Certificate for #12374 ⟨a, b | aaba=ab, bbba=a⟩

Back:⟨a, b | aaba=ab, bbba=a⟩

Completion settings:

[1] aaba=ab

Axiom: aaba=ab.

Referenced by [3], [4], [5], [7], [8].

[2] bbba=a

Axiom: bbba=a.

Defines rule #6.

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

[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 #4.

Referenced by [5].

[5] abbb=aaa

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

aab a ababa

Critical pair: aababb=abbaba.

Reduce LHS:

[1](aaba)bb
⇒ abbb

Reduce RHS:

[4](abba)ba
[2]⇒ aa(bbba)
⇒ aaa

Defines rule #5.

Referenced by [6], [7].

[6] aaaa=aa

Overlap of [5] abbb=aaa with [2] bbba=a:

a bbb bbba

Critical pair: aa=aaaa.

Flip LHS and RHS.

Defines rule #2.

Referenced by [8].

[7] aba=aab

Overlap of [5] abbb=aaa with [2] bbba=a:

ab bb bbba

Critical pair: aba=aaaba.

Reduce RHS:

[1]a(aaba)
⇒ aab

Defines rule #1.

[8] aaab=ab

Overlap of [6] aaaa=aa with [1] aaba=ab:

aa aa aaba

Critical pair: aaab=aaba.

Reduce RHS:

[1](aaba)
⇒ ab

Defines rule #3.