Certificate for #969 ⟨a, b | abaaaab=ba⟩

Completion settings:

[1] abaaaab=ba

Axiom: abaaaab=ba.

Referenced by [3].

[2] baaaa=c

Axiom: baaaa=c.

Referenced by [3], [4].

[3] ba=acb

Overlap of [1] abaaaab=ba with [2] baaaa=c:

a baaaab baaaa

Critical pair: acb=ba.

Flip LHS and RHS.

Defines rule #2.

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

[4] acacacacb=c

Overlap of [2] baaaa=c with [3] ba=acb:

baaaa ba

Critical pair: acbaaa=c.

Reduce LHS:

[3]ac(ba)aa
[3]⇒ acac(ba)a
[3]⇒ acacac(ba)
⇒ acacacacb

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

[5] acbcacacacb=bc

Overlap of [3] ba=acb with [4] acacacacb=c:

b a acacacacb

Critical pair: bc=acbcacacacb.

Flip LHS and RHS.

Referenced by [7].

[6] ca=acc

Overlap of [4] acacacacb=c with [3] ba=acb:

acacacac b ba

Critical pair: acacacacacb=ca.

Reduce LHS:

[4]ac(acacacacb)
⇒ acc

Flip LHS and RHS.

Defines rule #3.

Referenced by [7], [8].

[7] aaaacccccccccccccccbcccccccccccccccb=bc

Simplify [5] acbcacacacb=bc.

Reduce LHS:

[6]acb(ca)cacacb
[3]⇒ ac(ba)cccacacb
[6]⇒ a(ca)cbcccacacb
[6]⇒ aacccbcc(ca)cacb
[6]⇒ aacccbc(ca)cccacb
[6]⇒ aacccb(ca)cccccacb
[3]⇒ aaccc(ba)cccccccacb
[6]⇒ aacc(ca)cbcccccccacb
[6]⇒ aac(ca)cccbcccccccacb
[6]⇒ aa(ca)cccccbcccccccacb
[6]⇒ aaacccccccbcccccc(ca)cb
[6]⇒ aaacccccccbccccc(ca)cccb
[6]⇒ aaacccccccbcccc(ca)cccccb
[6]⇒ aaacccccccbccc(ca)cccccccb
[6]⇒ aaacccccccbcc(ca)cccccccccb
[6]⇒ aaacccccccbc(ca)cccccccccccb
[6]⇒ aaacccccccb(ca)cccccccccccccb
[3]⇒ aaaccccccc(ba)cccccccccccccccb
[6]⇒ aaacccccc(ca)cbcccccccccccccccb
[6]⇒ aaaccccc(ca)cccbcccccccccccccccb
[6]⇒ aaacccc(ca)cccccbcccccccccccccccb
[6]⇒ aaaccc(ca)cccccccbcccccccccccccccb
[6]⇒ aaacc(ca)cccccccccbcccccccccccccccb
[6]⇒ aaac(ca)cccccccccccbcccccccccccccccb
[6]⇒ aaa(ca)cccccccccccccbcccccccccccccccb
⇒ aaaacccccccccccccccbcccccccccccccccb

Referenced by [9].

[8] aaaacccccccccccccccb=c

Overlap of [4] acacacacb=c with [6] ca=acc:

a cacacacb ca

Critical pair: aacccacacb=c.

Reduce LHS:

[6]aacc(ca)cacb
[6]⇒ aac(ca)cccacb
[6]⇒ aa(ca)cccccacb
[6]⇒ aaacccccc(ca)cb
[6]⇒ aaaccccc(ca)cccb
[6]⇒ aaacccc(ca)cccccb
[6]⇒ aaaccc(ca)cccccccb
[6]⇒ aaacc(ca)cccccccccb
[6]⇒ aaac(ca)cccccccccccb
[6]⇒ aaa(ca)cccccccccccccb
⇒ aaaacccccccccccccccb

Defines rule #4.

Referenced by [9].

[9] ccccccccccccccccb=bc

Overlap of [7] aaaacccccccccccccccbcccccccccccccccb=bc with [8] aaaacccccccccccccccb=c:

aaaacccccccccccccccbcccccccccccccccb aaaacccccccccccccccb

Critical pair: ccccccccccccccccb=bc.

Defines rule #1.