Certificate for #20197 ⟨a, b | aba=a, aaba=bbb⟩

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 [2], [4], [5], [8].

[2] bbb=aa

Axiom: aaba=bbb.

Reduce LHS:

[1]	a(aba)
	⇒ aa

Flip LHS and RHS.

Defines rule #5.

Referenced by [3], [6].

[3] aab=baa

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

b bb bbb

Critical pair: baa=aab.

Flip LHS and RHS.

Referenced by [4], [5], [9].

[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].

[6] bbaa=aaaaa

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

bb b baaa

Critical pair: bbaa=aaaaa.

Referenced by [7].

[7] baa=aaaaaa

Simplify [4] abbaa=baa.

Reduce LHS:

[6]	a(bbaa)
	⇒ aaaaaa

Flip LHS and RHS.

Defines rule #2.

Referenced by [8], [9].

[8] aaaaaaa=aa

Overlap of [1] aba=a with [7] baa=aaaaaa:

a ba baa

Critical pair: aaaaaaa=aa.

Defines rule #1.

[9] aab=aaaaaa

Simplify [3] aab=baa.

Reduce RHS:

[7]	(baa)
	⇒ aaaaaa

Defines rule #4.