Certificate for #16421 ⟨a, b | aba=ab, bbaa=aa⟩

Back:⟨a, b | aba=ab, bbaa=aa⟩

Completion settings:

[1] aba=ab

Axiom: aba=ab.

Defines rule #2.

Referenced by [3], [5].

[2] bbaa=aa

Axiom: bbaa=aa.

Defines rule #5.

Referenced by [4], [7], [8].

[3] abba=abb

Overlap of [1] aba=ab with [1] aba=ab:

ab a aba

Critical pair: abab=abba.

Reduce LHS:

[1](aba)b
⇒ abb

Flip LHS and RHS.

Referenced by [4], [6].

[4] abb=aaa

Overlap of [3] abba=abb with [2] bbaa=aa:

a bba bbaa

Critical pair: aaa=abba.

Reduce RHS:

[3](abba)
⇒ abb

Flip LHS and RHS.

Defines rule #3.

Referenced by [5], [6].

[5] aaab=ab

Overlap of [1] aba=ab with [4] abb=aaa:

ab a abb

Critical pair: abaaa=abbb.

Reduce LHS:

[1](aba)aa
[1]⇒ (aba)a
[1]⇒ (aba)
⇒ ab

Reduce RHS:

[4](abb)b
⇒ aaab

Flip LHS and RHS.

Referenced by [7], [8].

[6] aaaa=aaa

Overlap of [3] abba=abb with [4] abb=aaa:

abba abb

Critical pair: aaaa=abb.

Reduce RHS:

[4](abb)
⇒ aaa

Defines rule #4.

Referenced by [8].

[7] bbab=ab

Overlap of [2] bbaa=aa with [5] aaab=ab:

bb aa aaab

Critical pair: bbab=aaab.

Reduce RHS:

[5](aaab)
⇒ ab

Defines rule #6.

[8] aab=ab

Overlap of [2] bbaa=aa with [5] aaab=ab:

bba a aaab

Critical pair: bbaab=aaaab.

Reduce LHS:

[2](bbaa)b
⇒ aab

Reduce RHS:

[6](aaaa)b
[5]⇒ (aaab)
⇒ ab

Defines rule #1.