Certificate for #1991 ⟨a, b | aaa=a, abba=b⟩

Completion settings:

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

[1] aaa=a

Axiom: aaa=a.

Defines rule #4.

Referenced by [3], [4].

[2] abba=b

Axiom: abba=b.

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

[3] aab=b

Overlap of [1] aaa=a with [2] abba=b:

aa a abba

Critical pair: aab=abba.

Reduce RHS:

[2]	(abba)
	⇒ b

Defines rule #3.

Referenced by [5], [8].

[4] baa=b

Overlap of [2] abba=b with [1] aaa=a:

abb a aaa

Critical pair: abba=baa.

Reduce LHS:

[2]	(abba)
	⇒ b

Flip LHS and RHS.

Referenced by [6].

[5] bba=ab

Overlap of [3] aab=b with [2] abba=b:

a ab abba

Critical pair: ab=bba.

Flip LHS and RHS.

Referenced by [7].

[6] ba=abb

Overlap of [2] abba=b with [4] baa=b:

ab ba baa

Critical pair: abb=ba.

Flip LHS and RHS.

Defines rule #2.

Referenced by [7].

[7] abbbb=ab

Simplify [5] bba=ab.

Reduce LHS:

[6]	b(ba)
[6]	⇒ (ba)bb
	⇒ abbbb

Referenced by [8].

[8] bbbb=b

Overlap of [3] aab=b with [7] abbbb=ab:

a ab abbbb

Critical pair: aab=bbbb.

Reduce LHS:

[3]	(aab)
	⇒ b

Flip LHS and RHS.

Defines rule #1.