Certificate for #12378 ⟨a, b | aaba=ba, aabb=a⟩

Back:⟨a, b | aaba=ba, aabb=a⟩

Completion settings:

[1] aaba=ba

Axiom: aaba=ba.

Referenced by [3], [5].

[2] aabb=a

Axiom: aabb=a.

Referenced by [3], [4], [5], [6], [7].

[3] bba=aa

Overlap of [1] aaba=ba with [1] aaba=ba:

aab a aaba

Critical pair: aabba=baaba.

Reduce LHS:

[2](aabb)a
⇒ aa

Reduce RHS:

[1]b(aaba)
⇒ bba

Flip LHS and RHS.

Defines rule #1.

Referenced by [4], [5].

[4] aaaa=aa

Overlap of [2] aabb=a with [3] bba=aa:

aa bb bba

Critical pair: aaaa=aa.

Referenced by [6].

[5] aba=baa

Overlap of [2] aabb=a with [3] bba=aa:

aab b bba

Critical pair: aabaa=aba.

Reduce LHS:

[1](aaba)a
⇒ baa

Flip LHS and RHS.

Defines rule #3.

[6] aaa=a

Overlap of [4] aaaa=aa with [2] aabb=a:

aa aa aabb

Critical pair: aaa=aabb.

Reduce RHS:

[2](aabb)
⇒ a

Defines rule #4.

Referenced by [7].

[7] abb=aa

Overlap of [6] aaa=a with [2] aabb=a:

a aa aabb

Critical pair: aa=abb.

Flip LHS and RHS.

Defines rule #2.