Certificate for #12498 ⟨a, b | abab=aa, abba=a⟩

Back:⟨a, b | abab=aa, abba=a⟩

Completion settings:

[1] abab=aa

Axiom: abab=aa.

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

[2] abba=a

Axiom: abba=a.

Defines rule #4.

Referenced by [4].

[3] abaa=aaab

Overlap of [1] abab=aa with [1] abab=aa:

ab ab abab

Critical pair: abaa=aaab.

Referenced by [6].

[4] aaba=aba

Overlap of [1] abab=aa with [2] abba=a:

ab ab abba

Critical pair: aba=aaba.

Flip LHS and RHS.

Referenced by [5], [6].

[5] aaa=aa

Overlap of [4] aaba=aba with [1] abab=aa:

a aba abab

Critical pair: aaa=abab.

Reduce RHS:

[1](abab)
⇒ aa

Defines rule #1.

Referenced by [6].

[6] aba=aab

Overlap of [4] aaba=aba with [5] aaa=aa:

aab a aaa

Critical pair: aabaa=abaaa.

Reduce LHS:

[4](aaba)a
[3]⇒ (abaa)
[5]⇒ (aaa)b
⇒ aab

Reduce RHS:

[3](abaa)a
[5]⇒ (aaa)ba
[4]⇒ (aaba)
⇒ aba

Flip LHS and RHS.

Defines rule #2.

Referenced by [7].

[7] aabb=aa

Overlap of [1] abab=aa with [6] aba=aab:

abab aba

Critical pair: aabb=aa.

Defines rule #3.