` -=== "ACE" +## The output - Coming soon! +Running the above script produces the following output: -## The output +??? info "Output from the Python script" + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: running for approx. 10.000s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 4 | 0.0% + #0: ToddCoxeter: time | run 0 = 21µs | all runs = 21µs | elapsed = 103µs + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 STOP + #0: ToddCoxeter: HLT 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 5,000,004 | 2,964,327 | 7,964,331 + #0: ToddCoxeter: diff 0.0.0 | +5,000,003 | +2,964,327 | +7,964,330 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,782,533 | 14,217,483 | 28.9% + #0: ToddCoxeter: diff 0.0.0 | +5,782,533 | +14,217,479 | +28.9% + #0: ToddCoxeter: phase 0.0 = 624ms | run 0 = 624ms | all runs = 624ms | elapsed = 624ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = partial, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 5,000,004 | 2,964,327 | 7,964,331 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,782,533 | 14,217,483 | 28.9% + #0: ToddCoxeter: time | run 0 = 624ms | all runs = 624ms | elapsed = 624ms + #0: ToddCoxeter: because a >= n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 5,000,004 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + #0: ToddCoxeter: large collapse, number of coincidences 100,002 >= 100,000 = large_collapse()! + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #1: ToddCoxeter: nodes | 1,909,907 | 6,057,700 | 7,964,331 + #1: ToddCoxeter: diff 0.1.0 | -3,090,097 | +3,093,373 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 0 | 7,639,628 | 0.0% + #1: ToddCoxeter: diff 0.1.0 | -5,782,533 | -6,577,855 | -28.9% + #1: ToddCoxeter: phase 0.1 = 375ms | run 0 = 1.001s | all runs = 1.001s | elapsed = 1.001s + #1: ToddCoxeter: lookahead progress | ~32.6% + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.2 | active | killed | defined + #0: ToddCoxeter: nodes | 95,046 | 7,869,285 | 7,964,331 + #0: ToddCoxeter: diff 0.1.1 | -1,814,861 | +1,811,585 | +0 + #0: ToddCoxeter: diff 0.1.0 | -4,904,958 | +4,904,958 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 380,177 | 7 | 100.0% + #0: ToddCoxeter: diff 0.1.1 | +380,177 | -7,639,621 | +100.0% + #0: ToddCoxeter: diff 0.1.0 | -5,402,356 | -14,217,476 | +71.1% + #0: ToddCoxeter: phase 0.1 = 491ms | run 0 = 1.116s | all runs = 1.116s | elapsed = 1.116s + #0: ToddCoxeter: lookahead_next() is now max(f x a = 190,092, m = 10,000) (-4,809,908) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 95,046 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.2 START + #0: ToddCoxeter: HLT 0.2.0 | active | killed | defined + #0: ToddCoxeter: nodes | 95,046 | 7,869,285 | 7,964,331 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 380,177 | 7 | 100.0% + #0: ToddCoxeter: time | run 0 = 1.116s | all runs = 1.116s | elapsed = 1.116s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.2 STOP + #0: ToddCoxeter: HLT 0.2.1 | active | killed | defined + #0: ToddCoxeter: nodes | 95,040 | 7,869,292 | 7,964,332 + #0: ToddCoxeter: diff 0.2.0 | -6 | +7 | +1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 380,160 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.0 | -17 | -7 | +0.0% + #0: ToddCoxeter: phase 0.2 = 19ms | run 0 = 1.136s | all runs = 1.136s | elapsed = 1.136s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.3.2 | active | killed | defined + #0: ToddCoxeter: nodes | 95,040 | 7,869,292 | 7,964,332 + #0: ToddCoxeter: diff 0.3.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.3.0 | -6 | +7 | +1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 380,160 | 0 | 100.0% + #0: ToddCoxeter: diff 0.3.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.3.0 | -17 | -7 | +0.0% + #0: ToddCoxeter: phase 0.3 = 19ms | run 0 = 1.136s | all runs = 1.136s | elapsed = 1.136s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 2 | 0 + #0: ToddCoxeter: time spent in phases | 491ms (43%) | - (0%) | 644ms (57%) | - (0%) + #0: ToddCoxeter: phase 0.3 = 20ms | run 0 = 1.137s | all runs = 1.137s | elapsed = 1.137s + The size of the group is 95040 + ``` -=== "Python" - - Running the above script produces the following output: - - ??? info "Output from the Python script" - - ``` - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 START (strategy() = hlt) - #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 START - #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined - #0: ToddCoxeter: nodes | 1 | 0 | 1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 0 | 4 | 0.0% - #0: ToddCoxeter: time | run 0 = 24µs | all runs = 24µs | elapsed = 84µs - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 STOP - #0: ToddCoxeter: HLT 0.0.1 | active | killed | defined - #0: ToddCoxeter: nodes | 5,000,004 | 2,964,327 | 7,964,331 - #0: ToddCoxeter: diff 0.0.0 | +5,000,003 | +2,964,327 | +7,964,330 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 5,782,533 | 14,217,483 | 28.9% - #0: ToddCoxeter: diff 0.0.0 | +5,782,533 | +14,217,479 | +28.9% - #0: ToddCoxeter: phase 0.0 = 489ms | run 0 = 489ms | all runs = 489ms | elapsed = 489ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined - #0: ToddCoxeter: nodes | 5,000,004 | 2,964,327 | 7,964,331 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 5,782,533 | 14,217,483 | 28.9% - #0: ToddCoxeter: time | run 0 = 489ms | all runs = 489ms | elapsed = 489ms - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 5,000,004 - #0: ToddCoxeter: n = lookahead_next() = 5,000,000 - #0: ToddCoxeter: large collapse, number of coincidences 100,002 >= 100,000 = large_collapse()! - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.1 STOP - #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined - #0: ToddCoxeter: nodes | 95,046 | 7,869,285 | 7,964,331 - #0: ToddCoxeter: diff 0.1.0 | -4,904,958 | +4,904,958 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 380,177 | 7 | 100.0% - #0: ToddCoxeter: diff 0.1.0 | -5,402,356 | -14,217,476 | +71.1% - #0: ToddCoxeter: phase 0.1 = 449ms | run 0 = 939ms | all runs = 939ms | elapsed = 939ms - #0: ToddCoxeter: lookahead_next() is now max(f x a = 190,092, m = 10,000) (-4,809,908) - #0: ToddCoxeter: because f x a < n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 95,046 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: m = lookahead_min() = 10,000 - #0: ToddCoxeter: n = lookahead_next() = 5,000,000 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.2 START - #0: ToddCoxeter: HLT 0.2.0 | active | killed | defined - #0: ToddCoxeter: nodes | 95,046 | 7,869,285 | 7,964,331 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 380,177 | 7 | 100.0% - #0: ToddCoxeter: time | run 0 = 939ms | all runs = 939ms | elapsed = 939ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.2 STOP - #0: ToddCoxeter: HLT 0.2.1 | active | killed | defined - #0: ToddCoxeter: nodes | 95,040 | 7,869,292 | 7,964,332 - #0: ToddCoxeter: diff 0.2.0 | -6 | +7 | +1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 380,160 | 0 | 100.0% - #0: ToddCoxeter: diff 0.2.0 | -17 | -7 | +0.0% - #0: ToddCoxeter: phase 0.2 = 13ms | run 0 = 952ms | all runs = 952ms | elapsed = 952ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.3.2 | active | killed | defined - #0: ToddCoxeter: nodes | 95,040 | 7,869,292 | 7,964,332 - #0: ToddCoxeter: diff 0.3.1 | +0 | +0 | +0 - #0: ToddCoxeter: diff 0.3.0 | -6 | +7 | +1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 380,160 | 0 | 100.0% - #0: ToddCoxeter: diff 0.3.1 | +0 | +0 | +0.0% - #0: ToddCoxeter: diff 0.3.0 | -17 | -7 | +0.0% - #0: ToddCoxeter: phase 0.3 = 13ms | run 0 = 952ms | all runs = 952ms | elapsed = 952ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 STOP (finished) - #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch - #0: ToddCoxeter: num. phases | 1 | 0 | 2 | 0 - #0: ToddCoxeter: time spent in phases | 449ms (47%) | - (0%) | 502ms (53%) | - (0%) - #0: ToddCoxeter: phase 0.3 = 14ms | run 0 = 953ms | all runs = 953ms | elapsed = 953ms - ``` - -=== "ACE" - - Coming soon! - -:simple-ticktick: The computed size of the group matches the size of the group -provided on [the ATLAS][atlas]: $95,040$. - -[atlas]: https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ -[libsemigroups_pybind11]: - https://libsemigroups.github.io/libsemigroups_pybind11/index.html +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $95,040$. diff --git a/docs/examples/atlas/mathieu/m22.md b/docs/examples/atlas/mathieu/m22.md index 160309e..f95e815 100644 --- a/docs/examples/atlas/mathieu/m22.md +++ b/docs/examples/atlas/mathieu/m22.md @@ -1,536 +1,578 @@ # Mathieu group M~22~ -> Order: $443,520$ +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). -> Presentation: -> $\langle{ a, b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^5 = [a,bab]^3 = (ababab^{−1})^5 = 1 }\rangle$ +## Claim -On this page, we verify that the above claimed presentation of the Mathieu group -M~22~ defines a group of order $443,520$. +$$ + \mathrm{M}_{22} = \langle a, b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^5 = [a,bab]^3 = (ababab^{−1})^5 = 1 \rangle +$$ -## The code +with -=== "Python" +$$ + |\mathrm{M}_{22}| = 443,520. +$$ - In [libsemigroups_pybind11][], the following script constructs the presentation - for M~11~ and runs the Todd-Coxeter algorithm. +On this page, we verify that the above claimed presentation of the Mathieu group +M~22~ defines a group of order $443,520$. - ```python - from libsemigroups_pybind11 import Presentation, ToddCoxeter, congruence_kind, presentation - from libsemigroups_pybind11.words import parse_relations +## The code - # Setup the presentation object with the empty and inverses, so it can represent a group - p = Presentation("abAB") - p.contains_empty_word(True) - presentation.add_inverse_rules(p, "ABab") +In [libsemigroups_pybind11][], the following script constructs the presentation +for M~11~ and runs the Todd-Coxeter algorithm. - # Add the defining relations - presentation.add_rule(p, parse_relations("a^2"), "") - presentation.add_rule(p, parse_relations("b^4"), "") - presentation.add_rule(p, parse_relations("(ab)^11"), "") - presentation.add_rule(p, parse_relations("(ab^2)^5"), "") - presentation.add_rule(p, parse_relations("(a,bab)^3"), "") - presentation.add_rule(p, parse_relations("(ababaB)^5"), "") +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") - # Run the Todd-Coxeter algorithm - tc = ToddCoxeter(congruence_kind.twosided, p) - tc.run() +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +presentation.add_rule(p, parse_relations("(a,bab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") - assert tc.number_of_classes() == 443520 - ``` -=== "ACE" +# Run the Todd-Coxeter algorithm +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.run() - Coming soon! +print(f"The size of the group is {tc.number_of_classes()}") +``` ## The output -=== "Python" +Running the above script produces the following output: - Running the above script produces the following output: - - ??? info "Output from the Python script" - - ``` - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 START (strategy() = hlt) - #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 START - #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined - #0: ToddCoxeter: nodes | 1 | 0 | 1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 0 | 4 | 0.0% - #0: ToddCoxeter: time | run 0 = 23µs | all runs = 23µs | elapsed = 78µs - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 STOP - #0: ToddCoxeter: HLT 0.0.1 | active | killed | defined - #0: ToddCoxeter: nodes | 5,000,018 | 1,632,158 | 6,632,176 - #0: ToddCoxeter: diff 0.0.0 | +5,000,017 | +1,632,158 | +6,632,175 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 5,679,076 | 14,320,996 | 28.4% - #0: ToddCoxeter: diff 0.0.0 | +5,679,076 | +14,320,992 | +28.4% - #0: ToddCoxeter: phase 0.0 = 418ms | run 0 = 418ms | all runs = 418ms | elapsed = 418ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined - #0: ToddCoxeter: nodes | 5,000,018 | 1,632,158 | 6,632,176 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 5,679,076 | 14,320,996 | 28.4% - #0: ToddCoxeter: time | run 0 = 418ms | all runs = 418ms | elapsed = 418ms - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 5,000,018 - #0: ToddCoxeter: n = lookahead_next() = 5,000,000 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.1 STOP - #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined - #0: ToddCoxeter: nodes | 2,423,713 | 4,208,463 | 6,632,176 - #0: ToddCoxeter: diff 0.1.0 | -2,576,305 | +2,576,305 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 7,815,443 | 1,879,409 | 80.6% - #0: ToddCoxeter: diff 0.1.0 | +2,136,367 | -12,441,587 | +52.2% - #0: ToddCoxeter: phase 0.1 = 398ms | run 0 = 816ms | all runs = 816ms | elapsed = 816ms - #0: ToddCoxeter: lookahead_next() is now max(f x a = 4,847,426, m = 10,000) (-152,574) - #0: ToddCoxeter: because f x a < n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 2,423,713 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: m = lookahead_min() = 10,000 - #0: ToddCoxeter: n = lookahead_next() = 5,000,000 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.2 START - #0: ToddCoxeter: HLT 0.2.0 | active | killed | defined - #0: ToddCoxeter: nodes | 2,423,713 | 4,208,463 | 6,632,176 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 7,815,443 | 1,879,409 | 80.6% - #0: ToddCoxeter: time | run 0 = 816ms | all runs = 816ms | elapsed = 816ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.2 STOP - #0: ToddCoxeter: HLT 0.2.1 | active | killed | defined - #0: ToddCoxeter: nodes | 4,847,474 | 4,320,458 | 9,167,932 - #0: ToddCoxeter: diff 0.2.0 | +2,423,761 | +111,995 | +2,535,756 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 10,422,837 | 8,967,059 | 53.8% - #0: ToddCoxeter: diff 0.2.0 | +2,607,394 | +7,087,650 | -26.9% - #0: ToddCoxeter: phase 0.2 = 78ms | run 0 = 894ms | all runs = 894ms | elapsed = 894ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.3 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.3.0 | active | killed | defined - #0: ToddCoxeter: nodes | 4,847,474 | 4,320,458 | 9,167,932 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 10,422,837 | 8,967,059 | 53.8% - #0: ToddCoxeter: time | run 0 = 894ms | all runs = 894ms | elapsed = 894ms - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,847,474 - #0: ToddCoxeter: n = lookahead_next() = 4,847,426 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.3.1 | active | killed | defined - #1: ToddCoxeter: nodes | 4,720,512 | 4,447,422 | 9,167,932 - #1: ToddCoxeter: diff 0.3.0 | -126,962 | +126,964 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 10,288,113 | 8,593,935 | 54.5% - #1: ToddCoxeter: diff 0.3.0 | -134,724 | -373,124 | +0.7% - #1: ToddCoxeter: phase 0.3 = 105ms | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s - #1: ToddCoxeter: lookahead progress | ~17.5% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.3 STOP - #0: ToddCoxeter: LOOKAHEAD 0.3.2 | active | killed | defined - #0: ToddCoxeter: nodes | 3,512,400 | 5,655,532 | 9,167,932 - #0: ToddCoxeter: diff 0.3.1 | -1,208,112 | +1,208,110 | +0 - #0: ToddCoxeter: diff 0.3.0 | -1,335,074 | +1,335,074 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 11,491,787 | 2,557,813 | 81.8% - #0: ToddCoxeter: diff 0.3.1 | +1,203,674 | -6,036,122 | +27.3% - #0: ToddCoxeter: diff 0.3.0 | +1,068,950 | -6,409,246 | +28.0% - #0: ToddCoxeter: phase 0.3 = 556ms | run 0 = 1.451s | all runs = 1.451s | elapsed = 1.451s - #0: ToddCoxeter: lookahead_next() is now 4,847,426 (+0) - #0: ToddCoxeter: because: - #0: ToddCoxeter: 1. n <= f x a = 7,024,800 - #0: ToddCoxeter: 2. a <= n - #0: ToddCoxeter: 3. l >= (l + a) / t = 1,211,868 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 3,512,400 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 1,335,074 - #0: ToddCoxeter: n = lookahead_next() = 4,847,426 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.4 START - #0: ToddCoxeter: HLT 0.4.0 | active | killed | defined - #0: ToddCoxeter: nodes | 3,512,400 | 5,655,532 | 9,167,932 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 11,491,787 | 2,557,813 | 81.8% - #0: ToddCoxeter: time | run 0 = 1.451s | all runs = 1.451s | elapsed = 1.451s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.4 STOP - #0: ToddCoxeter: HLT 0.4.1 | active | killed | defined - #0: ToddCoxeter: nodes | 4,847,449 | 5,703,629 | 10,551,078 - #0: ToddCoxeter: diff 0.4.0 | +1,335,049 | +48,097 | +1,383,146 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 12,934,269 | 6,455,527 | 66.7% - #0: ToddCoxeter: diff 0.4.0 | +1,442,482 | +3,897,714 | -15.1% - #0: ToddCoxeter: phase 0.4 = 62ms | run 0 = 1.513s | all runs = 1.513s | elapsed = 1.514s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.5 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.5.0 | active | killed | defined - #0: ToddCoxeter: nodes | 4,847,449 | 5,703,629 | 10,551,078 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 12,934,269 | 6,455,527 | 66.7% - #0: ToddCoxeter: time | run 0 = 1.513s | all runs = 1.513s | elapsed = 1.514s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,847,449 - #0: ToddCoxeter: n = lookahead_next() = 4,847,426 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.5.1 | active | killed | defined - #1: ToddCoxeter: nodes | 4,565,240 | 5,985,863 | 10,551,078 - #1: ToddCoxeter: diff 0.5.0 | -282,209 | +282,234 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 13,023,236 | 5,237,724 | 71.3% - #1: ToddCoxeter: diff 0.5.0 | +88,967 | -1,217,803 | +4.6% - #1: ToddCoxeter: phase 0.5 = 487ms | run 0 = 2.001s | all runs = 2.001s | elapsed = 2.001s - #1: ToddCoxeter: lookahead progress | ~77.2% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.5 STOP - #0: ToddCoxeter: LOOKAHEAD 0.5.2 | active | killed | defined - #0: ToddCoxeter: nodes | 4,134,106 | 6,416,972 | 10,551,078 - #0: ToddCoxeter: diff 0.5.1 | -431,134 | +431,109 | +0 - #0: ToddCoxeter: diff 0.5.0 | -713,343 | +713,343 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 13,557,552 | 2,978,872 | 82.0% - #0: ToddCoxeter: diff 0.5.1 | +534,316 | -2,258,852 | +10.7% - #0: ToddCoxeter: diff 0.5.0 | +623,283 | -3,476,655 | +15.3% - #0: ToddCoxeter: phase 0.5 = 642ms | run 0 = 2.156s | all runs = 2.156s | elapsed = 2.156s - #0: ToddCoxeter: lookahead_next() is now n x f = 9,694,852 (+4,847,426) - #0: ToddCoxeter: because: l < (l + a) / t = 1,211,862 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,134,106 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 713,343 - #0: ToddCoxeter: n = lookahead_next() = 4,847,426 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.6 START - #0: ToddCoxeter: HLT 0.6.0 | active | killed | defined - #0: ToddCoxeter: nodes | 4,134,106 | 6,416,972 | 10,551,078 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 13,557,552 | 2,978,872 | 82.0% - #0: ToddCoxeter: time | run 0 = 2.156s | all runs = 2.156s | elapsed = 2.156s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.6 STOP - #0: ToddCoxeter: HLT 0.6.1 | active | killed | defined - #0: ToddCoxeter: nodes | 9,694,873 | 6,657,105 | 16,351,978 - #0: ToddCoxeter: diff 0.6.0 | +5,560,767 | +240,133 | +5,800,900 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 19,575,370 | 19,204,122 | 50.5% - #0: ToddCoxeter: diff 0.6.0 | +6,017,818 | +16,225,250 | -31.5% - #0: ToddCoxeter: phase 0.6 = 425ms | run 0 = 2.582s | all runs = 2.582s | elapsed = 2.582s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.7 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.7.0 | active | killed | defined - #0: ToddCoxeter: nodes | 9,694,873 | 6,657,105 | 16,351,978 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 19,575,370 | 19,204,122 | 50.5% - #0: ToddCoxeter: time | run 0 = 2.582s | all runs = 2.582s | elapsed = 2.582s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 9,694,873 - #0: ToddCoxeter: n = lookahead_next() = 9,694,852 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.7.1 | active | killed | defined - #1: ToddCoxeter: nodes | 9,521,847 | 6,830,131 | 16,351,978 - #1: ToddCoxeter: diff 0.7.0 | -173,026 | +173,026 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 19,432,594 | 18,654,794 | 51.0% - #1: ToddCoxeter: diff 0.7.0 | -142,776 | -549,328 | +0.5% - #1: ToddCoxeter: phase 0.7 = 418ms | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s - #1: ToddCoxeter: lookahead progress | ~33.5% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.7 STOP - #0: ToddCoxeter: LOOKAHEAD 0.7.2 | active | killed | defined - #0: ToddCoxeter: nodes | 6,784,030 | 9,567,948 | 16,351,978 - #0: ToddCoxeter: diff 0.7.1 | -2,737,817 | +2,737,817 | +0 - #0: ToddCoxeter: diff 0.7.0 | -2,910,843 | +2,910,843 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 22,154,799 | 4,981,321 | 81.6% - #0: ToddCoxeter: diff 0.7.1 | +2,722,205 | -13,673,473 | +30.6% - #0: ToddCoxeter: diff 0.7.0 | +2,579,429 | -14,222,801 | +31.2% - #0: ToddCoxeter: phase 0.7 = 1.109s | run 0 = 3.691s | all runs = 3.691s | elapsed = 3.691s - #0: ToddCoxeter: lookahead_next() is now 9,694,852 (+0) - #0: ToddCoxeter: because: - #0: ToddCoxeter: 1. n <= f x a = 13,568,060 - #0: ToddCoxeter: 2. a <= n - #0: ToddCoxeter: 3. l >= (l + a) / t = 2,423,718 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 6,784,030 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 2,910,843 - #0: ToddCoxeter: n = lookahead_next() = 9,694,852 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.8 START - #0: ToddCoxeter: HLT 0.8.0 | active | killed | defined - #0: ToddCoxeter: nodes | 6,784,030 | 9,567,948 | 16,351,978 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 22,154,799 | 4,981,321 | 81.6% - #0: ToddCoxeter: time | run 0 = 3.691s | all runs = 3.691s | elapsed = 3.691s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.8 STOP - #0: ToddCoxeter: HLT 0.8.1 | active | killed | defined - #0: ToddCoxeter: nodes | 9,694,854 | 9,675,719 | 19,370,573 - #0: ToddCoxeter: diff 0.8.0 | +2,910,824 | +107,771 | +3,018,595 