Certificate for #20189 ⟨a, b | aba=a, aaab=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 [4], [7], [8].

[2] bbb=aaab

Axiom: aaab=bbb.

Flip LHS and RHS.

Defines rule #5.

Referenced by [3], [5].

[3] aaabb=baaab

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

b bb bbb

Critical pair: baaab=aaabb.

Flip LHS and RHS.

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

[4] abbaaab=baaab

Overlap of [1] aba=a with [3] aaabb=baaab:

ab a aaabb

Critical pair: abbaaab=aaabb.

Reduce RHS:

[3]	(aaabb)
	⇒ baaab

Referenced by [6].

[5] bbaaab=aaaaaab

Overlap of [3] aaabb=baaab with [2] bbb=aaab:

aaa bb bbb

Critical pair: aaaaaab=baaabb.

Reduce RHS:

[3]	b(aaabb)
	⇒ bbaaab

Flip LHS and RHS.

Referenced by [6].

[6] baaab=aaaaaaab

Simplify [4] abbaaab=baaab.

Reduce LHS:

[5]	a(bbaaab)
	⇒ aaaaaaab

Flip LHS and RHS.

Referenced by [7].

[7] baaa=aaaaaaa

Overlap of [6] baaab=aaaaaaab with [1] aba=a:

baa ab aba

Critical pair: baaa=aaaaaaaba.

Reduce RHS:

[1]	aaaaaa(aba)
	⇒ aaaaaaa

Defines rule #2.

Referenced by [8], [9].

[8] aaaaaaaa=aaa

Overlap of [1] aba=a with [7] baaa=aaaaaaa:

a ba baaa

Critical pair: aaaaaaaa=aaa.

Defines rule #1.

[9] aaabb=aaaaaaab

Simplify [3] aaabb=baaab.

Reduce RHS:

[7]	(baaa)b
	⇒ aaaaaaab

Defines rule #4.