Certificate for #20253 ⟨a, b | aba=b, aaaa=bbb

Completion settings:

[1] aba=b

Axiom: aba=b.

Referenced by [3], [4], [5], [6], [7], [8], [9], [10], [11].

[2] aaaa=bbb

Axiom: aaaa=bbb.

Defines rule #3.

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

[3] bba=abb

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

ab a aba

Critical pair: abb=bba.

Flip LHS and RHS.

Referenced by [5].

[4] abbbb=baaa

Overlap of [1] aba=b with [2] aaaa=bbb:

ab a aaaa

Critical pair: abbbb=baaa.

Referenced by [5].

[5] baaa=aaab

Overlap of [2] aaaa=bbb with [1] aba=b:

aaa a aba

Critical pair: aaab=bbbba.

Reduce RHS:

[3]bb(bba)
[3](bba)bb
[4](abbbb)
baaa

Flip LHS and RHS.

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

[6] bbbb=baa

Overlap of [1] aba=b with [5] baaa=aaab:

a ba baaa

Critical pair: aaaab=baa.

Reduce LHS:

[2](aaaa)b
bbbb

Referenced by [7], [11].

[7] baa=aab

Overlap of [5] baaa=aaab with [2] aaaa=bbb:

b aaa aaaa

Critical pair: bbbb=aaaba.

Reduce LHS:

[6](bbbb)
baa

Reduce RHS:

[1]aa(aba)
aab

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

[8] aaab=ba

Overlap of [1] aba=b with [7] baa=aab:

a ba baa

Critical pair: aaab=ba.

Referenced by [9].

[9] ba=ab

Overlap of [5] baaa=aaab with [7] baa=aab:

baaa baa

Critical pair: aaba=aaab.

Reduce LHS:

[1]a(aba)
ab

Reduce RHS:

[8](aaab)
ba

Flip LHS and RHS.

Defines rule #1.

Referenced by [10], [11].

[10] aab=b

Overlap of [7] baa=aab with [9] ba=ab:

baa ba

Critical pair: aba=aab.

Reduce LHS:

[1](aba)
b

Flip LHS and RHS.

Defines rule #2.

[11] bbbb=b

Simplify [6] bbbb=baa.

Reduce RHS:

[9](ba)a
[1](aba)
b

Defines rule #4.