Certificate for #16267 ⟨a, b | aab=ba, bbba=ba⟩

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], [5].

[2] aaaaaaaabbb=aab

Axiom: bbba=ba.

Reduce LHS:

[1]	bb(ba)
[1]	⇒ b(ba)ab
[1]	⇒ (ba)abab
[1]	⇒ aa(ba)bab
[1]	⇒ aaaab(ba)b
[1]	⇒ aaaa(ba)abb
[1]	⇒ aaaaaa(ba)bb
	⇒ aaaaaaaabbb

Reduce RHS:

[1]	(ba)
	⇒ aab

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

[3] aaaaaaaaaabb=aaaabb

Overlap of [1] ba=aab with [2] aaaaaaaabbb=aab:

b a aaaaaaaabbb

Critical pair: baab=aabaaaaaaabbb.

Reduce LHS:

[1]	(ba)ab
[1]	⇒ aa(ba)b
	⇒ aaaabb

Reduce RHS:

[1]	aa(ba)aaaaaabbb
[1]	⇒ aaaa(ba)aaaaabbb
[1]	⇒ aaaaaa(ba)aaaabbb
[1]	⇒ aaaaaaaa(ba)aaabbb
[1]	⇒ aaaaaaaaaa(ba)aabbb
[1]	⇒ aaaaaaaaaaaa(ba)abbb
[1]	⇒ aaaaaaaaaaaaaa(ba)bbb
[2]	⇒ aaaaaaaa(aaaaaaaabbb)b
	⇒ aaaaaaaaaabb

Flip LHS and RHS.

Referenced by [4].

[4] aaaabbb=aaaab

Overlap of [2] aaaaaaaabbb=aab with [1] ba=aab:

aaaaaaaabb b ba

Critical pair: aaaaaaaabbaab=aaba.

Reduce LHS:

[1]	aaaaaaaab(ba)ab
[1]	⇒ aaaaaaaa(ba)abab
[1]	⇒ aaaaaaaaaa(ba)bab
[3]	⇒ aa(aaaaaaaaaabb)ab
[1]	⇒ aaaaaab(ba)b
[1]	⇒ aaaaaa(ba)abb
[1]	⇒ aaaaaaaa(ba)bb
[3]	⇒ (aaaaaaaaaabb)b
	⇒ aaaabbb

Reduce RHS:

[1]	aa(ba)
	⇒ aaaab

Referenced by [5], [7].

[5] aaaaaaaabb=aabb

Overlap of [1] ba=aab with [4] aaaabbb=aaaab:

b a aaaabbb

Critical pair: baaaab=aabaaabbb.

Reduce LHS:

[1]	(ba)aaab
[1]	⇒ aa(ba)aab
[1]	⇒ aaaa(ba)ab
[1]	⇒ aaaaaa(ba)b
	⇒ aaaaaaaabb

Reduce RHS:

[1]	aa(ba)aabbb
[1]	⇒ aaaa(ba)abbb
[1]	⇒ aaaaaa(ba)bbb
[2]	⇒ (aaaaaaaabbb)b
	⇒ aabb

Referenced by [6], [7].

[6] aabbb=aab

Overlap of [2] aaaaaaaabbb=aab with [5] aaaaaaaabb=aabb:

aaaaaaaabbb aaaaaaaabb

Critical pair: aabbb=aab.

Defines rule #3.

Referenced by [7].

[7] aaaaaaaab=aab

Overlap of [5] aaaaaaaabb=aabb with [4] aaaabbb=aaaab:

aaaa aaaabb aaaabbb

Critical pair: aaaaaaaab=aabbb.

Reduce RHS:

[6]	(aabbb)
	⇒ aab

Defines rule #1.