Certificate for #1812 ⟨a, b, c | aba=ab, bbc=1⟩

Completion settings:

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

[1] aba=ab

Axiom: aba=ab.

Referenced by [4].

[2] bbc=1

Axiom: bbc=1.

Defines rule #5.

Referenced by [6], [8].

[3] ab=d

Axiom: ab=d.

Referenced by [4], [5], [6], [7], [10].

[4] aba=d

Simplify [1] aba=ab.

Reduce RHS:

[3]	(ab)
	⇒ d

Referenced by [5].

[5] da=d

Overlap of [4] aba=d with [3] ab=d:

aba ab

Critical pair: da=d.

Referenced by [7].

[6] a=dbc

Overlap of [3] ab=d with [2] bbc=1:

a b bbc

Critical pair: a=dbc.

Referenced by [9].

[7] db=dd

Overlap of [5] da=d with [3] ab=d:

d a ab

Critical pair: dd=db.

Flip LHS and RHS.

Defines rule #2.

Referenced by [8], [9].

[8] dddc=d

Overlap of [7] db=dd with [2] bbc=1:

d b bbc

Critical pair: d=ddbc.

Reduce RHS:

[7]	d(db)c
	⇒ dddc

Flip LHS and RHS.

Defines rule #1.

[9] a=ddc

Simplify [6] a=dbc.

Reduce RHS:

[7]	(db)c
	⇒ ddc

Defines rule #4.

Referenced by [10].

[10] ddcb=d

Overlap of [3] ab=d with [9] a=ddc:

ab a

Critical pair: ddcb=d.

Defines rule #3.