Certificate for #16264 ⟨a, b | aab=ba, bbab=bb⟩

Completion settings:

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

[1] ba=aab

Axiom: aab=ba.

Flip LHS and RHS.

Defines rule #1.

Referenced by [2], [3], [4], [5].

[2] aaaabbb=bb

Axiom: bbab=bb.

Reduce LHS:

[1]	b(ba)b
[1]	⇒ (ba)abb
[1]	⇒ aa(ba)bb
	⇒ aaaabbb

Referenced by [3], [4], [5].

[3] bbb=bb

Overlap of [1] ba=aab with [2] aaaabbb=bb:

b a aaaabbb

Critical pair: bbb=aabaaabbb.

Reduce RHS:

[1]	aa(ba)aabbb
[1]	⇒ aaaa(ba)abbb
[1]	⇒ aaaaaa(ba)bbb
[2]	⇒ aaaa(aaaabbb)b
[2]	⇒ (aaaabbb)
	⇒ bb

Defines rule #3.

Referenced by [5].

[4] aaaaaaaabb=aaaabb

Overlap of [2] aaaabbb=bb with [1] ba=aab:

aaaabb b ba

Critical pair: aaaabbaab=bba.

Reduce LHS:

[1]	aaaab(ba)ab
[1]	⇒ aaaa(ba)abab
[1]	⇒ aaaaaa(ba)bab
[1]	⇒ aaaaaaaab(ba)b
[1]	⇒ aaaaaaaa(ba)abb
[1]	⇒ aaaaaaaaaa(ba)bb
[2]	⇒ aaaaaaaa(aaaabbb)
	⇒ aaaaaaaabb

Reduce RHS:

[1]	b(ba)
[1]	⇒ (ba)ab
[1]	⇒ aa(ba)b
	⇒ aaaabb

Referenced by [5].

[5] aaaabb=bb

Overlap of [3] bbb=bb with [1] ba=aab:

bb b ba

Critical pair: bbaab=bba.

Reduce LHS:

[1]	b(ba)ab
[1]	⇒ (ba)abab
[1]	⇒ aa(ba)bab
[1]	⇒ aaaab(ba)b
[1]	⇒ aaaa(ba)abb
[1]	⇒ aaaaaa(ba)bb
[4]	⇒ (aaaaaaaabb)b
[2]	⇒ (aaaabbb)
	⇒ bb

Reduce RHS:

[1]	b(ba)
[1]	⇒ (ba)ab
[1]	⇒ aa(ba)b
	⇒ aaaabb

Flip LHS and RHS.

Defines rule #2.