Certificate for #1257 ⟨a, b | ababa=aaba⟩

Back:⟨a, b | ababa=aaba⟩

Completion settings:

[1] ababa=aaba

Axiom: ababa=aaba.

Referenced by [3].

[2] aaba=c

Axiom: aaba=c.

Defines rule #5.

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

[3] ababa=c

Simplify [1] ababa=aaba.

Reduce RHS:

[2](aaba)
⇒ c

Defines rule #6.

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

[4] abc=cba

Overlap of [3] ababa=c with [3] ababa=c:

ab aba ababa

Critical pair: abc=cba.

Defines rule #3.

Referenced by [5], [7].

[5] cbaba=caba

Overlap of [3] ababa=c with [2] aaba=c:

abab a aaba

Critical pair: ababc=caba.

Reduce LHS:

[4]ab(abc)
[4]⇒ (abc)ba
⇒ cbaba

Defines rule #4.

Referenced by [7].

[6] ac=cba

Overlap of [2] aaba=c with [3] ababa=c:

a aba ababa

Critical pair: ac=cba.

Defines rule #2.

[7] cbc=cc

Overlap of [3] ababa=c with [4] abc=cba:

abab a abc

Critical pair: ababcba=cbc.

Reduce LHS:

[4]ab(abc)ba
[4]⇒ (abc)baba
[5]⇒ (cbaba)ba
[3]⇒ c(ababa)
⇒ cc

Flip LHS and RHS.

Defines rule #1.