Certificate for #3100 ⟨a, b, c | ab=aa, bbcb=1⟩

Back:⟨a, b, c | ab=aa, bbcb=1⟩

Completion settings:

[1] ab=aa

Axiom: ab=aa.

Defines rule #1.

Referenced by [3], [5].

[2] bbcb=1

Axiom: bbcb=1.

Referenced by [3], [4], [6], [8].

[3] aaacb=a

Overlap of [1] ab=aa with [2] bbcb=1:

a b bbcb

Critical pair: a=aabcb.

Reduce RHS:

[1]a(ab)cb
⇒ aaacb

Flip LHS and RHS.

Referenced by [7].

[4] bcb=bbc

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

bbc b bbcb

Critical pair: bbc=bcb.

Flip LHS and RHS.

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

[5] aacb=aaac

Overlap of [1] ab=aa with [4] bcb=bbc:

a b bcb

Critical pair: abbc=aacb.

Reduce LHS:

[1](ab)bc
[1]⇒ a(ab)c
⇒ aaac

Flip LHS and RHS.

Referenced by [7].

[6] cb=bc

Overlap of [2] bbcb=1 with [4] bcb=bbc:

bbc b bcb

Critical pair: bbcbbc=cb.

Reduce LHS:

[2](bbcb)bc
⇒ bc

Flip LHS and RHS.

Defines rule #2.

[7] aaaac=a

Simplify [3] aaacb=a.

Reduce LHS:

[5]a(aacb)
⇒ aaaac

Defines rule #4.

[8] bbbc=1

Overlap of [2] bbcb=1 with [4] bcb=bbc:

b bcb bcb

Critical pair: bbbc=1.

Defines rule #3.