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 25,297,258 | 13,482,158 | 65.2% - #0: ToddCoxeter: diff 0.8.0 | +3,142,459 | +8,500,837 | -16.4% - #0: ToddCoxeter: phase 0.8 = 113ms | run 0 = 3.805s | all runs = 3.805s | elapsed = 3.805s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.9 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.9.0 | active | killed | defined - #0: ToddCoxeter: nodes | 9,694,854 | 9,675,719 | 19,370,573 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 25,297,258 | 13,482,158 | 65.2% - #0: ToddCoxeter: time | run 0 = 3.805s | all runs = 3.805s | elapsed = 3.805s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 9,694,854 - #0: ToddCoxeter: n = lookahead_next() = 9,694,852 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.9.1 | active | killed | defined - #1: ToddCoxeter: nodes | 9,601,434 | 9,769,139 | 19,370,573 - #1: ToddCoxeter: diff 0.9.0 | -93,420 | +93,420 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 25,215,676 | 13,190,060 | 65.7% - #1: ToddCoxeter: diff 0.9.0 | -81,582 | -292,098 | +0.4% - #1: ToddCoxeter: phase 0.9 = 196ms | run 0 = 4.001s | all runs = 4.001s | elapsed = 4.001s - #1: ToddCoxeter: lookahead progress | ~16.5% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.9.2 | active | killed | defined - #1: ToddCoxeter: nodes | 8,331,834 | 11,038,768 | 19,370,573 - #1: ToddCoxeter: diff 0.9.1 | -1,269,600 | +1,269,629 | +0 - #1: ToddCoxeter: diff 0.9.0 | -1,363,020 | +1,363,049 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 26,231,811 | 7,095,525 | 78.7% - #1: ToddCoxeter: diff 0.9.1 | +1,016,135 | -6,094,535 | +13.1% - #1: ToddCoxeter: diff 0.9.0 | +934,553 | -6,386,633 | +13.5% - #1: ToddCoxeter: phase 0.9 = 1.197s | run 0 = 5.001s | all runs = 5.001s | elapsed = 5.001s - #1: ToddCoxeter: lookahead progress | ~87.1% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.9 STOP - #0: ToddCoxeter: LOOKAHEAD 0.9.3 | active | killed | defined - #0: ToddCoxeter: nodes | 8,086,523 | 11,284,050 | 19,370,573 - #0: ToddCoxeter: diff 0.9.2 | -245,311 | +245,282 | +0 - #0: ToddCoxeter: diff 0.9.0 | -1,608,331 | +1,608,331 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 26,555,088 | 5,791,004 | 82.1% - #0: ToddCoxeter: diff 0.9.2 | +323,277 | -1,304,521 | +3.4% - #0: ToddCoxeter: diff 0.9.0 | +1,257,830 | -7,691,154 | +16.9% - #0: ToddCoxeter: phase 0.9 = 1.320s | run 0 = 5.125s | all runs = 5.125s | elapsed = 5.125s - #0: ToddCoxeter: lookahead_next() is now n x f = 19,389,704 (+9,694,852) - #0: ToddCoxeter: because: l < (l + a) / t = 2,423,713 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 8,086,523 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 1,608,331 - #0: ToddCoxeter: n = lookahead_next() = 9,694,852 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.10 START - #0: ToddCoxeter: HLT 0.10.0 | active | killed | defined - #0: ToddCoxeter: nodes | 8,086,523 | 11,284,050 | 19,370,573 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 26,555,088 | 5,791,004 | 82.1% - #0: ToddCoxeter: time | run 0 = 5.125s | all runs = 5.125s | elapsed = 5.125s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.10 STOP - #0: ToddCoxeter: HLT 0.10.1 | active | killed | defined - #0: ToddCoxeter: nodes | 19,389,750 | 11,773,835 | 31,163,585 - #0: ToddCoxeter: diff 0.10.0 | +11,303,227 | +489,785 | +11,793,012 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 38,786,740 | 38,772,260 | 50.0% - #0: ToddCoxeter: diff 0.10.0 | +12,231,652 | +32,981,256 | -32.1% - #0: ToddCoxeter: phase 0.10 = 829ms | run 0 = 5.955s | all runs = 5.955s | elapsed = 5.955s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.11 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.11.0 | active | killed | defined - #0: ToddCoxeter: nodes | 19,389,750 | 11,773,835 | 31,163,585 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 38,786,740 | 38,772,260 | 50.0% - #0: ToddCoxeter: time | run 0 = 5.955s | all runs = 5.955s | elapsed = 5.955s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 19,389,750 - #0: ToddCoxeter: n = lookahead_next() = 19,389,704 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.11.1 | active | killed | defined - #1: ToddCoxeter: nodes | 19,361,711 | 11,801,893 | 31,163,585 - #1: ToddCoxeter: diff 0.11.0 | -28,039 | +28,058 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 38,761,553 | 38,685,291 | 50.0% - #1: ToddCoxeter: diff 0.11.0 | -25,187 | -86,969 | +0.0% - #1: ToddCoxeter: phase 0.11 = 46ms | run 0 = 6.002s | all runs = 6.002s | elapsed = 6.002s - #1: ToddCoxeter: lookahead progress | ~4.6% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.11.2 | active | killed | defined - #1: ToddCoxeter: nodes | 18,900,346 | 12,263,243 | 31,163,585 - #1: ToddCoxeter: diff 0.11.1 | -461,365 | +461,350 | +0 - #1: ToddCoxeter: diff 0.11.0 | -489,404 | +489,408 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 38,376,840 | 37,224,544 | 50.8% - #1: ToddCoxeter: diff 0.11.1 | -384,713 | -1,460,747 | +0.7% - #1: ToddCoxeter: diff 0.11.0 | -409,900 | -1,547,716 | +0.8% - #1: ToddCoxeter: phase 0.11 = 1.047s | run 0 = 7.002s | all runs = 7.002s | elapsed = 7.002s - #1: ToddCoxeter: lookahead progress | ~38.4% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.11.3 | active | killed | defined - #1: ToddCoxeter: nodes | 14,438,129 | 16,725,554 | 31,163,585 - #1: ToddCoxeter: diff 0.11.2 | -4,462,217 | +4,462,311 | +0 - #1: ToddCoxeter: diff 0.11.0 | -4,951,621 | +4,951,719 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 42,716,669 | 15,035,847 | 74.0% - #1: ToddCoxeter: diff 0.11.2 | +4,339,829 | -22,188,697 | +23.2% - #1: ToddCoxeter: diff 0.11.0 | +3,929,929 | -23,736,413 | +24.0% - #1: ToddCoxeter: phase 0.11 = 2.047s | run 0 = 8.002s | all runs = 8.002s | elapsed = 8.002s - #1: ToddCoxeter: lookahead progress | ~72.2% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.11 STOP - #0: ToddCoxeter: LOOKAHEAD 0.11.4 | active | killed | defined - #0: ToddCoxeter: nodes | 13,427,510 | 17,736,075 | 31,163,585 - #0: ToddCoxeter: diff 0.11.3 | -1,010,619 | +1,010,521 | +0 - #0: ToddCoxeter: diff 0.11.0 | -5,962,240 | +5,962,240 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 43,882,513 | 9,827,527 | 81.7% - #0: ToddCoxeter: diff 0.11.3 | +1,165,844 | -5,208,320 | +7.7% - #0: ToddCoxeter: diff 0.11.0 | +5,095,773 | -28,944,733 | +31.7% - #0: ToddCoxeter: phase 0.11 = 2.208s | run 0 = 8.163s | all runs = 8.163s | elapsed = 8.163s - #0: ToddCoxeter: lookahead_next() is now 19,389,704 (+0) - #0: ToddCoxeter: because: - #0: ToddCoxeter: 1. n <= f x a = 26,855,020 - #0: ToddCoxeter: 2. a <= n - #0: ToddCoxeter: 3. l >= (l + a) / t = 4,847,437 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 13,427,510 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 5,962,240 - #0: ToddCoxeter: n = lookahead_next() = 19,389,704 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.12 START - #0: ToddCoxeter: HLT 0.12.0 | active | killed | defined - #0: ToddCoxeter: nodes | 13,427,510 | 17,736,075 | 31,163,585 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 43,882,513 | 9,827,527 | 81.7% - #0: ToddCoxeter: time | run 0 = 8.163s | all runs = 8.163s | elapsed = 8.163s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.12 STOP - #0: ToddCoxeter: HLT 0.12.1 | active | killed | defined - #0: ToddCoxeter: nodes | 19,389,776 | 17,959,888 | 37,349,664 - #0: ToddCoxeter: diff 0.12.0 | +5,962,266 | +223,813 | +6,186,079 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 50,324,580 | 27,234,524 | 64.9% - #0: ToddCoxeter: diff 0.12.0 | +6,442,067 | +17,406,997 | -16.8% - #0: ToddCoxeter: phase 0.12 = 230ms | run 0 = 8.393s | all runs = 8.393s | elapsed = 8.393s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.13 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.13.0 | active | killed | defined - #0: ToddCoxeter: nodes | 19,389,776 | 17,959,888 | 37,349,664 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 50,324,580 | 27,234,524 | 64.9% - #0: ToddCoxeter: time | run 0 = 8.393s | all runs = 8.393s | elapsed = 8.393s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 19,389,776 - #0: ToddCoxeter: n = lookahead_next() = 19,389,704 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.1 | active | killed | defined - #1: ToddCoxeter: nodes | 19,186,302 | 18,163,362 | 37,349,664 - #1: ToddCoxeter: diff 0.13.0 | -203,474 | +203,474 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 50,106,203 | 26,639,005 | 65.3% - #1: ToddCoxeter: diff 0.13.0 | -218,377 | -595,519 | +0.4% - #1: ToddCoxeter: phase 0.13 = 609ms | run 0 = 9.002s | all runs = 9.002s | elapsed = 9.002s - #1: ToddCoxeter: lookahead progress | ~20.7% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.2 | active | killed | defined - #1: ToddCoxeter: nodes | 18,905,330 | 18,444,334 | 37,349,664 - #1: ToddCoxeter: diff 0.13.1 | -280,972 | +280,972 | +0 - #1: ToddCoxeter: diff 0.13.0 | -484,446 | +484,446 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 49,829,308 | 25,792,012 | 65.9% - #1: ToddCoxeter: diff 0.13.1 | -276,895 | -846,993 | +0.6% - #1: ToddCoxeter: diff 0.13.0 | -495,272 | -1,442,512 | +1.0% - #1: ToddCoxeter: phase 0.13 = 1.610s | run 0 = 10.003s | all runs = 10.003s | elapsed = 10.003s - #1: ToddCoxeter: lookahead progress | ~54.1% - #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()! - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.3 | active | killed | defined - #1: ToddCoxeter: nodes | 11,154,426 | 26,195,391 | 37,349,664 - #1: ToddCoxeter: diff 0.13.2 | -7,750,904 | +7,751,057 | +0 - #1: ToddCoxeter: diff 0.13.0 | -8,235,350 | +8,235,503 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 0 | 44,617,704 | 0.0% - #1: ToddCoxeter: diff 0.13.2 | -49,829,308 | +18,825,692 | -65.9% - #1: ToddCoxeter: diff 0.13.0 | -50,324,580 | +17,383,180 | -64.9% - #1: ToddCoxeter: phase 0.13 = 2.610s | run 0 = 11.003s | all runs = 11.003s | elapsed = 11.003s - #1: ToddCoxeter: lookahead progress | ~66.6% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.4 | active | killed | defined - #1: ToddCoxeter: nodes | 2,148,036 | 35,201,729 | 37,349,664 - #1: ToddCoxeter: diff 0.13.3 | -9,006,390 | +9,006,338 | +0 - #1: ToddCoxeter: diff 0.13.0 | -17,241,740 | +17,241,841 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 0 | 8,592,144 | 0.0% - #1: ToddCoxeter: diff 0.13.3 | +0 | -36,025,560 | +0.0% - #1: ToddCoxeter: diff 0.13.0 | -50,324,580 | -18,642,380 | -64.9% - #1: ToddCoxeter: phase 0.13 = 3.610s | run 0 = 12.003s | all runs = 12.003s | elapsed = 12.003s - #1: ToddCoxeter: lookahead progress | ~66.6% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.13 STOP - #0: ToddCoxeter: LOOKAHEAD 0.13.5 | active | killed | defined - #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #0: ToddCoxeter: diff 0.13.4 | -1,704,516 | +1,704,415 | +0 - #0: ToddCoxeter: diff 0.13.0 | -18,946,256 | +18,946,256 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #0: ToddCoxeter: diff 0.13.4 | +1,774,080 | -8,592,144 | +100.0% - #0: ToddCoxeter: diff 0.13.0 | -48,550,500 | -27,234,524 | +35.1% - #0: ToddCoxeter: phase 0.13 = 3.928s | run 0 = 12.321s | all runs = 12.321s | elapsed = 12.321s - #0: ToddCoxeter: lookahead_next() is now max(f x a = 887,040, m = 10,000) (-18,502,664) - #0: ToddCoxeter: because f x a < n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 443,520 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: m = lookahead_min() = 10,000 - #0: ToddCoxeter: n = lookahead_next() = 19,389,704 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.14 START - #0: ToddCoxeter: HLT 0.14.0 | active | killed | defined - #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #0: ToddCoxeter: time | run 0 = 12.321s | all runs = 12.321s | elapsed = 12.321s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.14 STOP - #0: ToddCoxeter: HLT 0.14.1 | active | killed | defined - #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #0: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #0: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0.0% - #0: ToddCoxeter: phase 0.14 = 108ms | run 0 = 12.430s | all runs = 12.430s | elapsed = 12.430s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.15.2 | active | killed | defined - #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #0: ToddCoxeter: diff 0.15.1 | +0 | +0 | +0 - #0: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #0: ToddCoxeter: diff 0.15.1 | +0 | +0 | +0.0% - #0: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0.0% - #0: ToddCoxeter: phase 0.15 = 108ms | run 0 = 12.430s | all runs = 12.430s | elapsed = 12.430s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 STOP (finished) - #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch - #0: ToddCoxeter: num. phases | 7 | 0 | 8 | 0 - #0: ToddCoxeter: time spent in phases | 10.163s (82%) | - (0%) | 2.267s (18%) | - (0%) - #0: ToddCoxeter: phase 0.15 = 114ms | run 0 = 12.436s | all runs = 12.436s | elapsed = 12.436s - ``` - -=== "ACE" - - Coming soon! - -:simple-ticktick: The computed size of the group matches the size of the group -provided on [the ATLAS][atlas]: $443,520$. +??? info "Output from the Python script" + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: running for approx. 30.000s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 4 | 0.0% + #0: ToddCoxeter: time | run 0 = 19µs | all runs = 19µs | elapsed = 89µs + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 STOP + #0: ToddCoxeter: HLT 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 5,000,018 | 1,632,158 | 6,632,176 + #0: ToddCoxeter: diff 0.0.0 | +5,000,017 | +1,632,158 | +6,632,175 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,679,076 | 14,320,996 | 28.4% + #0: ToddCoxeter: diff 0.0.0 | +5,679,076 | +14,320,992 | +28.4% + #0: ToddCoxeter: phase 0.0 = 547ms | run 0 = 547ms | all runs = 547ms | elapsed = 547ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = partial, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 5,000,018 | 1,632,158 | 6,632,176 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,679,076 | 14,320,996 | 28.4% + #0: ToddCoxeter: time | run 0 = 547ms | all runs = 547ms | elapsed = 547ms + #0: ToddCoxeter: because a >= n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 5,000,018 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #1: ToddCoxeter: nodes | 2,644,921 | 3,987,547 | 6,632,176 + #1: ToddCoxeter: diff 0.1.0 | -2,355,097 | +2,355,389 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 7,553,691 | 3,025,993 | 71.4% + #1: ToddCoxeter: diff 0.1.0 | +1,874,615 | -11,295,003 | +43.0% + #1: ToddCoxeter: phase 0.1 = 452ms | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.001s + #1: ToddCoxeter: lookahead progress | ~60.5% + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.2 | active | killed | defined + #0: ToddCoxeter: nodes | 2,423,713 | 4,208,463 | 6,632,176 + #0: ToddCoxeter: diff 0.1.1 | -221,208 | +220,916 | +0 + #0: ToddCoxeter: diff 0.1.0 | -2,576,305 | +2,576,305 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 7,815,443 | 1,879,409 | 80.6% + #0: ToddCoxeter: diff 0.1.1 | +261,752 | -1,146,584 | +9.2% + #0: ToddCoxeter: diff 0.1.0 | +2,136,367 | -12,441,587 | +52.2% + #0: ToddCoxeter: phase 0.1 = 500ms | run 0 = 1.049s | all runs = 1.049s | elapsed = 1.049s + #0: ToddCoxeter: lookahead_next() is now max(f x a = 4,847,426, m = 10,000) (-152,574) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 2,423,713 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.2 START + #0: ToddCoxeter: HLT 0.2.0 | active | killed | defined + #0: ToddCoxeter: nodes | 2,423,713 | 4,208,463 | 6,632,176 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 7,815,443 | 1,879,409 | 80.6% + #0: ToddCoxeter: time | run 0 = 1.049s | all runs = 1.049s | elapsed = 1.049s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.2 STOP + #0: ToddCoxeter: HLT 0.2.1 | active | killed | defined + #0: ToddCoxeter: nodes | 4,847,474 | 4,320,458 | 9,167,932 + #0: ToddCoxeter: diff 0.2.0 | +2,423,761 | +111,995 | +2,535,756 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 10,422,837 | 8,967,059 | 53.8% + #0: ToddCoxeter: diff 0.2.0 | +2,607,394 | +7,087,650 | -26.9% + #0: ToddCoxeter: phase 0.2 = 89ms | run 0 = 1.138s | all runs = 1.138s | elapsed = 1.138s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.3 START (lookahead_extent() = partial, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.3.0 | active | killed | defined + #0: ToddCoxeter: nodes | 4,847,474 | 4,320,458 | 9,167,932 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 10,422,837 | 8,967,059 | 53.8% + #0: ToddCoxeter: time | run 0 = 1.138s | all runs = 1.138s | elapsed = 1.138s + #0: ToddCoxeter: because a >= n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,847,474 + #0: ToddCoxeter: n = lookahead_next() = 4,847,426 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.3 STOP + #0: ToddCoxeter: LOOKAHEAD 0.3.1 | active | killed | defined + #0: ToddCoxeter: nodes | 3,512,400 | 5,655,532 | 9,167,932 + #0: ToddCoxeter: diff 0.3.0 | -1,335,074 | +1,335,074 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 11,491,787 | 2,557,813 | 81.8% + #0: ToddCoxeter: diff 0.3.0 | +1,068,950 | -6,409,246 | +28.0% + #0: ToddCoxeter: phase 0.3 = 718ms | run 0 = 1.856s | all runs = 1.856s | elapsed = 1.857s + #0: ToddCoxeter: lookahead_next() is now 4,847,426 (+0) + #0: ToddCoxeter: because: + #0: ToddCoxeter: 1. n <= f x a = 7,024,800 + #0: ToddCoxeter: 2. a <= n + #0: ToddCoxeter: 3. l >= (l + a) / t = 1,211,868 + #0: ToddCoxeter: where: a = number_of_nodes_active() = 3,512,400 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: l = nodes killed in lookahead = 1,335,074 + #0: ToddCoxeter: n = lookahead_next() = 4,847,426 + #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.4 START + #0: ToddCoxeter: HLT 0.4.0 | active | killed | defined + #0: ToddCoxeter: nodes | 3,512,400 | 5,655,532 | 9,167,932 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 11,491,787 | 2,557,813 | 81.8% + #0: ToddCoxeter: time | run 0 = 1.857s | all runs = 1.857s | elapsed = 1.857s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.4 STOP + #0: ToddCoxeter: HLT 0.4.1 | active | killed | defined + #0: ToddCoxeter: nodes | 4,847,449 | 5,703,629 | 10,551,078 + #0: ToddCoxeter: diff 0.4.0 | +1,335,049 | +48,097 | +1,383,146 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 12,934,269 | 6,455,527 | 66.7% + #0: ToddCoxeter: diff 0.4.0 | +1,442,482 | +3,897,714 | -15.1% + #0: ToddCoxeter: phase 0.4 = 67ms | run 0 = 1.924s | all runs = 1.924s | elapsed = 1.924s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.5 START (lookahead_extent() = partial, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.5.0 | active | killed | defined + #0: ToddCoxeter: nodes | 4,847,449 | 5,703,629 | 10,551,078 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 12,934,269 | 6,455,527 | 66.7% + #0: ToddCoxeter: time | run 0 = 1.924s | all runs = 1.924s | elapsed = 1.924s + #0: ToddCoxeter: because a >= n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,847,449 + #0: ToddCoxeter: n = lookahead_next() = 4,847,426 + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.5.1 | active | killed | defined + #1: ToddCoxeter: nodes | 4,820,487 | 5,730,591 | 10,551,078 + #1: ToddCoxeter: diff 0.5.0 | -26,962 | +26,962 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 12,908,450 | 6,373,498 | 66.9% + #1: ToddCoxeter: diff 0.5.0 | -25,819 | -82,029 | +0.2% + #1: ToddCoxeter: phase 0.5 = 77ms | run 0 = 2.001s | all runs = 2.001s | elapsed = 2.001s + #1: ToddCoxeter: lookahead progress | ~11.0% + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.5 STOP + #0: ToddCoxeter: LOOKAHEAD 0.5.2 | active | killed | defined + #0: ToddCoxeter: nodes | 4,134,106 | 6,416,972 | 10,551,078 + #0: ToddCoxeter: diff 0.5.1 | -686,381 | +686,381 | +0 + #0: ToddCoxeter: diff 0.5.0 | -713,343 | +713,343 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 13,557,552 | 2,978,872 | 82.0% + #0: ToddCoxeter: diff 0.5.1 | +649,102 | -3,394,626 | +15.0% + #0: ToddCoxeter: diff 0.5.0 | +623,283 | -3,476,655 | +15.3% + #0: ToddCoxeter: phase 0.5 = 856ms | run 0 = 2.780s | all runs = 2.780s | elapsed = 2.780s + #0: ToddCoxeter: lookahead_next() is now n x f = 9,694,852 (+4,847,426) + #0: ToddCoxeter: because: l < (l + a) / t = 1,211,862 + #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,134,106 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: l = nodes killed in lookahead = 713,343 + #0: ToddCoxeter: n = lookahead_next() = 4,847,426 + #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.6 START + #0: ToddCoxeter: HLT 0.6.0 | active | killed | defined + #0: ToddCoxeter: nodes | 4,134,106 | 6,416,972 | 10,551,078 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 13,557,552 | 2,978,872 | 82.0% + #0: ToddCoxeter: time | run 0 = 2.780s | all runs = 2.780s | elapsed = 2.780s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.6.1 | active | killed | defined + #1: ToddCoxeter: nodes | 8,388,608 | 6,594,998 | 14,983,606 + #1: ToddCoxeter: diff 0.6.0 | +4,254,502 | +178,026 | +4,432,528 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 18,160,579 | 15,393,853 | 54.1% + #1: ToddCoxeter: diff 0.6.0 | +4,603,027 | +12,414,981 | -27.9% + #1: ToddCoxeter: phase 0.6 = 221ms | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.6 STOP + #0: ToddCoxeter: HLT 0.6.2 | active | killed | defined + #0: ToddCoxeter: nodes | 9,694,873 | 6,657,105 | 16,351,978 + #0: ToddCoxeter: diff 0.6.1 | +1,306,265 | +62,107 | +1,368,372 + #0: ToddCoxeter: diff 0.6.0 | +5,560,767 | +240,133 | +5,800,900 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 19,575,370 | 19,204,122 | 50.5% + #0: ToddCoxeter: diff 0.6.1 | +1,414,791 | +3,810,269 | -3.6% + #0: ToddCoxeter: diff 0.6.0 | +6,017,818 | +16,225,250 | -31.5% + #0: ToddCoxeter: phase 0.6 = 520ms | run 0 = 3.301s | all runs = 3.301s | elapsed = 3.301s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.7 START (lookahead_extent() = partial, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.7.0 | active | killed | defined + #0: ToddCoxeter: nodes | 9,694,873 | 6,657,105 | 16,351,978 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 19,575,370 | 19,204,122 | 50.5% + #0: ToddCoxeter: time | run 0 = 3.301s | all runs = 3.301s | elapsed = 3.301s + #0: ToddCoxeter: because a >= n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 9,694,873 + #0: ToddCoxeter: n = lookahead_next() = 9,694,852 + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.7.1 | active | killed | defined + #1: ToddCoxeter: nodes | 9,431,978 | 6,920,002 | 16,351,978 + #1: ToddCoxeter: diff 0.7.0 | -262,895 | +262,897 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 19,360,399 | 18,367,513 | 51.3% + #1: ToddCoxeter: diff 0.7.0 | -214,971 | -836,609 | +0.8% + #1: ToddCoxeter: phase 0.7 = 700ms | run 0 = 4.001s | all runs = 4.001s | elapsed = 4.001s + #1: ToddCoxeter: lookahead progress | ~41.5% + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.7 STOP + #0: ToddCoxeter: LOOKAHEAD 0.7.2 | active | killed | defined + #0: ToddCoxeter: nodes | 6,784,030 | 9,567,948 | 16,351,978 + #0: ToddCoxeter: diff 0.7.1 | -2,647,948 | +2,647,946 | +0 + #0: ToddCoxeter: diff 0.7.0 | -2,910,843 | +2,910,843 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 22,154,799 | 4,981,321 | 81.6% + #0: ToddCoxeter: diff 0.7.1 | +2,794,400 | -13,386,192 | +30.3% + #0: ToddCoxeter: diff 0.7.0 | +2,579,429 | -14,222,801 | +31.2% + #0: ToddCoxeter: phase 0.7 = 1.445s | run 0 = 4.746s | all runs = 4.746s | elapsed = 4.746s + #0: ToddCoxeter: lookahead_next() is now 9,694,852 (+0) + #0: ToddCoxeter: because: + #0: ToddCoxeter: 1. n <= f x a = 13,568,060 + #0: ToddCoxeter: 2. a <= n + #0: ToddCoxeter: 3. l >= (l + a) / t = 2,423,718 + #0: ToddCoxeter: where: a = number_of_nodes_active() = 6,784,030 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: l = nodes killed in lookahead = 2,910,843 + #0: ToddCoxeter: n = lookahead_next() = 9,694,852 + #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.8 START + #0: ToddCoxeter: HLT 0.8.0 | active | killed | defined + #0: ToddCoxeter: nodes | 6,784,030 | 9,567,948 | 16,351,978 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 22,154,799 | 4,981,321 | 81.6% + #0: ToddCoxeter: time | run 0 = 4.746s | all runs = 4.746s | elapsed = 4.746s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.8 STOP + #0: ToddCoxeter: HLT 0.8.1 | active | killed | defined + #0: ToddCoxeter: nodes | 9,694,854 | 9,675,719 | 19,370,573 + #0: ToddCoxeter: diff 0.8.0 | +2,910,824 | +107,771 | +3,018,595 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 25,297,258 | 13,482,158 | 65.2% + #0: ToddCoxeter: diff 0.8.0 | +3,142,459 | +8,500,837 | -16.4% + #0: ToddCoxeter: phase 0.8 = 130ms | run 0 = 4.876s | all runs = 4.876s | elapsed = 4.876s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.9 START (lookahead_extent() = partial, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.9.0 | active | killed | defined + #0: ToddCoxeter: nodes | 9,694,854 | 9,675,719 | 19,370,573 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 25,297,258 | 13,482,158 | 65.2% + #0: ToddCoxeter: time | run 0 = 4.876s | all runs = 4.876s | elapsed = 4.876s + #0: ToddCoxeter: because a >= n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 9,694,854 + #0: ToddCoxeter: n = lookahead_next() = 9,694,852 + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.9.1 | active | killed | defined + #1: ToddCoxeter: nodes | 9,648,811 | 9,721,762 | 19,370,573 + #1: ToddCoxeter: diff 0.9.0 | -46,043 | +46,043 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 25,256,260 | 13,338,984 | 65.4% + #1: ToddCoxeter: diff 0.9.0 | -40,998 | -143,174 | +0.2% + #1: ToddCoxeter: phase 0.9 = 125ms | run 0 = 5.001s | all runs = 5.001s | elapsed = 5.002s + #1: ToddCoxeter: lookahead progress | ~9.5% + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.9.2 | active | killed | defined + #1: ToddCoxeter: nodes | 9,403,427 | 9,967,160 | 19,370,573 + #1: ToddCoxeter: diff 0.9.1 | -245,384 | +245,398 | +0 + #1: ToddCoxeter: diff 0.9.0 | -291,427 | +291,441 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 25,010,253 | 12,603,455 | 66.5% + #1: ToddCoxeter: diff 0.9.1 | -246,007 | -735,529 | +1.1% + #1: ToddCoxeter: diff 0.9.0 | -287,005 | -878,703 | +1.3% + #1: ToddCoxeter: phase 0.9 = 1.126s | run 0 = 6.002s | all runs = 6.002s | elapsed = 6.002s + #1: ToddCoxeter: lookahead progress | ~63.4% + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.9 STOP + #0: ToddCoxeter: LOOKAHEAD 0.9.3 | active | killed | defined + #0: ToddCoxeter: nodes | 8,086,523 | 11,284,050 | 19,370,573 + #0: ToddCoxeter: diff 0.9.2 | -1,316,904 | +1,316,890 | +0 + #0: ToddCoxeter: diff 0.9.0 | -1,608,331 | +1,608,331 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 26,555,088 | 5,791,004 | 82.1% + #0: ToddCoxeter: diff 0.9.2 | +1,544,835 | -6,812,451 | +15.6% + #0: ToddCoxeter: diff 0.9.0 | +1,257,830 | -7,691,154 | +16.9% + #0: ToddCoxeter: phase 0.9 = 1.690s | run 0 = 6.566s | all runs = 6.566s | elapsed = 6.566s + #0: ToddCoxeter: lookahead_next() is now n x f = 19,389,704 (+9,694,852) + #0: ToddCoxeter: because: l < (l + a) / t = 2,423,713 + #0: ToddCoxeter: where: a = number_of_nodes_active() = 8,086,523 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: l = nodes killed in lookahead = 1,608,331 + #0: ToddCoxeter: n = lookahead_next() = 9,694,852 + #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.10 START + #0: ToddCoxeter: HLT 0.10.0 | active | killed | defined + #0: ToddCoxeter: nodes | 8,086,523 | 11,284,050 | 19,370,573 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 26,555,088 | 5,791,004 | 82.1% + #0: ToddCoxeter: time | run 0 = 6.566s | all runs = 6.566s | elapsed = 6.566s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.10.1 | active | killed | defined + #1: ToddCoxeter: nodes | 16,777,216 | 11,642,699 | 28,419,915 + #1: ToddCoxeter: diff 0.10.0 | +8,690,693 | +358,649 | +9,049,342 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 35,954,563 | 31,154,301 | 53.6% + #1: ToddCoxeter: diff 0.10.0 | +9,399,475 | +25,363,297 | -28.5% + #1: ToddCoxeter: phase 0.10 = 436ms | run 0 = 7.002s | all runs = 7.002s | elapsed = 7.002s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.10 STOP + #0: ToddCoxeter: HLT 0.10.2 | active | killed | defined + #0: ToddCoxeter: nodes | 19,389,750 | 11,773,835 | 31,163,585 + #0: ToddCoxeter: diff 0.10.1 | +2,612,534 | +131,136 | +2,743,670 + #0: ToddCoxeter: diff 0.10.0 | +11,303,227 | +489,785 | +11,793,012 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 38,786,740 | 38,772,260 | 50.0% + #0: ToddCoxeter: diff 0.10.1 | +2,832,177 | +7,617,959 | -3.6% + #0: ToddCoxeter: diff 0.10.0 | +12,231,652 | +32,981,256 | -32.1% + #0: ToddCoxeter: phase 0.10 = 1.051s | run 0 = 7.617s | all runs = 7.617s | elapsed = 7.617s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.11 START (lookahead_extent() = partial, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.11.0 | active | killed | defined + #0: ToddCoxeter: nodes | 19,389,750 | 11,773,835 | 31,163,585 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 38,786,740 | 38,772,260 | 50.0% + #0: ToddCoxeter: time | run 0 = 7.617s | all runs = 7.617s | elapsed = 7.617s + #0: ToddCoxeter: because a >= n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 19,389,750 + #0: ToddCoxeter: n = lookahead_next() = 19,389,704 + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.11.1 | active | killed | defined + #1: ToddCoxeter: nodes | 19,119,246 | 12,044,357 | 31,163,585 + #1: ToddCoxeter: diff 0.11.0 | -270,504 | +270,522 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 38,564,145 | 37,912,839 | 50.4% + #1: ToddCoxeter: diff 0.11.0 | -222,595 | -859,421 | +0.4% + #1: ToddCoxeter: phase 0.11 = 385ms | run 0 = 8.002s | all runs = 8.002s | elapsed = 8.002s + #1: ToddCoxeter: lookahead progress | ~12.6% + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.11.2 | active | killed | defined + #1: ToddCoxeter: nodes | 18,894,507 | 12,269,078 | 31,163,585 + #1: ToddCoxeter: diff 0.11.1 | -224,739 | +224,721 | +0 + #1: ToddCoxeter: diff 0.11.0 | -495,243 | +495,243 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 38,371,598 | 37,206,430 | 50.8% + #1: ToddCoxeter: diff 0.11.1 | -192,547 | -706,409 | +0.3% + #1: ToddCoxeter: diff 0.11.0 | -415,142 | -1,565,830 | +0.8% + #1: ToddCoxeter: phase 0.11 = 1.385s | run 0 = 9.003s | all runs = 9.003s | elapsed = 9.003s + #1: ToddCoxeter: lookahead progress | ~38.8% + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.11.3 | active | killed | defined + #1: ToddCoxeter: nodes | 15,851,580 | 15,312,085 | 31,163,585 + #1: ToddCoxeter: diff 0.11.2 | -3,042,927 | +3,043,007 | +0 + #1: ToddCoxeter: diff 0.11.0 | -3,538,170 | +3,538,250 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 41,333,919 | 22,072,401 | 65.2% + #1: ToddCoxeter: diff 0.11.2 | +2,962,321 | -15,134,029 | +14.4% + #1: ToddCoxeter: diff 0.11.0 | +2,547,179 | -16,699,859 | +15.2% + #1: ToddCoxeter: phase 0.11 = 2.386s | run 0 = 10.003s | all runs = 10.003s | elapsed = 10.003s + #1: ToddCoxeter: lookahead progress | ~62.6% + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.11 STOP + #0: ToddCoxeter: LOOKAHEAD 0.11.4 | active | killed | defined + #0: ToddCoxeter: nodes | 13,427,510 | 17,736,075 | 31,163,585 + #0: ToddCoxeter: diff 0.11.3 | -2,424,070 | +2,423,990 | +0 + #0: ToddCoxeter: diff 0.11.0 | -5,962,240 | +5,962,240 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 43,882,513 | 9,827,527 | 81.7% + #0: ToddCoxeter: diff 0.11.3 | +2,548,594 | -12,244,874 | +16.5% + #0: ToddCoxeter: diff 0.11.0 | +5,095,773 | -28,944,733 | +31.7% + #0: ToddCoxeter: phase 0.11 = 2.893s | run 0 = 10.511s | all runs = 10.511s | elapsed = 10.511s + #0: ToddCoxeter: lookahead_next() is now 19,389,704 (+0) + #0: ToddCoxeter: because: + #0: ToddCoxeter: 1. n <= f x a = 26,855,020 + #0: ToddCoxeter: 2. a <= n + #0: ToddCoxeter: 3. l >= (l + a) / t = 4,847,437 + #0: ToddCoxeter: where: a = number_of_nodes_active() = 13,427,510 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: l = nodes killed in lookahead = 5,962,240 + #0: ToddCoxeter: n = lookahead_next() = 19,389,704 + #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.12 START + #0: ToddCoxeter: HLT 0.12.0 | active | killed | defined + #0: ToddCoxeter: nodes | 13,427,510 | 17,736,075 | 31,163,585 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 43,882,513 | 9,827,527 | 81.7% + #0: ToddCoxeter: time | run 0 = 10.511s | all runs = 10.511s | elapsed = 10.511s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.12 STOP + #0: ToddCoxeter: HLT 0.12.1 | active | killed | defined + #0: ToddCoxeter: nodes | 19,389,776 | 17,959,888 | 37,349,664 + #0: ToddCoxeter: diff 0.12.0 | +5,962,266 | +223,813 | +6,186,079 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 50,324,580 | 27,234,524 | 64.9% + #0: ToddCoxeter: diff 0.12.0 | +6,442,067 | +17,406,997 | -16.8% + #0: ToddCoxeter: phase 0.12 = 267ms | run 0 = 10.778s | all runs = 10.778s | elapsed = 10.779s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.13 START (lookahead_extent() = partial, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.13.0 | active | killed | defined + #0: ToddCoxeter: nodes | 19,389,776 | 17,959,888 | 37,349,664 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 50,324,580 | 27,234,524 | 64.9% + #0: ToddCoxeter: time | run 0 = 10.778s | all runs = 10.778s | elapsed = 10.779s + #0: ToddCoxeter: because a >= n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 19,389,776 + #0: ToddCoxeter: n = lookahead_next() = 19,389,704 + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.13.1 | active | killed | defined + #1: ToddCoxeter: nodes | 19,314,200 | 18,035,464 | 37,349,664 + #1: ToddCoxeter: diff 0.13.0 | -75,576 | +75,576 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 50,245,486 | 27,011,314 | 65.0% + #1: ToddCoxeter: diff 0.13.0 | -79,094 | -223,210 | +0.2% + #1: ToddCoxeter: phase 0.13 = 224ms | run 0 = 11.003s | all runs = 11.003s | elapsed = 11.003s + #1: ToddCoxeter: lookahead progress | ~9.0% + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.13.2 | active | killed | defined + #1: ToddCoxeter: nodes | 19,102,330 | 18,247,350 | 37,349,664 + #1: ToddCoxeter: diff 0.13.1 | -211,870 | +211,886 | +0 + #1: ToddCoxeter: diff 0.13.0 | -287,446 | +287,462 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 50,032,914 | 26,376,406 | 65.5% + #1: ToddCoxeter: diff 0.13.1 | -212,572 | -634,908 | +0.4% + #1: ToddCoxeter: diff 0.13.0 | -291,666 | -858,118 | +0.6% + #1: ToddCoxeter: phase 0.13 = 1.225s | run 0 = 12.003s | all runs = 12.003s | elapsed = 12.003s + #1: ToddCoxeter: lookahead progress | ~31.3% + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.13.3 | active | killed | defined + #1: ToddCoxeter: nodes | 18,872,056 | 18,477,610 | 37,349,664 + #1: ToddCoxeter: diff 0.13.2 | -230,274 | +230,260 | +0 + #1: ToddCoxeter: diff 0.13.0 | -517,720 | +517,722 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 49,788,690 | 25,699,534 | 66.0% + #1: ToddCoxeter: diff 0.13.2 | -244,224 | -676,872 | +0.5% + #1: ToddCoxeter: diff 0.13.0 | -535,890 | -1,534,990 | +1.1% + #1: ToddCoxeter: phase 0.13 = 2.225s | run 0 = 13.004s | all runs = 13.004s | elapsed = 13.004s + #1: ToddCoxeter: lookahead progress | ~58.4% + #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()! + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.13.4 | active | killed | defined + #1: ToddCoxeter: nodes | 12,447,893 | 24,901,915 | 37,349,664 + #1: ToddCoxeter: diff 0.13.3 | -6,424,163 | +6,424,305 | +0 + #1: ToddCoxeter: diff 0.13.0 | -6,941,883 | +6,942,027 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 0 | 49,791,572 | 0.0% + #1: ToddCoxeter: diff 0.13.3 | -49,788,690 | +24,092,038 | -66.0% + #1: ToddCoxeter: diff 0.13.0 | -50,324,580 | +22,557,048 | -64.9% + #1: ToddCoxeter: phase 0.13 = 3.225s | run 0 = 14.004s | all runs = 14.004s | elapsed = 14.004s + #1: ToddCoxeter: lookahead progress | ~66.6% + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: LOOKAHEAD 0.13.5 | active | killed | defined + #1: ToddCoxeter: nodes | 5,024,869 | 32,324,910 | 37,349,664 + #1: ToddCoxeter: diff 0.13.4 | -7,423,024 | +7,422,995 | +0 + #1: ToddCoxeter: diff 0.13.0 | -14,364,907 | +14,365,022 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 0 | 20,099,476 | 0.0% + #1: ToddCoxeter: diff 0.13.4 | +0 | -29,692,096 | +0.0% + #1: ToddCoxeter: diff 0.13.0 | -50,324,580 | -7,135,048 | -64.9% + #1: ToddCoxeter: phase 0.13 = 4.226s | run 0 = 15.004s | all runs = 15.004s | elapsed = 15.004s + #1: ToddCoxeter: lookahead progress | ~66.6% + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.13 STOP + #0: ToddCoxeter: LOOKAHEAD 0.13.6 | active | killed | defined + #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 + #0: ToddCoxeter: diff 0.13.5 | -4,581,349 | +4,581,234 | +0 + #0: ToddCoxeter: diff 0.13.0 | -18,946,256 | +18,946,256 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% + #0: ToddCoxeter: diff 0.13.5 | +1,774,080 | -20,099,476 | +100.0% + #0: ToddCoxeter: diff 0.13.0 | -48,550,500 | -27,234,524 | +35.1% + #0: ToddCoxeter: phase 0.13 = 5.080s | run 0 = 15.858s | all runs = 15.858s | elapsed = 15.858s + #0: ToddCoxeter: lookahead_next() is now max(f x a = 887,040, m = 10,000) (-18,502,664) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 443,520 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 19,389,704 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.14 START + #0: ToddCoxeter: HLT 0.14.0 | active | killed | defined + #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% + #0: ToddCoxeter: time | run 0 = 15.858s | all runs = 15.858s | elapsed = 15.858s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.14.1 | active | killed | defined + #1: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 + #1: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% + #1: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.14 = 146ms | run 0 = 16.004s | all runs = 16.004s | elapsed = 16.005s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.14 STOP + #0: ToddCoxeter: HLT 0.14.2 | active | killed | defined + #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 + #0: ToddCoxeter: diff 0.14.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% + #0: ToddCoxeter: diff 0.14.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.14 = 151ms | run 0 = 16.009s | all runs = 16.009s | elapsed = 16.010s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.15.3 | active | killed | defined + #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 + #0: ToddCoxeter: diff 0.15.2 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% + #0: ToddCoxeter: diff 0.15.2 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.15 = 151ms | run 0 = 16.010s | all runs = 16.010s | elapsed = 16.010s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 7 | 0 | 8 | 0 + #0: ToddCoxeter: time spent in phases | 13.184s (82%) | - (0%) | 2.826s (18%) | - (0%) + #0: ToddCoxeter: phase 0.15 = 160ms | run 0 = 16.019s | all runs = 16.019s | elapsed = 16.019s + The size of the group is 443520 + ``` -[atlas]: https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ -[libsemigroups_pybind11]: - https://libsemigroups.github.io/libsemigroups_pybind11/index.html +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $443,520$. diff --git a/docs/examples/atlas/mathieu/m23/index.md b/docs/examples/atlas/mathieu/m23/index.md index 6d877fa..270bc16 100644 --- a/docs/examples/atlas/mathieu/m23/index.md +++ b/docs/examples/atlas/mathieu/m23/index.md @@ -1,12 +1,18 @@ # Mathieu group M~23~ -> Order: $10,200,960$ - -> Presentation: TODO - On this page, we provide links to verifications that the maximal subgroups of -the Mathieu group M~23~ define groups of the correct order. +the Mathieu group M~23~ define groups of the correct index. Two presentations +are given for M~23~, and two maximal subgroups have generating pairs specified. ## Maximal subgroups -The following are maximal subgroups that have been verified: +We attempted to verify the following maximal subgroups: + +- [Mathieu group M11 (Presentation 1)](p1_m11.md) + :material-checkbox-marked-circle-outline:{ style="color: #00c853" } +- [Mathieu group M22 (Presentation 1)](p1_m22.md) + :material-minus-circle-outline: +- [Mathieu group M11 (Presentation 2)](p2_m11.md) + :material-checkbox-marked-circle-outline:{ style="color: #00c853" } +- [Mathieu group M22 (Presentation 2)](p2_m22.md) + :material-minus-circle-outline: diff --git a/docs/examples/atlas/mathieu/m23/p1_m11.md b/docs/examples/atlas/mathieu/m23/p1_m11.md new file mode 100644 index 0000000..7c0316d --- /dev/null +++ b/docs/examples/atlas/mathieu/m23/p1_m11.md @@ -0,0 +1,159 @@ +# Mathieu group M~11~ as a subgroup of Mathieu group M~23~ (Presentation 1) + +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1](https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1). + +## Claim + +$$ + \mathrm{M}_{23} = \langle a, b \mid a^2 = b^4 = (ab)^{23} = (ab^2)^6 = [a, b]^6 = (abab^{−1}ab^2)^4 = (ab)^3ab^{−1}ab^2(abab^{−1})^2(ab)^3(ab^{−1})^3 = (abab^2)^3(ab^2ab^{−1})^2abab^2abab^{−1}ab^2 = 1\rangle +$$ + +and + +$$ + \mathrm{M}_{11} = {\mathrm{M}_{23}} \mathbin{/} \langle a, b \mid b = bbabbababbbabbabab = 1 \rangle +$$ + +with + +$$ + [\mathrm{M}_{23} : \mathrm{M}_{11}] = 1,288. +$$ + +## The code + +In [libsemigroups_pybind11][], the following script constructs the presentation +for M~23~, adds the generating pairs that define the subgroup M~11~, and runs +the Todd-Coxeter algorithm. + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^23"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("(a,b)^6"), "") +presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "") +presentation.add_rule(p, parse_relations("(ab)^3aBab^2(abaB)^2(ab)^3(aB)^3"), "") +presentation.add_rule(p, parse_relations("(abab^2)^3(ab^2aB)^2abab^2abaBab^2"), "") + + +# Add generating pairs for the maximal subgroup +tc = ToddCoxeter(congruence_kind.onesided, p) +tc.add_generating_pair("b", "") +tc.add_generating_pair("bbabbababbbabbabab", "") + +# Run the Todd-Coxeter algorithm +tc.strategy(tc.options.strategy.felsch) +tc.def_version(tc.options.def_version.two) +tc.def_policy(tc.options.def_policy.discard_all_if_no_space) +tc.use_relations_in_extra(True) +tc.def_max(10000) +tc.run() + +print(f"The index of the subgroup is {tc.number_of_classes()}") +``` + +## The output + +??? info "Output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 4, |R| = 12, |u| + |v| ∈ [2, 46], ∑(|u| + |v|) = 198 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 119 | 56 | 175 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 181 | 295 | 38.0% + #0: ToddCoxeter: time | run 0 = 248µs | all runs = 248µs | elapsed = 323µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #1: ToddCoxeter: nodes | 1,258,648 | 24,628 | 1,283,314 + #1: ToddCoxeter: diff 0.0.0 | +1,258,529 | +24,572 | +1,283,139 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 3,915,350 | 1,119,242 | 77.8% + #1: ToddCoxeter: diff 0.0.0 | +3,915,169 | +1,118,947 | +39.7% + #1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s + #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()! + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined + #1: ToddCoxeter: nodes | 273,941 | 2,129,053 | 2,402,478 + #1: ToddCoxeter: diff 0.0.1 | -984,707 | +2,104,425 | +1,119,164 + #1: ToddCoxeter: diff 0.0.0 | +273,822 | +2,128,997 | +2,402,303 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 0 | 1,095,764 | 0.0% + #1: ToddCoxeter: diff 0.0.1 | -3,915,350 | -23,478 | -77.8% + #1: ToddCoxeter: diff 0.0.0 | -181 | +1,095,469 | -38.0% + #1: ToddCoxeter: phase 0.0 = 2.000s | run 0 = 2.001s | all runs = 2.001s | elapsed = 2.001s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 2,401,190 | 2,402,478 + #0: ToddCoxeter: diff 0.0.2 | -272,653 | +272,137 | +0 + #0: ToddCoxeter: diff 0.0.0 | +1,169 | +2,401,134 | +2,402,303 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.0.2 | +5,152 | -1,095,764 | +100.0% + #0: ToddCoxeter: diff 0.0.0 | +4,971 | -295 | +62.0% + #0: ToddCoxeter: phase 0.0 = 2.013s | run 0 = 2.013s | all runs = 2.013s | elapsed = 2.013s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 2,401,190 | 2,402,478 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: time | run 0 = 2.013s | all runs = 2.013s | elapsed = 2.013s + #0: ToddCoxeter: triggered because there are skipped definitions (1,288 active nodes)! + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 2,401,190 | 2,402,478 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.1 = 391µs | run 0 = 2.014s | all runs = 2.014s | elapsed = 2.014s + #0: ToddCoxeter: lookahead_next() is now max(f x a = 2,576, m = 10,000) (-4,990,000) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 1,288 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.2 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 2,401,190 | 2,402,478 + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 418µs | run 0 = 2.014s | all runs = 2.014s | elapsed = 2.014s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 414µs (0%) | - (0%) | - (0%) | 2.013s (100%) + #0: ToddCoxeter: phase 0.2 = 437µs | run 0 = 2.014s | all runs = 2.014s | elapsed = 2.014s + The index of the subgroup is 1288 + ``` + +:material-checkbox-marked-circle-outline: The computed index is the same as +the claimed index: $1,288$. diff --git a/docs/examples/atlas/mathieu/m23/p1_m22.md b/docs/examples/atlas/mathieu/m23/p1_m22.md new file mode 100644 index 0000000..b6a71e5 --- /dev/null +++ b/docs/examples/atlas/mathieu/m23/p1_m22.md @@ -0,0 +1,272 @@ +# Mathieu group M~22~ as a subgroup of Mathieu group M~23~ (Presentation 1) + +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1](https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1). + +## Claim + +$$ + \mathrm{M}_{23} = \langle a, b \mid a^2 = b^4 = (ab)^{23} = (ab^2)^6 = [a, b]^6 = (abab^{−1}ab^2)^4 = (ab)^3ab^{−1}ab^2(abab^{−1})^2(ab)^3(ab^{−1})^3 = (abab^2)^3(ab^2ab^{−1})^2abab^2abab^{−1}ab^2 = 1\rangle +$$ + +and + +$$ + \mathrm{M}_{22} = {\mathrm{M}_{23}} \mathbin{/} \langle a, b \mid a = bababbab = 1 \rangle +$$ + +with + +$$ + [\mathrm{M}_{23} : \mathrm{M}_{22}] = 23. +$$ + +!!! warning + + The word $bababbab$ in plain text on the page + + [https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1](https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1) + + does not seem to generate a subgroup of the correct index. Expanding + the [GAP][] straightline programme at: + + [http://atlas.math.rwth-aachen.de/Atlas/v3/scripts/slp2.php%3Fformat=GAP&filename=M23G1-max1W1&filename_path=spor%252FM23&sub_dir=words](http://atlas.math.rwth-aachen.de/Atlas/v3/scripts/slp2.php%3Fformat=GAP&filename=M23G1-max1W1&filename_path=spor%252FM23&sub_dir=words) + + yields the word $bababbbab$ which does generate a subgroup of the correct index. + +## The code + +=== "Plain text" + + In [libsemigroups_pybind11][], the following script constructs the + presentation for M~23~, adds the generating pairs that define the subgroup + M~22~, and runs the Todd-Coxeter algorithm. This is based on the plain text + implementation from the ATLAS. + + ```python + from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, + ) + from libsemigroups_pybind11.words import parse_relations + + # Setup the presentation object with the empty word and inverses, so it can represent a group + p = Presentation("abAB") + p.contains_empty_word(True) + presentation.add_inverse_rules(p, "ABab") + + # Add the defining relations + presentation.add_rule(p, parse_relations("a^2"), "") + presentation.add_rule(p, parse_relations("b^4"), "") + presentation.add_rule(p, parse_relations("(ab)^23"), "") + presentation.add_rule(p, parse_relations("(ab^2)^6"), "") + presentation.add_rule(p, parse_relations("(a,b)^6"), "") + presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "") + presentation.add_rule(p, parse_relations("(ab)^3aBab^2(abaB)^2(ab)^3(aB)^3"), "") + presentation.add_rule(p, parse_relations("(abab^2)^3(ab^2aB)^2abab^2abaBab^2"), "") + + + # Add generating pairs for the maximal subgroup + tc = ToddCoxeter(congruence_kind.onesided, p) + tc.add_generating_pair("a", "") + tc.add_generating_pair("bababbab", "") + + # Run the Todd-Coxeter algorithm + tc.strategy(tc.options.strategy.felsch) + tc.def_version(tc.options.def_version.two) + tc.def_policy(tc.options.def_policy.discard_all_if_no_space) + tc.use_relations_in_extra(True) + tc.def_max(10000) + tc.run() + + print(f"The index of the group is {tc.number_of_classes()}") + ``` + +=== "Straightline program" + + In [libsemigroups_pybind11][], the following script constructs the + presentation for M~23~, adds the generating pairs that define the subgroup + M~22~, and runs the Todd-Coxeter algorithm. This is based on the + [GAP][] straightline program hosted on the ATLAS. + + ```python + from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, + ) + from libsemigroups_pybind11.words import parse_relations + + # Setup the presentation object with the empty word and inverses, so it can represent a group + p = Presentation("abAB") + p.contains_empty_word(True) + presentation.add_inverse_rules(p, "ABab") + + # Add the defining relations + presentation.add_rule(p, parse_relations("a^2"), "") + presentation.add_rule(p, parse_relations("b^4"), "") + presentation.add_rule(p, parse_relations("(ab)^23"), "") + presentation.add_rule(p, parse_relations("(ab^2)^6"), "") + presentation.add_rule(p, parse_relations("(a,b)^6"), "") + presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "") + presentation.add_rule(p, parse_relations("(ab)^3aBab^2(abaB)^2(ab)^3(aB)^3"), "") + presentation.add_rule(p, parse_relations("(abab^2)^3(ab^2aB)^2abab^2abaBab^2"), "") + + + # Add generating pairs for the maximal subgroup + tc = ToddCoxeter(congruence_kind.onesided, p) + tc.add_generating_pair("a", "") + tc.add_generating_pair("bababbbab", "") + + # Run the Todd-Coxeter algorithm + tc.strategy(tc.options.strategy.felsch) + tc.def_version(tc.options.def_version.two) + tc.def_policy(tc.options.def_policy.discard_all_if_no_space) + tc.use_relations_in_extra(True) + tc.def_max(10000) + tc.run() + + print(f"The index of the group is {tc.number_of_classes()}") + ``` + +## The output + +=== "Plain text" + + ??? info "Output from the Python script" + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 4, |R| = 12, |u| + |v| ∈ [2, 46], ∑(|u| + |v|) = 198 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 112 | 52 | 164 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 184 | 264 | 41.1% + #0: ToddCoxeter: time | run 0 = 239µs | all runs = 239µs | elapsed = 315µs + #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()! + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 1,176,014 | 1,177,302 + #0: ToddCoxeter: diff 0.0.0 | +1,176 | +1,175,962 | +1,177,138 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.0.0 | +4,968 | -264 | +58.9% + #0: ToddCoxeter: phase 0.0 = 980ms | run 0 = 980ms | all runs = 980ms | elapsed = 980ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 1,176,014 | 1,177,302 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: time | run 0 = 980ms | all runs = 980ms | elapsed = 980ms + #0: ToddCoxeter: triggered because there are skipped definitions (1,288 active nodes)! + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 1,176,014 | 1,177,302 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.1 = 436µs | run 0 = 980ms | all runs = 980ms | elapsed = 980ms + #0: ToddCoxeter: lookahead_next() is now max(f x a = 2,576, m = 10,000) (-4,990,000) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 1,288 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.2 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 1,176,014 | 1,177,302 + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 465µs | run 0 = 980ms | all runs = 980ms | elapsed = 980ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 460µs (0%) | - (0%) | - (0%) | 980ms (100%) + #0: ToddCoxeter: phase 0.2 = 498µs | run 0 = 980ms | all runs = 980ms | elapsed = 980ms + The index of the subgroup is 1288 + ``` + + :material-close-circle-outline: The computed index is $1,288$ which is not the + same as the claimed index: $23$. + +=== "Straightline program" + + ??? info "Output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 4, |R| = 12, |u| + |v| ∈ [2, 46], ∑(|u| + |v|) = 198 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 113 | 52 | 165 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 178 | 274 | 39.4% + #0: ToddCoxeter: time | run 0 = 251µs | all runs = 251µs | elapsed = 428µs + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 880,964 | 880,987 + #0: ToddCoxeter: diff 0.0.0 | -90 | +880,912 | +880,822 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 92 | 0 | 100.0% + #0: ToddCoxeter: diff 0.0.0 | -86 | -274 | +60.6% + #0: ToddCoxeter: phase 0.0 = 937ms | run 0 = 937ms | all runs = 937ms | elapsed = 937ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 880,964 | 880,987 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 92 | 0 | 100.0% + #0: ToddCoxeter: time | run 0 = 937ms | all runs = 937ms | elapsed = 937ms + #0: ToddCoxeter: triggered because there are skipped definitions (23 active nodes)! + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 880,964 | 880,987 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 92 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.1 = 44µs | run 0 = 937ms | all runs = 937ms | elapsed = 937ms + #0: ToddCoxeter: lookahead_next() is now max(f x a = 46, m = 10,000) (-4,990,000) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 23 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.2 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 880,964 | 880,987 + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 92 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 94µs | run 0 = 937ms | all runs = 937ms | elapsed = 937ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 87µs (0%) | - (0%) | - (0%) | 937ms (100%) + #0: ToddCoxeter: phase 0.2 = 119µs | run 0 = 937ms | all runs = 937ms | elapsed = 937ms + The index of the group is 23 + ``` + :material-checkbox-marked-circle-outline: The computed index is the same as + the claimed index: $23$. diff --git a/docs/examples/atlas/mathieu/m23/p2_m11.md b/docs/examples/atlas/mathieu/m23/p2_m11.md new file mode 100644 index 0000000..292a3b8 --- /dev/null +++ b/docs/examples/atlas/mathieu/m23/p2_m11.md @@ -0,0 +1,143 @@ +# Mathieu group M~11~ as a subgroup of Mathieu group M~23~ (Presentation 2) + +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1](https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1). + +## Claim + +$$ + \mathrm{M}_{23} = \langle a, b \mid a^2 = b^4 = (ab^2)^6 = (abab^{−1}ab^2)^4 = abababab^{−1}ab^2abab^{−1}abab^{−1}abababab^{−1}ab^{−1}ab^{−1} = abab^2abab^2abab^2ab^2ab^{−1}ab^2ab^{−1}abab^2abab^{−1}ab^2 = abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2ab^{−1}ab^2ab^2ab^{−1}ab^2ab^2 = ababababab^2abab^{−1}abab^2ababab^{−1}ab^2abab^2ab^2abab^{−1}ab^{−1}abab^2 = 1 \rangle +$$ + +and + +$$ + \mathrm{M}_{11} = {\mathrm{M}_{23}} \mathbin{/} \langle a, b \mid b = bbabbababbbabbabab = 1 \rangle +$$ + +with + +$$ + [\mathrm{M}_{23} : \mathrm{M}_{11}] = 23. +$$ + +## The code + +In [libsemigroups_pybind11][], the following script constructs the presentation +for M~23~, adds the generating pairs that define the subgroup M~11~, and runs +the Todd-Coxeter algorithm. + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("(a,b)^6"), "") +presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "") +presentation.add_rule(p, parse_relations("abababaBab^2abaBabaBabababaBaBaB"), "") +presentation.add_rule(p, parse_relations("abab^2abab^2abab^2ab^2aBab^2aBabab^2abaBab^2"), "") +presentation.add_rule( + p, parse_relations("abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2aBab^2ab^2aBab^2ab^2"), "" +) +presentation.add_rule( + p, parse_relations("ababababab^2abaBabab^2ababaBab^2abab^2ab^2abaBaBabab^2"), "" +) + + +# Add generating pairs for the maximal subgroup +tc = ToddCoxeter(congruence_kind.onesided, p) +tc.add_generating_pair("b", "") +tc.add_generating_pair("bbabbababbbabbabab", "") + +# Run the Todd-Coxeter algorithm +tc.strategy(tc.options.strategy.felsch) +tc.def_version(tc.options.def_version.two) +tc.def_policy(tc.options.def_policy.discard_all_if_no_space) +tc.use_relations_in_extra(True) +tc.def_max(10000) +tc.run() + +print(f"The index of the subgroup is {tc.number_of_classes()}") +``` + +## The output + +??? info "Output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 4, |R| = 13, |u| + |v| ∈ [2, 48], ∑(|u| + |v|) = 248 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 149 | 68 | 217 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 221 | 375 | 37.1% + #0: ToddCoxeter: time | run 0 = 390µs | all runs = 390µs | elapsed = 457µs + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 25,400 | 26,688 + #0: ToddCoxeter: diff 0.0.0 | +1,139 | +25,332 | +26,471 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.0.0 | +4,931 | -375 | +62.9% + #0: ToddCoxeter: phase 0.0 = 33ms | run 0 = 33ms | all runs = 33ms | elapsed = 33ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 25,400 | 26,688 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: time | run 0 = 33ms | all runs = 33ms | elapsed = 33ms + #0: ToddCoxeter: triggered because there are skipped definitions (1,288 active nodes)! + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 25,400 | 26,688 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.1 = 474µs | run 0 = 33ms | all runs = 33ms | elapsed = 33ms + #0: ToddCoxeter: lookahead_next() is now max(f x a = 2,576, m = 10,000) (-4,990,000) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 1,288 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.2 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 25,400 | 26,688 + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 498µs | run 0 = 33ms | all runs = 33ms | elapsed = 33ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 495µs (1%) | - (0%) | - (0%) | 33ms (97%) + #0: ToddCoxeter: phase 0.2 = 519µs | run 0 = 33ms | all runs = 33ms | elapsed = 34ms + The index of the subgroup is 1288 + ``` + +:material-checkbox-marked-circle-outline: The computed index is the same as +the claimed index: $1,288$. diff --git a/docs/examples/atlas/mathieu/m23/p2_m22.md b/docs/examples/atlas/mathieu/m23/p2_m22.md new file mode 100644 index 0000000..1720e64 --- /dev/null +++ b/docs/examples/atlas/mathieu/m23/p2_m22.md @@ -0,0 +1,258 @@ +# Mathieu group M~22~ as a subgroup of Mathieu group M~23~ (Presentation 2) + +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1](https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1). + +## Claim + +$$ + \mathrm{M}_{23} = \langle a, b \mid a^2 = b^4 = (ab^2)^6 = (abab^{−1}ab^2)^4 = abababab^{−1}ab^2abab^{−1}abab^{−1}abababab^{−1}ab^{−1}ab^{−1} = abab^2abab^2abab^2ab^2ab^{−1}ab^2ab^{−1}abab^2abab^{−1}ab^2 = abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2ab^{−1}ab^2ab^2ab^{−1}ab^2ab^2 = ababababab^2abab^{−1}abab^2ababab^{−1}ab^2abab^2ab^2abab^{−1}ab^{−1}abab^2 = 1 \rangle +$$ + +and + +$$ + \mathrm{M}_{22} = {\mathrm{M}_{23}} \mathbin{/} \langle a, b \mid a = bababbab = 1 \rangle +$$ + +with + +$$ + [\mathrm{M}_{23} : \mathrm{M}_{22}] = 23. +$$ + +!!! warning + + The word $bababbab$ in plain text on the page + + [https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1](https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1) + + does not seem to generate a subgroup of the correct index. Expanding + the [GAP][] straightline programme at: + + [http://atlas.math.rwth-aachen.de/Atlas/v3/scripts/slp2.php%3Fformat=GAP&filename=M23G1-max1W1&filename_path=spor%252FM23&sub_dir=words](http://atlas.math.rwth-aachen.de/Atlas/v3/scripts/slp2.php%3Fformat=GAP&filename=M23G1-max1W1&filename_path=spor%252FM23&sub_dir=words) + + yields the word $bababbbab$ which does generate a subgroup of the correct index. + +=== "Plain text" + + In [libsemigroups_pybind11][], the following script constructs the + presentation for M~23~, adds the generating pairs that define the subgroup + M~22~, and runs the Todd-Coxeter algorithm. This is based on the plain text + implementation from the ATLAS. + + ```python + from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, + ) + from libsemigroups_pybind11.words import parse_relations + + # Setup the presentation object with the empty word and inverses, so it can represent a group + p = Presentation("abAB") + p.contains_empty_word(True) + presentation.add_inverse_rules(p, "ABab") + + # Add the defining relations + presentation.add_rule(p, parse_relations("a^2"), "") + presentation.add_rule(p, parse_relations("b^4"), "") + presentation.add_rule(p, parse_relations("(ab^2)^6"), "") + presentation.add_rule(p, parse_relations("(a,b)^6"), "") + presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "") + presentation.add_rule(p, parse_relations("abababaBab^2abaBabaBabababaBaBaB"), "") + presentation.add_rule(p, parse_relations("abab^2abab^2abab^2ab^2aBab^2aBabab^2abaBab^2"), "") + presentation.add_rule( + p, parse_relations("abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2aBab^2ab^2aBab^2ab^2"), "" + ) + presentation.add_rule( + p, parse_relations("ababababab^2abaBabab^2ababaBab^2abab^2ab^2abaBaBabab^2"), "" + ) + + + # Add generating pairs for the maximal subgroup + tc = ToddCoxeter(congruence_kind.onesided, p) + tc.add_generating_pair("a", "") + tc.add_generating_pair("bababbab", "") + + # Run the Todd-Coxeter algorithm + tc.strategy(tc.options.strategy.felsch) + tc.def_version(tc.options.def_version.two) + tc.def_policy(tc.options.def_policy.discard_all_if_no_space) + tc.use_relations_in_extra(True) + tc.def_max(10000) + tc.run() + + print(f"The index of the subgroup is {tc.number_of_classes()}") + ``` + +=== "Straightline program" + + In [libsemigroups_pybind11][], the following script constructs the + presentation for M~23~, adds the generating pairs that define the subgroup + M~22~, and runs the Todd-Coxeter algorithm. This is based on the + [GAP][] straightline program hosted on the ATLAS. + + ```python + from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, + ) + from libsemigroups_pybind11.words import parse_relations + + # Setup the presentation object with the empty word and inverses, so it can represent a group + p = Presentation("abAB") + p.contains_empty_word(True) + presentation.add_inverse_rules(p, "ABab") + + # Add the defining relations + presentation.add_rule(p, parse_relations("a^2"), "") + presentation.add_rule(p, parse_relations("b^4"), "") + presentation.add_rule(p, parse_relations("(ab^2)^6"), "") + presentation.add_rule(p, parse_relations("(a,b)^6"), "") + presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "") + presentation.add_rule(p, parse_relations("abababaBab^2abaBabaBabababaBaBaB"), "") + presentation.add_rule(p, parse_relations("abab^2abab^2abab^2ab^2aBab^2aBabab^2abaBab^2"), "") + presentation.add_rule( + p, parse_relations("abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2aBab^2ab^2aBab^2ab^2"), "" + ) + presentation.add_rule( + p, parse_relations("ababababab^2abaBabab^2ababaBab^2abab^2ab^2abaBaBabab^2"), "" + ) + + + # Add generating pairs for the maximal subgroup + tc = ToddCoxeter(congruence_kind.onesided, p) + tc.add_generating_pair("a", "") + tc.add_generating_pair("bababbbab", "") + + # Run the Todd-Coxeter algorithm + tc.strategy(tc.options.strategy.felsch) + tc.def_version(tc.options.def_version.two) + tc.def_policy(tc.options.def_policy.discard_all_if_no_space) + tc.use_relations_in_extra(True) + tc.def_max(10000) + tc.run() + + print(f"The index of the subgroup is {tc.number_of_classes()}") + ``` + +## The output + +=== "Plain text" + + ??? info "Output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 4, |R| = 13, |u| + |v| ∈ [2, 48], ∑(|u| + |v|) = 248 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 153 | 53 | 206 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 231 | 381 | 37.7% + #0: ToddCoxeter: time | run 0 = 349µs | all runs = 349µs | elapsed = 420µs + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 14,407 | 15,695 + #0: ToddCoxeter: diff 0.0.0 | +1,135 | +14,354 | +15,489 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.0.0 | +4,921 | -381 | +62.3% + #0: ToddCoxeter: phase 0.0 = 51ms | run 0 = 51ms | all runs = 51ms | elapsed = 51ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.1.2 | active | killed | defined + #0: ToddCoxeter: nodes | 1,288 | 14,407 | 15,695 + #0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.1.0 | +1,135 | +14,354 | +15,489 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 5,152 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.1.0 | +4,921 | -381 | +62.3% + #0: ToddCoxeter: phase 0.1 = 51ms | run 0 = 51ms | all runs = 51ms | elapsed = 51ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | - (0%) | 51ms (99%) + #0: ToddCoxeter: phase 0.1 = 51ms | run 0 = 51ms | all runs = 51ms | elapsed = 51ms + The index of the subgroup is 1288 + ``` + + :material-close-circle-outline: The computed index is $1,288$ which is not the + same as the claimed index: $23$. + +=== "Plain text" + + ??? info "Output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 4, |R| = 13, |u| + |v| ∈ [2, 48], ∑(|u| + |v|) = 248 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 144 | 63 | 207 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 212 | 364 | 36.8% + #0: ToddCoxeter: time | run 0 = 394µs | all runs = 394µs | elapsed = 2ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 40,048 | 40,071 + #0: ToddCoxeter: diff 0.0.0 | -121 | +39,985 | +39,864 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 92 | 0 | 100.0% + #0: ToddCoxeter: diff 0.0.0 | -120 | -364 | +63.2% + #0: ToddCoxeter: phase 0.0 = 52ms | run 0 = 53ms | all runs = 53ms | elapsed = 54ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 40,048 | 40,071 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 92 | 0 | 100.0% + #0: ToddCoxeter: time | run 0 = 53ms | all runs = 53ms | elapsed = 54ms + #0: ToddCoxeter: triggered because there are skipped definitions (23 active nodes)! + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 40,048 | 40,071 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 92 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.1 = 31µs | run 0 = 53ms | all runs = 53ms | elapsed = 54ms + #0: ToddCoxeter: lookahead_next() is now max(f x a = 46, m = 10,000) (-4,990,000) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 23 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.2 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 40,048 | 40,071 + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 92 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 59µs | run 0 = 53ms | all runs = 53ms | elapsed = 54ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 54µs (0%) | - (0%) | - (0%) | 52ms (99%) + #0: ToddCoxeter: phase 0.2 = 83µs | run 0 = 53ms | all runs = 53ms | elapsed = 55ms + The index of the subgroup is 23 + ``` + + :material-close-circle-outline: The computed index is the same as + the claimed index: $23$. diff --git a/docs/examples/atlas/mathieu/m24/index.md b/docs/examples/atlas/mathieu/m24/index.md index 99166da..1157692 100644 --- a/docs/examples/atlas/mathieu/m24/index.md +++ b/docs/examples/atlas/mathieu/m24/index.md @@ -1,12 +1,158 @@ # Mathieu group M~24~ -> Order: $244,823,040$ +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/spor/M24/mag/M24G1-P1.M](https://brauer.maths.qmul.ac.uk/Atlas/spor/M24/mag/M24G1-P1.M). +This is a link to a Magma file that contains a presentation (amongst other +things). This presentation differs from the presentation provided in plain-text +on the ATLAS webpage due to the inclusion of an additional relation. The +comments in the Magma file claim that this additional relation is redundant, and +we do not verify this claim. -> Presentation: TODO +## Claim -On this page, we provide links to verifications that the maximal subgroups of -the Mathieu group M~24~ define groups of the correct order. +$$ + \mathrm{M}_{24} = \langle a, b \mid a^2 = b^3 = (ab)^{23} = [a, b]^{12} = [a, bab]^5 = (ababab^{−1})^3(abab^{−1}ab^{−1})^3 = [a,b^{-1}abab]^5 = (ab(abab^{−1})^3)^4 = 1 \rangle +$$ -## Maximal subgroups +with -The following are maximal subgroups that have been verified: +$$ + [\mathrm{M}_{23} : \langle ab \rangle] = 10,644,480. +$$ + +## The code + +In [libsemigroups_pybind11][], the following script constructs the presentation +for M~24~, adds the generating pairs that define the subgroup +$\langle ab \rangle$, and runs the Todd-Coxeter algorithm. + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^23"), "") +presentation.add_rule(p, parse_relations("(a,b)^12"), "") +presentation.add_rule(p, parse_relations("(a,bab)^5"), "") +presentation.add_rule(p, parse_relations("(ababaB)^3(abaBaB)^3"), "") +presentation.add_rule(p, parse_relations("(a,Babab)^5"), "") # the claimed redundant relation +presentation.add_rule(p, parse_relations("(ab(abaB)^3)^4"), "") + +# Make the presentation a bit shorter +presentation.replace_word_with_new_generator(p, presentation.longest_subword_reducing_length(p)) +presentation.replace_word_with_new_generator(p, presentation.longest_subword_reducing_length(p)) +presentation.sort_each_rule(p) +presentation.sort_rules(p) + +# Add generating pairs for the subgroup +tc = ToddCoxeter(congruence_kind.onesided, p) +tc.add_generating_pair("ab", "") + +# Run the Todd-Coxeter algorithm +tc.strategy(tc.options.strategy.felsch) +tc.def_version(tc.options.def_version.two) +tc.def_policy(tc.options.def_policy.discard_all_if_no_space) +tc.use_relations_in_extra(True) +tc.def_max(10000) +tc.run() + +print(f"The index of the subgroup is {tc.number_of_classes()}") +``` + +## The output + +The truncated output of the enumeration is below: + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 6, |R| = 14, |u| + |v| ∈ [2, 45], ∑(|u| + |v|) = 178 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 101 | 35 | 136 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 157 | 449 | 25.9% + #0: ToddCoxeter: time | run 0 = 180µs | all runs = 180µs | elapsed = 344µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #1: ToddCoxeter: nodes | 1,415,661 | 175,779 | 1,591,465 + #1: ToddCoxeter: diff 0.0.0 | +1,415,560 | +175,744 | +1,591,329 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 5,654,857 | 2,839,109 | 66.6% + #1: ToddCoxeter: diff 0.0.0 | +5,654,700 | +2,838,660 | +40.7% + #1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined + #1: ToddCoxeter: nodes | 2,802,886 | 379,414 | 3,182,324 + #1: ToddCoxeter: diff 0.0.1 | +1,387,225 | +203,635 | +1,590,859 + #1: ToddCoxeter: diff 0.0.0 | +2,802,785 | +379,379 | +3,182,188 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 11,202,682 | 5,614,634 | 66.6% + #1: ToddCoxeter: diff 0.0.1 | +5,547,825 | +2,775,525 | +0.0% + #1: ToddCoxeter: diff 0.0.0 | +11,202,525 | +5,614,185 | +40.7% + #1: ToddCoxeter: phase 0.0 = 2.000s | run 0 = 2.000s | all runs = 2.000s | elapsed = 2.001s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined + #1: ToddCoxeter: nodes | 4,194,304 | 561,668 | 4,755,972 + #1: ToddCoxeter: diff 0.0.2 | +1,391,418 | +182,254 | +1,573,648 + #1: ToddCoxeter: diff 0.0.0 | +4,194,203 | +561,633 | +4,755,836 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 16,763,867 | 8,401,957 | 66.6% + #1: ToddCoxeter: diff 0.0.2 | +5,561,185 | +2,787,323 | -0.0% + #1: ToddCoxeter: diff 0.0.0 | +16,763,710 | +8,401,508 | +40.7% + #1: ToddCoxeter: phase 0.0 = 3.000s | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s + ++++++++++++++++++++++++++++++++ + [... lines omitted ...] + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.65 | active | killed | defined + #0: ToddCoxeter: nodes | 10,644,480 | 52,857,227 | 63,501,707 + #0: ToddCoxeter: diff 0.1.64 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 63,866,880 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.64 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.1 = 1min4s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min37s + #0: ToddCoxeter: lookahead_next() is now max(f x a = 21,288,960, m = 10,000) (+16,288,960) + #0: ToddCoxeter: because a > n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 10,644,480 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.66 | active | killed | defined + #0: ToddCoxeter: nodes | 10,644,480 | 52,857,227 | 63,501,707 + #0: ToddCoxeter: diff 0.2.65 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 63,866,880 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.65 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 1min4s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min37s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 1min4s (29%) | - (0%) | - (0%) | 2min33s (70%) + #0: ToddCoxeter: phase 0.2 = 1min4s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min38s + The index of the subgroup is 10644480 + ``` + +:material-checkbox-marked-circle-outline: The computed index is the same as the +claimed index: $10,644,480$. diff --git a/docs/examples/atlas/misc/t.md b/docs/examples/atlas/misc/t.md index 512ded2..7ef0b33 100644 --- a/docs/examples/atlas/misc/t.md +++ b/docs/examples/atlas/misc/t.md @@ -1,28 +1,136 @@ # Tits group T -> Order: $17,971,200$ +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TF42/](https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TF42/). -> Presentation: TODO +## Claim -On this page, we verify that the above claimed presentation of the Tits group T -defines a group of order $17,971,200$. +$$ + \mathrm{T} = \langle a, b \mid a^2 = b^3 = (ab)^{13} = [a, b]^5 = [a, bab]^4 = ((ab)^4ab^{−1})^6 = 1 \rangle +$$ + +with + +$$ + |\mathrm{T}| = 17,971,200. +$$ + +On this page, we verify that the above claimed presentation of the Tits group +T~3~ defines a group of order $17,971,200$. ## The code -=== "Python" +In [libsemigroups_pybind11][], the following script constructs the presentation +for T and runs the Todd-Coxeter algorithm. + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^13"), "") +presentation.add_rule(p, parse_relations("(a,b)^5"), "") +presentation.add_rule(p, parse_relations("(a,bab)^4"), "") +presentation.add_rule(p, parse_relations("(ababababaB)^6"), "") - Coming soon! +presentation.balance(p, "abAB", "ABab") +presentation.replace_subword(p, "A", "a") +presentation.replace_subword(p, "B", "bb") +p.alphabet("ab") -=== "ACE" +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.strategy(ToddCoxeter.options.strategy.felsch) - Coming soon! +print(f"The size of the group is {tc.number_of_classes()}") +``` ## The output -=== "Python" +The truncated output of the enumeration is below: - Coming soon! +??? info "Truncated output from the Python script" -=== "ACE" + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 2, |R| = 10, |u| + |v| ∈ [2, 75], ∑(|u| + |v|) = 190 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 2 | 0.0% + #0: ToddCoxeter: time | run 0 = 28µs | all runs = 28µs | elapsed = 99µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #1: ToddCoxeter: nodes | 931,110 | 56,827 | 987,966 + #1: ToddCoxeter: diff 0.0.0 | +931,109 | +56,827 | +987,965 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,636,776 | 225,444 | 87.9% + #1: ToddCoxeter: diff 0.0.0 | +1,636,776 | +225,442 | +87.9% + #1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined + #1: ToddCoxeter: nodes | 1,689,659 | 107,031 | 1,796,700 + #1: ToddCoxeter: diff 0.0.1 | +758,549 | +50,204 | +808,734 + #1: ToddCoxeter: diff 0.0.0 | +1,689,658 | +107,031 | +1,796,699 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 2,979,269 | 400,049 | 88.2% + #1: ToddCoxeter: diff 0.0.1 | +1,342,493 | +174,605 | +0.3% + #1: ToddCoxeter: diff 0.0.0 | +2,979,269 | +400,047 | +88.2% + #1: ToddCoxeter: phase 0.0 = 2.000s | run 0 = 2.000s | all runs = 2.000s | elapsed = 2.000s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined + #1: ToddCoxeter: nodes | 2,422,503 | 156,074 | 2,578,589 + #1: ToddCoxeter: diff 0.0.2 | +732,844 | +49,043 | +781,889 + #1: ToddCoxeter: diff 0.0.0 | +2,422,502 | +156,074 | +2,578,588 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 4,279,477 | 565,529 | 88.3% + #1: ToddCoxeter: diff 0.0.2 | +1,300,208 | +165,480 | +0.2% + #1: ToddCoxeter: diff 0.0.0 | +4,279,477 | +565,527 | +88.3% + #1: ToddCoxeter: phase 0.0 = 3.001s | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.4 | active | killed | defined + #1: ToddCoxeter: nodes | 3,109,354 | 202,556 | 3,311,914 + #1: ToddCoxeter: diff 0.0.3 | +686,851 | +46,482 | +733,325 + #1: ToddCoxeter: diff 0.0.0 | +3,109,353 | +202,556 | +3,311,913 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 5,499,715 | 718,993 | 88.4% + #1: ToddCoxeter: diff 0.0.3 | +1,220,238 | +153,464 | +0.1% + #1: ToddCoxeter: diff 0.0.0 | +5,499,715 | +718,991 | +88.4% + #1: ToddCoxeter: phase 0.0 = 4.001s | run 0 = 4.001s | all runs = 4.001s | elapsed = 4.001s + ++++++++++++++++++++++++++++++++ + [... lines omitted ...] + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.53 | active | killed | defined + #0: ToddCoxeter: nodes | 17,971,200 | 3,439,895 | 21,411,095 + #0: ToddCoxeter: diff 0.2.52 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 35,942,400 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.52 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 51.384s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min37s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 51.384s (24%) | - (0%) | - (0%) | 2min46s (76%) + #0: ToddCoxeter: phase 0.2 = 51.594s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min37s + The size of the group is 17971200 + ``` - Coming soon! +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $17,971,200$. diff --git a/docs/examples/atlas/monster-sections/fi22/index.md b/docs/examples/atlas/monster-sections/fi22/index.md index 042bca6..1ce8261 100644 --- a/docs/examples/atlas/monster-sections/fi22/index.md +++ b/docs/examples/atlas/monster-sections/fi22/index.md @@ -2,11 +2,9 @@ > Order: $64,561,751,654,400$ -> Presentation: TODO +> Presentation: +> $\langle a, b \mid a^2 = b^{13} = (ab)^{11} = (ab^2)^{21} = [a, b]^3 = [a, b^2]^3 = [a, b^3]^3 = [a, b^4]^2 = [a, b^5]^3 = [a, bab^2]^3 = [a, b^{−1}ab{−2}]^2 = [a, bab^5]^2 = [a, b^2ab^5]^2 = 1 \rangle$ -On this page, we provide links to verifications that the maximal subgroups of -the Fischer group Fi~22~ define groups of the correct order. +!!! info "Work in Progress" -## Maximal subgroups - -The following are maximal subgroups that have been verified: + This page is a work in progress. diff --git a/docs/examples/atlas/monster-sections/he/index.md b/docs/examples/atlas/monster-sections/he/index.md index c2f88a7..75b5801 100644 --- a/docs/examples/atlas/monster-sections/he/index.md +++ b/docs/examples/atlas/monster-sections/he/index.md @@ -2,11 +2,9 @@ > Order: $4,030,387,200$ -> Presentation: TODO +> Presentation: +> $\langle a, b \mid a^2 = b^7 = (ab)^{17} = [a, b]^6 = [a, b^3]^5 = [a, babab^{−1}abab] = (ab)^4ab^2ab^{−3}ababab^{−1}ab^3ab^{−2}ab^2 = 1 \rangle$ -On this page, we provide links to verifications that the maximal subgroups of -the Held group He define groups of the correct order. +!!! info "Work in Progress" -## Maximal subgroups - -The following are maximal subgroups that have been verified: + This page is a work in progress. diff --git a/docs/examples/atlas/pariahs/j1.md b/docs/examples/atlas/pariahs/j1.md index ef2c3d5..e7d9242 100644 --- a/docs/examples/atlas/pariahs/j1.md +++ b/docs/examples/atlas/pariahs/j1.md @@ -1,28 +1,127 @@ # Janko group J~1~ -> Order: $175,560$ +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J1/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J1/). -> Presentation: TODO +## Claim -On this page, we verify that the above claimed presentation of the Janko group J~1~ -defines a group of order $175,560$. +$$ + \mathrm{J}_1 = \langle a, b \mid a^2 = b^3 = (ab)^7 = (ab(abab^{−1})^3)^5 = (ab(abab^{−1})^6abab(ab^{−1})^2)^2 = 1 \rangle +$$ + +with + +$$ + |\mathrm{J}_1| = 175,560. +$$ + +On this page, we verify that the above claimed presentation of the Janko group +J~1~ defines a group of order $175,560$. ## The code -=== "Python" +In [libsemigroups_pybind11][], the following script constructs the presentation +for J~1~ and runs the Todd-Coxeter algorithm. + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^7"), "") +presentation.add_rule(p, parse_relations("(ab(abaB)^3)^5"), "") +presentation.add_rule(p, parse_relations("(ab(abaB)^6abab(aB)^2)^2"), "") - Coming soon! +presentation.balance(p, "abAB", "ABab") +presentation.replace_subword(p, "A", "a") +presentation.replace_subword(p, "B", "bb") +p.alphabet("ab") -=== "ACE" +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.strategy(ToddCoxeter.options.strategy.felsch) - Coming soon! +print(f"The size of the group is {tc.number_of_classes()}") +``` ## The output -=== "Python" +Running the above script produces the following output: - Coming soon! +??? info "Output from the Python script" -=== "ACE" + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 2, |R| = 9, |u| + |v| ∈ [2, 87], ∑(|u| + |v|) = 206 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 2 | 0.0% + #0: ToddCoxeter: time | run 0 = 33µs | all runs = 33µs | elapsed = 99µs + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 STOP + #0: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #0: ToddCoxeter: nodes | 175,560 | 263,717 | 439,277 + #0: ToddCoxeter: diff 0.0.0 | +175,559 | +263,717 | +439,276 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 351,120 | 0 | 100.0% + #0: ToddCoxeter: diff 0.0.0 | +351,120 | -2 | +100.0% + #0: ToddCoxeter: phase 0.0 = 815ms | run 0 = 815ms | all runs = 815ms | elapsed = 815ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt) + #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined + #0: ToddCoxeter: nodes | 175,560 | 263,717 | 439,277 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 351,120 | 0 | 100.0% + #0: ToddCoxeter: time | run 0 = 815ms | all runs = 815ms | elapsed = 815ms + #0: ToddCoxeter: triggered because there are skipped definitions (175,560 active nodes)! + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: LOOKAHEAD 0.1 STOP + #0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined + #0: ToddCoxeter: nodes | 175,560 | 263,717 | 439,277 + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 351,120 | 0 | 100.0% + #0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.1 = 111ms | run 0 = 926ms | all runs = 926ms | elapsed = 926ms + #0: ToddCoxeter: lookahead_next() is now max(f x a = 351,120, m = 10,000) (-4,648,880) + #0: ToddCoxeter: because f x a < n + #0: ToddCoxeter: where: a = number_of_nodes_active() = 175,560 + #0: ToddCoxeter: f = lookahead_growth_factor() = 2 + #0: ToddCoxeter: m = lookahead_min() = 10,000 + #0: ToddCoxeter: n = lookahead_next() = 5,000,000 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.2 | active | killed | defined + #0: ToddCoxeter: nodes | 175,560 | 263,717 | 439,277 + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 351,120 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 111ms | run 0 = 926ms | all runs = 926ms | elapsed = 926ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 111ms (12%) | - (0%) | - (0%) | 815ms (88%) + #0: ToddCoxeter: phase 0.2 = 112ms | run 0 = 927ms | all runs = 927ms | elapsed = 927ms + The size of the group is 175560 + ``` - Coming soon! +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $175,560$. diff --git a/docs/examples/atlas/pariahs/j3.md b/docs/examples/atlas/pariahs/j3.md index 4962867..0c377e0 100644 --- a/docs/examples/atlas/pariahs/j3.md +++ b/docs/examples/atlas/pariahs/j3.md @@ -1,28 +1,128 @@ # Janko group J~3~ -> Order: $50,232,960$ +The claims on this page come from +[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J3/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J3/). -> Presentation: TODO +## Claim -On this page, we verify that the above claimed presentation of the Janko group J~3~ -defines a group of order $50,232,960$. +$$ + \mathrm{J}_3 = \langle a, b \mid a^2 = b^3 = (ab)^{19} = [a, b]^9 = ((ab)^6(ab^{−1})^5)^2 = ((ababab^{−1})^2abab^{−1}ab^{−1}abab^{−1})^2 = abab(abab^{−1})^3abab(abab^{−1})^4ab^{−1}(abab^{−1})^3 = (ababababab^{−1}abab^{−1})^4 = 1 \rangle +$$ + +with + +$$ + |\mathrm{J}_3| = 50,232,960. +$$ + +On this page, we verify that the above claimed presentation of the Janko group +J~3~ defines a group of order $50,232,960$. ## The code -=== "Python" +In [libsemigroups_pybind11][], the following script constructs the presentation +for J~1~ and runs the Todd-Coxeter algorithm. + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Setup the presentation object with the empty word and inverses, so it can represent a group +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(a,b)^9"), "") +presentation.add_rule(p, parse_relations("(ab)^19"), "") +presentation.add_rule(p, parse_relations("((ab)^6(aB)^5)^2"), "") +presentation.add_rule(p, parse_relations("((ababaB)^2abaBaBabaB)^2"), "") +presentation.add_rule(p, parse_relations("abab(abaB)^3abab(abaB)^4aB(abaB)^3"), "") +presentation.add_rule(p, parse_relations("((ab)^3(abaB)^2)^4"), "") - Coming soon! +presentation.balance(p, "abAB", "ABab") +presentation.replace_subword(p, "A", "a") +presentation.replace_subword(p, "B", "bb") +p.alphabet("ab") -=== "ACE" +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.strategy(ToddCoxeter.options.strategy.felsch) - Coming soon! +print(f"The size of the group is {tc.number_of_classes()}") +``` ## The output -=== "Python" +The truncated output of the enumeration is below: - Coming soon! +??? info "Truncated output from the Python script" -=== "ACE" + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 2, |R| = 12, |u| + |v| ∈ [2, 70], ∑(|u| + |v|) = 350 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 2 | 0.0% + #0: ToddCoxeter: time | run 0 = 24µs | all runs = 24µs | elapsed = 100µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #1: ToddCoxeter: nodes | 1,123,584 | 18,085 | 1,141,712 + #1: ToddCoxeter: diff 0.0.0 | +1,123,583 | +18,085 | +1,141,711 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,891,405 | 355,763 | 84.2% + #1: ToddCoxeter: diff 0.0.0 | +1,891,405 | +355,761 | +84.2% + #1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined + #1: ToddCoxeter: nodes | 2,248,417 | 39,195 | 2,287,629 + #1: ToddCoxeter: diff 0.0.1 | +1,124,833 | +21,110 | +1,145,917 + #1: ToddCoxeter: diff 0.0.0 | +2,248,416 | +39,195 | +2,287,628 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 3,785,925 | 710,909 | 84.2% + #1: ToddCoxeter: diff 0.0.1 | +1,894,520 | +355,146 | +0.0% + #1: ToddCoxeter: diff 0.0.0 | +3,785,925 | +710,907 | +84.2% + #1: ToddCoxeter: phase 0.0 = 2.000s | run 0 = 2.000s | all runs = 2.000s | elapsed = 2.000s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined + #1: ToddCoxeter: nodes | 3,346,541 | 61,314 | 3,407,869 + #1: ToddCoxeter: diff 0.0.2 | +1,098,124 | +22,119 | +1,120,240 + #1: ToddCoxeter: diff 0.0.0 | +3,346,540 | +61,314 | +3,407,868 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 5,635,856 | 1,057,226 | 84.2% + #1: ToddCoxeter: diff 0.0.2 | +1,849,931 | +346,317 | +0.0% + #1: ToddCoxeter: diff 0.0.0 | +5,635,856 | +1,057,224 | +84.2% + #1: ToddCoxeter: phase 0.0 = 3.001s | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s + ++++++++++++++++++++++++++++++++ + [... lines omitted ...] + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.2.317 | active | killed | defined + #1: ToddCoxeter: nodes | 50,232,960 | 110,530,444 | 160,763,404 + #1: ToddCoxeter: diff 0.2.316 | +0 | +0 | +0 + #1: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 100,465,920 | 0 | 100.0% + #1: ToddCoxeter: diff 0.2.316 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.2 = 5min16s | run 0 = 19min52s | all runs = 19min52s | elapsed = 19min52s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 5min15s (26%) | - (0%) | - (0%) | 14min36s (74%) + #0: ToddCoxeter: phase 0.2 = 5min16s | run 0 = 19min52s | all runs = 19min52s | elapsed = 19min52s + The size of the group is 50232960 + ``` - Coming soon! +:material-checkbox-marked-circle-outline: The computed size of the group matches +the size of the group provided on the ATLAS: $50,232,960$. diff --git a/docs/examples/atlas/pariahs/j4/index.md b/docs/examples/atlas/pariahs/j4/index.md index 5907a8d..d5d0c9e 100644 --- a/docs/examples/atlas/pariahs/j4/index.md +++ b/docs/examples/atlas/pariahs/j4/index.md @@ -2,11 +2,6 @@ > Order: $86,775,571,046,077,562,880$ -> Presentation: TODO +!!! tip "Work in Progress" -On this page, we provide links to verifications that the maximal subgroups of -the Janko group J~4~ define groups of the correct order. - -## Maximal subgroups - -The following are maximal subgroups that have been verified: + This page is a work in progress. diff --git a/docs/examples/atlas/pariahs/ru.md b/docs/examples/atlas/pariahs/ru.md new file mode 100644 index 0000000..ae24cd5 --- /dev/null +++ b/docs/examples/atlas/pariahs/ru.md @@ -0,0 +1,141 @@ +# Rudvalis group Ru + +The claims on this page come from +[Bradley, Curtis and Malik (2010)](https://doi.org/10.1112/jlms/jdq039). + +## Claim + +$$ + \mathrm{Ru} = \langle a, b, c, t \mid a^7 = b^3 = c^2 = t^2 = (cb)^{-1}bc = (ct)^{-1}tc = (ac)^6 = ((ca)^{-1}ac)^4 = (bt)^3 = b^{-1}aba^{-2} = (abc)^7 = (ab^2t)^3 = (bacat)^{-1}acat(aca)^{-1}baca = (bt)^{-1}(ca)^{-2}acabt(acab)^{-1}(ca)^2b = ta^{-3}ta^3b^{-1}ta^{-1}tab(aca)^{-2}(a^{-3}c^{-1})^2a^3ca^5ca \rangle +$$ + +and + +$$ + \mathrm{A}_{8} = \mathrm{Ru} \mathbin{/} \langle a, b, c \rangle +$$ + +with + +$$ + [\mathrm{Ru} : \mathrm{A}_8] = 7,238,400. +$$ + +On this page, we verify that the above subgroup of the Rudvalis group Ru defines +a group of the correct index. + +## The code + +In [libsemigroups_pybind11][], the following script constructs the presentation +for M~23~, adds the generating pairs that define the subgroup M~11~, and runs +the Todd-Coxeter algorithm. + +```python +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +p = Presentation("abctABCT").contains_empty_word(True) +presentation.add_inverse_rules(p, "ABCTabct") +presentation.add_rule(p, parse_relations("a^7"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("c^2"), "") +presentation.add_rule(p, parse_relations("t^2"), "") +presentation.add_rule(p, parse_relations("BCbc"), "") +presentation.add_rule(p, parse_relations("TCtc"), "") +presentation.add_rule(p, parse_relations("(ac)^6"), "") +presentation.add_rule(p, parse_relations("(ACac)^4"), "") +presentation.add_rule(p, parse_relations("(bt)^3"), "") +presentation.add_rule(p, parse_relations("BabA^2"), "") +presentation.add_rule(p, parse_relations("(abc)^7"), "") +presentation.add_rule(p, parse_relations("(ab^2t)^3"), "") +presentation.add_rule(p, parse_relations("TACABacatACAbaca"), "") +presentation.add_rule(p, parse_relations("TB(AC)^2acabtBACA(ca)^2b"), "") +presentation.add_rule(p, parse_relations("tA^3ta^3BtAtabACA^2(A^3C)^2a^3ca^5ca"), "") +presentation.balance(p, p.alphabet(), "ABCTabct") + +tc = ToddCoxeter(congruence_kind.onesided, p) + +tc.add_generating_pair("a", "") +tc.add_generating_pair("b", "") +tc.add_generating_pair("c", "") + +tc.strategy(ToddCoxeter.options.strategy.felsch).use_relations_in_extra(True) +tc.run() + +print(f"The index of the subgroup is {tc.number_of_classes()}") +``` + +## The output + +The truncated output of the enumeration is below: + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = felsch) + #0: ToddCoxeter: |A| = 8, |R| = 23, |u| + |v| ∈ [2, 37], ∑(|u| + |v|) = 183 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.0 START + #0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 21 | 18 | 39 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 68 | 100 | 40.5% + #0: ToddCoxeter: time | run 0 = 146µs | all runs = 146µs | elapsed = 236µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined + #1: ToddCoxeter: nodes | 537,051 | 187,750 | 724,832 + #1: ToddCoxeter: diff 0.0.0 | +537,030 | +187,732 | +724,793 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 2,308,258 | 1,988,150 | 53.7% + #1: ToddCoxeter: diff 0.0.0 | +2,308,190 | +1,988,050 | +13.2% + #1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined + #1: ToddCoxeter: nodes | 1,188,023 | 273,755 | 1,461,793 + #1: ToddCoxeter: diff 0.0.1 | +650,972 | +86,005 | +736,961 + #1: ToddCoxeter: diff 0.0.0 | +1,188,002 | +273,737 | +1,461,754 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 5,066,218 | 4,437,966 | 53.3% + #1: ToddCoxeter: diff 0.0.1 | +2,757,960 | +2,449,816 | -0.4% + #1: ToddCoxeter: diff 0.0.0 | +5,066,150 | +4,437,866 | +12.8% + #1: ToddCoxeter: phase 0.0 = 2.000s | run 0 = 2.001s | all runs = 2.001s | elapsed = 2.001s + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined + #1: ToddCoxeter: nodes | 1,782,973 | 360,686 | 2,143,668 + #1: ToddCoxeter: diff 0.0.2 | +594,950 | +86,931 | +681,875 + #1: ToddCoxeter: diff 0.0.0 | +1,782,952 | +360,668 | +2,143,629 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 7,622,880 | 6,640,904 | 53.4% + #1: ToddCoxeter: diff 0.0.2 | +2,556,662 | +2,202,938 | +0.1% + #1: ToddCoxeter: diff 0.0.0 | +7,622,812 | +6,640,804 | +13.0% + #1: ToddCoxeter: phase 0.0 = 3.001s | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s + #0: ToddCoxeter: large collapse, number of coincidences 100,003 >= 100,000 = large_collapse()! + ++++++++++++++++++++++++++++++++ + [... lines omitted ...] + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: FELSCH 0.2.26 | active | killed | defined + #0: ToddCoxeter: nodes | 7,238,400 | 23,613,836 | 30,852,236 + #0: ToddCoxeter: diff 0.2.25 | +0 | +0 | +0 + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 57,907,200 | 0 | 100.0% + #0: ToddCoxeter: diff 0.2.25 | +0 | +0 | +0.0% + #0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0% + #0: ToddCoxeter: phase 0.2 = 24.099s | run 0 = 3min6s | all runs = 3min6s | elapsed = 3min6s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1 + #0: ToddCoxeter: time spent in phases | 24.099s (13%) | - (0%) | - (0%) | 2min42s (87%) + #0: ToddCoxeter: phase 0.2 = 24.384s | run 0 = 3min6s | all runs = 3min6s | elapsed = 3min6s + The index of the subgroup is 7238400 + ``` + +:material-checkbox-marked-circle-outline: The computed index is the same as +the claimed index: $7,238,400$. diff --git a/docs/examples/atlas/pariahs/ru/index.md b/docs/examples/atlas/pariahs/ru/index.md deleted file mode 100644 index 8bc0864..0000000 --- a/docs/examples/atlas/pariahs/ru/index.md +++ /dev/null @@ -1,12 +0,0 @@ -# Rudvalis group Ru - -> Order: $145,926,144,000$ - -> Presentation: TODO - -On this page, we provide links to verifications that the maximal subgroups of -the Rudvalis group Ru define groups of the correct order. - -## Maximal subgroups - -The following are maximal subgroups that have been verified: diff --git a/docs/examples/index.md b/docs/examples/index.md index e258f97..58e58fe 100644 --- a/docs/examples/index.md +++ b/docs/examples/index.md @@ -1,13 +1,11 @@ # Examples -This page contains various links that explain how the -[Semigroups package for GAP][] and/or [libsemigroups_pybind11][] have been used -to conduct mathematical research. +This page contains various links that explain how the [Semigroups package for +GAP][Semigroups] and/or [libsemigroups_pybind11][] have been used for research +and scholarship. Please use the menu at the left or the links below: - [Short presentations for transformation monoids](short.md) -- [Verifying presentations in the ATLAS of Finite Groups](atlas/index.md) - -[Semigroups package for GAP]: https://semigroups.github.io/Semigroups/ -[libsemigroups_pybind11]: https://libsemigroups.github.io/libsemigroups_pybind11/index.html +- [Verifying presentations for sporadic simple groups](atlas/index.md) +- [Finding complete rewriting systems for 1-relation monoids](1-relation/index.md) diff --git a/docs/examples/short.md b/docs/examples/short.md index 293451a..a08b7cc 100644 --- a/docs/examples/short.md +++ b/docs/examples/short.md @@ -2,27 +2,24 @@ On this page we describe the computational claims made in the paper: -* T. Aird, J. D. Mitchell, M. T. Whyte, Short presentations for -transformation monoids, -[https://doi.org/10.48550/arXiv.2406.19294](https://doi.org/10.48550/arXiv.2406.19294) +- T. Aird, J. D. Mitchell, M. T. Whyte, Short presentations for transformation + monoids, + [https://doi.org/10.48550/arXiv.2406.19294](https://doi.org/10.48550/arXiv.2406.19294) -and how they can be verified using -[libsemigroups_pybind11][] and the [Semigroups package for GAP][]. - -[libsemigroups_pybind11]: https://libsemigroups.github.io/libsemigroups_pybind11/index.html +and how they can be verified using [libsemigroups_pybind11][] and the +[Semigroups package for GAP][Semigroups]. Below we give the commands that permit the computation of the presentations for the symmetric inverse monoid, full transformation monoid, and partial -transformation monoid from the [paper -above](https://doi.org/10.48550/arXiv.2406.19294) for the small values of $n$ -not covered by the main theorems in the paper. +transformation monoid from the +[paper above](https://doi.org/10.48550/arXiv.2406.19294) for the small values of +$n$ not covered by the main theorems in the paper. ## The symmetric inverse monoid First we include [Python][] code that implements the presentation for the -symmetric inverse monoid of degree $n\geq 3$ from Theorem -1.4 of [the paper](https://doi.org/10.48550/arXiv.2406.19294). - +symmetric inverse monoid of degree $n\geq 3$ from Theorem 1.4 of +[the paper](https://doi.org/10.48550/arXiv.2406.19294). === "Python" @@ -36,7 +33,7 @@ symmetric inverse monoid of degree $n\geq 3$ from Theorem p = to(examples.symmetric_group_Car56(n), rtype=(Presentation, str)) alpha = p.alphabet()[-1] + p.alphabet() p.alphabet(p.alphabet() + "z") - + presentation.add_rule(p, "abaz", "zaba") presentation.add_rule(p, alpha + "z", "zz" + alpha) presentation.add_rule(p, "azazaza", "zaz") @@ -178,47 +175,31 @@ presentation.to_gap_string(theorem_1_4_symm_inv(3), "S") ### $n = 4$ -Here we give the [GAP][] and [Python][] code for verifying that the -presentation -\begin{equation}\label{eq-t-4} -\begin{aligned} -\langle a_2, a_3, a_4, z \mid &\ - a_2^2=\varepsilon, - a_3^2=\varepsilon, - a_4^2=\varepsilon, - (a_2a_3)^3=\varepsilon, - (a_3a_4)^3=\varepsilon, - (a_4a_2)^3=\varepsilon,\\\\ - &\ (a_2a_3a_2a_4)^2=\varepsilon, - (a_3a_4a_3a_2)^2=\varepsilon, - (a_4a_2a_4a_3)^2=\varepsilon, \\\\ -&\ (a_4a_2a_3a_2z)^2=(za_4a_2a_3a_2)^2, \\\\ -&\ (a_2a_3a_2z)^3a_2a_3a_2=za_2a_3a_2z,\\\\ -&\ a_3a_4a_3za_3a_4a_3=(za_3)^2, \\\\ -&\ a_2z= z(za_3)^2 -\rangle -\end{aligned} -\end{equation} -defines the full transformation monoid of degree $4$. By Theorem 1.5 in [the -paper](https://doi.org/10.48550/arXiv.2406.19294) this presentation has the smallest number of non-$S_n$ -relations, i.e. relations containing the letter $\zeta$ which represents the -rank $n-1$ idempotent: -\begin{pmatrix} -1 & 2 & 3 & 4 \\\\ -1 & 1 & 3 & 4 -\end{pmatrix} +Here we give the [GAP][] and [Python][] code for verifying that the presentation +\begin{equation}\label{eq-t-4} \begin{aligned} \langle a_2, a_3, a_4, z \mid &\ +a_2^2=\varepsilon, a_3^2=\varepsilon, a_4^2=\varepsilon, (a_2a_3)^3=\varepsilon, +(a_3a_4)^3=\varepsilon, (a_4a_2)^3=\varepsilon,\\\\ &\ +(a_2a_3a_2a_4)^2=\varepsilon, (a_3a_4a_3a_2)^2=\varepsilon, +(a_4a_2a_4a_3)^2=\varepsilon, \\\\ &\ (a_4a_2a_3a_2z)^2=(za_4a_2a_3a_2)^2, \\\\ +&\ (a_2a_3a_2z)^3a_2a_3a_2=za_2a_3a_2z,\\\\ &\ a_3a_4a_3za_3a_4a_3=(za_3)^2, +\\\\ &\ a_2z= z(za_3)^2 \rangle \end{aligned} \end{equation} defines the full +transformation monoid of degree $4$. By Theorem 1.5 in +[the paper](https://doi.org/10.48550/arXiv.2406.19294) this presentation has the +smallest number of non-$S_n$ relations, i.e. relations containing the letter +$\zeta$ which represents the rank $n-1$ idempotent: \begin{pmatrix} 1 & 2 & 3 & +4 \\\\ 1 & 1 & 3 & 4 \end{pmatrix} Note that the presentation with generators $a_2, a_3, a_4$ and the relations in the presentation in \eqref{eq-t-4} not containing $z$ defines the symmetric group of degree $4$ (this is Carmichael's presentation from: -* Carmichael, Robert D. *Introduction to the Theory of Groups of Finite Order*. - New York: Dover Publications, 1956). +- Carmichael, Robert D. _Introduction to the Theory of Groups of Finite Order_. + New York: Dover Publications, 1956). -In the [GAP][] code below we show that the presentation defines a -monoid of the same size as $\mathcal{T}_4$ and that the function mapping -$a_2, a_3, a_4, z$ to the permutations $(1, 2), (1, 3), (1, 4)$ and the -transformation $\zeta$ is indeed an isomorphism. +In the [GAP][] code below we show that the presentation defines a monoid of the +same size as $\mathcal{T}_4$ and that the function mapping $a_2, a_3, a_4, z$ to +the permutations $(1, 2), (1, 3), (1, 4)$ and the transformation $\zeta$ is +indeed an isomorphism. === "Python" @@ -262,19 +243,19 @@ transformation $\zeta$ is indeed an isomorphism. [b*c*b*z*b*c*b, z*b*z*b], [a*z, z*z*b*z*b]]; M := F / R; Size(M); - SemigroupIsomorphismByImages(M, - FullTransformationMonoid(4), + SemigroupIsomorphismByImages(M, + FullTransformationMonoid(4), GeneratorsOfMonoid(M), - [AsTransformation((1, 2)), - AsTransformation((1, 3)), + [AsTransformation((1, 2)), + AsTransformation((1, 3)), AsTransformation((1, 4)), Transformation([1, 1, 3, 4])]); ``` ### $n = 5$ -As stated in [the paper](https://doi.org/10.48550/arXiv.2406.19294), the presentation from Theorem 1.5 defines $T_n$ -when $n = 5$. +As stated in [the paper](https://doi.org/10.48550/arXiv.2406.19294), the +presentation from Theorem 1.5 defines $T_n$ when $n = 5$. ```python from libsemigroups_pybind11 import Presentation, presentation, to @@ -362,7 +343,7 @@ def theorem_1_5_full_transf(n: int) -> Presentation: Presentation, presentation, congruence_kind, - ToddCoxeter, + ToddCoxeter, to ) from libsemigroups_pybind11.presentation import examples @@ -440,11 +421,10 @@ def theorem_1_5_full_transf(n: int) -> Presentation: ### $n = 2$ The below is some [Python][] to verify that the presentation -\begin{equation}\label{PT-2} - \langle a_2, \zeta, \eta \mid a_2^2 = 1,\quad a_2\zeta = \zeta,\quad \zeta a_2 \eta a_2 - = \zeta,\quad \eta a_2 \zeta a_2 = \eta,\quad a_2 \eta a_2 \eta a_2 = \zeta \eta \rangle -\end{equation} -defines $PT_2$. +\begin{equation}\label{PT-2} \langle a_2, \zeta, \eta \mid a_2^2 = 1,\quad +a_2\zeta = \zeta,\quad \zeta a_2 \eta a_2 = \zeta,\quad \eta a_2 \zeta a_2 = +\eta,\quad a_2 \eta a_2 \eta a_2 = \zeta \eta \rangle \end{equation} defines +$PT_2$. === "Python" @@ -457,7 +437,7 @@ defines $PT_2$. p = Presentation("azn").contains_empty_word(True) p.rules = ["aa", "", "az", "z", "zana", "z", "naza", "n", "anana", "zn"] - + tc = ToddCoxeter(congruence_kind.twosided, p) tc.number_of_classes() # returns 9 ``` @@ -552,9 +532,9 @@ def partial_transf_monoid_456(n: int) -> Presentation: "n" + A + A[0], A + A[0] + "n", A[-1] + A[0] + A[1] + A[0] + "z" + A[-1] + A[0] + A[1] + A[0] + "z", "z" + A[-1] + A[0] + A[1] + A[0] + "z" + A[-1] + A[0] + A[1] + A[0], - (A[0] + A[1] + A[0] + "z") * 3 + A[0] + A[1] + A[0], + (A[0] + A[1] + A[0] + "z") * 3 + A[0] + A[1] + A[0], "z" + A[0] + A[1] + A[0] + "z", - A[1] + A[2:] + A[1] + A[0] + "z", + A[1] + A[2:] + A[1] + A[0] + "z", "z" + A[1] + A[2:] + A[1], "zbnb", "bnbz", "zn", "anana", @@ -700,7 +680,3 @@ def partial_transf_monoid_456(n: int) -> Presentation: S := F / R; Size(S); ``` - -[GAP]: https://gap-system.org -[Semigroups package for GAP]: https://semigroups.github.io/Semigroups/ -[Python]: https://www.python.org diff --git a/docs/gap/first-steps.md b/docs/gap/first-steps.md index 4504721..e5ac619 100644 --- a/docs/gap/first-steps.md +++ b/docs/gap/first-steps.md @@ -8,7 +8,7 @@ in learning more about GAP itself, see also the ## The GAP prompt -After installing GAP and the `Semigroups` package as described in the +After installing GAP and the [Semigroups][] package as described in the [Installation](install.md) section, you should be able to start GAP from your terminal simply by typing `gap`++enter++[^1]. If GAP has been installed correctly, you should see the GAP banner followed by the _GAP prompt_ `gap> ` similar to @@ -101,7 +101,7 @@ GAP reference manual for more details about the break loop and the main loop. ### The help system -GAP and the `Semigroups` package implement many different functions. What a function does may not +GAP and the [Semigroups][] package implement many different functions. What a function does may not always be clear from its name, and it may be necessary to read the _documentation_ describing its functionality. This can be done directly from the GAP REPL via the _help system_. To learn more about a function, simply prepend a question mark `?` to its name. @@ -856,7 +856,7 @@ fi; Note that with the `#!gap break;` statements, only a single counterexample is printed, hence the search was stopped early. -The `Semigroups` package offers a simpler way of checking if a multiplication table +The [Semigroups][] package offers a simpler way of checking if a multiplication table defines a semigroup, which we will cover in [the next section](generated.md#multiplication-tables). diff --git a/docs/gap/fp.md b/docs/gap/fp.md index 2ebab0e..2bebfc9 100644 --- a/docs/gap/fp.md +++ b/docs/gap/fp.md @@ -611,8 +611,6 @@ This says that the symmetric inverse monoid has 18 right congruences with up to removed has 26 such congruences. So, these monoids are not isomorphic, and so the relation $a^2=1$ is not redundant. -[libsemigroups]: https://libsemigroups.github.io/libsemigroups/ -[GAP]: https://gap-system.org [Semigroups]: https://semigroups.github.io/Semigroups/ [ACE]: https://gap-packages.github.io/ace/ [KBMAG]: https://gap-packages.github.io/kbmag/ diff --git a/docs/gap/generated.md b/docs/gap/generated.md index f9acbe3..f9aaf54 100644 --- a/docs/gap/generated.md +++ b/docs/gap/generated.md @@ -1,12 +1,12 @@ # Semigroups and monoids defined by generators -This section provides information about how to use the `Semigroups` package for +This section provides information about how to use the [Semigroups][] package for GAP to compute with a semigroup or monoid defined by a collection of generators. We assume some basic familiarity with the GAP programming language, see the [GAP: First Steps](first-steps.md) section for a basic overview if you do not yet feel comfortable with GAP. -All code examples in this section assume that the `Semigroups` package +All code examples in this section assume that the [Semigroups][] package is loaded. To do so, simply add ```gap LoadPackage("Semigroups"); @@ -15,8 +15,8 @@ at the start of your gap script, or execute this command at the start of your GAP session. !!! failure "If Semigroups fails to load" - If you encounter an error when loading the `Semigroups` package, - it may be due to the kernel module (the `C++` component powering `Semigroups`) + If you encounter an error when loading the [Semigroups][] package, + it may be due to the kernel module (the `C++` component powering [Semigroups][]) not being compiled. Execute the command `#!gap SetInfoLevel(InfoPackageLoading, 4);` and attempt to load the package again. If you see an error similar to the following: @@ -42,7 +42,7 @@ Any GAP object equipped with an associative multiplication can in principle be used to construct a semigroup in GAP. However, there are certain kinds of semigroups for which there exist the algorithms that are several orders of magnitude faster than those applicable in the general case. -In particular, the `Semigroups` package implements efficient algorithms for +In particular, the [Semigroups][] package implements efficient algorithms for the so called [_acting semigroups_](https://semigroups.github.io/Semigroups/doc/chap6_mj.html#X7A3AC74C7FF85825)[^1] and has an efficient implementation of the @@ -50,9 +50,9 @@ and has an efficient implementation of the which can be applied to subsemigroups of certain semigroups. In this section we give a brief description of the main types of -elements which can be efficiently computed with via the `Semigroups` package. +elements which can be efficiently computed with via the [Semigroups][] package. See [Chapter 6](https://semigroups.github.io/Semigroups/doc/chap6_mj.html) -of the `Semigroups` package documentation for more details. +of the [Semigroups][] package documentation for more details. ### Transformations @@ -205,7 +205,7 @@ functions respectively. i.e. partial functions with domain and codomain in $\{1, \ldots, n\}$. However, for any fixed $n$, the semigroup of all partial transformations of degree $n$ is isomorphic to the semigroup of all total transformations - of degree $n+1$, which is what the `Semigroups` package utilizes. + of degree $n+1$, which is what the [Semigroups][] package utilizes. See the documentation of the [`PartialTransformationMonoid`](https://semigroups.github.io/Semigroups/doc/chap7_mj.html#X808A27F87E5AC598) for more details. @@ -354,7 +354,7 @@ Bipartitions in GAP can be specified using the function by specifying the blocks of the bipartition. Bipartitions can be multiplied using the `*` operator (see [Chapter 3](https://semigroups.github.io/Semigroups/doc/chap3_mj.html#X7C18DB427C9C0917) -of the `Semigroups` manual for a definition of bipartition multiplication). +of the [Semigroups][] manual for a definition of bipartition multiplication). Furthermore, bipartitions admit an involution [`Star`](https://semigroups.github.io/Semigroups/doc/chap3_mj.html#X7CE00E0C79F62745), which swaps all the points $i$ with $-i$ in the partition. @@ -474,13 +474,13 @@ functions respectively. bipartitions of distinct degrees _cannot_ be multiplied together, and attempting to do so at the moment will cause erroneous output. - This issue will be resolved in a future release of the `Semigroups` + This issue will be resolved