Certificate for #1470 ⟨a, b | aaa=bb, abab=1⟩

Back:⟨a, b | aaa=bb, abab=1⟩

Completion settings:

[1] bb=aaa

Axiom: aaa=bb.

Flip LHS and RHS.

Defines rule #3.

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

[2] abab=1

Axiom: abab=1.

Referenced by [4], [7].

[3] aaab=baaa

Overlap of [1] bb=aaa with [1] bb=aaa:

b b bb

Critical pair: baaa=aaab.

Flip LHS and RHS.

Referenced by [5], [7].

[4] abaaaa=b

Overlap of [2] abab=1 with [1] bb=aaa:

aba b bb

Critical pair: abaaaa=b.

Referenced by [5], [6].

[5] aab=baaaaaaa

Overlap of [3] aaab=baaa with [4] abaaaa=b:

aa ab abaaaa

Critical pair: aab=baaaaaaa.

Referenced by [6].

[6] ab=baaaaaaaaaaa

Overlap of [5] aab=baaaaaaa with [4] abaaaa=b:

a ab abaaaa

Critical pair: ab=baaaaaaaaaaa.

Defines rule #2.

Referenced by [7].

[7] aaaaaaaaaaaaaaa=1

Overlap of [2] abab=1 with [6] ab=baaaaaaaaaaa:

abab ab

Critical pair: baaaaaaaaaaaab=1.

Reduce LHS:

[3]baaaaaaaaa(aaab)
[3]⇒ baaaaaa(aaab)aaa
[3]⇒ baaa(aaab)aaaaaa
[3]⇒ b(aaab)aaaaaaaaa
[1]⇒ (bb)aaaaaaaaaaaa
⇒ aaaaaaaaaaaaaaa

Defines rule #1.