Certificate for #298 ⟨a, b | aaabbbba=1⟩

Back:⟨a, b | aaabbbba=1⟩

Completion settings:

[1] aaabbbba=1

Axiom: aaabbbba=1.

Referenced by [4].

[2] aaaa=c

Axiom: aaaa=c.

Defines rule #7.

Referenced by [5], [7], [12].

[3] bbbb=d

Axiom: bbbb=d.

Defines rule #8.

Referenced by [4], [6].

[4] aaada=1

Overlap of [1] aaabbbba=1 with [3] bbbb=d:

aaa bbbba bbbb

Critical pair: aaada=1.

Referenced by [7], [8], [9], [11].

[5] ca=ac

Overlap of [2] aaaa=c with [2] aaaa=c:

a aaa aaaa

Critical pair: ac=ca.

Flip LHS and RHS.

Defines rule #1.

[6] db=bd

Overlap of [3] bbbb=d with [3] bbbb=d:

b bbb bbbb

Critical pair: bd=db.

Flip LHS and RHS.

Defines rule #5.

Referenced by [10].

[7] cda=a

Overlap of [2] aaaa=c with [4] aaada=1:

a aaa aaada

Critical pair: a=cda.

Flip LHS and RHS.

Referenced by [9].

[8] aada=aaad

Overlap of [4] aaada=1 with [4] aaada=1:

aaad a aaada

Critical pair: aaad=aada.

Flip LHS and RHS.

Referenced by [11].

[9] cd=1

Overlap of [7] cda=a with [4] aaada=1:

cd a aaada

Critical pair: cd=aaada.

Reduce RHS:

[4](aaada)
⇒ 1

Defines rule #3.

Referenced by [10], [12].

[10] cbd=b

Overlap of [9] cd=1 with [6] db=bd:

c d db

Critical pair: cbd=b.

Referenced by [13].

[11] da=ad

Overlap of [8] aada=aaad with [8] aada=aaad:

aad a aada

Critical pair: aadaaad=aaadada.

Reduce LHS:

[8](aada)aad
[4]⇒ (aaada)ad
⇒ ad

Reduce RHS:

[4](aaada)da
⇒ da

Flip LHS and RHS.

Defines rule #4.

Referenced by [12].

[12] dc=1

Overlap of [11] da=ad with [2] aaaa=c:

d a aaaa

Critical pair: dc=adaaa.

Reduce RHS:

[11]a(da)aa
[11]⇒ aa(da)a
[11]⇒ aaa(da)
[2]⇒ (aaaa)d
[9]⇒ (cd)
⇒ 1

Defines rule #6.

Referenced by [13].

[13] cb=bc

Overlap of [10] cbd=b with [12] dc=1:

cb d dc

Critical pair: cb=bc.

Defines rule #2.