Certificate for #12323 ⟨a, b | aaab=bb, bbba=b⟩

Back:⟨a, b | aaab=bb, bbba=b⟩

Completion settings:

[1] aaab=bb

Axiom: aaab=bb.

Defines rule #3.

Referenced by [3], [7].

[2] bbba=b

Axiom: bbba=b.

Referenced by [3], [4], [5], [6], [7].

[3] baab=bbbbb

Overlap of [2] bbba=b with [1] aaab=bb:

bbb a aaab

Critical pair: bbbbb=baab.

Flip LHS and RHS.

Referenced by [4], [5].

[4] bab=bbbbbbb

Overlap of [2] bbba=b with [3] baab=bbbbb:

bb ba baab

Critical pair: bbbbbbb=bab.

Flip LHS and RHS.

Referenced by [6], [8].

[5] bbbbbbbbb=bb

Overlap of [3] baab=bbbbb with [3] baab=bbbbb:

baa b baab

Critical pair: baabbbbb=bbbbbaab.

Reduce LHS:

[3](baab)bbbb
⇒ bbbbbbbbb

Reduce RHS:

[2]bb(bbba)ab
[2]⇒ (bbba)b
⇒ bb

Referenced by [6].

[6] bba=bbbbbbb

Overlap of [4] bab=bbbbbbb with [2] bbba=b:

ba b bbba

Critical pair: bab=bbbbbbbbba.

Reduce LHS:

[4](bab)
⇒ bbbbbbb

Reduce RHS:

[5](bbbbbbbbb)a
⇒ bba

Flip LHS and RHS.

Referenced by [7], [8].

[7] bbbbbbbb=b

Overlap of [1] aaab=bb with [6] bba=bbbbbbb:

aaa b bba

Critical pair: aaabbbbbbb=bbba.

Reduce LHS:

[1](aaab)bbbbbb
⇒ bbbbbbbb

Reduce RHS:

[2](bbba)
⇒ b

Defines rule #1.

Referenced by [8].

[8] ba=bbbbbb

Overlap of [4] bab=bbbbbbb with [6] bba=bbbbbbb:

ba b bba

Critical pair: babbbbbbb=bbbbbbbba.

Reduce LHS:

[4](bab)bbbbbb
[7]⇒ (bbbbbbbb)bbbbb
⇒ bbbbbb

Reduce RHS:

[7](bbbbbbbb)a
⇒ ba

Flip LHS and RHS.

Defines rule #2.