Certificate for #15716 ⟨a, b | aab=ba, babab=b⟩

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 #2.

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

[2] aaaaaabbb=b

Axiom: babab=b.

Reduce LHS:

[1]	(ba)bab
[1]	⇒ aab(ba)b
[1]	⇒ aa(ba)abb
[1]	⇒ aaaa(ba)bb
	⇒ aaaaaabbb

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

[3] aaaaaabb=bb

Overlap of [1] ba=aab with [2] aaaaaabbb=b:

b a aaaaaabbb

Critical pair: bb=aabaaaaabbb.

Reduce RHS:

[1]	aa(ba)aaaabbb
[1]	⇒ aaaa(ba)aaabbb
[1]	⇒ aaaaaa(ba)aabbb
[1]	⇒ aaaaaaaa(ba)abbb
[1]	⇒ aaaaaaaaaa(ba)bbb
[2]	⇒ aaaaaa(aaaaaabbb)b
	⇒ aaaaaabb

Flip LHS and RHS.

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

[4] aabbb=aab

Overlap of [2] aaaaaabbb=b with [1] ba=aab:

aaaaaabb b ba

Critical pair: aaaaaabbaab=ba.

Reduce LHS:

[3]	(aaaaaabb)aab
[1]	⇒ b(ba)ab
[1]	⇒ (ba)abab
[1]	⇒ aa(ba)bab
[1]	⇒ aaaab(ba)b
[1]	⇒ aaaa(ba)abb
[1]	⇒ aaaaaa(ba)bb
[3]	⇒ aa(aaaaaabb)b
	⇒ aabbb

Reduce RHS:

[1]	(ba)
	⇒ aab

Referenced by [6].

[5] bbb=b

Overlap of [2] aaaaaabbb=b with [3] aaaaaabb=bb:

aaaaaabbb aaaaaabb

Critical pair: bbb=b.

Defines rule #3.

Referenced by [6].

[6] aaaaaab=b

Overlap of [3] aaaaaabb=bb with [4] aabbb=aab:

aaaa aabb aabbb

Critical pair: aaaaaab=bbb.

Reduce RHS:

[5]	(bbb)
	⇒ b

Defines rule #1.