Certificate for #4187 ⟨a, b | bab=aaa, bbb=b⟩

Completion settings:

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

[1] bab=aaa

Axiom: bab=aaa.

Defines rule #6.

Referenced by [3], [4], [5], [6], [7], [8].

[2] bbb=b

Axiom: bbb=b.

Defines rule #8.

Referenced by [4], [5].

[3] aaaab=baaaa

Overlap of [1] bab=aaa with [1] bab=aaa:

ba b bab

Critical pair: baaaa=aaaab.

Flip LHS and RHS.

Defines rule #4.

Referenced by [7], [8].

[4] aaabb=aaa

Overlap of [1] bab=aaa with [2] bbb=b:

ba b bbb

Critical pair: bab=aaabb.

Reduce LHS:

[1]	(bab)
	⇒ aaa

Flip LHS and RHS.

Defines rule #7.

[5] bbaaa=aaa

Overlap of [2] bbb=b with [1] bab=aaa:

bb b bab

Critical pair: bbaaa=bab.

Reduce RHS:

[1]	(bab)
	⇒ aaa

Defines rule #5.

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

[6] aaabaaa=baaaa

Overlap of [1] bab=aaa with [5] bbaaa=aaa:

ba b bbaaa

Critical pair: baaaa=aaabaaa.

Flip LHS and RHS.

Defines rule #3.

Referenced by [8].

[7] abaaaa=baaaaaaa

Overlap of [5] bbaaa=aaa with [3] aaaab=baaaa:

bba aa aaaab

Critical pair: bbabaaaa=aaaaab.

Reduce LHS:

[1]	b(bab)aaaa
	⇒ baaaaaaa

Reduce RHS:

[3]	a(aaaab)
	⇒ abaaaa

Flip LHS and RHS.

Defines rule #2.

Referenced by [8].

[8] aaaaaaaaaaaaa=aaaaa

Overlap of [6] aaabaaa=baaaa with [3] aaaab=baaaa:

aaabaa a aaaab

Critical pair: aaabaabaaaa=baaaaaaab.

Reduce LHS:

[7]	aaaba(abaaaa)
[1]	⇒ aaa(bab)aaaaaaa
	⇒ aaaaaaaaaaaaa

Reduce RHS:

[3]	baaa(aaaab)
[6]	⇒ b(aaabaaa)a
[5]	⇒ (bbaaa)aa
	⇒ aaaaa

Defines rule #1.