Certificate for #12306 ⟨a, b | aaab=bb, abba=a⟩

Completion settings:

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

[1] bb=aaab

Axiom: aaab=bb.

Flip LHS and RHS.

Defines rule #3.

Referenced by [2], [3], [7].

[2] aaaaba=a

Axiom: abba=a.

Reduce LHS:

[1]	a(bb)a
	⇒ aaaaba

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

[3] baaab=aaaaaab

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

b b bb

Critical pair: baaab=aaabb.

Reduce RHS:

[1]	aaa(bb)
	⇒ aaaaaab

Referenced by [4], [6].

[4] aaaaaaaaaab=aaab

Overlap of [2] aaaaba=a with [3] baaab=aaaaaab:

aaaa ba baaab

Critical pair: aaaaaaaaaab=aaab.

Referenced by [5].

[5] aaaba=aaaaaaa

Overlap of [4] aaaaaaaaaab=aaab with [2] aaaaba=a:

aaaaaa aaaab aaaaba

Critical pair: aaaaaaa=aaaba.

Flip LHS and RHS.

Referenced by [6].

[6] baaaaaaa=aaa

Overlap of [3] baaab=aaaaaab with [5] aaaba=aaaaaaa:

b aaab aaaba

Critical pair: baaaaaaa=aaaaaaba.

Reduce RHS:

[2]	aa(aaaaba)
	⇒ aaa

Referenced by [7], [8], [9].

[7] baaa=aaaaaa

Overlap of [1] bb=aaab with [6] baaaaaaa=aaa:

b b baaaaaaa

Critical pair: baaa=aaabaaaaaaa.

Reduce RHS:

[6]	aaa(baaaaaaa)
	⇒ aaaaaa

Referenced by [8].

[8] aaaaaaaa=a

Overlap of [6] baaaaaaa=aaa with [2] aaaaba=a:

baaaa aaa aaaaba

Critical pair: baaaaa=aaaaba.

Reduce LHS:

[7]	(baaa)aa
	⇒ aaaaaaaa

Reduce RHS:

[2]	(aaaaba)
	⇒ a

Defines rule #1.

Referenced by [9].

[9] ba=aaaa

Overlap of [6] baaaaaaa=aaa with [8] aaaaaaaa=a:

b aaaaaaa aaaaaaaa

Critical pair: ba=aaaa.

Defines rule #2.