Certificate for #15520 ⟨a, b | aaa=bb, aabbb=b⟩

Completion settings:

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

[1] aaa=bb

Axiom: aaa=bb.

Defines rule #4.

Referenced by [3], [4], [6].

[2] aabbb=b

Axiom: aabbb=b.

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

[3] bba=abb

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

a aa aaa

Critical pair: abb=bba.

Flip LHS and RHS.

Referenced by [5], [6].

[4] ab=bbbbb

Overlap of [1] aaa=bb with [2] aabbb=b:

a aa aabbb

Critical pair: ab=bbbbb.

Defines rule #2.

Referenced by [5].

[5] ba=bbbbbbbbbbbbbbb

Overlap of [2] aabbb=b with [3] bba=abb:

aab bb bba

Critical pair: aababb=ba.

Reduce LHS:

[4]	a(ab)abb
[3]	⇒ abbb(bba)bb
[3]	⇒ ab(bba)bbbb
[4]	⇒ (ab)abbbbbb
[3]	⇒ bbb(bba)bbbbbb
[3]	⇒ b(bba)bbbbbbbb
[4]	⇒ b(ab)bbbbbbbbb
	⇒ bbbbbbbbbbbbbbb

Flip LHS and RHS.

Referenced by [6], [8].

[6] bbbbbbbbbbbbb=bbb

Overlap of [5] ba=bbbbbbbbbbbbbbb with [1] aaa=bb:

b a aaa

Critical pair: bbb=bbbbbbbbbbbbbbbaa.

Reduce RHS:

[3]	bbbbbbbbbbbbb(bba)a
[3]	⇒ bbbbbbbbbbb(bba)bba
[3]	⇒ bbbbbbbbb(bba)bbbba
[3]	⇒ bbbbbbb(bba)bbbbbba
[3]	⇒ bbbbb(bba)bbbbbbbba
[3]	⇒ bbb(bba)bbbbbbbbbba
[3]	⇒ b(bba)bbbbbbbbbbbba
[3]	⇒ babbbbbbbbbbbb(bba)
[3]	⇒ babbbbbbbbbb(bba)bb
[3]	⇒ babbbbbbbb(bba)bbbb
[3]	⇒ babbbbbb(bba)bbbbbb
[3]	⇒ babbbb(bba)bbbbbbbb
[3]	⇒ babb(bba)bbbbbbbbbb
[3]	⇒ ba(bba)bbbbbbbbbbbb
[2]	⇒ b(aabbb)bbbbbbbbbbb
	⇒ bbbbbbbbbbbbb

Flip LHS and RHS.

Referenced by [7].

[7] bbbbbbbbbbb=b

Overlap of [2] aabbb=b with [6] bbbbbbbbbbbbb=bbb:

aa bbb bbbbbbbbbbbbb

Critical pair: aabbb=bbbbbbbbbbb.

Reduce LHS:

[2]	(aabbb)
	⇒ b

Flip LHS and RHS.

Defines rule #1.

Referenced by [8].

[8] ba=bbbbb

Simplify [5] ba=bbbbbbbbbbbbbbb.

Reduce RHS:

[7]	(bbbbbbbbbbb)bbbb
	⇒ bbbbb

Defines rule #3.