Certificate for #891 ⟨a, b | aa=a, abba=b⟩

Back:⟨a, b | aa=a, abba=b⟩

Completion settings:

[1] aa=a

Axiom: aa=a.

Defines rule #1.

Referenced by [3], [4].

[2] abba=b

Axiom: abba=b.

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

[3] ab=b

Overlap of [1] aa=a with [2] abba=b:

a a abba

Critical pair: ab=abba.

Reduce RHS:

[2](abba)
⇒ b

Defines rule #2.

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

[4] bba=ba

Overlap of [2] abba=b with [1] aa=a:

abb a aa

Critical pair: abba=ba.

Reduce LHS:

[3](ab)ba
⇒ bba

Referenced by [5], [6].

[5] bbb=ba

Overlap of [2] abba=b with [2] abba=b:

abb a abba

Critical pair: abbb=bbba.

Reduce LHS:

[3](ab)bb
⇒ bbb

Reduce RHS:

[4]b(bba)
[4]⇒ (bba)
⇒ ba

Referenced by [7].

[6] ba=b

Overlap of [2] abba=b with [3] ab=b:

abba ab

Critical pair: bba=b.

Reduce LHS:

[4](bba)
⇒ ba

Defines rule #3.

Referenced by [7].

[7] bb=b

Overlap of [2] abba=b with [3] ab=b:

abb a ab

Critical pair: abbb=bb.

Reduce LHS:

[3](ab)bb
[5]⇒ (bbb)
[6]⇒ (ba)
⇒ b

Flip LHS and RHS.

Defines rule #4.