Certificate for #26601 ⟨a, b | aa=1, ababbbbb=b⟩

Completion settings:

[1] aa=1

Axiom: aa=1.

Defines rule #3.

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

[2] ababbbbb=b

Axiom: ababbbbb=b.

Referenced by [3], [4], [7], [9].

[3] babbbbb=ab

Overlap of [1] aa=1 with [2] ababbbbb=b:

a a ababbbbb

Critical pair: ab=babbbbb.

Flip LHS and RHS.

Referenced by [4], [5], [8], [9], [10].

[4] babbbbab=b

Overlap of [3] babbbbb=ab with [3] babbbbb=ab:

babbbb b babbbbb

Critical pair: babbbbab=ababbbbb.

Reduce RHS:

[2](ababbbbb)
⇒ b

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

[5] babbbab=bbbbb

Overlap of [4] babbbbab=b with [3] babbbbb=ab:

babbb bab babbbbb

Critical pair: babbbab=bbbbb.

Referenced by [7].

[6] bbbbab=babbbb

Overlap of [4] babbbbab=b with [4] babbbbab=b:

babbb bab babbbbab

Critical pair: babbbb=bbbbab.

Flip LHS and RHS.

Referenced by [7], [8], [11].

[7] bbab=abbbbbbbb

Overlap of [2] ababbbbb=b with [6] bbbbab=babbbb:

ababb bbb bbbbab

Critical pair: ababbbabbbb=bbab.

Reduce LHS:

[5]a(babbbab)bbb
⇒ abbbbbbbb

Flip LHS and RHS.

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

[8] abab=bbbbbbbbbbbb

Overlap of [3] babbbbb=ab with [6] bbbbab=babbbb:

bab bbbb bbbbab

Critical pair: babbabbbb=abab.

Reduce LHS:

[7]ba(bbab)bbb
[1]⇒ b(aa)bbbbbbbbbbb
⇒ bbbbbbbbbbbb

Flip LHS and RHS.

Referenced by [11].

[9] bab=abbbbbbbbbbbb

Overlap of [2] ababbbbb=b with [7] bbab=abbbbbbbb:

ababbb bb bbab

Critical pair: ababbbabbbbbbbb=bab.

Reduce LHS:

[3]ababb(babbbbb)bbb
[7]⇒ aba(bbab)bbb
[1]⇒ ab(aa)bbbbbbbbbbb
⇒ abbbbbbbbbbbb

Flip LHS and RHS.

Defines rule #2.

Referenced by [10], [11].

[10] abbbbbbbbbbbbbbbb=ab

Overlap of [3] babbbbb=ab with [9] bab=abbbbbbbbbbbb:

babbbbb bab

Critical pair: abbbbbbbbbbbbbbbb=ab.

Referenced by [11].

[11] bbbbbbbbbbbbbbbb=b

Overlap of [4] babbbbab=b with [9] bab=abbbbbbbbbbbb:

babbbbab bab

Critical pair: abbbbbbbbbbbbbbbab=b.

Reduce LHS:

[6]abbbbbbbbbbb(bbbbab)
[6]⇒ abbbbbbbb(bbbbab)bbb
[6]⇒ abbbbb(bbbbab)bbbbbb
[6]⇒ abb(bbbbab)bbbbbbbbb
[7]⇒ ab(bbab)bbbbbbbbbbbb
[10]⇒ ab(abbbbbbbbbbbbbbbb)bbbb
[8]⇒ (abab)bbbb
⇒ bbbbbbbbbbbbbbbb

Defines rule #1.