Certificate for #16061 ⟨a, b | aaa=bb, abab=ab⟩

Completion settings:

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1

[1] aaa=bb

Axiom: aaa=bb.

Defines rule #6.

Referenced by [3], [4], [8].

[2] abab=ab

Axiom: abab=ab.

Defines rule #5.

Referenced by [4], [5].

[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 #4.

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

[4] babbb=bbb

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

aa a abab

Critical pair: aaab=bbbab.

Reduce LHS:

[1]	(aaa)b
	⇒ bbb

Reduce RHS:

[3]	b(bba)b
	⇒ babbb

Flip LHS and RHS.

Defines rule #3.

Referenced by [6], [7], [10].

[5] abaabb=aabb

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

aba b bba

Critical pair: abaabb=abba.

Reduce RHS:

[3]	a(bba)
	⇒ aabb

Defines rule #7.

[6] abbbbb=bbbb

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

b ba babbb

Critical pair: bbbb=abbbbb.

Flip LHS and RHS.

Referenced by [8].

[7] baabbbb=abbbb

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

babb b bba

Critical pair: babbabb=bbbba.

Reduce LHS:

[3]	ba(bba)bb
	⇒ baabbbb

Reduce RHS:

[3]	bb(bba)
[3]	⇒ (bba)bb
	⇒ abbbb

Referenced by [9].

[8] aabbbb=bbbbbbb

Overlap of [1] aaa=bb with [6] abbbbb=bbbb:

aa a abbbbb

Critical pair: aabbbb=bbbbbbb.

Referenced by [9].

[9] abbbb=bbbbbbbb

Simplify [7] baabbbb=abbbb.

Reduce LHS:

[8]	b(aabbbb)
	⇒ bbbbbbbb

Flip LHS and RHS.

Defines rule #2.

Referenced by [10].

[10] bbbbbbbbb=bbbb

Overlap of [4] babbb=bbb with [9] abbbb=bbbbbbbb:

b abbb abbbb

Critical pair: bbbbbbbbb=bbbb.

Defines rule #1.