Certificate for #7421 ⟨a, b, c | ab=1, acba=bc⟩

Back:⟨a, b, c | ab=1, acba=bc⟩

Completion settings:

[1] ab=1

Axiom: ab=1.

Defines rule #1.

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

[2] acba=bc

Axiom: acba=bc.

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

[3] bcb=acb

Overlap of [2] acba=bc with [1] ab=1:

acb a ab

Critical pair: acb=bcb.

Flip LHS and RHS.

Referenced by [4].

[4] aacb=cb

Overlap of [1] ab=1 with [3] bcb=acb:

a b bcb

Critical pair: aacb=cb.

Referenced by [5], [6].

[5] cba=c

Overlap of [4] aacb=cb with [2] acba=bc:

a acb acba

Critical pair: abc=cba.

Reduce LHS:

[1](ab)c
⇒ c

Flip LHS and RHS.

Defines rule #4.

Referenced by [6], [7].

[6] aac=c

Overlap of [4] aacb=cb with [5] cba=c:

aa cb cba

Critical pair: aac=cba.

Reduce RHS:

[5](cba)
⇒ c

Defines rule #3.

[7] bc=ac

Overlap of [2] acba=bc with [5] cba=c:

a cba cba

Critical pair: ac=bc.

Flip LHS and RHS.

Defines rule #2.