Certificate for #670 ⟨a, b | aabababba=1⟩

Completion settings:

• Auxiliary generators:
  1. c = aaa
  2. d = bababb
• Reduction order:
  • Right-to-left recursive path with deg(c) = deg(d) = deg(a) = 0, c < d < a; deg(b) = 1

Axiom: aabababba=1.

Referenced by [4].

Axiom: aaa=c.

Defines rule #5.

Referenced by [5], [6], [10], [14], [15], [19], [22], [24], [25], [27].

Axiom: bababb=d.

Defines rule #13.

Referenced by [4], [11], [15], [17].

Overlap of [1] aabababba=1 with [3] bababb=d:

Critical pair: aada=1.

Referenced by [6], [7], [8], [9], [10].

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

Critical pair: ac=ca.

Flip LHS and RHS.

Defines rule #3.

Referenced by [16], [19], [20], [24], [29].

Overlap of [2] aaa=c with [4] aada=1:

Critical pair: a=cda.

Flip LHS and RHS.

Referenced by [8].

Overlap of [4] aada=1 with [4] aada=1:

Critical pair: aad=ada.

Flip LHS and RHS.

Referenced by [9], [10].

Overlap of [6] cda=a with [4] aada=1:

Critical pair: cd=aada.

Reduce RHS:

Defines rule #2.

Referenced by [14], [21], [24].

Overlap of [4] aada=1 with [7] ada=aad:

Critical pair: aadaad=da.

Reduce LHS:

Flip LHS and RHS.

Defines rule #4.

Referenced by [10], [11], [12], [23], [26], [28].

Overlap of [9] da=ad with [2] aaa=c:

Critical pair: dc=adaa.

Reduce RHS:

Defines rule #1.

Referenced by [13], [15], [18], [23], [26], [28].

Overlap of [3] bababb=d with [3] bababb=d:

Critical pair: bababd=dababb.

Reduce RHS:

Defines rule #9.

Referenced by [12], [13].

Overlap of [11] bababd=adbabb with [9] da=ad:

Critical pair: bababad=adbabba.

Defines rule #11.

Overlap of [11] bababd=adbabb with [10] dc=1:

Critical pair: babab=adbabbc.

Flip LHS and RHS.

Referenced by [14].

Overlap of [2] aaa=c with [13] adbabbc=babab:

Critical pair: aababab=cdbabbc.

Reduce RHS:

Flip LHS and RHS.

Defines rule #8.

Referenced by [15], [16], [20].

Overlap of [3] bababb=d with [14] babbc=aababab:

Critical pair: baaababab=dc.

Reduce LHS:

Reduce RHS:

Referenced by [17].

Overlap of [14] babbc=aababab with [5] ca=ac:

Critical pair: babbac=aabababa.

Defines rule #10.

Referenced by [24].

Overlap of [15] bcbabab=1 with [3] bababb=d:

Critical pair: bcbad=abb.

Referenced by [18].

Overlap of [17] bcbad=abb with [10] dc=1:

Critical pair: bcba=abbc.

Defines rule #6.

Referenced by [19], [20].

Overlap of [18] bcba=abbc with [2] aaa=c:

Critical pair: bcbc=abbcaa.

Reduce RHS:

Flip LHS and RHS.

Referenced by [21].

Overlap of [18] bcba=abbc with [14] babbc=aababab:

Critical pair: bcaababab=abbcbbc.

Reduce LHS:

Flip LHS and RHS.

Referenced by [27].

Overlap of [19] abbaac=bcbc with [8] cd=1:

Critical pair: abbaa=bcbcd.

Reduce RHS:

Referenced by [22].

Overlap of [2] aaa=c with [21] abbaa=bcb:

Critical pair: aabcb=cbbaa.

Flip LHS and RHS.

Referenced by [23].

Overlap of [10] dc=1 with [22] cbbaa=aabcb:

Critical pair: daabcb=bbaa.

Reduce LHS:

Flip LHS and RHS.

Defines rule #7.

Referenced by [24].

Overlap of [16] babbac=aabababa with [5] ca=ac:

Critical pair: babbaac=aabababaa.

Reduce LHS:

Flip LHS and RHS.

Referenced by [25].

Overlap of [2] aaa=c with [24] aabababaa=bbcbc:

Critical pair: abbcbc=cbababaa.

Flip LHS and RHS.

Referenced by [26].

Overlap of [10] dc=1 with [25] cbababaa=abbcbc:

Critical pair: dabbcbc=bababaa.

Reduce LHS:

Flip LHS and RHS.

Defines rule #12.

Overlap of [2] aaa=c with [20] abbcbbc=baacbabab:

Critical pair: aabaacbabab=cbbcbbc.

Flip LHS and RHS.

Referenced by [28].

Overlap of [10] dc=1 with [27] cbbcbbc=aabaacbabab:

Critical pair: daabaacbabab=bbcbbc.

Reduce LHS:

Flip LHS and RHS.

Defines rule #14.

Referenced by [29].

Overlap of [28] bbcbbc=aadbaacbabab with [5] ca=ac:

Critical pair: bbcbbac=aadbaacbababa.

Defines rule #15.