Question 13202: Find an Additive Inverse in an Already Qualified Ring

Original question and givens

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

commute=Yes

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

commute=Yes

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

commute=Yes

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

commute=Yes

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

commute=Yes

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

commute=Yes

additive unordered pair=[1, 3];a+b=5;b+a=5

commute=Yes

additive unordered pair=[1, 4];a+b=0;b+a=0

commute=Yes

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

commute=Yes

additive unordered pair=[2, 3];a+b=4;b+a=4

commute=Yes

additive unordered pair=[2, 4];a+b=1;b+a=1

commute=Yes

additive unordered pair=[2, 5];a+b=0;b+a=0

commute=Yes

additive unordered pair=[3, 4];a+b=2;b+a=2

commute=Yes

additive unordered pair=[3, 5];a+b=1;b+a=1

commute=Yes

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

commute=Yes

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

two-sided identity=Yes

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

two-sided identity=No

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

two-sided identity=No

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

two-sided identity=No

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

two-sided identity=No

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

two-sided identity=No

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

additive inverse=0

element=1;actual additive zero=0;two-sided additive inverse candidates=[4]

additive inverse=4

element=2;actual additive zero=0;two-sided additive inverse candidates=[5]

additive inverse=5

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

additive inverse=3

element=4;actual additive zero=0;two-sided additive inverse candidates=[1]

additive inverse=1

element=5;actual additive zero=0;two-sided additive inverse candidates=[2]

additive inverse=2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

two-sided identity=No

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

two-sided identity=Yes

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

two-sided identity=No

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

two-sided identity=No

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

two-sided identity=No

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

two-sided identity=No

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

left values=[3, 0, 3, 0, 0, 3];right values=[3, 0, 3, 0, 0, 3];all equal=Yes

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

left values=[3, 0, 3, 0, 0, 3];right values=[3, 0, 3, 0, 0, 3];all equal=Yes

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

left values=[3, 0, 3, 0, 0, 3];right values=[3, 0, 3, 0, 0, 3];all equal=Yes

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

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

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

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

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

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

left values=[0, 3, 0, 3, 3, 0];right values=[0, 3, 0, 3, 3, 0];all equal=Yes

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

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

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

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

selected element=0;additive inverse=0;actual zero=0

left sum=0;right sum=0

Reverse checks

• S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S179 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S175 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(0, 0, 0, 0, 0)

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.