# Question 13231: Keep Additive Zero and Multiplicative One Distinct

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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.

R0025-13231 · q-20261014-advance-20261027-r01-finite-unital-ring-0081



## Original question and givens

Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 4, 2, 0], [0, 3, 4, 1, 2, 5], [0, 4, 2, 2, 4, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).


- G001 — Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 4, 2, 0], [0, 3, 4, 1, 2, 5], [0, 4, 2, 2, 4, 0], [0, 5, 0, 5, 0, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).





## 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-finite-unital-ring](https://ocme.evemisslab.com/en/mko/mko-finite-unital-ring/)





## Step-by-step derivation

S001 · impliesBefore

same carrier Fin(6);addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 0, 3, 4], [2, 5, 4, 1, 0, 3], [3, 0, 1, 4, 5, 2], [4, 3, 0, 5, 2, 1], [5, 4, 3, 2, 1, 0]];multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 4, 4, 2, 0], [0, 3, 4, 1, 2, 5], [0, 4, 2, 2, 4, 0], [0, 5, 0, 5, 0, 5]]

After

both operations typed and closed;unital ring does not mean field

Conditions: none

Bind both tables to the same complete finite carrier.This protocol requires a unit distinct from zero,not multiplicative commutativity or invertibility of every nonzero element.Neither group nor ring axioms are inferred from samples.

Step identifiers and dependenciesRule: ring_shared_typed_carrier · Uses: G001





S002 · impliesBefore

axiom=addition_associativity;a=0;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S001





S003 · impliesBefore

axiom=addition_associativity;a=0;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[1, 2, 5, 0, 3, 4];right values=[1, 2, 5, 0, 3, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S002





S004 · impliesBefore

axiom=addition_associativity;a=0;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 5, 4, 1, 0, 3];right values=[2, 5, 4, 1, 0, 3];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S003





S005 · impliesBefore

axiom=addition_associativity;a=0;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[3, 0, 1, 4, 5, 2];right values=[3, 0, 1, 4, 5, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S004





S006 · impliesBefore

axiom=addition_associativity;a=0;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 3, 0, 5, 2, 1];right values=[4, 3, 0, 5, 2, 1];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S005





S007 · impliesBefore

axiom=addition_associativity;a=0;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 4, 3, 2, 1, 0];right values=[5, 4, 3, 2, 1, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S006





S008 · impliesBefore

axiom=addition_associativity;a=1;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[1, 2, 5, 0, 3, 4];right values=[1, 2, 5, 0, 3, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S007





S009 · impliesBefore

axiom=addition_associativity;a=1;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 5, 4, 1, 0, 3];right values=[2, 5, 4, 1, 0, 3];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S008





S010 · impliesBefore

axiom=addition_associativity;a=1;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 4, 3, 2, 1, 0];right values=[5, 4, 3, 2, 1, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S009





S011 · impliesBefore

axiom=addition_associativity;a=1;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S010





S012 · impliesBefore

axiom=addition_associativity;a=1;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[3, 0, 1, 4, 5, 2];right values=[3, 0, 1, 4, 5, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S011





S013 · impliesBefore

axiom=addition_associativity;a=1;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 3, 0, 5, 2, 1];right values=[4, 3, 0, 5, 2, 1];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S012





S014 · impliesBefore

axiom=addition_associativity;a=2;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 5, 4, 1, 0, 3];right values=[2, 5, 4, 1, 0, 3];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S013





S015 · impliesBefore

axiom=addition_associativity;a=2;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 4, 3, 2, 1, 0];right values=[5, 4, 3, 2, 1, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S014





S016 · impliesBefore

axiom=addition_associativity;a=2;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 3, 0, 5, 2, 1];right values=[4, 3, 0, 5, 2, 1];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S015





S017 · impliesBefore

axiom=addition_associativity;a=2;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[1, 2, 5, 0, 3, 4];right values=[1, 2, 5, 0, 3, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S016





S018 · impliesBefore

axiom=addition_associativity;a=2;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S017





