Certificate for #16063 ⟨a, b | aaa=bb, abab=bb⟩

Back:⟨a, b | aaa=bb, abab=bb⟩

Completion settings:

[1] aaa=bb

Axiom: aaa=bb.

Defines rule #6.

Referenced by [3], [4].

[2] abab=bb

Axiom: abab=bb.

Defines rule #5.

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

[3] bba=abb

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

a aa aaa

Critical pair: abb=bba.

Flip LHS and RHS.

Defines rule #3.

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

[4] aabb=babbb

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

aa a abab

Critical pair: aabb=bbbab.

Reduce RHS:

[3]b(bba)b
⇒ babbb

Defines rule #4.

Referenced by [5], [6].

[5] babbbbbb=babb

Overlap of [2] abab=bb with [3] bba=abb:

aba b bba

Critical pair: abaabb=bbba.

Reduce LHS:

[4]ab(aabb)
[3]⇒ a(bba)bbb
[4]⇒ (aabb)bbb
⇒ babbbbbb

Reduce RHS:

[3]b(bba)
⇒ babb

Defines rule #2.

[6] abbbbbbb=abbb

Overlap of [4] aabb=babbb with [3] bba=abb:

aab b bba

Critical pair: aababb=babbbba.

Reduce LHS:

[2]a(abab)b
⇒ abbb

Reduce RHS:

[3]babb(bba)
[3]⇒ ba(bba)bb
[4]⇒ b(aabb)bb
[3]⇒ (bba)bbbbb
⇒ abbbbbbb

Flip LHS and RHS.

Referenced by [7].

[7] bbbbbbb=bbb

Overlap of [6] abbbbbbb=abbb with [3] bba=abb:

abbbbb bb bba

Critical pair: abbbbbabb=abbba.

Reduce LHS:

[3]abbb(bba)bb
[3]⇒ ab(bba)bbbb
[2]⇒ (abab)bbbbb
⇒ bbbbbbb

Reduce RHS:

[3]ab(bba)
[2]⇒ (abab)b
⇒ bbb

Defines rule #1.