Certificate for #19657 ⟨a, b | aba=a, aabba=bb⟩

Completion settings:

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

[1] aba=a

Axiom: aba=a.

Defines rule #4.

Referenced by [3], [4].

[2] aabba=bb

Axiom: aabba=bb.

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

[3] abbb=bb

Overlap of [1] aba=a with [2] aabba=bb:

ab a aabba

Critical pair: abbb=aabba.

Reduce RHS:

[2]	(aabba)
	⇒ bb

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

[4] bbba=bb

Overlap of [2] aabba=bb with [1] aba=a:

aabb a aba

Critical pair: aabba=bbba.

Reduce LHS:

[2]	(aabba)
	⇒ bb

Flip LHS and RHS.

Referenced by [6].

[5] bbbbb=bb

Overlap of [2] aabba=bb with [3] abbb=bb:

aabb a abbb

Critical pair: aabbbb=bbbbb.

Reduce LHS:

[3]	a(abbb)b
[3]	⇒ (abbb)
	⇒ bb

Flip LHS and RHS.

Defines rule #1.

[6] bba=abb

Overlap of [3] abbb=bb with [4] bbba=bb:

a bbb bbba

Critical pair: abb=bba.

Flip LHS and RHS.

Referenced by [7], [8].

[7] abb=bbbb

Overlap of [6] bba=abb with [2] aabba=bb:

bb a aabba

Critical pair: bbbb=abbabba.

Reduce RHS:

[6]	a(bba)bba
[3]	⇒ a(abbb)ba
[3]	⇒ (abbb)a
[6]	⇒ (bba)
	⇒ abb

Flip LHS and RHS.

Defines rule #2.

Referenced by [8].

[8] bba=bbbb

Simplify [6] bba=abb.

Reduce RHS:

[7]	(abb)
	⇒ bbbb

Defines rule #3.