S019 · impliesBefore

axiom=addition_associativity;a=2;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[3, 0, 1, 4, 5, 2];right values=[3, 0, 1, 4, 5, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S018





S020 · impliesBefore

axiom=addition_associativity;a=3;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[3, 0, 1, 4, 5, 2];right values=[3, 0, 1, 4, 5, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S019





S021 · impliesBefore

axiom=addition_associativity;a=3;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S020





S022 · impliesBefore

axiom=addition_associativity;a=3;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[1, 2, 5, 0, 3, 4];right values=[1, 2, 5, 0, 3, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S021





S023 · impliesBefore

axiom=addition_associativity;a=3;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 3, 0, 5, 2, 1];right values=[4, 3, 0, 5, 2, 1];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S022





S024 · impliesBefore

axiom=addition_associativity;a=3;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 4, 3, 2, 1, 0];right values=[5, 4, 3, 2, 1, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S023





S025 · impliesBefore

axiom=addition_associativity;a=3;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 5, 4, 1, 0, 3];right values=[2, 5, 4, 1, 0, 3];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S024





S026 · impliesBefore

axiom=addition_associativity;a=4;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 3, 0, 5, 2, 1];right values=[4, 3, 0, 5, 2, 1];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S025





S027 · impliesBefore

axiom=addition_associativity;a=4;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[3, 0, 1, 4, 5, 2];right values=[3, 0, 1, 4, 5, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S026





S028 · impliesBefore

axiom=addition_associativity;a=4;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S027





S029 · impliesBefore

axiom=addition_associativity;a=4;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 4, 3, 2, 1, 0];right values=[5, 4, 3, 2, 1, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S028





S030 · impliesBefore

axiom=addition_associativity;a=4;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 5, 4, 1, 0, 3];right values=[2, 5, 4, 1, 0, 3];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S029





S031 · impliesBefore

axiom=addition_associativity;a=4;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[1, 2, 5, 0, 3, 4];right values=[1, 2, 5, 0, 3, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S030





S032 · impliesBefore

axiom=addition_associativity;a=5;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 4, 3, 2, 1, 0];right values=[5, 4, 3, 2, 1, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S031





S033 · impliesBefore

axiom=addition_associativity;a=5;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 3, 0, 5, 2, 1];right values=[4, 3, 0, 5, 2, 1];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S032





S034 · impliesBefore

axiom=addition_associativity;a=5;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[3, 0, 1, 4, 5, 2];right values=[3, 0, 1, 4, 5, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S033





S035 · impliesBefore

axiom=addition_associativity;a=5;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 5, 4, 1, 0, 3];right values=[2, 5, 4, 1, 0, 3];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S034





S036 · impliesBefore

axiom=addition_associativity;a=5;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[1, 2, 5, 0, 3, 4];right values=[1, 2, 5, 0, 3, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S035





S037 · impliesBefore

axiom=addition_associativity;a=5;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S036





S038 · impliesBefore

additive unordered pair=[0, 1];a+b=1;b+a=1

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S037





S039 · impliesBefore

additive unordered pair=[0, 2];a+b=2;b+a=2

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S038





S040 · impliesBefore

additive unordered pair=[0, 3];a+b=3;b+a=3

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S039





S041 · impliesBefore

additive unordered pair=[0, 4];a+b=4;b+a=4

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S040





S042 · impliesBefore

additive unordered pair=[0, 5];a+b=5;b+a=5

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S041





S043 · impliesBefore

additive unordered pair=[1, 2];a+b=5;b+a=5

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S042





S044 · impliesBefore

additive unordered pair=[1, 3];a+b=0;b+a=0

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S043





S045 · impliesBefore

additive unordered pair=[1, 4];a+b=3;b+a=3

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S044





S046 · impliesBefore

additive unordered pair=[1, 5];a+b=4;b+a=4

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S045





S047 · impliesBefore

additive unordered pair=[2, 3];a+b=1;b+a=1

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S046





S048 · impliesBefore

additive unordered pair=[2, 4];a+b=0;b+a=0

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S047





S049 · impliesBefore

additive unordered pair=[2, 5];a+b=3;b+a=3

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S048





S050 · impliesBefore

additive unordered pair=[3, 4];a+b=5;b+a=5

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S049





S051 · impliesBefore

additive unordered pair=[3, 5];a+b=2;b+a=2

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S050





S052 · impliesBefore

additive unordered pair=[4, 5];a+b=1;b+a=1

After

commute=Yes

Conditions: none

Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically.

