Question 13246: Keep Additive Zero and Multiplicative One Distinct

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0025-13246 · q-20261014-advance-20261027-r01-finite-unital-ring-0096

Original question and givens

Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 5, 5], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 5, 0, 2, 2]] 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, 5, 4, 2, 0, 3], [2, 4, 5, 1, 3, 0], [3, 2, 1, 0, 5, 4], [4, 0, 3, 5, 2, 1], [5, 3, 0, 4, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 2, 0, 5, 5], [0, 3, 0, 3, 3, 0], [0, 4, 5, 3, 1, 2], [0, 5, 5, 0, 2, 2]] 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

Step-by-step derivation

S001 · implies

Before

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

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 dependencies

Rule: ring_shared_typed_carrier · Uses: G001

S002 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S001

S003 · implies

Before

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

After

left values=[1, 5, 4, 2, 0, 3];right values=[1, 5, 4, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S002

S004 · implies

Before

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

After

left values=[2, 4, 5, 1, 3, 0];right values=[2, 4, 5, 1, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S003

S005 · implies

Before

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

After

left values=[3, 2, 1, 0, 5, 4];right values=[3, 2, 1, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S004

S006 · implies

Before

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

After

left values=[4, 0, 3, 5, 2, 1];right values=[4, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S005

S007 · implies

Before

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

After

left values=[5, 3, 0, 4, 1, 2];right values=[5, 3, 0, 4, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S006

S008 · implies

Before

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

After

left values=[1, 5, 4, 2, 0, 3];right values=[1, 5, 4, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S007

S009 · implies

Before

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

After

left values=[5, 3, 0, 4, 1, 2];right values=[5, 3, 0, 4, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S008

S010 · implies

Before

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

After

left values=[4, 0, 3, 5, 2, 1];right values=[4, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S009

S011 · implies

Before

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

After

left values=[2, 4, 5, 1, 3, 0];right values=[2, 4, 5, 1, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S010

S012 · implies

Before

axiom=addition_associativity;a=1;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S011

S013 · implies

Before

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

After

left values=[3, 2, 1, 0, 5, 4];right values=[3, 2, 1, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S012

S014 · implies

Before

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

After

left values=[2, 4, 5, 1, 3, 0];right values=[2, 4, 5, 1, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S013

S015 · implies

Before

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

After

left values=[4, 0, 3, 5, 2, 1];right values=[4, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S014

S016 · implies

Before

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

After

left values=[5, 3, 0, 4, 1, 2];right values=[5, 3, 0, 4, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S015

S017 · implies

Before

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

After

left values=[1, 5, 4, 2, 0, 3];right values=[1, 5, 4, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S016

S018 · implies

Before

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

After

left values=[3, 2, 1, 0, 5, 4];right values=[3, 2, 1, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S017

S019 · implies

Before

axiom=addition_associativity;a=2;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S018

S020 · implies

Before

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

After

left values=[3, 2, 1, 0, 5, 4];right values=[3, 2, 1, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S019

S021 · implies

Before

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

After

left values=[2, 4, 5, 1, 3, 0];right values=[2, 4, 5, 1, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S020

S022 · implies

Before

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

After

left values=[1, 5, 4, 2, 0, 3];right values=[1, 5, 4, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S021

S023 · implies

Before

axiom=addition_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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S022

S024 · implies

Before

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

After

left values=[5, 3, 0, 4, 1, 2];right values=[5, 3, 0, 4, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S023

S025 · implies

Before

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

After

left values=[4, 0, 3, 5, 2, 1];right values=[4, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S024

S026 · implies

Before

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

After

left values=[4, 0, 3, 5, 2, 1];right values=[4, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S025

S027 · implies

Before

axiom=addition_associativity;a=4;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S026

S028 · implies

Before

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

After

left values=[3, 2, 1, 0, 5, 4];right values=[3, 2, 1, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S027

S029 · implies

Before

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

After

left values=[5, 3, 0, 4, 1, 2];right values=[5, 3, 0, 4, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S028

S030 · implies

Before

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

After

left values=[2, 4, 5, 1, 3, 0];right values=[2, 4, 5, 1, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S029

S031 · implies

Before

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

After

left values=[1, 5, 4, 2, 0, 3];right values=[1, 5, 4, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S030

S032 · implies

Before

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

After

left values=[5, 3, 0, 4, 1, 2];right values=[5, 3, 0, 4, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S031

S033 · implies

Before

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

After

left values=[3, 2, 1, 0, 5, 4];right values=[3, 2, 1, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S032

