Certificate for #12267 ⟨a, b | aaab=ab, bbba=b⟩

Back:⟨a, b | aaab=ab, bbba=b⟩

Completion settings:

[1] aaab=ab

Axiom: aaab=ab.

Defines rule #3.

Referenced by [3].

[2] bbba=b

Axiom: bbba=b.

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

[3] baab=bb

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

bbb a aaab

Critical pair: bbbab=baab.

Reduce LHS:

[2](bbba)b
⇒ bb

Flip LHS and RHS.

Referenced by [4], [7].

[4] bab=bbbb

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

bb ba baab

Critical pair: bbbb=bab.

Flip LHS and RHS.

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

[5] bbbbbb=bb

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

bb ba bab

Critical pair: bbbbbb=bb.

Referenced by [6].

[6] bba=bbbb

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

ba b bbba

Critical pair: bab=bbbbbba.

Reduce LHS:

[4](bab)
⇒ bbbb

Reduce RHS:

[5](bbbbbb)a
⇒ bba

Flip LHS and RHS.

Referenced by [7], [8].

[7] bbbbb=b

Overlap of [3] baab=bb with [6] bba=bbbb:

baa b bba

Critical pair: baabbbb=bbba.

Reduce LHS:

[3](baab)bbb
⇒ bbbbb

Reduce RHS:

[2](bbba)
⇒ b

Defines rule #1.

Referenced by [8].

[8] ba=bbb

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

ba b bba

Critical pair: babbbb=bbbbba.

Reduce LHS:

[4](bab)bbb
[7]⇒ (bbbbb)bb
⇒ bbb

Reduce RHS:

[7](bbbbb)a
⇒ ba

Flip LHS and RHS.

Defines rule #2.