Certificate for #15696 ⟨a, b | aab=ba, ababb=b⟩

Completion settings:

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

[1] aab=ba

Axiom: aab=ba.

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

[2] ababb=b

Axiom: ababb=b.

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

[3] bbab=ab

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

a ab ababb

Critical pair: ab=baabb.

Reduce RHS:

[1]	b(aab)b
	⇒ bbab

Flip LHS and RHS.

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

[4] babab=aba

Overlap of [1] aab=ba with [3] bbab=ab:

aa b bbab

Critical pair: aaab=babab.

Reduce LHS:

[1]	a(aab)
	⇒ aba

Flip LHS and RHS.

Referenced by [5], [6].

[5] ab=baa

Overlap of [2] ababb=b with [3] bbab=ab:

abab b bbab

Critical pair: ababab=bbab.

Reduce LHS:

[4]	a(babab)
[1]	⇒ (aab)a
	⇒ baa

Reduce RHS:

[3]	(bbab)
	⇒ ab

Flip LHS and RHS.

Defines rule #2.

Referenced by [6], [8].

[6] bbb=baaa

Overlap of [3] bbab=ab with [2] ababb=b:

bb ab ababb

Critical pair: bbb=ababb.

Reduce RHS:

[5]	(ab)abb
[1]	⇒ ba(aab)b
[4]	⇒ (babab)
[5]	⇒ (ab)a
	⇒ baaa

Referenced by [7], [9].

[7] baaaa=ba

Overlap of [3] bbab=ab with [3] bbab=ab:

bba b bbab

Critical pair: bbaab=abbab.

Reduce LHS:

[1]	bb(aab)
[6]	⇒ (bbb)a
	⇒ baaaa

Reduce RHS:

[3]	a(bbab)
[1]	⇒ (aab)
	⇒ ba

Referenced by [8].

[8] baaa=b

Overlap of [2] ababb=b with [5] ab=baa:

ababb ab

Critical pair: baaabb=b.

Reduce LHS:

[5]	baa(ab)b
[5]	⇒ ba(ab)aab
[7]	⇒ ba(baaaa)b
[5]	⇒ b(ab)ab
[5]	⇒ bbaa(ab)
[5]	⇒ bba(ab)aa
[3]	⇒ (bbab)aaaa
[7]	⇒ a(baaaa)
[5]	⇒ (ab)a
	⇒ baaa

Defines rule #1.

Referenced by [9].

[9] bbb=b

Simplify [6] bbb=baaa.

Reduce RHS:

[8]	(baaa)
	⇒ b

Defines rule #3.