S034 · implies

Before

axiom=addition_associativity;a=5;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S033

S035 · implies

Before

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

After

left values=[4, 0, 3, 5, 2, 1];right values=[4, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S034

S036 · implies

Before

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

After

left values=[1, 5, 4, 2, 0, 3];right values=[1, 5, 4, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S035

S037 · implies

Before

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

After

left values=[2, 4, 5, 1, 3, 0];right values=[2, 4, 5, 1, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S036

S038 · implies

Before

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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S037

S039 · implies

Before

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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S038

S040 · implies

Before

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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S039

S041 · implies

Before

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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S040

S042 · implies

Before

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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S041

S043 · implies

Before

additive unordered pair=[1, 2];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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S042

S044 · implies

Before

additive unordered pair=[1, 3];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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S043

S045 · implies

Before

additive unordered pair=[1, 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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S044

S046 · implies

Before

additive unordered pair=[1, 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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S045

S047 · implies

Before

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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S046

S048 · implies

Before

additive unordered pair=[2, 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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S047

S049 · implies

Before

additive unordered pair=[2, 5];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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S048

S050 · implies

Before

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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S049

S051 · implies

Before

additive unordered pair=[3, 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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S050

S052 · implies

Before

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 dependencies

Rule: ring_additive_commutativity_all_pairs · Uses: G001, S051

S053 · implies

Before

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 dependencies

Rule: ring_derive_operation_identity_labels · Uses: G001, S052

S054 · implies

Before

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

Rule: ring_derive_operation_identity_labels · Uses: G001, S053

S055 · implies

Before

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

Rule: ring_derive_operation_identity_labels · Uses: G001, S054

S056 · implies

Before

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

Rule: ring_derive_operation_identity_labels · Uses: G001, S055

S057 · implies

Before

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

Rule: ring_derive_operation_identity_labels · Uses: G001, S056

S058 · implies

Before

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

Rule: ring_derive_operation_identity_labels · Uses: G001, S057

S059 · implies

Before

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 dependencies

Rule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S058

S060 · implies

Before

element=1;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 dependencies

Rule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S059

S061 · implies

Before

element=2;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 dependencies

Rule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S060

S062 · implies

Before

element=3;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 dependencies

Rule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S061

S063 · implies

Before

element=4;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 dependencies

Rule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S062

S064 · implies

Before

element=5;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 dependencies

Rule: ring_additive_inverse_not_multiplicative_inverse · Uses: G001, S063

S065 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S064

S066 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S065

S067 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S066

S068 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S067

S069 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S068

S070 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S069

S071 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S070

S072 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S071

S073 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S072

S074 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S073

S075 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S074

S076 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S075

S077 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S076

S078 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S077

S079 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S078

S080 · implies

Before

axiom=multiplication_associativity;a=2;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S079

S081 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S080

S082 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S081

S083 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S082

S084 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S083

S085 · implies

Before

axiom=multiplication_associativity;a=3;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S084

S086 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S085

S087 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S086

S088 · implies

Before

axiom=multiplication_associativity;a=3;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S087

S089 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S088

S090 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S089

S091 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S090

S092 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S091

S093 · implies

Before

axiom=multiplication_associativity;a=4;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S092

S094 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S093

S095 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S094

S096 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S095

S097 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S096

S098 · implies

Before

axiom=multiplication_associativity;a=5;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S097

S099 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S098

S100 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S099

S101 · implies

Before

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 dependencies

Rule: ring_derive_operation_identity_labels · Uses: G001, S100

S102 · implies

Before

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 dependencies

Rule: ring_derive_operation_identity_labels · Uses: G001, S101

S103 · implies

Before

operation=multiplication;candidate identity=2;left products=[0, 2, 2, 0, 5, 5];right products=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_derive_operation_identity_labels · Uses: G001, S102

S104 · implies

Before

operation=multiplication;candidate identity=3;left products=[0, 3, 0, 3, 3, 0];right products=[0, 3, 0, 3, 3, 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 dependencies

Rule: ring_derive_operation_identity_labels · Uses: G001, S103

S105 · implies

Before

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

Rule: ring_derive_operation_identity_labels · Uses: G001, S104

S106 · implies

Before

operation=multiplication;candidate identity=5;left products=[0, 5, 5, 0, 2, 2];right products=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_derive_operation_identity_labels · Uses: G001, S105

S107 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S106

S108 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S107

S109 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S108

S110 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S109

S111 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S110

S112 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S111

S113 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S112

S114 · implies

Before

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

After

left values=[1, 5, 4, 2, 0, 3];right values=[1, 5, 4, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S113