Step identifiers and dependenciesRule: ring_additive_commutativity_all_pairs · Uses: G001, S051





S053 · impliesBefore

operation=addition;candidate identity=0;left products=[0, 1, 2, 3, 4, 5];right products=[0, 1, 2, 3, 4, 5]

After

two-sided identity=Yes

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S052





S054 · impliesBefore

operation=addition;candidate identity=1;left products=[1, 2, 5, 0, 3, 4];right products=[1, 2, 5, 0, 3, 4]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S053





S055 · impliesBefore

operation=addition;candidate identity=2;left products=[2, 5, 4, 1, 0, 3];right products=[2, 5, 4, 1, 0, 3]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S054





S056 · impliesBefore

operation=addition;candidate identity=3;left products=[3, 0, 1, 4, 5, 2];right products=[3, 0, 1, 4, 5, 2]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S055





S057 · impliesBefore

operation=addition;candidate identity=4;left products=[4, 3, 0, 5, 2, 1];right products=[4, 3, 0, 5, 2, 1]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S056





S058 · impliesBefore

operation=addition;candidate identity=5;left products=[5, 4, 3, 2, 1, 0];right products=[5, 4, 3, 2, 1, 0]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S057





S059 · impliesBefore

element=0;actual additive zero=0;two-sided additive inverse candidates=[0]

After

additive inverse=0

Conditions: none

This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it.

Step identifiers and dependenciesRule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S058





S060 · impliesBefore

element=1;actual additive zero=0;two-sided additive inverse candidates=[3]

After

additive inverse=3

Conditions: none

This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it.

Step identifiers and dependenciesRule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S059





S061 · impliesBefore

element=2;actual additive zero=0;two-sided additive inverse candidates=[4]

After

additive inverse=4

Conditions: none

This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it.

Step identifiers and dependenciesRule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S060





S062 · impliesBefore

element=3;actual additive zero=0;two-sided additive inverse candidates=[1]

After

additive inverse=1

Conditions: none

This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it.

Step identifiers and dependenciesRule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S061





S063 · impliesBefore

element=4;actual additive zero=0;two-sided additive inverse candidates=[2]

After

additive inverse=2

Conditions: none

This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it.

Step identifiers and dependenciesRule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S062





S064 · impliesBefore

element=5;actual additive zero=0;two-sided additive inverse candidates=[5]

After

additive inverse=5

Conditions: none

This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it.

Step identifiers and dependenciesRule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S063





S065 · impliesBefore

axiom=multiplication_associativity;a=0;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S064





S066 · impliesBefore

axiom=multiplication_associativity;a=0;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S065





S067 · impliesBefore

axiom=multiplication_associativity;a=0;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S066





S068 · impliesBefore

axiom=multiplication_associativity;a=0;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S067





S069 · impliesBefore

axiom=multiplication_associativity;a=0;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S068





S070 · impliesBefore

axiom=multiplication_associativity;a=0;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S069





S071 · impliesBefore

axiom=multiplication_associativity;a=1;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S070





S072 · impliesBefore

axiom=multiplication_associativity;a=1;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S071





S073 · impliesBefore

axiom=multiplication_associativity;a=1;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S072





S074 · impliesBefore

axiom=multiplication_associativity;a=1;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S073





S075 · impliesBefore

axiom=multiplication_associativity;a=1;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S074





S076 · impliesBefore

axiom=multiplication_associativity;a=1;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S075





S077 · impliesBefore

axiom=multiplication_associativity;a=2;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S076





S078 · impliesBefore

axiom=multiplication_associativity;a=2;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S077