in a future release of the [Semigroups][] package, a fix was implemented in [Pull request #1185](https://github.com/semigroups/Semigroups/pull/1185). !!! info See [Chapter 3](https://semigroups.github.io/Semigroups/doc/chap3_mj.html) - of the `Semigroups` manual for more details about bipartitions. + of the [Semigroups][] manual for more details about bipartitions. ### Matrices over semirings @@ -902,7 +902,7 @@ The entries `false` and `true` get mapped to `0` and `1` respectively. A boolean matrix $A$ of dimension $n$ can equivalently be viewed as a binary relation $R$ of the set $\{1, \ldots, n\}$, where $(i, j) \in R$ if and only -if $A_{i, j} = 1$. The `Semigroups` package implements +if $A_{i, j} = 1$. The [Semigroups][] package implements functions for checking if a boolean matrix, when viewed as a binary relation in this manner, is reflexive, symmetric, antisymmetric or transitive via the functions @@ -1089,7 +1089,7 @@ functions, though not all semirings support all of these functions. !!! info See [Chapter 5](https://semigroups.github.io/Semigroups/doc/chap5_mj.html) - of the `Semigroups` manual for more details about matrices over semirings, + of the [Semigroups][] manual for more details about matrices over semirings, as well as [Chapter 24](https://docs.gap-system.org/doc/ref/chap24_mj.html#X812CCAB278643A59) of the GAP reference manual for more details about general methods for matrices. Note, however that not all methods may be implemented for @@ -1098,7 +1098,7 @@ functions, though not all semirings support all of these functions. ## Constructing semigroups In this section we briefly outline different methods for constructing -semigroups using GAP and the `Semigroups` package. We will occasionally +semigroups using GAP and the [Semigroups][] package. We will occasionally use some functions to showcase properties of these semigroups, such as `Size`, `Elements`, `IsRegularSemigroup` and so on. These functions will be explained in more detail in the [Analyzing semigroups](#analyzing-semigroups) @@ -1106,7 +1106,7 @@ section later on. ### Built-in semigroups and monoids -GAP and the `Semigroups` package provide a range of built-in standard semigroups, +GAP and the [Semigroups][] package provide a range of built-in standard semigroups, such as the * _full transformation semigroup_ $\mathcal{T}_n$ consisting of all degree $n$ @@ -1188,8 +1188,8 @@ whether or not the semigroup is regular: !!! info See [Chapter 7](https://semigroups.github.io/Semigroups/doc/chap7_mj.html) - of the `Semigroups` manual for a list of **many** more standard semigroups - and monoids implemented by the `Semigroups` package. + of the [Semigroups][] manual for a list of **many** more standard semigroups + and monoids implemented by the [Semigroups][] package. ### Finitely generated subsemigroups and submonoids @@ -1265,7 +1265,7 @@ section, in theory any collection of elements with associative multiplication can be used to construct a finitely generated semigroup using the `Semigroup` function. However, using the kinds of elements described in the [Kinds of elements](#kinds-of-elements) section is guaranteed to utilize -the fast algorithms implemented in `Semigroups`. For an extreme example +the fast algorithms implemented in [Semigroups][]. For an extreme example consider: === "GAP REPL" @@ -1302,13 +1302,13 @@ consider: We are able to enumerate a semigroup of 3 million elements in 4.5 seconds. Of course the runtime will depend on the machine and so on, but the point is that -this computation takes seconds instead of days with `Semigroups`. -Trying to run the computation without the `Semigroups` package loaded +this computation takes seconds instead of days with [Semigroups][]. +Trying to run the computation without the [Semigroups][] package loaded will cause the `Size` computation to run significantly longer (we have not -been able to run it to completion without the `Semigroups` package loaded). +been able to run it to completion without the [Semigroups][] package loaded). In certain cases, we may even be able to determine that a semigroup is -infinite using the `Semigroups` package: +infinite using the [Semigroups][] package: === "GAP REPL" ```gap-repl @@ -1479,7 +1479,7 @@ we can use the [`Monoid`](https://docs.gap-system.org/doc/ref/chap51_mj.html#X7F !!! info See [Chapter 6](https://semigroups.github.io/Semigroups/doc/chap6_mj.html) - of the `Semigroups` manual for more info on how the `Semigroups` package + of the [Semigroups][] manual for more info on how the [Semigroups][] package handles finitely generated subsemigroups as well as [Section 51.1](https://docs.gap-system.org/doc/ref/chap51_mj.html#X80AF5F307DBDC2B4) and @@ -1608,7 +1608,7 @@ The [`Smallsemi`](https://gap-packages.github.io/smallsemi/doc/chap0_mj.html) package contains a database of all semigroups of order up to $8$ up to isomorphism and anti-isomorphism. To use the `Smallsemi` package we need to load it first, just as -we did with the `Semigroups` package. +we did with the [Semigroups][] package. We can get the $n$-th semigroup of order $m$ using the [`SmallSemigroup`](https://gap-packages.github.io/smallsemi/doc/chap4_mj.html#X8538248D78185960) function with arguments `m` and `n`. @@ -1763,7 +1763,7 @@ object. ``` There are two ways of visualizing these graphs using the -`Semigroups` package: +[Semigroups][] package: 1. We can visualize these graphs without the edge coloring using the [`DotRightCayleyDigraph`](https://semigroups.github.io/Semigroups/doc/chap16_mj.html#X7E38369D7E8BEA4C) and @@ -1860,7 +1860,7 @@ There are two ways of visualizing these graphs using the !!! info See [Chapter 16](https://semigroups.github.io/Semigroups/doc/chap16_mj.html) - of the `Semigroups` manual for more information about visualizing semigroups + of the [Semigroups][] manual for more information about visualizing semigroups and their Cayley graphs, as well as [Chapter 9](https://docs.gap-system.org/pkg/digraphs/doc/chap9_mj.html) of the `Digraphs` package manual. @@ -1990,8 +1990,8 @@ We can !!! info See [Chapter 12](https://semigroups.github.io/Semigroups/doc/chap12_mj.html) - of the `Semigroups` manual for a comprehensive list of semigroup - properties that can be checked by the `Semigroups` package, as well as + of the [Semigroups][] manual for a comprehensive list of semigroup + properties that can be checked by the [Semigroups][] package, as well as [Section 51.4](https://docs.gap-system.org/doc/ref/chap51_mj.html#X78274024827F306D) of the GAP reference manual. @@ -2096,7 +2096,7 @@ The egg-box diagram of $\mathcal{T}_3$. !!! info See [Chapter 10](https://semigroups.github.io/Semigroups/doc/chap10_mj.html) - of the `Semigroups` manual for a deeper look at Green's relation related + of the [Semigroups][] manual for a deeper look at Green's relation related functionality. ### Isomorphisms @@ -2139,7 +2139,7 @@ and return `fail` if the semigroups are not isomorphic. !!! info See [Chapter 14](https://semigroups.github.io/Semigroups/doc/chap14_mj.html) - of the `Semigroups` manual for more information about + of the [Semigroups][] manual for more information about semigroup isomorphisms and homomorphisms. ### Lattices of congruences @@ -2221,7 +2221,7 @@ Result of `#!gap Splash(DotString(latt));`. !!! info See [Chapter 13](https://semigroups.github.io/Semigroups/doc/chap13_mj.html) - of the `Semigroups` manual for more information about + of the [Semigroups][] manual for more information about semigroup congruences. diff --git a/docs/gap/index.md b/docs/gap/index.md index e3e5386..3b5c220 100644 --- a/docs/gap/index.md +++ b/docs/gap/index.md @@ -1,16 +1,17 @@ -# `Semigroups` in GAP +# [Semigroups][] in [GAP][] -[GAP][] is a computer algebra system for discrete algebra. While the core focus of GAP is +[GAP][] is a computer algebra system for discrete algebra. While the core focus of [GAP][] is group theory, several extensions to the system are provided by -[_packages_](https://docs.gap-system.org/doc/ref/chap76_mj.html). An index of GAP packages that are shipped together -with gap is available online in the [GAP Packages](https://www.gap-system.org/packages/) section of the GAP system +[_packages_](https://docs.gap-system.org/doc/ref/chap76_mj.html). An index of [GAP][] packages that are shipped together +with gap is available online in the [GAP Packages](https://www.gap-system.org/packages/) section of the [GAP][] system website. -In this worksheet we will be primarily focused on using the [`Semigroups`](https://semigroups.github.io/Semigroups/) -package for computational investigations of semigroups. The worksheet does not assume prior familiarity with the -GAP programming language, though we recommend the +In this worksheet we will be primarily focused on using the [Semigroups][] +package for computational investigations of semigroups. The worksheet does not +assume prior familiarity with the [GAP][] programming language, though we recommend +the [GAP software carpentries lesson](https://carpentries-incubator.github.io/gap-lesson/index.html) to those interested -in learning more about GAP. +in learning more about [GAP][]. Click on the menu at the top left for links to the different parts of this tutorial, or select from one of the options below: @@ -20,5 +21,3 @@ tutorial, or select from one of the options below: * [Semigroups and monoids defined by generators](generated.md) * [Finitely presented semigroups and monoids](fp.md) * [Worksheet](worksheet.md) - -[GAP]: https://www.gap-system.org/ diff --git a/docs/gap/install.md b/docs/gap/install.md index 250425e..1676418 100644 --- a/docs/gap/install.md +++ b/docs/gap/install.md @@ -1,6 +1,6 @@ -# Installing GAP and the `Semigroups` package +# Installing [GAP][] and the [Semigroups][] package -In this section we detail the installation of GAP and `Semigroups` on various platforms. +In this section we detail the installation of [GAP][] and [Semigroups][] on various platforms. We present two options for doing so: - [Option 1: Standard install](#option-1-standard-install): The standard install path. It is slightly more involved @@ -14,12 +14,12 @@ but we recommend the [Unix shell](https://swcarpentry.github.io/shell-novice/) s lesson to those interested in learning more. !!! warning - Please do not install GAP using your systems package manager - e.g. do **not** install it via `#!bash apt-get install gap` or `#!bash brew install gap`. The version of GAP packaged by + Please do not install [GAP][] using your systems package manager + e.g. do **not** install it via `#!bash apt-get install gap` or `#!bash brew install gap`. The version of [GAP][] packaged by your operating system and installed by these tools can be several years out of date and will likely cause issues further down the line. - You can remove a system installed version of GAP on an Ubuntu distribution of Linux by running + You can remove a system installed version of [GAP][] on an Ubuntu distribution of Linux by running ```bash sudo apt-get remove gap ``` @@ -30,11 +30,11 @@ lesson to those interested in learning more. ## Option 1: Standard install -This section outlines the standard installation process for GAP and `Semigroups`. It is split into two parts: +This section outlines the standard installation process for [GAP][] and [Semigroups][]. It is split into two parts: the [System based prerequisites](#system-based-prerequisites) section where we outline steps required to set up -your operating system for installing GAP and `Semigroups`, and +your operating system for installing [GAP][] and [Semigroups][], and [Common installation steps](#common-installation-steps) where we outline a method -for installing GAP and `Semigroups` that should work on all systems, provided the prerequisites are installed. +for installing [GAP][] and [Semigroups][] that should work on all systems, provided the prerequisites are installed. ### System based prerequisites @@ -42,7 +42,7 @@ Please select the appropriate tab for your operating system below and follow the === "Windows" - GAP is primarily developed for Linux-based operating systems. In order to make + [GAP][] is primarily developed for Linux-based operating systems. In order to make it run on Windows, it is generally easier to install a Linux subsystem which provides a Linux shell than to try and make it run on Windows directly. Windows provides such a feature via the Windows subsystem for Linux (WSL). @@ -77,7 +77,7 @@ Please select the appropriate tab for your operating system below and follow the ``` If all went well, you now have an Ubuntu Linux subsystem available via the - `#!powershell ubuntu` command. To install GAP and `Semigroups`, execute the `#!powershell ubuntu` + `#!powershell ubuntu` command. To install [GAP][] and [Semigroups][], execute the `#!powershell ubuntu` command in your terminal and then proceed as described in the [Common installation steps](#common-installation-steps) section, following the instructions for a Linux system. @@ -142,16 +142,16 @@ Please select the appropriate tab for your operating system below and follow the Before proceeding with the install instructions in this section, please make sure you have completed the steps outlined in the [System based prerequisites](#system-based-prerequisites) section for your operating system above. -In order to install GAP we follow the +In order to install [GAP][] we follow the [installation instructions](https://www.gap-system.org/install/linux/) from the -GAP website. -We will install GAP in the "home" directory. You are free to use any other +[GAP][] website. +We will install [GAP][] in the "home" directory. You are free to use any other directory, if you do so, substitute the `~` in the following commands with whatever base directory you are using. 1. Type `#!bash cd ~`++enter++ in your terminal. This will ensure you are in the home directory, where the rest of the install will occur. -2. Download the current GAP distribution, at time of writing it is 4.15.1. If this has changed, then +2. Download the current [GAP][] distribution, at time of writing it is 4.15.1. If this has changed, then substitute the appropriate version number in the `#!bash export` command below: ```bash export GAP_VERSION=4.15.1 @@ -181,7 +181,7 @@ whatever base directory you are using. ``` replacing the 8 in the `-j8` by the number of cores you want to use. If all goes well, make should finish with without errors and running - `#!bash ./gap`++enter++ gives the GAP prompt (you can quit from the GAP prompt by + `#!bash ./gap`++enter++ gives the [GAP][] prompt (you can quit from the [GAP][] prompt by typing `#!gap quit;`++enter++). 5. We are not quite done yet, since some of the optional packages need to be compiled before they can be used. In order to do so run @@ -189,8 +189,8 @@ whatever base directory you are using. cd pkg ../bin/BuildPackages.sh ``` -6. To test the package compilation, type `#!bash ../gap`++enter++ to open a GAP - prompt and within the GAP prompt write +6. To test the package compilation, type `#!bash ../gap`++enter++ to open a [GAP][] + prompt and within the [GAP][] prompt write ```gap SetInfoLevel(InfoPackageLoading, 4); LoadPackage("Semigroups"); @@ -207,8 +207,8 @@ whatever base directory you are using. #I Semigroups: return from LoadPackage, package is not available fail ``` - If this is the case then type `#!gap quit;`++enter++ in the GAP session and perform the following steps to - compile the `Semigroups` package manually: + If this is the case then type `#!gap quit;`++enter++ in the [GAP][] session and perform the following steps to + compile the [Semigroups][] package manually: ```bash cd semigroups ./configure @@ -217,7 +217,7 @@ whatever base directory you are using. compilation of the semigroups package is quite memory intensive, and may get killed if the computer runs out of memory. If so try compiling again setting a lower number of threads when running make, e.g. `#!bash make -j2` or even just `#!bash make`. -8. To run GAP from an arbitrary folder on your computer you currently need to write +8. To run [GAP][] from an arbitrary folder on your computer you currently need to write `#!bash ~/gap-$GAP_VERSION/gap`++enter++. To avoid having to do this execute the following: ```bash mkdir -p ~/.local/bin @@ -234,10 +234,10 @@ whatever base directory you are using. echo 'if ! [[ "$PATH" =~ "$HOME/.local/bin:" ]] then PATH="$HOME/.local/bin:$PATH"; fi' >> ~/.zshrc source ~/.zshrc ``` - Test this works by typing `#!bash gap`++enter++ in the terminal, if all went well it should open a GAP prompt. + Test this works by typing `#!bash gap`++enter++ in the terminal, if all went well it should open a [GAP][] prompt. -If all went well you should now be able to type `#!bash gap`++enter++ in your terminal, which should give you a GAP prompt, -and running `#!gap LoadPackage("Semigroups");` in the GAP prompt should successfully load the `Semigroups` package. +If all went well you should now be able to type `#!bash gap`++enter++ in your terminal, which should give you a [GAP][] prompt, +and running `#!gap LoadPackage("Semigroups");` in the [GAP][] prompt should successfully load the [Semigroups][] package. ## Option 2: Experimental install using Julia @@ -245,12 +245,12 @@ and running `#!gap LoadPackage("Semigroups");` in the GAP prompt should successf The content in this section is experimental and may not work on all systems. In this case we advise proceeding with the steps of the [Option 1: Standard install](#option-1-standard-install) section instead. -It has recently become possible to install GAP using the +It has recently become possible to install [GAP][] using the [Julia](https://julialang.org/) programming language. This installation path is currently experimental but can be much quicker than the method detailed in the [Option 1: Standard install](#option-1-standard-install) section. -If you already installed GAP and `Semigroups` using the instructions from +If you already installed [GAP][] and [Semigroups][] using the instructions from the section [Option 1: Standard install](#option-1-standard-install), then skip this section. @@ -296,13 +296,13 @@ We follow the Julia install instructions from ```julia using GAP; GAP.prompt();` ``` - This should start a GAP prompt. In the GAP prompt type + This should start a [GAP][] prompt. In the [GAP][] prompt type ```gap SetInfoLevel(InfoPackageLoading, 4); LoadPackage("Semigroups"); ``` - To check that the `Semigroups` package is available. - Exit the GAP prompt by typing `#!gap quit;`++enter++, which should + To check that the [Semigroups][] package is available. + Exit the [GAP][] prompt by typing `#!gap quit;`++enter++, which should return you to the Julia prompt. Exit the Julia prompt too via `#!julia exit()`++enter++. 3. We will now create a script which will allow us to start the gap prompt by @@ -324,5 +324,5 @@ We follow the Julia install instructions from ```julia using GAP; GAP.Setup.create_gap_sh("~/.local/bin/", "gap"); ``` - Then `#!julia exit()` the Julia prompt. You should now be able to start a GAP + Then `#!julia exit()` the Julia prompt. You should now be able to start a [GAP][] session by typing `#!bahs gap`++enter++ in your terminal. diff --git a/docs/gap/worksheet.md b/docs/gap/worksheet.md index bbaf010..da3def5 100644 --- a/docs/gap/worksheet.md +++ b/docs/gap/worksheet.md @@ -1,35 +1,35 @@ # Worksheet - This worksheet contains some exercises to be completed in [GAP][] with the use -of the [Semigroups package for GAP][] package as part of the [GAP][] workshop -session of [NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). +of the [Semigroups package for GAP][Semigroups] package as part of the [GAP][] +workshop session of +[NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). 1. Let $J_n$ denote the Jones monoid of degree $n$, and ![An element of the Jones monoid](../images/bipartition.png) - 1. Find a collection of bipartitions $\{x_1, x_2, \dots, x_j\}$ such that the tensor product $x_1 \otimes x_2 \otimes \dots \otimes x_j$ is equal - to $x$. What is the largest possible value of $j$? For the definition of - the tensor product of bipartitions, see the documentation of the function - [`TensorBipartition`](https://semigroups.github.io/Semigroups/doc/chap3_mj.html#X7F7D823084F79C7D). + to $x$. What is the largest possible value of $j$? For the definition of + the tensor product of bipartitions, see the documentation of the + function + [`TensorBipartition`](https://semigroups.github.io/Semigroups/doc/chap3_mj.html#X7F7D823084F79C7D). 2. How many elements does $J_{14}$ contain? 3. How many idempotent elements does $J_{14}$ contain? - 4. How many idempotent elements does the Green's - $\mathscr{D}$-class of $x$ in $J_{14}$ contain? + 4. How many idempotent elements does the Green's $\mathscr{D}$-class of $x$ + in $J_{14}$ contain? 5. How many idempotent elements $e$ such that $7$ belongs to the same part as a negative number does the Green's $\mathscr{D}$-class of $x$ in $J_{14}$ contain? - 6. Same question as the previous part but for *irreducible* idempotents + 6. Same question as the previous part but for _irreducible_ idempotents instead. ??? hint - + You might find the following functions helpful: * [`DClass`](https://semigroups.github.io/Semigroups/doc/chap10_mj.html#X81B7AD4C7C552867) @@ -39,10 +39,10 @@ session of [NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). * [`NrIdempotents`](https://semigroups.github.io/Semigroups/doc/chap11_mj.html#X7CFC4DB387452320) * [`Size`](https://docs.gap-system.org/doc/ref/chap30_mj.html#X858ADA3B7A684421) -2. Recall that the commuting graph of a semigroup $S$ is the graph with - nodes the elements of $S$ and an edge $(x, y)$ if $xy = yx$ holds. - Let $\mathcal{T}_n$ be the full transformation monoid on $n \leq 5$ points. - Show that the clique numbers of the commuting graph of $\mathcal{T}_n$ are +2. Recall that the commuting graph of a semigroup $S$ is the graph with nodes + the elements of $S$ and an edge $(x, y)$ if $xy = yx$ holds. Let + $\mathcal{T}_n$ be the full transformation monoid on $n \leq 5$ points. Show + that the clique numbers of the commuting graph of $\mathcal{T}_n$ are $2 ^ {n - 1}$. ??? hint @@ -55,35 +55,35 @@ session of [NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). function from the [`Digraphs` package for GAP](https://docs.gap-system.org/pkg/digraphs/doc/chap0_mj.html). The [`Digraphs` package for GAP](https://docs.gap-system.org/pkg/digraphs/doc/chap0_mj.html) - is required by the - [`Semigroups` package for GAP](https://semigroups.github.io/Semigroups/doc/chap0_mj.html) - so you don't have to install - anything new to do this. + is required by the + [Semigroups package for GAP][Semigroups] so you don't have to install anything new to do this. 3. Let $S$ be the monoid defined by the presentation $\langle a, b, c, d, e, f, g \mid abcd = a^3ea^2, ef= dg\rangle$. - 1. Show that $S$ is infinite. - 2. Create a homomorphism from the free semigroup $\{a, b, c, d, + 2. Create a homomorphism from the free semigroup + $\{a, b, c, d, e, f, g\}^+$ to $S$. 