Certificate for #19773 ⟨a, b | aba=b, bbbbb=aa⟩

Back:⟨a, b | aba=b, bbbbb=aa⟩

Completion settings:

[1] aba=b

Axiom: aba=b.

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

[2] aa=bbbbb

Axiom: bbbbb=aa.

Flip LHS and RHS.

Defines rule #3.

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

[3] bba=abb

Overlap of [1] aba=b with [1] aba=b:

ab a aba

Critical pair: abb=bba.

Flip LHS and RHS.

Referenced by [5], [6].

[4] ba=abbbbbb

Overlap of [1] aba=b with [2] aa=bbbbb:

ab a aa

Critical pair: abbbbbb=ba.

Flip LHS and RHS.

Referenced by [7].

[5] abbbbbb=ab

Overlap of [2] aa=bbbbb with [1] aba=b:

a a aba

Critical pair: ab=bbbbbba.

Reduce RHS:

[3]bbbb(bba)
[3]⇒ bb(bba)bb
[3]⇒ (bba)bbbb
⇒ abbbbbb

Flip LHS and RHS.

Referenced by [6], [7].

[6] bbbbbb=b

Overlap of [5] abbbbbb=ab with [3] bba=abb:

abbbb bb bba

Critical pair: abbbbabb=aba.

Reduce LHS:

[3]abb(bba)bb
[3]⇒ a(bba)bbbb
[5]⇒ a(abbbbbb)
[2]⇒ (aa)b
⇒ bbbbbb

Reduce RHS:

[1](aba)
⇒ b

Defines rule #1.

[7] ba=ab

Simplify [4] ba=abbbbbb.

Reduce RHS:

[5](abbbbbb)
⇒ ab

Defines rule #2.