# Question 23165: Complete Left Cosets Are Equal or Disjoint and Cover the Group [Return to catalogue page 116](https://ocme.evemisslab.com/en/questions/page/116/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations/#q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0040) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0045-23165 · q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0040 ## Original question and givens Use the complete finite Cayley table=[[12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], identity=2 and subset=[0, 2, 4, 6, 8, 10, 12]. Row g/column h gives g*h. First certify every group axiom, then audit the entire candidate for identity, product and inverse closure; cardinality alone is not subgroup qualification. Give (representative_coset_rows, equal_sizes, flattened_elements, partition_certificate, certificate). - G001 — Use the complete finite Cayley table=[[12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], identity=2 and subset=[0, 2, 4, 6, 8, 10, 12]. Row g/column h gives g*h. First certify every group axiom, then audit the entire candidate for identity, product and inverse closure; cardinality alone is not subgroup qualification. Give (representative_coset_rows, equal_sizes, flattened_elements, partition_certificate, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-lagrange-theorem-finite-groups](https://ocme.evemisslab.com/en/mko/mko-lagrange-theorem-finite-groups/) ## Step-by-step derivation S001 · impliesBefore source operation=lagrange-left-coset-partition;original snapshot and mathematics retained After explicit ordered source output fields=[representative_coset_rows, equal_sizes, flattened_elements, partition_certificate, certificate] Conditions: none Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL closed Cayley table=[[12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]];ACTUAL declared group identity=2 After ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]];two-sided group identity=Yes Conditions: none Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points. Step identifiers and dependenciesRule: groupaction_group_identity_both_sides_not_action_identity · Uses: G001, S001 S003 · impliesBefore ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S002 S004 · impliesBefore ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S003 S005 · impliesBefore ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S004 S006 · impliesBefore ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S005 S007 · impliesBefore ORIGINAL group pair(g,h)=[0, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S006 S008 · impliesBefore ORIGINAL group pair(g,h)=[0, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S007 S009 · impliesBefore ORIGINAL group pair(g,h)=[0, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S008 S010 · impliesBefore ORIGINAL group pair(g,h)=[0, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S009 S011 · impliesBefore ORIGINAL group pair(g,h)=[0, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S010 S012 · impliesBefore ORIGINAL group pair(g,h)=[0, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S011 S013 · impliesBefore ORIGINAL group pair(g,h)=[0, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S012 S014 · impliesBefore ORIGINAL group pair(g,h)=[0, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S013 S015 · impliesBefore ORIGINAL group pair(g,h)=[0, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S014 S016 · impliesBefore ORIGINAL group pair(g,h)=[0, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S015 S017 · impliesBefore ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S016 S018 · impliesBefore ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S017 S019 · impliesBefore ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S018 S020 · impliesBefore ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S019 S021 · impliesBefore ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S020 S022 · impliesBefore ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S021 S023 · impliesBefore ORIGINAL group pair(g,h)=[1, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S022 S024 · impliesBefore ORIGINAL group pair(g,h)=[1, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S023 S025 · impliesBefore ORIGINAL group pair(g,h)=[1, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S024 S026 · impliesBefore ORIGINAL group pair(g,h)=[1, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S025 S027 · impliesBefore ORIGINAL group pair(g,h)=[1, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S026 S028 · impliesBefore ORIGINAL group pair(g,h)=[1, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S027 S029 · impliesBefore ORIGINAL group pair(g,h)=[1, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S028 S030 · impliesBefore ORIGINAL group pair(g,h)=[1, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S029 S031 · impliesBefore ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S030 S032 · impliesBefore ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S031 S033 · impliesBefore ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S032 S034 · impliesBefore ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S033 S035 · impliesBefore ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S034 S036 · impliesBefore ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S035 S037 · impliesBefore ORIGINAL group pair(g,h)=[2, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S036 S038 · impliesBefore ORIGINAL group pair(g,h)=[2, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S037 S039 · impliesBefore ORIGINAL group pair(g,h)=[2, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S038 S040 · impliesBefore ORIGINAL group pair(g,h)=[2, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S039 S041 · impliesBefore ORIGINAL group pair(g,h)=[2, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S040 S042 · impliesBefore ORIGINAL group pair(g,h)=[2, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S041 S043 · impliesBefore ORIGINAL group pair(g,h)=[2, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S042 S044 · impliesBefore ORIGINAL group pair(g,h)=[2, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S043 S045 · impliesBefore ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S044 S046 · impliesBefore ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S045 S047 · impliesBefore ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S046 S048 · impliesBefore ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S047 S049 · impliesBefore ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S048 S050 · impliesBefore ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S049 S051 · impliesBefore ORIGINAL group pair(g,h)=[3, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S050 S052 · impliesBefore ORIGINAL group pair(g,h)=[3, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S051 S053 · impliesBefore ORIGINAL group pair(g,h)=[3, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S052 S054 · impliesBefore ORIGINAL group pair(g,h)=[3, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S053 S055 · impliesBefore ORIGINAL group pair(g,h)=[3, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S054 S056 · impliesBefore ORIGINAL group pair(g,h)=[3, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S055 S057 · impliesBefore ORIGINAL group pair(g,h)=[3, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S056 S058 · impliesBefore ORIGINAL group pair(g,h)=[3, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S057 S059 · impliesBefore ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S058 S060 · impliesBefore ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S059 S061 · impliesBefore ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S060 S062 · impliesBefore ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S061 S063 · impliesBefore ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S062 S064 · impliesBefore ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S063 S065 · impliesBefore ORIGINAL group pair(g,h)=[4, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S064 S066 · impliesBefore ORIGINAL group pair(g,h)=[4, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S065 S067 · impliesBefore ORIGINAL group pair(g,h)=[4, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S066 S068 · impliesBefore ORIGINAL group pair(g,h)=[4, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S067 S069 · impliesBefore ORIGINAL group pair(g,h)=[4, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S068 S070 · impliesBefore ORIGINAL group pair(g,h)=[4, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S069 S071 · impliesBefore ORIGINAL group pair(g,h)=[4, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S070 S072 · impliesBefore ORIGINAL group pair(g,h)=[4, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S071 S073 · impliesBefore ORIGINAL group pair(g,h)=[5, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S072 S074 · impliesBefore ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S073 S075 · impliesBefore ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S074 S076 · impliesBefore ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S075 S077 · impliesBefore ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S076 S078 · impliesBefore ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S077 S079 · impliesBefore ORIGINAL group pair(g,h)=[5, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S078 S080 · impliesBefore ORIGINAL group pair(g,h)=[5, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S079 S081 · impliesBefore ORIGINAL group pair(g,h)=[5, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S080 S082 · impliesBefore ORIGINAL group pair(g,h)=[5, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S081 S083 · impliesBefore ORIGINAL group pair(g,h)=[5, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S082 S084 · impliesBefore ORIGINAL group pair(g,h)=[5, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S083 S085 · impliesBefore ORIGINAL group pair(g,h)=[5, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S084 S086 · impliesBefore ORIGINAL group pair(g,h)=[5, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S085 S087 · impliesBefore ORIGINAL group pair(g,h)=[6, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S086 S088 · impliesBefore ORIGINAL group pair(g,h)=[6, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S087 S089 · impliesBefore