3. Partition the first $1000$ elements of the free semigroup - $\{a, b, c, d, e, f, g\}^+$ so that words belong to the same - part if and only if they represent the same element of $S$. + $\{a, b, c, d, e, f, g\}^+$ so that words belong to the same part if and + only if they represent the same element of $S$. ??? hint - + You might find the following helpful: * [`SemigroupHomomorphismByImages`](https://semigroups.github.io/Semigroups/doc/chap14_mj.html#X817596438369885B) * [`Enumerator`](https://docs.gap-system.org/doc/ref/chap30_mj.html#X7EF8910F82B45EC7) 4. Is the monoid defined by the presentation + $$ \langle x_2, \ldots, x_n\mid x_i^2 = (x_ix_j) ^3 = (x_ix_jx_k)^4 = 1\quad i, j, k \text{ distinct}\rangle - $$ + $$ + the symmetric group for $n\geq 2$? ??? hint @@ -91,31 +91,31 @@ session of [NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). You might want to use [`NumberOfRightCongruences`](https://semigroups.github.io/Semigroups/doc/chap13_mj.html#X7AE16F237E862934). -5. Determine which of the relations in the presentation are redundant - and which are not: +5. Determine which of the relations in the presentation are redundant and which + are not: + $$ - \langle a, b \mid + \langle a, b \mid a^4=1, b^2= b, ba^3ba= a^2(ab)^2, (ba^2)^2= (a^2b)^2, (ba)^2a^2= aba^3b, a(ab)^4= (ab)^4 \rangle. $$ + What is the minimal set of the relations in this presentation that define the same monoid? 6. Let $S$ be the Catalan monoid of degree $3$. - 1. Draw the egg-box diagram of $S$. 2. Draw the left and right Cayley graphs of $S$. - 3. Show that $S$ has two non-trivial non-universal non-Rees - congruences. + 3. Show that $S$ has two non-trivial non-universal non-Rees congruences. 4. Show that the lattice of left and right congruences of $S$ are isomorphic. - 5. What are the maximal subsemigroups of $S$? Show that none of the - maximal subsemigroup is isomorphic to any of the others. + 5. What are the maximal subsemigroups of $S$? Show that none of the maximal + subsemigroup is isomorphic to any of the others. ??? hint @@ -128,8 +128,8 @@ session of [NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). * [`CongruencesOfSemigroup`](https://semigroups.github.io/Semigroups/doc/chap13_mj.html#X7E8D5BA27CB5A4DA) * [`MaximalSubsemigroups`](https://semigroups.github.io/Semigroups/doc/chap11_mj.html#X860A10E387C19150) -7. Let $S$ be the dual of the full transformation monoid on 5 points. - Find a transformation representation of $S$ on $32$ points. +7. Let $S$ be the dual of the full transformation monoid on 5 points. Find a + transformation representation of $S$ on $32$ points. ??? hint @@ -138,19 +138,17 @@ session of [NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). * [`DualSemigroup`](https://semigroups.github.io/Semigroups/doc/chap8_mj.html#X79F2643C8642A3B0) * [`SmallerDegreeTransformationRepresentation`](https://semigroups.github.io/Semigroups/doc/chap14_mj.html#X794E5DA4872989E4) -8. The translational hull $\Omega(S)$ of a semigroup $S$ is the set of - all bitranslations $(\lambda, \rho)$ under componentwise - composition. Any element $s$ of $S$ induces an inner bitranslation - $(\lambda_s, \rho_s)$ of $S$, where $\lambda_s(x) = sx$ and - $\rho_s(x) = xs$ for all $x \in S$. This gives a homomorphism - $\pi_S$ from $S$ to $\Omega(S)$. If this is injective, then $S$ is - *weakly reductive*. Restricting the bitranslations to an ideal $I$ - of $S$ gives a homomorphism $\pi_{S|I}$ from $S$ to $\Omega(I)$. If - $I$ is weakly reductive, then $I$ is densely embedded in $S$ if and - only if $\pi_{S|I}$ is an isomorphism. - - 1. Determine the size of $\Omega(I)$ for each ideal $I$ of $J_7$ - (the Jones monoid of degree 7). +8. The translational hull $\Omega(S)$ of a semigroup $S$ is the set of all + bitranslations $(\lambda, \rho)$ under componentwise composition. Any + element $s$ of $S$ induces an inner bitranslation $(\lambda_s, \rho_s)$ of + $S$, where $\lambda_s(x) = sx$ and $\rho_s(x) = xs$ for all $x \in S$. This + gives a homomorphism $\pi_S$ from $S$ to $\Omega(S)$. If this is injective, + then $S$ is _weakly reductive_. Restricting the bitranslations to an ideal + $I$ of $S$ gives a homomorphism $\pi_{S|I}$ from $S$ to $\Omega(I)$. If $I$ + is weakly reductive, then $I$ is densely embedded in $S$ if and only if + $\pi_{S|I}$ is an isomorphism. + 1. Determine the size of $\Omega(I)$ for each ideal $I$ of $J_7$ (the Jones + monoid of degree 7). 2. Which of the ideals of $J_7$ are weakly reductive? @@ -163,9 +161,6 @@ session of [NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). ??? hint - You might want to look at: + You might want to look at: [Chapter 18](https://semigroups.github.io/Semigroups/doc/chap18_mj.html) - of the `Semigroups` package manual. - -[GAP]: https://www.gap-system.org/ -[Semigroups package for GAP]: https://semigroups.github.io/Semigroups/ + of the [Semigroups package for GAP][Semigroups] manual. diff --git a/docs/index.md b/docs/index.md index 8dc5922..de0b00d 100644 --- a/docs/index.md +++ b/docs/index.md @@ -1,13 +1,15 @@ +--- +hide: + - navigation +--- + # `Semigroups` - Zero to ε -The GAP package `Semgroups` and Python package `libsemigroups_pybind11` -tutorials for the -[NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html) -workshop session on applications of GAP and other computational tools -to semigroup theory. -The aim of these tutorials is to -help you get started using tools for computational semigroup theory developed -at the University of St Andrews. +Welcome to this tutorial for the [Semigroups package for GAP][Semigroups] and +the Python package [libsemigroups_pybind11][]. These were first used at +[NBSAN 40](https://manchester-semigroups.github.io/NBSAN/next.html). The aim of +these tutorials is to help you get started using tools for computational +semigroup theory developed at the University of St Andrews. Click on the menu at the top left for links to the different parts of this tutorial, or select from one of the options below: @@ -15,8 +17,17 @@ tutorial, or select from one of the options below: - [Semigroups in GAP](gap/index.md) - [Semigroups in Python](py/index.md) +Some examples of more advanced usage of [libsemigroups][] can be found here on +the [examples](examples/index.md) page. + The QR code for this page is:
![a qr code](images/qr.png)
+ +## Acknowledgments + +The authors of this website would like to thank the University of St Andrews for +funding Joseph Edwards, through the Impact and Innovation Fund, to work on this +website. diff --git a/docs/py/first-steps.md b/docs/py/first-steps.md index 6f0ee82..75696fa 100644 --- a/docs/py/first-steps.md +++ b/docs/py/first-steps.md @@ -1,6 +1,6 @@ -# First steps --- libsemigroups_pybind11 +# First steps --- [libsemigroups_pybind11][] -Now that you've successfully installed `libsemigroups_pybind11`, this page +Now that you've successfully installed [libsemigroups_pybind11][], this page explains a few ways to start using it; see [Installation](install.md). If you are familiar with [Jupyter](https://jupyter.org/) notebooks and/or are @@ -17,7 +17,7 @@ terminal application depends on your operating system: - on macOS, open **Terminal**; - on Linux, open the terminal application provided by your desktop environment. -If you installed `libsemigroups_pybind11` in a conda or mamba environment, first +If you installed [libsemigroups_pybind11][] in a conda or mamba environment, first activate that environment: ```console @@ -34,7 +34,7 @@ $ ipython ``` You should now see a prompt beginning with `In [1]:`. Import -`libsemigroups_pybind11` as follows: +[libsemigroups_pybind11][] as follows: ```python import libsemigroups_pybind11 @@ -69,7 +69,7 @@ A [Jupyter](https://jupyter.org/) notebook is often more convenient than a terminal session because you can keep code, output, notes, and plots together in one place. -If you installed `libsemigroups_pybind11` in a conda or mamba environment, +If you installed [libsemigroups_pybind11][] in a conda or mamba environment, activate it first: ```console @@ -90,7 +90,7 @@ import libsemigroups_pybind11 ``` Run the cell. If it finishes without an error, then the notebook can see your -installation of `libsemigroups_pybind11`. +installation of [libsemigroups_pybind11][]. You can use the same imports as in [IPython](https://ipython.org/): @@ -117,8 +117,8 @@ share a notebook with someone else. Examples include The details differ from service to service, but the general idea is: 1. Create a new Python notebook. -2. Install `libsemigroups_pybind11` in the notebook. -3. Import `libsemigroups_pybind11` in a later cell. +2. Install [libsemigroups_pybind11][] in the notebook. +3. Import [libsemigroups_pybind11][] in a later cell. For services that allow `pip` installs from a notebook cell, the first cell can usually be: @@ -158,4 +158,4 @@ import libsemigroups_pybind11 dir(libsemigroups_pybind11) ``` -This should print a list of names provided by `libsemigroups_pybind11`. +This should print a list of names provided by [libsemigroups_pybind11][]. diff --git a/docs/py/fp.md b/docs/py/fp.md index dbcec8e..54a3bd5 100644 --- a/docs/py/fp.md +++ b/docs/py/fp.md @@ -390,5 +390,3 @@ This says that the symmetric inverse monoid has 18 right congruences with up to 5 classes, but the monoid defined by the presentation with the first relation removed `p.rules[2:]` has 26 such congruences. So, these monoids are not isomorphic, and so the first relation in `p` is not redundant. - -[libsemigroups_pybind11]: https://libsemigroups.github.io/libsemigroups_pybind11/index.html diff --git a/docs/py/generated.md b/docs/py/generated.md index e756b2c..3eff18b 100644 --- a/docs/py/generated.md +++ b/docs/py/generated.md @@ -500,5 +500,3 @@ finitely generated semigroups. On the next page, you will learn about computing with finitely presented semigroups. - -[libsemigroups_pybind11]: https://libsemigroups.github.io/libsemigroups_pybind11/ diff --git a/docs/py/index.md b/docs/py/index.md index 38c1e4e..1d5ec8c 100644 --- a/docs/py/index.md +++ b/docs/py/index.md @@ -1,25 +1,25 @@ # Semigroups in Python This part of the tutorial relates to using the functionality of -[libsemigroups][] in Python via the Python package -[libsemigroups_pybind11][]. +[libsemigroups][] in Python via the Python package [libsemigroups_pybind11][]. -Unlike the [Semigroups package for GAP][], [libsemigroups_pybind11][] provides -only direct low-level access to the functionality of [libsemigroups][]. This -means that [libsemigroups_pybind11][] does not provide any higher-level +Unlike the [Semigroups package for GAP][Semigroups], [libsemigroups_pybind11][] +provides only direct low-level access to the functionality of [libsemigroups][]. +This means that [libsemigroups_pybind11][] does not provide any higher-level constructs (like semigroups, homomorphisms, and so on), but gives fine-grained -control of the specific algorithms from [libsemigroups][]. For an overview of the -algorithms available in [libsemigroups][] and/or [libsemigroups_pybind11][], see the -[libsemigroups documentation][] or the -[libsemigroups_pybind11 documentation][]. +control of the specific algorithms from [libsemigroups][]. For an overview of +the algorithms available in [libsemigroups][] and/or [libsemigroups_pybind11][], +see the +[`libsemigroups` documentation](https://libsemigroups.github.io/libsemigroups/) +or the [libsemigroups_pybind11 documentation][]. -The higher-level constructs in the Semigroups package for [GAP][] make some choices -for you about the best way to compute a particular thing. For example, if you -ask [GAP][] for the size of a finitely presented semigroup, it will run in parallel -some variants of Todd-Coxeter, Knuth-Bendix, and will try to compute the small -overlap class of the presentation. This works well in many cases, but for some -hard cases may not be enough, which is where the fine-grained control offered -by [libsemigroups_pybind11][] comes into play. +The higher-level constructs in the Semigroups package for [GAP][] make some +choices for you about the best way to compute a particular thing. For example, +if you ask [GAP][] for the size of a finitely presented semigroup, it will run +in parallel some variants of Todd-Coxeter, Knuth-Bendix, and will try to compute +the small overlap class of the presentation. This works well in many cases, but +for some hard cases may not be enough, which is where the fine-grained control +offered by [libsemigroups_pybind11][] comes into play. This part of this tutorial consists of: @@ -28,10 +28,3 @@ This part of this tutorial consists of: - [Semigroups and monoids defined by generators](generated.md) - [Finitely presented semigroups and monoids](fp.md) - [Worksheet](worksheet.md) - -[libsemigroups]: https://libsemigroups.github.io/libsemigroups/ -[libsemigroups_pybind11]: https://libsemigroups.github.io/libsemigroups_pybind11/ -[Semigroups package for GAP]: https://semigroups.github.io/Semigroups/ -[GAP]: https://www.gap-system.org/ -[libsemigroups documentation]: https://libsemigroups.github.io/libsemigroups/ -[libsemigroups_pybind11 documentation]: https://libsemigroups.github.io/libsemigroups_pybind11/ diff --git a/docs/py/install.md b/docs/py/install.md index 8c720b9..530d1f6 100644 --- a/docs/py/install.md +++ b/docs/py/install.md @@ -1,11 +1,16 @@ # Installation in Python -`libsemigroups_pybind11` is compatible with Python versions TODO +The [libsemigroups_pybind11][] Python package can be installed using `pip` or +any package manager that is able to access conda-forge (such as +[conda](https://docs.conda.io/projects/conda/en/stable/), +[Mamba](https://mamba.readthedocs.io/en/latest/#) or +[Pixi](https://pixi.prefix.dev/latest/)). The package can be installed for +Python 3.9+. ## Installing with pip -It's possible to install `libsemigroups_pybind11` using `pip` by doing -one of the following (depending on your system and setup): +It's possible to install [libsemigroups_pybind11][] using `pip` by doing one of +the following (depending on your system and setup): ```console $ pip install libsemigroups_pybind11 @@ -21,13 +26,13 @@ $ python -m pip install libsemigroups_pybind11 ## Installing with mamba -This installation method assumes that you have anaconda or miniconda -installed. See the [getting started](http://bit.ly/33B0Vfs) and +This installation method assumes that you have anaconda or miniconda installed. +See the [getting started](http://bit.ly/33B0Vfs) and [miniconda download page](https://conda.io/miniconda.html) on the [conda](https://conda.io/) website. -It might be a good idea to create and activate a conda/mamba environment -to contain the installation of the `libsemigroups_pybind11`: +It might be a good idea to create and activate a conda/mamba environment to +contain the installation of the [libsemigroups_pybind11][]: ```console $ mamba create --name myenv_name libsemigroups_pybind11 @@ -59,21 +64,21 @@ Save the file and then in the terminal: mamba env create -f environment.yml --name libsemigroups_pybind11 ``` -Every time you want to use `libsemigroups_pybind11` make sure type the -following into the terminal: +Every time you want to use [libsemigroups_pybind11][] make sure type the following +into the terminal: ```bash mamba activate libsemigroups_pybind11 ``` -## Adding libsemigroups as a dependency to your project using uv +## Adding `libsemigroups` as a dependency to your project using uv -If you manage a Python project with [uv](https://docs.astral.sh/uv), you -can add libsemigroups as a dependency as follows: +If you manage a Python project with [uv](https://docs.astral.sh/uv), you can add +`libsemigroups` as a dependency as follows: ```console $ uv add libsemigroups_pybind11 ``` -For more information, see uv's guide on [managing -dependencies](https://docs.astral.sh/uv/concepts/projects/dependencies/). +For more information, see uv's guide on +[managing dependencies](https://docs.astral.sh/uv/concepts/projects/dependencies/). diff --git a/docs/py/worksheet.md b/docs/py/worksheet.md index 2d7c5ea..35ef726 100644 --- a/docs/py/worksheet.md +++ b/docs/py/worksheet.md @@ -1,6 +1,6 @@ # Worksheet -This page contains some exercises for computing some things using `libsemigroups_pybind11`. +This page contains some exercises for computing some things using [libsemigroups_pybind11][]. 2. Recall that the commuting graph of a semigroup $S$ is the graph with nodes @@ -11,7 +11,7 @@ This page contains some exercises for computing some things using `libsemigroups ??? hint You can define $\mathcal{T}_n$ as a finitely presented monoid in - `libsemigroups_pybind11` by doing: + [libsemigroups_pybind11][] by doing: ```python from libsemigroups_pybind11.presentation import examples @@ -96,7 +96,7 @@ This page contains some exercises for computing some things using `libsemigroups ??? hint - You can define $J_{14}$ as a finitely presented monoid in `libsemigroups_pybind11` by doing: + You can define $J_{14}$ as a finitely presented monoid in [libsemigroups_pybind11][] by doing: ```python from libsemigroups_pybind11.presentation import examples diff --git a/docs/stylesheets/extra.css b/docs/stylesheets/extra.css index 8f73613..dea2783 100644 --- a/docs/stylesheets/extra.css +++ b/docs/stylesheets/extra.css @@ -34,12 +34,20 @@ background-color: #b8e186; } +.correct-index-alt { + background: linear-gradient(135deg, #b8e186 50%, #ffed4b 50%); +} + +.some-correct-index { + background: linear-gradient(135deg, #b8e186 50%, #f1b6da 50%); +} + .incorrect-size { background-color: #f1b6da; } -.could-not-verify { - background-color: #cccccc; +.work-in-progress { + background-color: #dddddd; } .no-presentation { diff --git a/includes/links.md b/includes/links.md new file mode 100644 index 0000000..b9841b6 --- /dev/null +++ b/includes/links.md @@ -0,0 +1,6 @@ +[GAP]: https://www.gap-system.org/ +[libsemigroups]: https://libsemigroups.github.io/libsemigroups/ +[libsemigroups_pybind11]: https://libsemigroups.github.io/libsemigroups_pybind11/index.html +[libsemigroups_pybind11 documentation]: https://libsemigroups.github.io/libsemigroups_pybind11/ +[Python]: https://www.python.org/ +[Semigroups]: https://semigroups.github.io/Semigroups/ diff --git a/mkdocs.yml b/mkdocs.yml index 18b827c..1e68a0e 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -1,55 +1,8 @@ site_name: Semigroups - Zero to ε +site_url: https://libsemigroups.github.io/tutorial/ -nav: - - Home: index.md - - Semigroups in GAP: - - gap/index.md - - "GAP: Installation": gap/install.md - - "GAP: First steps": gap/first-steps.md - - "GAP: Semigroups defined by generators": gap/generated.md - - "GAP: Finitely presented semigroups": gap/fp.md - - "GAP: Worksheet": gap/worksheet.md - - Semigroups in Python: - - py/index.md - - "Python: Installation": py/install.md - - "Python: First steps": py/first-steps.md - - "Python: Semigroups defined by generators": py/generated.md - - "Python: Finitely presented semigroups": py/fp.md - - "Python: Worksheet": py/worksheet.md - - Examples: - - examples/index.md - - examples/short.md - - "Verifying presentations in the ATLAS of Finite Groups": - - examples/atlas/index.md - - "Mathieu groups": - - examples/atlas/mathieu/m11.md - - examples/atlas/mathieu/m12.md - - examples/atlas/mathieu/m22.md - - "Mathieu group M23": - - examples/atlas/mathieu/m23/index.md - - "Mathieu group M24": - - examples/atlas/mathieu/m24/index.md - - "Leech lattice groups": - - examples/atlas/leech-lattice/hs.md - - examples/atlas/leech-lattice/j2.md - - "Conway group Co2": - - examples/atlas/leech-lattice/co2/index.md - - "McLaughlin group McL": - - examples/atlas/leech-lattice/mcl/index.md - - "Monster sections": - - "Held group He": - - examples/atlas/monster-sections/he/index.md - - "Fischer group Fi22": - - examples/atlas/monster-sections/fi22/index.md - - "Pariahs": - - examples/atlas/pariahs/j1.md - - examples/atlas/pariahs/j3.md - - "Rudvalis group Ru": - - examples/atlas/pariahs/ru/index.md - - "Janko group J4": - - examples/atlas/pariahs/j4/index.md - - "Miscellaneous": - - examples/atlas/misc/t.md +watch: + - includes theme: name: material @@ -72,6 +25,8 @@ theme: - navigation.top - navigation.indexes - navigation.footer + - navigation.instants + - navigation.prune - toc.integrate - content.code.copy - content.code.select @@ -82,6 +37,7 @@ plugins: # - optimize # - privacy - search + - awesome-nav markdown_extensions: - attr_list @@ -96,7 +52,10 @@ markdown_extensions: use_pygments: true pygments_lang_class: true - pymdownx.inlinehilite - - pymdownx.snippets + - pymdownx.snippets: + auto_append: + - includes/links.md + check_paths: true - pymdownx.superfences - pymdownx.tabbed: alternate_style: true @@ -108,7 +67,7 @@ markdown_extensions: emoji_generator: !!python/name:material.extensions.emoji.to_svg - pymdownx.tilde -copyright: Copyright © 2026 Reinis Cirpons, Joe Edwards and James Mitchell +copyright: Copyright © 2026 Reinis Cirpons, Joseph Edwards and James Mitchell extra: social: