Certificate for #1965 ⟨a, b, c | abc=ba, aca=1⟩

Back:⟨a, b, c | abc=ba, aca=1⟩

Completion settings:

[1] ba=abc

Axiom: abc=ba.

Flip LHS and RHS.

Defines rule #2.

Referenced by [4].

[2] aca=1

Axiom: aca=1.

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

[3] ca=ac

Overlap of [2] aca=1 with [2] aca=1:

ac a aca

Critical pair: ac=ca.

Flip LHS and RHS.

Defines rule #3.

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

[4] aabccc=b

Overlap of [1] ba=abc with [2] aca=1:

b a aca

Critical pair: b=abcca.

Reduce RHS:

[3]abc(ca)
[3]⇒ ab(ca)c
[1]⇒ a(ba)cc
⇒ aabccc

Flip LHS and RHS.

Referenced by [5].

[5] bccc=cb

Overlap of [3] ca=ac with [4] aabccc=b:

c a aabccc

Critical pair: cb=acabccc.

Reduce RHS:

[2](aca)bccc
⇒ bccc

Flip LHS and RHS.

Defines rule #1.

[6] aac=1

Overlap of [2] aca=1 with [3] ca=ac:

a ca ca

Critical pair: aac=1.

Defines rule #4.