# Question 23464: An Order-p Witness Must Exclude Every Smaller Positive Return

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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.

R0045-23464 · q-20261014-advance-20261112-r01-cauchy-theorem-finite-groups-0064



## Original question and givens

Use complete finite Cayley table=[[10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=8. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, certificate).


- G001 — Use complete finite Cayley table=[[10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8]] with identity=2, row g/column h meaning g*h. Certify closure, associativity, identity and inverses. Element order is the least positive exponent returning to this same identity, not just one exponent that does so. Prime=2, supplied element=8. Give (group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, 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-cauchy-theorem-finite-groups](https://ocme.evemisslab.com/en/mko/mko-cauchy-theorem-finite-groups/)





## Step-by-step derivation

S001 · impliesBefore

source operation=cauchy-order-p-witness;original snapshot and mathematics retained

After

explicit ordered source output fields=[group_order, prime, element, positive_power_rows, p_power_is_identity, no_smaller_positive_identity_power, 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=[[10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8]];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]];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(12)

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, 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]]

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(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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(12)

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]]

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(12)

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, 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(12)

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, 0, 0, Yes], [11, 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(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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(12)

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, 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]]

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)=[1, 0];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[1, 1];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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, 2];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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, 3];check ALL k in Fin(12)

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]]

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, 4];check ALL k in Fin(12)

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, 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, S018





S020 · impliesBefore

ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S019





S021 · impliesBefore

ORIGINAL group pair(g,h)=[1, 6];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S020





S022 · impliesBefore

ORIGINAL group pair(g,h)=[1, 7];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S021





S023 · impliesBefore

ORIGINAL group pair(g,h)=[1, 8];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S022





S024 · impliesBefore

ORIGINAL group pair(g,h)=[1, 9];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S023





S025 · impliesBefore

ORIGINAL group pair(g,h)=[1, 10];check ALL k in Fin(12)

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, 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]]

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, 11];check ALL k in Fin(12)

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, 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]]

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)=[2, 0];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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)=[2, 1];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[2, 2];check ALL k in Fin(12)

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]]

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)=[2, 3];check ALL k in Fin(12)

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, 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, S029





S031 · impliesBefore

ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S030





S032 · impliesBefore

ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S031





S033 · impliesBefore

ORIGINAL group pair(g,h)=[2, 6];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S032





S034 · impliesBefore

ORIGINAL group pair(g,h)=[2, 7];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S033





S035 · impliesBefore

ORIGINAL group pair(g,h)=[2, 8];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S034





S036 · impliesBefore

ORIGINAL group pair(g,h)=[2, 9];check ALL k in Fin(12)

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, 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]]

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, 10];check ALL k in Fin(12)

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, 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]]

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, 11];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[3, 0];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[3, 1];check ALL k in Fin(12)

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]]

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)=[3, 2];check ALL k in Fin(12)

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, 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, S040





S042 · impliesBefore

ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S041





S043 · impliesBefore

ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S042





S044 · impliesBefore

ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S043





S045 · impliesBefore

ORIGINAL group pair(g,h)=[3, 6];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S044





S046 · impliesBefore

ORIGINAL group pair(g,h)=[3, 7];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S045





S047 · impliesBefore

ORIGINAL group pair(g,h)=[3, 8];check ALL k in Fin(12)

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, 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]]

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, 9];check ALL k in Fin(12)

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, 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]]

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, 10];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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, 11];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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)=[4, 0];check ALL k in Fin(12)

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]]

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)=[4, 1];check ALL k in Fin(12)

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, 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, S051





S053 · impliesBefore

ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S052





S054 · impliesBefore

ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S053





S055 · impliesBefore

ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S054





S056 · impliesBefore

ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S055





S057 · impliesBefore

ORIGINAL group pair(g,h)=[4, 6];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S056





S058 · impliesBefore

ORIGINAL group pair(g,h)=[4, 7];check ALL k in Fin(12)

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, 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]]

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, 8];check ALL k in Fin(12)

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, 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]]

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, 9];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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, 10];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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, 11];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[5, 0];check ALL k in Fin(12)

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, 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, S062





S064 · impliesBefore

ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S063





S065 · impliesBefore

ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S064





S066 · impliesBefore

ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S065





S067 · impliesBefore

ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S066





S068 · impliesBefore

ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S067





S069 · impliesBefore

ORIGINAL group pair(g,h)=[5, 6];check ALL k in Fin(12)

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, 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]]

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)=[5, 7];check ALL k in Fin(12)

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, 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]]

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)=[5, 8];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[5, 9];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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, 10];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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, 11];check ALL k in Fin(12)

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]]

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)=[6, 0];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S074





S076 · impliesBefore

ORIGINAL group pair(g,h)=[6, 1];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S075





S077 · impliesBefore

ORIGINAL group pair(g,h)=[6, 2];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S076





