Certificate for #12360 ⟨a, b | aaba=ab, abbb=a⟩

Back:⟨a, b | aaba=ab, abbb=a⟩

Completion settings:

[1] aaba=ab

Axiom: aaba=ab.

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

[2] abbb=a

Axiom: abbb=a.

Defines rule #4.

Referenced by [5].

[3] ababa=abb

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

aab a aaba

Critical pair: aabab=ababa.

Reduce LHS:

[1](aaba)b
⇒ abb

Flip LHS and RHS.

Referenced by [4], [5].

[4] abba=aabb

Overlap of [1] aaba=ab with [3] ababa=abb:

a aba ababa

Critical pair: aabb=abba.

Flip LHS and RHS.

Defines rule #3.

Referenced by [5].

[5] aaa=a

Overlap of [1] aaba=ab with [3] ababa=abb:

aab a ababa

Critical pair: aababb=abbaba.

Reduce LHS:

[1](aaba)bb
[2]⇒ (abbb)
⇒ a

Reduce RHS:

[4](abba)ba
[2]⇒ a(abbb)a
⇒ aaa

Flip LHS and RHS.

Defines rule #1.

Referenced by [6].

[6] aba=aab

Overlap of [5] aaa=a with [1] aaba=ab:

a aa aaba

Critical pair: aab=aba.

Flip LHS and RHS.

Defines rule #2.