Certificate for #5087 ⟨a, b | aaa=bb, abbb=b⟩

Completion settings:

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1

[1] aaa=bb

Axiom: aaa=bb.

Defines rule #4.

Referenced by [3], [4].

[2] abbb=b

Axiom: abbb=b.

Referenced by [4], [5], [6], [7], [8], [9], [10].

[3] bba=abb

Overlap of [1] aaa=bb with [1] aaa=bb:

a aa aaa

Critical pair: abb=bba.

Flip LHS and RHS.

Referenced by [5], [7], [8], [9].

[4] aab=bbbbb

Overlap of [1] aaa=bb with [2] abbb=b:

aa a abbb

Critical pair: aab=bbbbb.

Referenced by [7], [9].

[5] ababb=ba

Overlap of [2] abbb=b with [3] bba=abb:

ab bb bba

Critical pair: ababb=ba.

Referenced by [6], [7].

[6] bab=abb

Overlap of [5] ababb=ba with [2] abbb=b:

ab abb abbb

Critical pair: abb=bab.

Flip LHS and RHS.

Referenced by [8], [9].

[7] baa=bbbbb

Overlap of [5] ababb=ba with [3] bba=abb:

aba bb bba

Critical pair: abaabb=baa.

Reduce LHS:

[4]	ab(aab)b
[2]	⇒ (abbb)bbbb
	⇒ bbbbb

Flip LHS and RHS.

Referenced by [8].

[8] ba=bbbbbbb

Overlap of [6] bab=abb with [3] bba=abb:

ba b bba

Critical pair: baabb=abbba.

Reduce LHS:

[7]	(baa)bb
	⇒ bbbbbbb

Reduce RHS:

[2]	(abbb)a
	⇒ ba

Flip LHS and RHS.

Defines rule #3.

[9] ab=bbbbbbb

Overlap of [6] bab=abb with [6] bab=abb:

ba b bab

Critical pair: baabb=abbab.

Reduce LHS:

[4]	b(aab)b
	⇒ bbbbbbb

Reduce RHS:

[3]	a(bba)b
[2]	⇒ a(abbb)
	⇒ ab

Flip LHS and RHS.

Defines rule #2.

Referenced by [10].

[10] bbbbbbbbb=b

Overlap of [2] abbb=b with [9] ab=bbbbbbb:

abbb ab

Critical pair: bbbbbbbbb=b.

Defines rule #1.