S078 · impliesBefore

ORIGINAL group pair(g,h)=[6, 3];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S077





S079 · impliesBefore

ORIGINAL group pair(g,h)=[6, 4];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S078





S080 · impliesBefore

ORIGINAL group pair(g,h)=[6, 5];check ALL k in Fin(12)

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, 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]]

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)=[6, 6];check ALL k in Fin(12)

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, 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]]

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)=[6, 7];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[6, 8];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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)=[6, 9];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[6, 10];check ALL k in Fin(12)

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]]

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)=[6, 11];check ALL k in Fin(12)

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, 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, S085





S087 · impliesBefore

ORIGINAL group pair(g,h)=[7, 0];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S086





S088 · impliesBefore

ORIGINAL group pair(g,h)=[7, 1];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S087





S089 · impliesBefore

ORIGINAL group pair(g,h)=[7, 2];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S088





S090 · impliesBefore

ORIGINAL group pair(g,h)=[7, 3];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S089





S091 · impliesBefore

ORIGINAL group pair(g,h)=[7, 4];check ALL k in Fin(12)

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, 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]]

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)=[7, 5];check ALL k in Fin(12)

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, 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]]

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)=[7, 6];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[7, 7];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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)=[7, 8];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[7, 9];check ALL k in Fin(12)

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]]

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)=[7, 10];check ALL k in Fin(12)

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, 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, S096





S098 · impliesBefore

ORIGINAL group pair(g,h)=[7, 11];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S097





S099 · impliesBefore

ORIGINAL group pair(g,h)=[8, 0];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S098





S100 · impliesBefore

ORIGINAL group pair(g,h)=[8, 1];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S099





S101 · impliesBefore

ORIGINAL group pair(g,h)=[8, 2];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S100





S102 · impliesBefore

ORIGINAL group pair(g,h)=[8, 3];check ALL k in Fin(12)

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, 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]]

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)=[8, 4];check ALL k in Fin(12)

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, 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]]

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)=[8, 5];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[8, 6];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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)=[8, 7];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[8, 8];check ALL k in Fin(12)

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]]

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)=[8, 9];check ALL k in Fin(12)

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, 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, S107





S109 · impliesBefore

ORIGINAL group pair(g,h)=[8, 10];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S108





S110 · impliesBefore

ORIGINAL group pair(g,h)=[8, 11];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S109





S111 · impliesBefore

ORIGINAL group pair(g,h)=[9, 0];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S110





S112 · impliesBefore

ORIGINAL group pair(g,h)=[9, 1];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S111





S113 · impliesBefore

ORIGINAL group pair(g,h)=[9, 2];check ALL k in Fin(12)

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, 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]]

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)=[9, 3];check ALL k in Fin(12)

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, 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]]

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)=[9, 4];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[9, 5];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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)=[9, 6];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[9, 7];check ALL k in Fin(12)

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]]

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)=[9, 8];check ALL k in Fin(12)

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, 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, S118





S120 · impliesBefore

ORIGINAL group pair(g,h)=[9, 9];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S119





S121 · impliesBefore

ORIGINAL group pair(g,h)=[9, 10];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S120





S122 · impliesBefore

ORIGINAL group pair(g,h)=[9, 11];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S121





S123 · impliesBefore

ORIGINAL group pair(g,h)=[10, 0];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S122





S124 · impliesBefore

ORIGINAL group pair(g,h)=[10, 1];check ALL k in Fin(12)

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, 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]]

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)=[10, 2];check ALL k in Fin(12)

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, 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]]

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)=[10, 3];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[10, 4];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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)=[10, 5];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[10, 6];check ALL k in Fin(12)

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]]

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)=[10, 7];check ALL k in Fin(12)

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, 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, S129





S131 · impliesBefore

ORIGINAL group pair(g,h)=[10, 8];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S130





S132 · impliesBefore

ORIGINAL group pair(g,h)=[10, 9];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S131





S133 · impliesBefore

ORIGINAL group pair(g,h)=[10, 10];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S132





S134 · impliesBefore

ORIGINAL group pair(g,h)=[10, 11];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S133





S135 · impliesBefore

ORIGINAL group pair(g,h)=[11, 0];check ALL k in Fin(12)

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, 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]]

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)=[11, 1];check ALL k in Fin(12)

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, 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]]

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)=[11, 2];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, 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]]

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)=[11, 3];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, 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]]

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)=[11, 4];check ALL k in Fin(12)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, 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]]

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)=[11, 5];check ALL k in Fin(12)

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]]

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)=[11, 6];check ALL k in Fin(12)

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, 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, S140





S142 · impliesBefore

ORIGINAL group pair(g,h)=[11, 7];check ALL k in Fin(12)

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, 0, 0, Yes], [11, 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, S141





S143 · impliesBefore

ORIGINAL group pair(g,h)=[11, 8];check ALL k in Fin(12)