S079 · impliesBefore

axiom=multiplication_associativity;a=2;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S078





S080 · impliesBefore

axiom=multiplication_associativity;a=2;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S079





S081 · impliesBefore

axiom=multiplication_associativity;a=2;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S080





S082 · impliesBefore

axiom=multiplication_associativity;a=2;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S081





S083 · impliesBefore

axiom=multiplication_associativity;a=3;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S082





S084 · impliesBefore

axiom=multiplication_associativity;a=3;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S083





S085 · impliesBefore

axiom=multiplication_associativity;a=3;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S084





S086 · impliesBefore

axiom=multiplication_associativity;a=3;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S085





S087 · impliesBefore

axiom=multiplication_associativity;a=3;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S086





S088 · impliesBefore

axiom=multiplication_associativity;a=3;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S087





S089 · impliesBefore

axiom=multiplication_associativity;a=4;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S088





S090 · impliesBefore

axiom=multiplication_associativity;a=4;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S089





S091 · impliesBefore

axiom=multiplication_associativity;a=4;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S090





S092 · impliesBefore

axiom=multiplication_associativity;a=4;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S091





S093 · impliesBefore

axiom=multiplication_associativity;a=4;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S092





S094 · impliesBefore

axiom=multiplication_associativity;a=4;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S093





S095 · impliesBefore

axiom=multiplication_associativity;a=5;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S094





S096 · impliesBefore

axiom=multiplication_associativity;a=5;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S095





S097 · impliesBefore

axiom=multiplication_associativity;a=5;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S096





S098 · impliesBefore

axiom=multiplication_associativity;a=5;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S097





S099 · impliesBefore

axiom=multiplication_associativity;a=5;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S098





S100 · impliesBefore

axiom=multiplication_associativity;a=5;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S099





S101 · impliesBefore

operation=multiplication;candidate identity=0;left products=[0, 0, 0, 0, 0, 0];right products=[0, 0, 0, 0, 0, 0]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S100





S102 · impliesBefore

operation=multiplication;candidate identity=1;left products=[0, 1, 2, 3, 4, 5];right products=[0, 1, 2, 3, 4, 5]

After

two-sided identity=Yes

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S101





S103 · impliesBefore

operation=multiplication;candidate identity=2;left products=[0, 2, 4, 4, 2, 0];right products=[0, 2, 4, 4, 2, 0]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S102





S104 · impliesBefore

operation=multiplication;candidate identity=3;left products=[0, 3, 4, 1, 2, 5];right products=[0, 3, 4, 1, 2, 5]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S103





S105 · impliesBefore

operation=multiplication;candidate identity=4;left products=[0, 4, 2, 2, 4, 0];right products=[0, 4, 2, 2, 4, 0]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S104





S106 · impliesBefore

operation=multiplication;candidate identity=5;left products=[0, 5, 0, 5, 0, 5];right products=[0, 5, 0, 5, 0, 5]

After

two-sided identity=No

Conditions: none

Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance.

Step identifiers and dependenciesRule: ring_derive_operation_identity_labels · Uses: G001, S105





S107 · impliesBefore

axiom=left_distributivity;a=0;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S106





S108 · impliesBefore

axiom=left_distributivity;a=0;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S107





S109 · impliesBefore

axiom=left_distributivity;a=0;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S108





S110 · impliesBefore

axiom=left_distributivity;a=0;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S109





S111 · impliesBefore

axiom=left_distributivity;a=0;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S110





S112 · impliesBefore

axiom=left_distributivity;a=0;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S111





S113 · impliesBefore

axiom=left_distributivity;a=1;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S112





S114 · impliesBefore

axiom=left_distributivity;a=1;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[1, 2, 5, 0, 3, 4];right values=[1, 2, 5, 0, 3, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S113





S115 · impliesBefore

axiom=left_distributivity;a=1;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 5, 4, 1, 0, 3];right values=[2, 5, 4, 1, 0, 3];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S114





S116 · impliesBefore

axiom=left_distributivity;a=1;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[3, 0, 1, 4, 5, 2];right values=[3, 0, 1, 4, 5, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S115





