Certificate for #712 ⟨a, b | ababaabba=1⟩

Completion settings:

[1] ababaabba=1

Axiom: ababaabba=1.

Referenced by [3].

[2] baab=c

Axiom: baab=c.

Defines rule #5.

Referenced by [3], [4], [5], [6], [7].

[3] abacba=1

Overlap of [1] ababaabba=1 with [2] baab=c:

aba baabba baab

Critical pair: abacba=1.

Referenced by [5], [6], [8], [9], [10].

[4] baac=caab

Overlap of [2] baab=c with [2] baab=c:

baa b baab

Critical pair: baac=caab.

Referenced by [14].

[5] cacba=ba

Overlap of [2] baab=c with [3] abacba=1:

ba ab abacba

Critical pair: ba=cacba.

Flip LHS and RHS.

Referenced by [7].

[6] abacc=ab

Overlap of [3] abacba=1 with [2] baab=c:

abac ba baab

Critical pair: abacc=ab.

Referenced by [8].

[7] cacc=c

Overlap of [5] cacba=ba with [2] baab=c:

cac ba baab

Critical pair: cacc=baab.

Reduce RHS:

[2](baab)
c

Referenced by [11].

[8] bacc=b

Overlap of [3] abacba=1 with [6] abacc=ab:

abacb a abacc

Critical pair: abacbab=bacc.

Reduce LHS:

[3](abacba)b
b

Flip LHS and RHS.

Referenced by [9], [12].

[9] abacb=cc

Overlap of [3] abacba=1 with [8] bacc=b:

abac ba bacc

Critical pair: abacb=cc.

Referenced by [10].

[10] cca=1

Overlap of [3] abacba=1 with [9] abacb=cc:

abacba abacb

Critical pair: cca=1.

Defines rule #2.

Referenced by [11], [12], [15], [16], [17], [18].

[11] cac=1

Overlap of [7] cacc=c with [10] cca=1:

cac c cca

Critical pair: cac=cca.

Reduce RHS:

[10](cca)
⇒ 1

Referenced by [13].

[12] bac=bca

Overlap of [8] bacc=b with [10] cca=1:

bac c cca

Critical pair: bac=bca.

Referenced by [14].

[13] ac=ca

Overlap of [11] cac=1 with [11] cac=1:

ca c cac

Critical pair: ca=ac.

Flip LHS and RHS.

Defines rule #1.

Referenced by [14], [15], [17].

[14] bcaa=caab

Overlap of [4] baac=caab with [13] ac=ca:

ba ac ac

Critical pair: baca=caab.

Reduce LHS:

[12](bac)a
bcaa

Referenced by [15], [17].

[15] caabc=ba

Overlap of [14] bcaa=caab with [13] ac=ca:

bca a ac

Critical pair: bcaca=caabc.

Reduce LHS:

[13]bc(ac)a
[10]b(cca)a
ba

Flip LHS and RHS.

Referenced by [16], [17].

[16] abc=cba

Overlap of [10] cca=1 with [15] caabc=ba:

c ca caabc

Critical pair: cba=abc.

Flip LHS and RHS.

Referenced by [18].

[17] baaa=aaab

Overlap of [15] caabc=ba with [14] bcaa=caab:

caa bc bcaa

Critical pair: caacaab=baaa.

Reduce LHS:

[13]ca(ac)aab
[13]c(ac)aaab
[10](cca)aaab
aaab

Flip LHS and RHS.

Defines rule #4.

[18] bc=cccba

Overlap of [10] cca=1 with [16] abc=cba:

cc a abc

Critical pair: cccba=bc.

Flip LHS and RHS.

Defines rule #3.