Certificate for #16124 ⟨a, b | aab=aa, baba=bb⟩

Back:⟨a, b | aab=aa, baba=bb⟩

Completion settings:

[1] aab=aa

Axiom: aab=aa.

Defines rule #4.

Referenced by [3], [4].

[2] baba=bb

Axiom: baba=bb.

Defines rule #5.

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

[3] aaaa=aa

Overlap of [1] aab=aa with [2] baba=bb:

aa b baba

Critical pair: aabb=aaaba.

Reduce LHS:

[1](aab)b
[1]⇒ (aab)
⇒ aa

Reduce RHS:

[1]a(aab)a
⇒ aaaa

Flip LHS and RHS.

Defines rule #6.

[4] bbab=bba

Overlap of [2] baba=bb with [1] aab=aa:

bab a aab

Critical pair: babaa=bbab.

Reduce LHS:

[2](baba)a
⇒ bba

Flip LHS and RHS.

Referenced by [6], [7], [8], [9].

[5] bbba=babb

Overlap of [2] baba=bb with [2] baba=bb:

ba ba baba

Critical pair: babb=bbba.

Flip LHS and RHS.

Referenced by [8], [9].

[6] bbaa=bbb

Overlap of [4] bbab=bba with [2] baba=bb:

b bab baba

Critical pair: bbb=bbaa.

Flip LHS and RHS.

Referenced by [7], [9].

[7] bbbb=bbb

Overlap of [4] bbab=bba with [4] bbab=bba:

bba b bbab

Critical pair: bbabba=bbabab.

Reduce LHS:

[4](bbab)ba
[4]⇒ (bbab)a
[6]⇒ (bbaa)
⇒ bbb

Reduce RHS:

[4](bbab)ab
[6]⇒ (bbaa)b
⇒ bbbb

Flip LHS and RHS.

Defines rule #1.

Referenced by [8].

[8] babbb=babb

Overlap of [7] bbbb=bbb with [4] bbab=bba:

bb bb bbab

Critical pair: bbbba=bbbab.

Reduce LHS:

[7](bbbb)a
[5]⇒ (bbba)
⇒ babb

Reduce RHS:

[5](bbba)b
⇒ babbb

Flip LHS and RHS.

Defines rule #2.

[9] bba=babb

Overlap of [4] bbab=bba with [6] bbaa=bbb:

bba b bbaa

Critical pair: bbabbb=bbabaa.

Reduce LHS:

[4](bbab)bb
[4]⇒ (bbab)b
[4]⇒ (bbab)
⇒ bba

Reduce RHS:

[4](bbab)aa
[6]⇒ (bbaa)a
[5]⇒ (bbba)
⇒ babb

Defines rule #3.