ORIGINAL group pair(g,h)=[6, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S088 S090 · impliesBefore ORIGINAL group pair(g,h)=[6, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S089 S091 · impliesBefore ORIGINAL group pair(g,h)=[6, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S090 S092 · impliesBefore ORIGINAL group pair(g,h)=[6, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S091 S093 · impliesBefore ORIGINAL group pair(g,h)=[6, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S092 S094 · impliesBefore ORIGINAL group pair(g,h)=[6, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S093 S095 · impliesBefore ORIGINAL group pair(g,h)=[6, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S094 S096 · impliesBefore ORIGINAL group pair(g,h)=[6, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S095 S097 · impliesBefore ORIGINAL group pair(g,h)=[6, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S096 S098 · impliesBefore ORIGINAL group pair(g,h)=[6, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S097 S099 · impliesBefore ORIGINAL group pair(g,h)=[6, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S098 S100 · impliesBefore ORIGINAL group pair(g,h)=[6, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S099 S101 · impliesBefore ORIGINAL group pair(g,h)=[7, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S100 S102 · impliesBefore ORIGINAL group pair(g,h)=[7, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S101 S103 · impliesBefore ORIGINAL group pair(g,h)=[7, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S102 S104 · impliesBefore ORIGINAL group pair(g,h)=[7, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S103 S105 · impliesBefore ORIGINAL group pair(g,h)=[7, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S104 S106 · impliesBefore ORIGINAL group pair(g,h)=[7, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S105 S107 · impliesBefore ORIGINAL group pair(g,h)=[7, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S106 S108 · impliesBefore ORIGINAL group pair(g,h)=[7, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S107 S109 · impliesBefore ORIGINAL group pair(g,h)=[7, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S108 S110 · impliesBefore ORIGINAL group pair(g,h)=[7, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S109 S111 · impliesBefore ORIGINAL group pair(g,h)=[7, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S110 S112 · impliesBefore ORIGINAL group pair(g,h)=[7, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S111 S113 · impliesBefore ORIGINAL group pair(g,h)=[7, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S112 S114 · impliesBefore ORIGINAL group pair(g,h)=[7, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S113 S115 · impliesBefore ORIGINAL group pair(g,h)=[8, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S114 S116 · impliesBefore ORIGINAL group pair(g,h)=[8, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S115 S117 · impliesBefore ORIGINAL group pair(g,h)=[8, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S116 S118 · impliesBefore ORIGINAL group pair(g,h)=[8, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S117 S119 · impliesBefore ORIGINAL group pair(g,h)=[8, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S118 S120 · impliesBefore ORIGINAL group pair(g,h)=[8, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S119 S121 · impliesBefore ORIGINAL group pair(g,h)=[8, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S120 S122 · impliesBefore ORIGINAL group pair(g,h)=[8, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S121 S123 · impliesBefore ORIGINAL group pair(g,h)=[8, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S122 S124 · impliesBefore ORIGINAL group pair(g,h)=[8, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S123 S125 · impliesBefore ORIGINAL group pair(g,h)=[8, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S124 S126 · impliesBefore ORIGINAL group pair(g,h)=[8, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S125 S127 · impliesBefore ORIGINAL group pair(g,h)=[8, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S126 S128 · impliesBefore ORIGINAL group pair(g,h)=[8, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S127 S129 · impliesBefore ORIGINAL group pair(g,h)=[9, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S128 S130 · impliesBefore ORIGINAL group pair(g,h)=[9, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S129 S131 · impliesBefore ORIGINAL group pair(g,h)=[9, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S130 S132 · impliesBefore ORIGINAL group pair(g,h)=[9, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S131 S133 · impliesBefore ORIGINAL group pair(g,h)=[9, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S132 S134 · impliesBefore ORIGINAL group pair(g,h)=[9, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S133 S135 · impliesBefore ORIGINAL group pair(g,h)=[9, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S134 S136 · impliesBefore ORIGINAL group pair(g,h)=[9, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S135 S137 · impliesBefore ORIGINAL group pair(g,h)=[9, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S136 S138 · impliesBefore ORIGINAL