Certificate for #15953 ⟨a, b | aaa=aa, aaba=ab⟩

Back:⟨a, b | aaa=aa, aaba=ab⟩

Completion settings:

[1] aaa=aa

Axiom: aaa=aa.

Defines rule #5.

Referenced by [6], [7].

[2] aaba=ab

Axiom: aaba=ab.

Referenced by [4].

[3] ab=c

Axiom: ab=c.

Defines rule #2.

Referenced by [4], [5], [6], [9].

[4] aaba=c

Simplify [2] aaba=ab.

Reduce RHS:

[3](ab)
⇒ c

Referenced by [5].

[5] aca=c

Overlap of [4] aaba=c with [3] ab=c:

a aba ab

Critical pair: aca=c.

Referenced by [7], [8].

[6] aac=ac

Overlap of [1] aaa=aa with [3] ab=c:

aa a ab

Critical pair: aac=aab.

Reduce RHS:

[3]a(ab)
⇒ ac

Referenced by [7].

[7] ac=c

Overlap of [1] aaa=aa with [5] aca=c:

aa a aca

Critical pair: aac=aaca.

Reduce LHS:

[6](aac)
⇒ ac

Reduce RHS:

[6](aac)a
[5]⇒ (aca)
⇒ c

Defines rule #1.

Referenced by [8].

[8] ca=c

Overlap of [5] aca=c with [7] ac=c:

aca ac

Critical pair: ca=c.

Defines rule #3.

Referenced by [9].

[9] cb=cc

Overlap of [8] ca=c with [3] ab=c:

c a ab

Critical pair: cc=cb.

Flip LHS and RHS.

Defines rule #4.