Certificate for #16470 ⟨a, b | aba=bb, bbbb=aa⟩

Completion settings:

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

[1] bb=aba

Axiom: aba=bb.

Flip LHS and RHS.

Defines rule #4.

Referenced by [2], [3], [4], [7].

[2] abaaba=aa

Axiom: bbbb=aa.

Reduce LHS:

[1]	(bb)bb
[1]	⇒ aba(bb)
	⇒ abaaba

Referenced by [4], [5], [6], [7], [11].

[3] abab=baba

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

b b bb

Critical pair: baba=abab.

Flip LHS and RHS.

Defines rule #5.

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

[4] babaab=aaa

Overlap of [3] abab=baba with [3] abab=baba:

ab ab abab

Critical pair: abbaba=babaab.

Reduce LHS:

[1]	a(bb)aba
[2]	⇒ a(abaaba)
	⇒ aaa

Flip LHS and RHS.

Referenced by [6], [9].

[5] aaaba=abaaa

Overlap of [2] abaaba=aa with [2] abaaba=aa:

aba aba abaaba

Critical pair: abaaa=aaaba.

Flip LHS and RHS.

Referenced by [7].

[6] aab=aaaa

Overlap of [2] abaaba=aa with [3] abab=baba:

aba aba abab

Critical pair: abababa=aab.

Reduce LHS:

[3]	(abab)aba
[4]	⇒ (babaab)a
	⇒ aaaa

Flip LHS and RHS.

Defines rule #3.

Referenced by [8], [9], [11].

[7] abaa=aaaaa

Overlap of [3] abab=baba with [2] abaaba=aa:

ab ab abaaba

Critical pair: abaa=babaaaba.

Reduce RHS:

[5]	bab(aaaba)
[3]	⇒ b(abab)aaa
[1]	⇒ (bb)abaaaa
[2]	⇒ (abaaba)aaa
	⇒ aaaaa

Referenced by [8], [9], [11].

[8] baaaaa=aaaaaaa

Overlap of [6] aab=aaaa with [3] abab=baba:

a ab abab

Critical pair: ababa=aaaaab.

Reduce LHS:

[3]	(abab)a
[7]	⇒ b(abaa)
	⇒ baaaaa

Reduce RHS:

[6]	aaa(aab)
	⇒ aaaaaaa

Referenced by [9], [10].

[9] aaaaaaaaa=aaa

Simplify [4] babaab=aaa.

Reduce LHS:

[7]	b(abaa)b
[8]	⇒ (baaaaa)b
[6]	⇒ aaaaa(aab)
	⇒ aaaaaaaaa

Referenced by [10].

[10] baaa=aaaaa

Overlap of [8] baaaaa=aaaaaaa with [9] aaaaaaaaa=aaa:

b aaaaa aaaaaaaaa

Critical pair: baaa=aaaaaaaaaaa.

Reduce RHS:

[9]	(aaaaaaaaa)aa
	⇒ aaaaa

Referenced by [12].

[11] aaaaaaaa=aa

Overlap of [2] abaaba=aa with [7] abaa=aaaaa:

abaaba abaa

Critical pair: aaaaaba=aa.

Reduce LHS:

[6]	aaa(aab)a
	⇒ aaaaaaaa

Defines rule #1.

Referenced by [12].

[12] baa=aaaa

Overlap of [10] baaa=aaaaa with [11] aaaaaaaa=aa:

b aaa aaaaaaaa

Critical pair: baa=aaaaaaaaaa.

Reduce RHS:

[11]	(aaaaaaaa)aa
	⇒ aaaa

Defines rule #2.