group pair(g,h)=[9, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S137 S139 · impliesBefore ORIGINAL group pair(g,h)=[9, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S138 S140 · impliesBefore ORIGINAL group pair(g,h)=[9, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S139 S141 · impliesBefore ORIGINAL group pair(g,h)=[9, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S140 S142 · impliesBefore ORIGINAL group pair(g,h)=[9, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S141 S143 · impliesBefore ORIGINAL group pair(g,h)=[10, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S142 S144 · impliesBefore ORIGINAL group pair(g,h)=[10, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S143 S145 · impliesBefore ORIGINAL group pair(g,h)=[10, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S144 S146 · impliesBefore ORIGINAL group pair(g,h)=[10, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S145 S147 · impliesBefore ORIGINAL group pair(g,h)=[10, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S146 S148 · impliesBefore ORIGINAL group pair(g,h)=[10, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S147 S149 · impliesBefore ORIGINAL group pair(g,h)=[10, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S148 S150 · impliesBefore ORIGINAL group pair(g,h)=[10, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S149 S151 · impliesBefore ORIGINAL group pair(g,h)=[10, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S150 S152 · impliesBefore ORIGINAL group pair(g,h)=[10, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S151 S153 · impliesBefore ORIGINAL group pair(g,h)=[10, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S152 S154 · impliesBefore ORIGINAL group pair(g,h)=[10, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S153 S155 · impliesBefore ORIGINAL group pair(g,h)=[10, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S154 S156 · impliesBefore ORIGINAL group pair(g,h)=[10, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S155 S157 · impliesBefore ORIGINAL group pair(g,h)=[11, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S156 S158 · impliesBefore ORIGINAL group pair(g,h)=[11, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S157 S159 · impliesBefore ORIGINAL group pair(g,h)=[11, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S158 S160 · impliesBefore ORIGINAL group pair(g,h)=[11, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S159 S161 · impliesBefore ORIGINAL group pair(g,h)=[11, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S160 S162 · impliesBefore ORIGINAL group pair(g,h)=[11, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S161 S163 · impliesBefore ORIGINAL group pair(g,h)=[11, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S162 S164 · impliesBefore ORIGINAL group pair(g,h)=[11, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S163 S165 · impliesBefore ORIGINAL group pair(g,h)=[11, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S164 S166 · impliesBefore ORIGINAL group pair(g,h)=[11, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S165 S167 · impliesBefore ORIGINAL group pair(g,h)=[11, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S166 S168 · impliesBefore ORIGINAL group pair(g,h)=[11, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S167 S169 · impliesBefore ORIGINAL group pair(g,h)=[11, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S168 S170 · impliesBefore ORIGINAL group pair(g,h)=[11, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S169 S171 · impliesBefore ORIGINAL group pair(g,h)=[12, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S170 S172 · impliesBefore ORIGINAL group pair(g,h)=[12, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S171 S173 · impliesBefore ORIGINAL group pair(g,h)=[12, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S172 S174 · impliesBefore ORIGINAL group pair(g,h)=[12, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S173 S175 · impliesBefore ORIGINAL group pair(g,h)=[12, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S174 S176 · impliesBefore ORIGINAL group pair(g,h)=[12, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S175 S177 · impliesBefore ORIGINAL group pair(g,h)=[12, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S176 S178 · impliesBefore ORIGINAL group pair(g,h)=[12, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S177 S179 · impliesBefore ORIGINAL group pair(g,h)=[12, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S178 S180 · impliesBefore ORIGINAL group pair(g,h)=[12, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S179 S181 · impliesBefore ORIGINAL group pair(g,h)=[12, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S180 S182 · impliesBefore ORIGINAL group pair(g,h)=[12, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S181 S183 · impliesBefore ORIGINAL group pair(g,h)=[12, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S182 S184 · impliesBefore ORIGINAL group pair(g,h)=[12, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S183 S185 · impliesBefore ORIGINAL group pair(g,h)=[13, 0];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 12, 12, Yes], [4, 13, 13, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes], [12, 7, 7, Yes], [13, 