Certificate for #19554 ⟨a, b | aab=b, abbaa=ba⟩

Completion settings:

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

[1] aab=b

Axiom: aab=b.

Defines rule #4.

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

[2] abbaa=ba

Axiom: abbaa=ba.

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

[3] bbaa=aba

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

a ab abbaa

Critical pair: aba=bbaa.

Flip LHS and RHS.

Referenced by [6], [8].

[4] bab=abbb

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

abb aa aab

Critical pair: abbb=bab.

Flip LHS and RHS.

Defines rule #2.

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

[5] bbbbb=bb

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

abba a aab

Critical pair: abbab=baab.

Reduce LHS:

[4]	ab(bab)
[4]	⇒ a(bab)bb
[1]	⇒ (aab)bbbb
	⇒ bbbbb

Reduce RHS:

[1]	b(aab)
	⇒ bb

Defines rule #1.

Referenced by [6], [8].

[6] abbbba=aba

Overlap of [5] bbbbb=bb with [3] bbaa=aba:

bbb bb bbaa

Critical pair: bbbaba=bbaa.

Reduce LHS:

[4]	bb(bab)a
[4]	⇒ b(bab)bba
[5]	⇒ ba(bbbbb)a
[4]	⇒ (bab)ba
	⇒ abbbba

Reduce RHS:

[3]	(bbaa)
	⇒ aba

Referenced by [7].

[7] bbbba=ba

Overlap of [1] aab=b with [6] abbbba=aba:

a ab abbbba

Critical pair: aaba=bbbba.

Reduce LHS:

[1]	(aab)a
	⇒ ba

Flip LHS and RHS.

Defines rule #3.

Referenced by [8].

[8] baa=abba

Overlap of [7] bbbba=ba with [3] bbaa=aba:

bb bba bbaa

Critical pair: bbaba=baa.

Reduce LHS:

[4]	b(bab)a
[4]	⇒ (bab)bba
[5]	⇒ a(bbbbb)a
	⇒ abba

Flip LHS and RHS.

Defines rule #5.