S117 · impliesBefore

axiom=left_distributivity;a=1;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 3, 0, 5, 2, 1];right values=[4, 3, 0, 5, 2, 1];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S116





S118 · impliesBefore

axiom=left_distributivity;a=1;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 4, 3, 2, 1, 0];right values=[5, 4, 3, 2, 1, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S117





S119 · impliesBefore

axiom=left_distributivity;a=2;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S118





S120 · impliesBefore

axiom=left_distributivity;a=2;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 4, 0, 0, 4, 2];right values=[2, 4, 0, 0, 4, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S119





S121 · impliesBefore

axiom=left_distributivity;a=2;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 0, 2, 2, 0, 4];right values=[4, 0, 2, 2, 0, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S120





S122 · impliesBefore

axiom=left_distributivity;a=2;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 0, 2, 2, 0, 4];right values=[4, 0, 2, 2, 0, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S121





S123 · impliesBefore

axiom=left_distributivity;a=2;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 4, 0, 0, 4, 2];right values=[2, 4, 0, 0, 4, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S122





S124 · impliesBefore

axiom=left_distributivity;a=2;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S123





S125 · impliesBefore

axiom=left_distributivity;a=3;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S124





S126 · impliesBefore

axiom=left_distributivity;a=3;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[3, 4, 5, 0, 1, 2];right values=[3, 4, 5, 0, 1, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S125





S127 · impliesBefore

axiom=left_distributivity;a=3;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 5, 2, 3, 0, 1];right values=[4, 5, 2, 3, 0, 1];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S126





S128 · impliesBefore

axiom=left_distributivity;a=3;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[1, 0, 3, 2, 5, 4];right values=[1, 0, 3, 2, 5, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S127





S129 · impliesBefore

axiom=left_distributivity;a=3;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 1, 0, 5, 4, 3];right values=[2, 1, 0, 5, 4, 3];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S128





S130 · impliesBefore

axiom=left_distributivity;a=3;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 2, 1, 4, 3, 0];right values=[5, 2, 1, 4, 3, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S129





S131 · impliesBefore

axiom=left_distributivity;a=4;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S130





S132 · impliesBefore

axiom=left_distributivity;a=4;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 2, 0, 0, 2, 4];right values=[4, 2, 0, 0, 2, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S131





S133 · impliesBefore

axiom=left_distributivity;a=4;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 0, 4, 4, 0, 2];right values=[2, 0, 4, 4, 0, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S132





S134 · impliesBefore

axiom=left_distributivity;a=4;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[2, 0, 4, 4, 0, 2];right values=[2, 0, 4, 4, 0, 2];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S133





S135 · impliesBefore

axiom=left_distributivity;a=4;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[4, 2, 0, 0, 2, 4];right values=[4, 2, 0, 0, 2, 4];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S134





S136 · impliesBefore

axiom=left_distributivity;a=4;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S135





S137 · impliesBefore

axiom=left_distributivity;a=5;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S136





S138 · impliesBefore

axiom=left_distributivity;a=5;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 0, 5, 0, 5, 0];right values=[5, 0, 5, 0, 5, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S137





S139 · impliesBefore

axiom=left_distributivity;a=5;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S138





S140 · impliesBefore

axiom=left_distributivity;a=5;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 0, 5, 0, 5, 0];right values=[5, 0, 5, 0, 5, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S139





S141 · impliesBefore

axiom=left_distributivity;a=5;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S140





S142 · impliesBefore

axiom=left_distributivity;a=5;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[5, 0, 5, 0, 5, 0];right values=[5, 0, 5, 0, 5, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S141





S143 · impliesBefore

axiom=right_distributivity;a=0;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S142





S144 · impliesBefore

axiom=right_distributivity;a=0;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S143





S145 · impliesBefore

axiom=right_distributivity;a=0;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S144





S146 · impliesBefore

axiom=right_distributivity;a=0;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S145





S147 · impliesBefore

axiom=right_distributivity;a=0;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S146