8, 8, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S184 S186 · impliesBefore ORIGINAL group pair(g,h)=[13, 1];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 12, 12, Yes], [3, 13, 13, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes], [12, 8, 8, Yes], [13, 9, 9, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S185 S187 · impliesBefore ORIGINAL group pair(g,h)=[13, 2];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 12, 12, Yes], [2, 13, 13, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes], [12, 9, 9, Yes], [13, 10, 10, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S186 S188 · impliesBefore ORIGINAL group pair(g,h)=[13, 3];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 12, 12, Yes], [1, 13, 13, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes], [12, 10, 10, Yes], [13, 11, 11, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S187 S189 · impliesBefore ORIGINAL group pair(g,h)=[13, 4];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 13, 13, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes], [12, 11, 11, Yes], [13, 12, 12, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S188 S190 · impliesBefore ORIGINAL group pair(g,h)=[13, 5];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S189 S191 · impliesBefore ORIGINAL group pair(g,h)=[13, 6];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 12, 12, Yes], [12, 13, 13, Yes], [13, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S190 S192 · impliesBefore ORIGINAL group pair(g,h)=[13, 7];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 12, 12, Yes], [11, 13, 13, Yes], [12, 0, 0, Yes], [13, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S191 S193 · impliesBefore ORIGINAL group pair(g,h)=[13, 8];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 12, 12, Yes], [10, 13, 13, Yes], [11, 0, 0, Yes], [12, 1, 1, Yes], [13, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S192 S194 · impliesBefore ORIGINAL group pair(g,h)=[13, 9];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 12, 12, Yes], [9, 13, 13, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes], [12, 2, 2, Yes], [13, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S193 S195 · impliesBefore ORIGINAL group pair(g,h)=[13, 10];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 12, 12, Yes], [8, 13, 13, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes], [12, 3, 3, Yes], [13, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S194 S196 · impliesBefore ORIGINAL group pair(g,h)=[13, 11];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 12, 12, Yes], [7, 13, 13, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes], [12, 4, 4, Yes], [13, 5, 5, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S195 S197 · impliesBefore ORIGINAL group pair(g,h)=[13, 12];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 12, 12, Yes], [6, 13, 13, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes], [12, 5, 5, Yes], [13, 6, 6, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S196 S198 · impliesBefore ORIGINAL group pair(g,h)=[13, 13];check ALL k in Fin(14) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 12, 12, Yes], [5, 13, 13, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes], [12, 6, 6, Yes], [13, 7, 7, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S197 S199 · impliesBefore ORIGINAL group element g=0;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 12, 12, No], [1, 13, 13, No], [2, 0, 0, No], [3, 1, 1, No], [4, 2, 2, Yes], [5, 3, 3, No], [6, 4, 4, No], [7, 5, 5, No], [8, 6, 6, No], [9, 7, 7, No], [10, 8, 8, No], [11, 9, 9, No], [12, 10, 10, No], [13, 11, 11, No]] After first actual two-sided inverse witness=4;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S198 S200 · impliesBefore ORIGINAL group element g=1;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 13, 13, No], [1, 0, 0, No], [2, 1, 1, No], [3, 2, 2, Yes], [4, 3, 3, No], [5, 4, 4, No], [6, 5, 5, No], [7, 6, 6, No], [8, 7, 7, No], [9, 8, 8, No], [10, 9, 9, No], [11, 10, 10, No], [12, 11, 11, No], [13, 12, 12, No]] After first actual two-sided inverse witness=3;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S199 S201 · impliesBefore ORIGINAL group element g=2;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, No], [1, 1, 1, No], [2, 2, 2, Yes], [3, 3, 3, No], [4, 4, 4, No], [5, 5, 5, No], [6, 6, 6, No], [7, 7, 7, No], [8, 8, 8, No], [9, 9, 9, No], [10, 10, 10, No], [11, 11, 11, No], [12, 12, 12, No], [13, 13, 13, No]] After first actual two-sided inverse witness=2;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S200 S202 · impliesBefore ORIGINAL group element g=3;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, No], [1, 2, 2, Yes], [2, 3, 3, No], [3, 4, 4, No], [4, 5, 5, No], [5, 6, 6, No], [6, 7, 7, No], [7, 8, 8, No], [8, 9, 9, No], [9, 10, 10, No], [10, 11, 11, No], [11, 12, 12, No], [12, 13, 13, No], [13, 0, 0, No]] After first actual two-sided inverse witness=1;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S201 S203 · impliesBefore ORIGINAL group element g=4;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, Yes], [1, 3, 3, No], [2, 4, 4, No], [3, 5, 5, No], [4, 6, 6, No], [5, 7, 7, No], [6, 8, 8, No], [7, 9, 9, No], [8, 10, 10, No], [9, 