Certificate for #21092 ⟨a, b | bb=aa, aaab=aba⟩

Completion settings:

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

[1] bb=aa

Axiom: bb=aa.

Defines rule #5.

Referenced by [3], [5], [8], [9].

[2] aaab=aba

Axiom: aaab=aba.

Referenced by [4].

[3] aab=baa

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

b b bb

Critical pair: baa=aab.

Flip LHS and RHS.

Defines rule #4.

Referenced by [4], [5], [6], [7], [8], [9].

[4] abaa=aba

Simplify [2] aaab=aba.

Reduce LHS:

[3]	a(aab)
	⇒ abaa

Referenced by [5], [6], [7], [8], [9].

[5] abab=aaaaa

Overlap of [4] abaa=aba with [3] aab=baa:

ab aa aab

Critical pair: abbaa=abab.

Reduce LHS:

[1]	a(bb)aa
	⇒ aaaaa

Flip LHS and RHS.

Referenced by [6].

[6] aaaaaa=aaaaa

Overlap of [4] abaa=aba with [3] aab=baa:

aba a aab

Critical pair: ababaa=abaab.

Reduce LHS:

[4]	ab(abaa)
[5]	⇒ (abab)a
	⇒ aaaaaa

Reduce RHS:

[4]	(abaa)b
[5]	⇒ (abab)
	⇒ aaaaa

Defines rule #1.

Referenced by [8].

[7] baaaa=baaa

Overlap of [3] aab=baa with [4] abaa=aba:

a ab abaa

Critical pair: aaba=baaaa.

Reduce LHS:

[3]	(aab)a
	⇒ baaa

Flip LHS and RHS.

Defines rule #2.

Referenced by [8], [9].

[8] baba=aaaaa

Overlap of [7] baaaa=baaa with [3] aab=baa:

baa aa aab

Critical pair: baabaa=baaab.

Reduce LHS:

[3]	b(aab)aa
[1]	⇒ (bb)aaaa
[6]	⇒ (aaaaaa)
	⇒ aaaaa

Reduce RHS:

[3]	ba(aab)
[4]	⇒ b(abaa)
	⇒ baba

Flip LHS and RHS.

Referenced by [9].

[9] aba=baaa

Overlap of [1] bb=aa with [8] baba=aaaaa:

b b baba

Critical pair: baaaaa=aaaba.

Reduce LHS:

[7]	(baaaa)a
[7]	⇒ (baaaa)
	⇒ baaa

Reduce RHS:

[3]	a(aab)a
[4]	⇒ (abaa)a
[4]	⇒ (abaa)
	⇒ aba

Flip LHS and RHS.

Defines rule #3.