Certificate for #13119 ⟨a, b | bbb=aaa, abba=b⟩

Completion settings:

[1] bbb=aaa

Axiom: bbb=aaa.

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

[2] abba=b

Axiom: abba=b.

Referenced by [4], [5], [6], [8].

[3] aaab=baaa

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

b bb bbb

Critical pair: baaa=aaab.

Flip LHS and RHS.

Defines rule #3.

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

[4] baab=aaaaaaa

Overlap of [2] abba=b with [3] aaab=baaa:

abb a aaab

Critical pair: abbbaaa=baab.

Reduce LHS:

[1]a(bbb)aaa
⇒ aaaaaaa

Flip LHS and RHS.

Defines rule #6.

Referenced by [5], [7].

[5] bab=abaaaaaaa

Overlap of [2] abba=b with [4] baab=aaaaaaa:

ab ba baab

Critical pair: abaaaaaaa=bab.

Flip LHS and RHS.

Defines rule #5.

Referenced by [6].

[6] bb=aabaaaaaaaaaaaaaa

Overlap of [2] abba=b with [5] bab=abaaaaaaa:

ab ba bab

Critical pair: ababaaaaaaa=bb.

Reduce LHS:

[5]a(bab)aaaaaaa
⇒ aabaaaaaaaaaaaaaa

Flip LHS and RHS.

Defines rule #4.

Referenced by [7], [8].

[7] aaaaaaaaaaaaaaaaaaaaa=aaa

Overlap of [1] bbb=aaa with [6] bb=aabaaaaaaaaaaaaaa:

bbb bb

Critical pair: aabaaaaaaaaaaaaaab=aaa.

Reduce LHS:

[3]aabaaaaaaaaaaa(aaab)
[3]⇒ aabaaaaaaaa(aaab)aaa
[3]⇒ aabaaaaa(aaab)aaaaaa
[3]⇒ aabaa(aaab)aaaaaaaaa
[4]⇒ aa(baab)aaaaaaaaaaaa
⇒ aaaaaaaaaaaaaaaaaaaaa

Defines rule #1.

[8] baaaaaaaaaaaaaaaaaa=b

Overlap of [2] abba=b with [6] bb=aabaaaaaaaaaaaaaa:

a bba bb

Critical pair: aaabaaaaaaaaaaaaaaa=b.

Reduce LHS:

[3](aaab)aaaaaaaaaaaaaaa
⇒ baaaaaaaaaaaaaaaaaa

Defines rule #2.