Certificate for #18950 ⟨a, b | aab=a, bbbabb=a⟩

Completion settings:

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

[1] aab=a

Axiom: aab=a.

Referenced by [3], [5], [6], [7], [8].

[2] bbbabb=a

Axiom: bbbabb=a.

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

[3] abbabb=aaa

Overlap of [1] aab=a with [2] bbbabb=a:

aa b bbbabb

Critical pair: aaa=abbabb.

Flip LHS and RHS.

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

[4] ababb=bbbaa

Overlap of [2] bbbabb=a with [2] bbbabb=a:

bbba bb bbbabb

Critical pair: bbbaa=ababb.

Flip LHS and RHS.

Referenced by [5].

[5] bbbaa=aaaa

Overlap of [1] aab=a with [3] abbabb=aaa:

a ab abbabb

Critical pair: aaaa=ababb.

Reduce RHS:

[4]	(ababb)
	⇒ bbbaa

Flip LHS and RHS.

Referenced by [6], [8].

[6] ab=aaaaa

Overlap of [2] bbbabb=a with [3] abbabb=aaa:

bbb abb abbabb

Critical pair: bbbaaa=aabb.

Reduce LHS:

[5]	(bbbaa)a
	⇒ aaaaa

Reduce RHS:

[1]	(aab)b
	⇒ ab

Flip LHS and RHS.

Defines rule #2.

Referenced by [7].

[7] aaaaaa=a

Overlap of [3] abbabb=aaa with [2] bbbabb=a:

abba bb bbbabb

Critical pair: abbaa=aaababb.

Reduce LHS:

[6]	(ab)baa
[1]	⇒ aaa(aab)aa
	⇒ aaaaaa

Reduce RHS:

[1]	a(aab)abb
[1]	⇒ a(aab)b
[1]	⇒ (aab)
	⇒ a

Defines rule #1.

[8] bbba=aaa

Overlap of [5] bbbaa=aaaa with [1] aab=a:

bbb aa aab

Critical pair: bbba=aaaab.

Reduce RHS:

[1]	aa(aab)
	⇒ aaa

Defines rule #3.