Certificate for #6747 ⟨a, b | aba=a, abba=bb⟩

Completion settings:

Reduction order:
- Left-to-right shortlex with b < a

[1] aba=a

Axiom: aba=a.

Defines rule #3.

Referenced by [3], [4].

[2] abba=bb

Axiom: abba=bb.

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

[3] abbb=bb

Overlap of [1] aba=a with [2] abba=bb:

ab a abba

Critical pair: abbb=abba.

Reduce RHS:

[2]	(abba)
	⇒ bb

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

[4] bbba=bb

Overlap of [2] abba=bb with [1] aba=a:

abb a aba

Critical pair: abba=bbba.

Reduce LHS:

[2]	(abba)
	⇒ bb

Flip LHS and RHS.

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

[5] abb=bba

Overlap of [3] abbb=bb with [4] bbba=bb:

a bbb bbba

Critical pair: abb=bba.

Referenced by [6], [7], [8], [9].

[6] bbab=bb

Overlap of [3] abbb=bb with [4] bbba=bb:

ab bb bbba

Critical pair: abbb=bbba.

Reduce LHS:

[5]	(abb)b
	⇒ bbab

Reduce RHS:

[4]	(bbba)
	⇒ bb

Referenced by [7].

[7] bbbb=bb

Overlap of [2] abba=bb with [5] abb=bba:

abb a abb

Critical pair: abbbba=bbbb.

Reduce LHS:

[5]	(abb)bba
[6]	⇒ (bbab)ba
[4]	⇒ (bbba)
	⇒ bb

Flip LHS and RHS.

Defines rule #4.

Referenced by [8].

[8] bba=bbb

Overlap of [3] abbb=bb with [7] bbbb=bb:

a bbb bbbb

Critical pair: abb=bbb.

Reduce LHS:

[5]	(abb)
	⇒ bba

Defines rule #1.

Referenced by [9].

[9] abb=bbb

Simplify [5] abb=bba.

Reduce RHS:

[8]	(bba)
	⇒ bbb

Defines rule #2.