Certificate for #19701 ⟨a, b | aba=a, bbbbb=aa⟩

Completion settings:

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

[1] aba=a

Axiom: aba=a.

Defines rule #3.

Referenced by [4], [5], [10].

[2] bbbbb=aa

Axiom: bbbbb=aa.

Defines rule #5.

Referenced by [3], [6].

[3] aab=baa

Overlap of [2] bbbbb=aa with [2] bbbbb=aa:

b bbbb bbbbb

Critical pair: baa=aab.

Flip LHS and RHS.

Referenced by [4], [5], [7], [8], [9], [11].

[4] abbaa=baa

Overlap of [1] aba=a with [3] aab=baa:

ab a aab

Critical pair: abbaa=aab.

Reduce RHS:

[3]	(aab)
	⇒ baa

Referenced by [7].

[5] baaa=aa

Overlap of [3] aab=baa with [1] aba=a:

a ab aba

Critical pair: aa=baaa.

Flip LHS and RHS.

Referenced by [6], [9].

[6] bbbbaa=aaaaa

Overlap of [2] bbbbb=aa with [5] baaa=aa:

bbbb b baaa

Critical pair: bbbbaa=aaaaa.

Referenced by [8].

[7] abbbaa=bbaa

Overlap of [4] abbaa=baa with [3] aab=baa:

abb aa aab

Critical pair: abbbaa=baab.

Reduce RHS:

[3]	b(aab)
	⇒ bbaa

Referenced by [8].

[8] bbbaa=aaaaaa

Overlap of [7] abbbaa=bbaa with [3] aab=baa:

abbb aa aab

Critical pair: abbbbaa=bbaab.

Reduce LHS:

[6]	a(bbbbaa)
	⇒ aaaaaa

Reduce RHS:

[3]	bb(aab)
	⇒ bbbaa

Flip LHS and RHS.

Referenced by [9].

[9] baa=aaaaaaaa

Overlap of [3] aab=baa with [8] bbbaa=aaaaaa:

aa b bbbaa

Critical pair: aaaaaaaa=baabbaa.

Reduce RHS:

[3]	b(aab)baa
[3]	⇒ bb(aab)aa
[5]	⇒ bb(baaa)a
[5]	⇒ b(baaa)
	⇒ baa

Flip LHS and RHS.

Defines rule #2.

Referenced by [10], [11].

[10] aaaaaaaaa=aa

Overlap of [1] aba=a with [9] baa=aaaaaaaa:

a ba baa

Critical pair: aaaaaaaaa=aa.

Defines rule #1.

[11] aab=aaaaaaaa

Simplify [3] aab=baa.

Reduce RHS:

[9]	(baa)
	⇒ aaaaaaaa

Defines rule #4.