Certificate for #7546 ⟨a, b | aaa=1, bbaabb=a⟩

Completion settings:

Auxiliary generators:
1. c = aa
Reduction order:
- Left-to-right recursive path with deg(b) = deg(c) = 0, b < c; deg(a) = 1

[1] aaa=1

Axiom: aaa=1.

Referenced by [4].

[2] bbaabb=a

Axiom: bbaabb=a.

Referenced by [5].

[3] aa=c

Axiom: aa=c.

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

[4] ca=1

Overlap of [1] aaa=1 with [3] aa=c:

aaa aa

Critical pair: ca=1.

Referenced by [7].

[5] a=bbcbb

Overlap of [2] bbaabb=a with [3] aa=c:

bb aabb aa

Critical pair: bbcbb=a.

Flip LHS and RHS.

Referenced by [6], [7], [13].

[6] bbcbbbbcbb=c

Overlap of [3] aa=c with [5] a=bbcbb:

aa a

Critical pair: bbcbba=c.

Reduce LHS:

[5]	bbcbb(a)
	⇒ bbcbbbbcbb

Referenced by [8], [9], [10].

[7] cbbcbb=1

Simplify [4] ca=1.

Reduce LHS:

[5]	c(a)
	⇒ cbbcbb

Referenced by [8], [9], [10], [12].

[8] bbcbb=cc

Overlap of [7] cbbcbb=1 with [6] bbcbbbbcbb=c:

c bbcbb bbcbbbbcbb

Critical pair: cc=bbcbb.

Flip LHS and RHS.

Defines rule #2.

Referenced by [9], [10], [11], [13].

[9] cbbcbc=bcbbcc

Overlap of [7] cbbcbb=1 with [6] bbcbbbbcbb=c:

cbbcb b bbcbbbbcbb

Critical pair: cbbcbc=bcbbbbcbb.

Reduce RHS:

[8]	bcbb(bbcbb)
	⇒ bcbbcc

Referenced by [12].

[10] ccc=1

Overlap of [6] bbcbbbbcbb=c with [6] bbcbbbbcbb=c:

bbcbb bbcbb bbcbbbbcbb

Critical pair: bbcbbc=cbbcbb.

Reduce LHS:

[8]	(bbcbb)c
	⇒ ccc

Reduce RHS:

[7]	(cbbcbb)
	⇒ 1

Defines rule #1.

[11] ccbcbb=bbcbcc

Overlap of [8] bbcbb=cc with [8] bbcbb=cc:

bbcb b bbcbb

Critical pair: bbcbcc=ccbcbb.

Flip LHS and RHS.

Defines rule #4.

[12] cbbcb=bcbbc

Overlap of [9] cbbcbc=bcbbcc with [7] cbbcbb=1:

cbbcb c cbbcbb

Critical pair: cbbcb=bcbbccbbcbb.

Reduce RHS:

[7]	bcbbc(cbbcbb)
	⇒ bcbbc

Defines rule #3.

[13] a=cc

Simplify [5] a=bbcbb.

Reduce RHS:

[8]	(bbcbb)
	⇒ cc

Defines rule #5.