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, 0, 0, Yes], [10, 1, 1, Yes], [11, 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, S142





S144 · impliesBefore

ORIGINAL group pair(g,h)=[11, 9];check ALL k in Fin(12)

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, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 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, S143





S145 · impliesBefore

ORIGINAL group pair(g,h)=[11, 10];check ALL k in Fin(12)

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, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 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, S144





S146 · impliesBefore

ORIGINAL group pair(g,h)=[11, 11];check ALL k in Fin(12)

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, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 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, S145





S147 · impliesBefore

ORIGINAL group element g=0;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 10, 10, No], [1, 11, 11, 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]]

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, S146





S148 · impliesBefore

ORIGINAL group element g=1;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 11, 11, 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]]

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, S147





S149 · 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]]

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, S148





S150 · 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, 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, S149





S151 · 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, 0, 0, No], [11, 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, S150





S152 · 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, 0, 0, No], [10, 1, 1, No], [11, 2, 2, Yes]]

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, S151





S153 · 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, 0, 0, No], [9, 1, 1, No], [10, 2, 2, Yes], [11, 3, 3, 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, S152





S154 · 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, 0, 0, No], [8, 1, 1, No], [9, 2, 2, Yes], [10, 3, 3, No], [11, 4, 4, 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, S153





S155 · 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, 0, 0, No], [7, 1, 1, No], [8, 2, 2, Yes], [9, 3, 3, No], [10, 4, 4, No], [11, 5, 5, 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, S154





S156 · 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, 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]]

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, S155





S157 · 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, 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]]

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, S156





S158 · 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, 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]]

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, S157





S159 · 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, S158





S160 · impliesBefore

CURRENT ORIGINAL complete Cayley table=[[10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8]];actual identity=2;order=12

After

ALL group axioms qualified;actual inverse list=[4, 3, 2, 1, 0, 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, S159





S161 · impliesBefore

Cauchy ORIGINAL prime divisor:ORIGINAL integer=2;complete possible trial divisors=[]

After

ALL [divisor,remainder] rows=[];prime=Yes;a composite>1 has a factor at most its square root

Conditions: none

Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification.

Step identifiers and dependenciesRule: finite_prime_ALL_trial_divisors_through_square_root · Uses: G001, S160





S162 · impliesBefore

CURRENT qualified group order=12;ORIGINAL prime p=2;divisibility remainder=0

After

product-e p-tuples have count |G|^(p-1)=12^1;cyclic rotation preserves product e by conjugation;rotation orbits have size1 or p;fixed tuples are(g,...,g) with g^p=e

Conditions: none

Choose the first p−1 elements freely;the last is uniquely the inverse product.Rotation conjugates the original product,so preserves e without commuting factors.Primality makes periods1 or p;the total count is divisible by p,so the fixed-solution count is too.Including the identity gives at least p solutions and a nonidentity one whose exact order divides p and hence equals p.This is a general argument,not enumeration of astronomical tuples.

Step identifiers and dependenciesRule: cauchy_prime_divisibility_and_general_rotation_count_not_enumeration · Uses: G001, S161





S163 · impliesBefore

Cauchy ORIGINAL supplied witness:ORIGINAL element=8;positive exponent=1;previous power=2;right multiply ORIGINAL g via table

After

g^1=8;identity=2;is_identity=No;all earlier POSITIVE powers are nonidentity

Conditions: none

Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.

Step identifiers and dependenciesRule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S162





S164 · impliesBefore

Cauchy ORIGINAL supplied witness:ORIGINAL element=8;positive exponent=2;previous power=8;right multiply ORIGINAL g via table

After

g^2=2;identity=2;is_identity=Yes;all earlier POSITIVE powers are nonidentity

Conditions: none

Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.

Step identifiers and dependenciesRule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S163





S165 · impliesBefore

Cauchy ORIGINAL supplied witness:CURRENT certified group order=12,element=8;ALL powers through first return=[[1, 8], [2, 2]]

After

exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order

Conditions: none

Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not.

Step identifiers and dependenciesRule: cauchy_finite_order_exists_and_cyclic_subgroup_divisibility · Uses: G001, S164









## Reverse checks


- 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.
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- S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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- S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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- 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.
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- 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.
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- 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.
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- 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.
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- 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.
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- 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.
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- 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.
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- 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.
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- 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.
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- 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.
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- 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.
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- 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

(12, 2, 8, [[1, 8], [2, 2]], Yes, Yes, g_to_p_is_identity_and_no_smaller_positive_power_is)

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: e2b4e6470bca0eadf5869d357da6b4dbe58257288909cbd7a10b32e953cf8089
- candidate record SHA-256: bde0fcb6cad0e5e8306f229a22d0571d6989174daed5980bbdf7afe801647358
- 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-20261112-r01-cauchy-theorem-finite-groups-0064.json)