S115 · implies

Before

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

After

left values=[2, 4, 5, 1, 3, 0];right values=[2, 4, 5, 1, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S114

S116 · implies

Before

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

After

left values=[3, 2, 1, 0, 5, 4];right values=[3, 2, 1, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S115

S117 · implies

Before

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

After

left values=[4, 0, 3, 5, 2, 1];right values=[4, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S116

S118 · implies

Before

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

After

left values=[5, 3, 0, 4, 1, 2];right values=[5, 3, 0, 4, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S117

S119 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S118

S120 · implies

Before

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

After

left values=[2, 5, 5, 2, 0, 0];right values=[2, 5, 5, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S119

S121 · implies

Before

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

After

left values=[2, 5, 5, 2, 0, 0];right values=[2, 5, 5, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S120

S122 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S121

S123 · implies

Before

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

After

left values=[5, 0, 0, 5, 2, 2];right values=[5, 0, 0, 5, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S122

S124 · implies

Before

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

After

left values=[5, 0, 0, 5, 2, 2];right values=[5, 0, 0, 5, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S123

S125 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S124

S126 · implies

Before

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

After

left values=[3, 0, 3, 0, 0, 3];right values=[3, 0, 3, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S125

S127 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S126

S128 · implies

Before

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

After

left values=[3, 0, 3, 0, 0, 3];right values=[3, 0, 3, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S127

S129 · implies

Before

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

After

left values=[3, 0, 3, 0, 0, 3];right values=[3, 0, 3, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S128

S130 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S129

S131 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S130

S132 · implies

Before

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

After

left values=[4, 2, 1, 5, 0, 3];right values=[4, 2, 1, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S131

S133 · implies

Before

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

After

left values=[5, 1, 2, 4, 3, 0];right values=[5, 1, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S132

S134 · implies

Before

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

After

left values=[3, 5, 4, 0, 2, 1];right values=[3, 5, 4, 0, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S133

S135 · implies

Before

axiom=left_distributivity;a=4;b=4;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S134

S136 · implies

Before

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

After

left values=[2, 3, 0, 1, 4, 5];right values=[2, 3, 0, 1, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S135

S137 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S136

S138 · implies

Before

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

After

left values=[5, 2, 2, 5, 0, 0];right values=[5, 2, 2, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S137

S139 · implies

Before

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

After

left values=[5, 2, 2, 5, 0, 0];right values=[5, 2, 2, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S138

S140 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S139

S141 · implies

Before

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

After

left values=[2, 0, 0, 2, 5, 5];right values=[2, 0, 0, 2, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S140

S142 · implies

Before

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

After

left values=[2, 0, 0, 2, 5, 5];right values=[2, 0, 0, 2, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S141

S143 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S142

S144 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S143

S145 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S144

S146 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S145

S147 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S146

S148 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S147

S149 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S148

S150 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S149

S151 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S150

S152 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S151

S153 · implies

Before

axiom=right_distributivity;a=1;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S152

S154 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S153

S155 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S154

S156 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S155

S157 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S156

S158 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S157

S159 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S158

S160 · implies

Before

axiom=right_distributivity;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S159

S161 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S160

S162 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S161

S163 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S162

S164 · implies

Before

axiom=right_distributivity;a=3;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S163

S165 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S164

S166 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S165

S167 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S166

S168 · implies

Before

axiom=right_distributivity;a=4;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S167

S169 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S168

S170 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S169

S171 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S170

S172 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S171

S173 · implies

Before

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

After

left values=[0, 5, 5, 0, 2, 2];right values=[0, 5, 5, 0, 2, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S172

S174 · implies

Before

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

After

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S173

S175 · implies

Before

axiom=right_distributivity;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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S174

S176 · implies

Before

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

After

left values=[0, 4, 5, 3, 1, 2];right values=[0, 4, 5, 3, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S175

S177 · implies

Before

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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S176

S178 · implies

Before

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

After

left values=[0, 2, 2, 0, 5, 5];right values=[0, 2, 2, 0, 5, 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 dependencies

Rule: ring_complete_axiom_third_element · Uses: G001, S177

S179 · implies

Before

additive associativity first failure=null;additive commute=Yes;zero=0;inverses=[0, 4, 5, 3, 1, 2];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 dependencies

Rule: 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: 64a3214905a40d543d89ffa79190c8355ae82c215e02862d80e4b43bd974858f
candidate record SHA-256: 23ec1b1b4356c849f5c837663a34646aace2c2c60aaca457477820fa0499bddf
semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON