Certificate for #2609 ⟨a, b | ababba=baba⟩

Completion settings:

Auxiliary generators:
1. c = ba
2. d = bcc
Reduction order:
- Left-to-right recursive path with deg(c) = 0; deg(a) = deg(d) = deg(b) = 1, a < d < b

[1] ababba=baba

Axiom: ababba=baba.

Referenced by [4].

[2] ba=c

Axiom: ba=c.

Defines rule #6.

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

[3] bcc=d

Axiom: bcc=d.

Defines rule #2.

Referenced by [6], [7], [8], [9], [10].

[4] ababba=cc

Simplify [1] ababba=baba.

Reduce RHS:

[2]	(ba)ba
[2]	⇒ c(ba)
	⇒ cc

Referenced by [5].

[5] acbc=cc

Overlap of [4] ababba=cc with [2] ba=c:

a babba ba

Critical pair: acbba=cc.

Reduce LHS:

[2]	acb(ba)
	⇒ acbc

Defines rule #5.

Referenced by [6], [7].

[6] ccbc=d

Overlap of [2] ba=c with [5] acbc=cc:

b a acbc

Critical pair: bcc=ccbc.

Reduce LHS:

[3]	(bcc)
	⇒ d

Flip LHS and RHS.

Defines rule #3.

Referenced by [8], [9], [10].

[7] acd=ccc

Overlap of [5] acbc=cc with [3] bcc=d:

ac bc bcc

Critical pair: acd=ccc.

Defines rule #4.

[8] bd=dbc

Overlap of [3] bcc=d with [6] ccbc=d:

b cc ccbc

Critical pair: bd=dbc.

Defines rule #7.

[9] bcd=dcbc

Overlap of [3] bcc=d with [6] ccbc=d:

bc c ccbc

Critical pair: bcd=dcbc.

Defines rule #8.

[10] ccd=dc

Overlap of [6] ccbc=d with [3] bcc=d:

cc bc bcc

Critical pair: ccd=dc.

Defines rule #1.