11, 11, No], [10, 12, 12, No], [11, 13, 13, No], [12, 0, 0, No], [13, 1, 1, No]] After first actual two-sided inverse witness=0;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S202 S204 · impliesBefore ORIGINAL group element g=5;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, No], [1, 4, 4, No], [2, 5, 5, No], [3, 6, 6, No], [4, 7, 7, No], [5, 8, 8, No], [6, 9, 9, No], [7, 10, 10, No], [8, 11, 11, No], [9, 12, 12, No], [10, 13, 13, No], [11, 0, 0, No], [12, 1, 1, No], [13, 2, 2, Yes]] After first actual two-sided inverse witness=13;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S203 S205 · impliesBefore ORIGINAL group element g=6;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 4, 4, No], [1, 5, 5, No], [2, 6, 6, No], [3, 7, 7, No], [4, 8, 8, No], [5, 9, 9, No], [6, 10, 10, No], [7, 11, 11, No], [8, 12, 12, No], [9, 13, 13, No], [10, 0, 0, No], [11, 1, 1, No], [12, 2, 2, Yes], [13, 3, 3, No]] After first actual two-sided inverse witness=12;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S204 S206 · impliesBefore ORIGINAL group element g=7;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 5, 5, No], [1, 6, 6, No], [2, 7, 7, No], [3, 8, 8, No], [4, 9, 9, No], [5, 10, 10, No], [6, 11, 11, No], [7, 12, 12, No], [8, 13, 13, No], [9, 0, 0, No], [10, 1, 1, No], [11, 2, 2, Yes], [12, 3, 3, No], [13, 4, 4, No]] After first actual two-sided inverse witness=11;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S205 S207 · impliesBefore ORIGINAL group element g=8;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 6, 6, No], [1, 7, 7, No], [2, 8, 8, No], [3, 9, 9, No], [4, 10, 10, No], [5, 11, 11, No], [6, 12, 12, No], [7, 13, 13, No], [8, 0, 0, No], [9, 1, 1, No], [10, 2, 2, Yes], [11, 3, 3, No], [12, 4, 4, No], [13, 5, 5, No]] After first actual two-sided inverse witness=10;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S206 S208 · impliesBefore ORIGINAL group element g=9;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 7, 7, No], [1, 8, 8, No], [2, 9, 9, No], [3, 10, 10, No], [4, 11, 11, No], [5, 12, 12, No], [6, 13, 13, No], [7, 0, 0, No], [8, 1, 1, No], [9, 2, 2, Yes], [10, 3, 3, No], [11, 4, 4, No], [12, 5, 5, No], [13, 6, 6, No]] After first actual two-sided inverse witness=9;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S207 S209 · impliesBefore ORIGINAL group element g=10;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 8, 8, No], [1, 9, 9, No], [2, 10, 10, No], [3, 11, 11, No], [4, 12, 12, No], [5, 13, 13, No], [6, 0, 0, No], [7, 1, 1, No], [8, 2, 2, Yes], [9, 3, 3, No], [10, 4, 4, No], [11, 5, 5, No], [12, 6, 6, No], [13, 7, 7, No]] After first actual two-sided inverse witness=8;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S208 S210 · impliesBefore ORIGINAL group element g=11;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 9, 9, No], [1, 10, 10, No], [2, 11, 11, No], [3, 12, 12, No], [4, 13, 13, No], [5, 0, 0, No], [6, 1, 1, No], [7, 2, 2, Yes], [8, 3, 3, No], [9, 4, 4, No], [10, 5, 5, No], [11, 6, 6, No], [12, 7, 7, No], [13, 8, 8, No]] After first actual two-sided inverse witness=7;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S209 S211 · impliesBefore ORIGINAL group element g=12;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 10, 10, No], [1, 11, 11, No], [2, 12, 12, No], [3, 13, 13, No], [4, 0, 0, No], [5, 1, 1, No], [6, 2, 2, Yes], [7, 3, 3, No], [8, 4, 4, No], [9, 5, 5, No], [10, 6, 6, No], [11, 7, 7, No], [12, 8, 8, No], [13, 9, 9, No]] After first actual two-sided inverse witness=6;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S210 S212 · impliesBefore ORIGINAL group element g=13;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 11, 11, No], [1, 12, 12, No], [2, 13, 13, No], [3, 0, 0, No], [4, 1, 1, No], [5, 2, 2, Yes], [6, 3, 3, No], [7, 4, 4, No], [8, 5, 5, No], [9, 6, 6, No], [10, 7, 7, No], [11, 8, 8, No], [12, 9, 9, No], [13, 10, 10, No]] After first actual two-sided inverse witness=5;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S211 S213 · impliesBefore structurally closed COMPLETE group candidate;two-sided identity=Yes;ALL-triple associativity=Yes;ALL-element inverses=Yes After group_valid=Yes is the CONJUNCTION;False remains a reported candidate result,not a silent repair Conditions: none Group qualification is the conjunction of these complete obligations.A permutation-looking action table cannot repair a bad group table;do not replace the identity or modify the table. Step identifiers and dependenciesRule: groupaction_group_axiom_conjunction_candidate_may_fail · Uses: G001, S212 S214 · impliesBefore CURRENT ORIGINAL complete Cayley table=[[12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]];actual identity=2;order=14 After ALL group axioms qualified;actual inverse list=[4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5];no point carrier or group-action assumption is used Conditions: none Certify closure,identity,associativity and two-sided inverses in the group table itself.Do not import an old8-element action-table limit or assume cyclicity/commutativity.The frozen pure helper receives only this question's table and trace. Step identifiers and dependenciesRule: group_CURRENT_complete_certified_Cayley_NOT_action_wrapper · Uses: G001, S213 S215 · impliesBefore ORIGINAL subset=[0, 2, 4, 6, 8, 10, 12];actual group identity=2 After contains_identity=Yes;ALL [g,h,product,in_candidate] rows=[[0, 0, 12, Yes], [0, 2, 0, Yes], [0, 4, 2, Yes], [0, 6, 4, Yes], [0, 8, 6, Yes], [0, 10, 8, Yes], [0, 12, 10, Yes], [2, 0, 0, Yes], [2, 2, 2, Yes], [2, 4, 4, Yes], [2, 6, 6, Yes], [2, 8, 8, Yes], [2, 10, 10, Yes], [2, 12, 12, Yes], [4, 0, 2, Yes], [4, 2, 4, Yes], [4, 4, 6, Yes], [4, 6, 8, Yes], [4, 8, 10, Yes], [4, 10, 12, Yes], [4, 12, 0, Yes], [6, 0, 4, Yes], [6, 2, 6, Yes], [6, 4, 8, Yes], [6, 6, 10, Yes], [6, 8, 12, Yes], [6, 10, 0, Yes], [6, 12, 2, Yes], [8, 0, 6, Yes], [8, 2, 8, Yes], [8, 4, 10, Yes], [8, 6, 12, Yes], [8, 8, 0, Yes], [8, 10, 2, Yes], [8, 12, 4, Yes], [10, 0, 8, Yes], [10, 2, 10, Yes], [10, 4, 12, Yes], [10, 6, 0, Yes], [10, 8, 2, Yes], [10, 10, 4, Yes], [10, 12, 6, Yes], [12, 0, 10, Yes], [12, 2, 12, Yes], [12, 4, 0, Yes], [12, 6, 2, Yes], [12, 8, 4, Yes], [12, 10, 6, Yes], [12, 12, 8, Yes]];ALL [g,inverse,in_candidate] rows=[[0, 4, Yes], [2, 2, Yes], [4, 0, Yes], [6, 12, Yes], [8, 10, Yes], [10, 8, Yes], [12, 6, Yes]];product_closed=Yes,inverse_closed=Yes,subgroup=Yes Conditions: none A complete candidate may fail.Identity,product and inverse closure must all hold;divisor-sized cardinality replaces none of them. Step identifiers and dependenciesRule: lagrange_candidate_ALL_subgroup_obligations_not_size · Uses: G001, S214 S216 · impliesBefore ORIGINAL group representative g=0;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=4 After ALL [h,g*h] rows=[[0, 12], [2, 0], [4, 2], [6, 4], [8, 6], [10, 8], [12, 10]];LEFT coset=[0, 2, 4, 6, 8, 10, 12];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 2], [2, 4], [4, 6], [6, 8], [8, 10], [10, 12], [12, 0]];least representative for this distinct set=0 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S215 S217 · impliesBefore ORIGINAL group representative g=1;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=3 After ALL [h,g*h] rows=[[0, 13], [2, 1], [4, 3], [6, 5], [8, 7], [10, 9], [12, 11]];LEFT coset=[1, 3, 5, 7, 9, 11, 13];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 2], [3, 4], [5, 6], [7, 8], [9, 10], [11, 12], [13, 0]];least representative for this distinct set=1 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S216 S218 · impliesBefore ORIGINAL group representative g=2;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=2 After ALL [h,g*h] rows=[[0, 0], [2, 2], [4, 4], [6, 6], [8, 8], [10, 10], [12, 12]];LEFT coset=[0, 2, 4, 6, 8, 10, 12];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 0], [2, 2], [4, 4], [6, 6], [8, 8], [10, 10], [12, 12]];least representative for this distinct set=0 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S217 S219 · impliesBefore ORIGINAL group representative g=3;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=1 After ALL [h,g*h] rows=[[0, 1], [2, 3], [4, 5], [6, 7], [8, 9], [10, 11], [12, 13]];LEFT coset=[1, 3, 5, 7, 9, 11, 13];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 0], [3, 2], [5, 4], [7, 6], [9, 8], [11, 10], [13, 12]];least representative for this distinct set=1 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S218 S220 · impliesBefore ORIGINAL group representative g=4;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=0 After ALL [h,g*h] rows=[[0, 2], [2, 4], [4, 6], [6, 8], [8, 10], [10, 12], [12, 0]];LEFT coset=[0, 2, 4, 6, 8, 10, 12];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 12], [2, 0], [4, 2], [6, 4], [8, 6], [10, 8], [12, 10]];least representative for this distinct set=0 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S219 S221 · impliesBefore ORIGINAL group representative g=5;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=13 After ALL [h,g*h] rows=[[0, 3], [2, 5], [4, 7], [6, 9], [8, 11], [10, 13], [12, 1]];LEFT coset=[1, 3, 5, 7, 9, 11, 13];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 12], [3, 0], [5, 2], [7, 4], [9, 6], [11, 8], [13, 10]];least representative for this distinct set=1 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S220 S222 · impliesBefore ORIGINAL group representative g=6;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=12 After ALL [h,g*h] rows=[[0, 4], [2, 6], [4, 8], [6, 10], [8, 12], [10, 0], [12, 2]];LEFT coset=[0, 2, 4, 6, 8, 10, 12];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 10], [2, 12], [4, 0], [6, 2], [8, 4], [10, 6], [12, 8]];least representative for this distinct set=0 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S221 S223 · impliesBefore ORIGINAL group representative g=7;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=11 After ALL [h,g*h] rows=[[0, 5], [2, 7], [4, 9], [6, 11], [8, 13], [10, 1], [12, 3]];LEFT coset=[1, 3, 5, 7, 9, 11, 13];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 10], [3, 12], [5, 0], [7, 2], [9, 4], [11, 6], [13, 8]];least representative for this distinct set=1 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S222 S224 · impliesBefore ORIGINAL group representative g=8;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=10 After ALL [h,g*h] rows=[[0, 6], [2, 8], [4, 10], [6, 12], [8, 0], [10, 2], [12, 4]];LEFT coset=[0, 2, 4, 6, 8, 10, 12];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 8], [2, 10], [4, 12], [6, 0], [8, 2], [10, 4], [12, 6]];least representative for this distinct set=0 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S223 S225 · impliesBefore ORIGINAL group representative g=9;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=9 After ALL [h,g*h] rows=[[0, 7], [2, 9], [4, 11], [6, 13], [8, 1], [10, 3], [12, 5]];LEFT coset=[1, 3, 5, 7, 9, 11, 13];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 8], [3, 10], [5, 12], [7, 0], [9, 2], [11, 4], [13, 6]];least representative for this distinct set=1 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S224 S226 · impliesBefore ORIGINAL group representative g=10;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=8 After ALL [h,g*h] rows=[[0, 8], [2, 10], [4, 12], [6, 0], [8, 2], [10, 4], [12, 6]];LEFT coset=[0, 2, 4, 6, 8, 10, 12];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 6], [2, 8], [4, 10], [6, 12], [8, 0], [10, 2], [12, 4]];least representative for this distinct set=0 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S225 S227 · impliesBefore ORIGINAL group representative g=11;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=7 After ALL [h,g*h] rows=[[0, 9], [2, 11], [4, 13], [6, 1], [8, 3], [10, 5], [12, 7]];LEFT coset=[1, 3, 5, 7, 9, 11, 13];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 6], [3, 8], [5, 10], [7, 12], [9, 0], [11, 2], [13, 4]];least representative for this distinct set=1 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S226 S228 · impliesBefore ORIGINAL group representative g=12;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=6 After ALL [h,g*h] rows=[[0, 10], [2, 12], [4, 0], [6, 2], [8, 4], [10, 6], [12, 8]];LEFT coset=[0, 2, 4, 6, 8, 10, 12];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 4], [2, 6], [4, 8], [6, 10], [8, 12], [10, 0], [12, 2]];least representative for this distinct set=0 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S227 S229 · impliesBefore ORIGINAL group representative g=13;SAME subgroup H=[0, 2, 4, 6, 8, 10, 12];inverse of g=5 After ALL [h,g*h] rows=[[0, 11], [2, 13], [4, 1], [6, 3], [8, 5], [10, 7], [12, 9]];LEFT coset=[1, 3, 5, 7, 9, 11, 13];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 4], [3, 6], [5, 8], [7, 10], [9, 12], [11, 0], [13, 2]];least representative for this distinct set=1 Conditions: none Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set. Step identifiers and dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S228 S230 · impliesBefore ALL distinct LEFT cosets sorted by SOURCE English element-list order=[[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]] After every size=7;flattened elements=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];partition covers ORIGINAL group exactly once=Yes Conditions: none If gh=kh1,then k_inverse*g=h1*h_inverse lies in H and the two cosets contain each other;otherwise they are disjoint.The identity puts each g in gH,so they cover G.Sort only by the coset element list,not the representative field. Step identifiers and dependenciesRule: lagrange_LEFT_cosets_equal_or_disjoint_cover_with_source_order · Uses: G001, S229 ## Reverse checks - S230 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S229 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S228 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S224 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S223 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S222 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S221 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S220 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S219 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S218 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S217 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S216 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S215 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S214 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S213 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S212 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S210 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S209 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S204 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S203 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S202 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S199 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S198 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S193 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S185 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S179 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S175 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[0, [0, 2, 4, 6, 8, 10, 12]], [1, [1, 3, 5, 7, 9, 11, 13]]], Yes, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], Yes, left_cosets_equal_or_disjoint_and_cover_G) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 15889c9fdf259a9e6f929136ae71bf95d0796c00346bbc178387957d7ce4891d - candidate record SHA-256: fe9ec095000514712c694eeaea033a553542ea27e0e5f38fd1f4c2369c759ae3 - semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438 - question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0040.json)