S148 · impliesBefore

axiom=right_distributivity;a=0;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S147





S149 · impliesBefore

axiom=right_distributivity;a=1;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S148





S150 · impliesBefore

axiom=right_distributivity;a=1;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S149





S151 · impliesBefore

axiom=right_distributivity;a=1;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S150





S152 · impliesBefore

axiom=right_distributivity;a=1;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S151





S153 · impliesBefore

axiom=right_distributivity;a=1;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S152





S154 · impliesBefore

axiom=right_distributivity;a=1;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S153





S155 · impliesBefore

axiom=right_distributivity;a=2;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S154





S156 · impliesBefore

axiom=right_distributivity;a=2;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S155





S157 · impliesBefore

axiom=right_distributivity;a=2;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S156





S158 · impliesBefore

axiom=right_distributivity;a=2;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S157





S159 · impliesBefore

axiom=right_distributivity;a=2;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S158





S160 · impliesBefore

axiom=right_distributivity;a=2;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S159





S161 · impliesBefore

axiom=right_distributivity;a=3;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

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





S162 · impliesBefore

axiom=right_distributivity;a=3;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

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





S163 · impliesBefore

axiom=right_distributivity;a=3;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

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





S164 · impliesBefore

axiom=right_distributivity;a=3;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

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





S165 · impliesBefore

axiom=right_distributivity;a=3;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

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





S166 · impliesBefore

axiom=right_distributivity;a=3;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S165





S167 · impliesBefore

axiom=right_distributivity;a=4;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S166





S168 · impliesBefore

axiom=right_distributivity;a=4;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S167





S169 · impliesBefore

axiom=right_distributivity;a=4;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S168





S170 · impliesBefore

axiom=right_distributivity;a=4;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S169





S171 · impliesBefore

axiom=right_distributivity;a=4;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S170





S172 · impliesBefore

axiom=right_distributivity;a=4;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S171





S173 · impliesBefore

axiom=right_distributivity;a=5;b=0;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 5, 0, 5, 0, 5];right values=[0, 5, 0, 5, 0, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S172





S174 · impliesBefore

axiom=right_distributivity;a=5;b=1;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 4, 2, 2, 4, 0];right values=[0, 4, 2, 2, 4, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S173





S175 · impliesBefore

axiom=right_distributivity;a=5;b=2;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 3, 4, 1, 2, 5];right values=[0, 3, 4, 1, 2, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S174





S176 · impliesBefore

axiom=right_distributivity;a=5;b=3;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 2, 4, 4, 2, 0];right values=[0, 2, 4, 4, 2, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S175





S177 · impliesBefore

axiom=right_distributivity;a=5;b=4;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S176





S178 · impliesBefore

axiom=right_distributivity;a=5;b=5;all c=[0, 1, 2, 3, 4, 5]

After

left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=Yes

Conditions: none

For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure.

Step identifiers and dependenciesRule: ring_complete_axiom_third_element · Uses: G001, S177





S179 · impliesBefore

additive associativity first failure=null;additive commute=Yes;zero=0;inverses=[0, 3, 4, 1, 2, 5];multiplication first failure=null;one=1;left distributivity first failure=null;right distributivity first failure=null

After

additive Abelian=Yes;multiplication associative=Yes;left distributive=Yes;right distributive=Yes;distinct identities=Yes;qualified=Yes;priority failure=null

Conditions: none

Retain every individual flag,then report the first failure by the declared priority:additive structure,multiplicative associativity,left distributivity,right distributivity,distinct identities.One passing obligation cannot replace another.

Step identifiers and dependenciesRule: ring_all_obligations_and_failure_priority · Uses: G001, S178









## Reverse checks


- S179 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S175 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(0, 1, Yes)

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: e623662db7b79e39f44bd3806b77acc9038640da96e0d92f6a7830bf6c563601
- candidate record SHA-256: 500e7c2bac06f9ec50124b1a53f613cd9fb041f95b75566c594ee5039e3dce62
- semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
- question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261027-r01-finite-unital-ring-0081.json)