Certificate for #145 ⟨a, b | aababba=1⟩

Completion settings:

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

Axiom: aababba=1.

Referenced by [4].

Axiom: aaa=c.

Defines rule #5.

Referenced by [5], [6], [11], [15], [16], [19].

Axiom: babb=d.

Defines rule #13.

Referenced by [4], [9], [16].

Overlap of [1] aababba=1 with [3] babb=d:

Critical pair: aada=1.

Referenced by [6], [7], [8], [10], [11].

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

Critical pair: ac=ca.

Flip LHS and RHS.

Defines rule #3.

Referenced by [17], [20].

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 [10], [11].

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

Critical pair: cd=aada.

Reduce RHS:

Defines rule #2.

Referenced by [15], [23].

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

Critical pair: babd=dabb.

Referenced by [12].

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

Critical pair: aadaad=da.

Reduce LHS:

Flip LHS and RHS.

Defines rule #4.

Referenced by [11], [12], [13].

Overlap of [10] da=ad with [2] aaa=c:

Critical pair: dc=adaa.

Reduce RHS:

Defines rule #1.

Referenced by [14], [16], [21].

Simplify [9] babd=dabb.

Reduce RHS:

Defines rule #7.

Referenced by [13], [14].

Overlap of [12] babd=adbb with [10] da=ad:

Critical pair: babad=adbba.

Defines rule #10.

Overlap of [12] babd=adbb with [11] dc=1:

Critical pair: bab=adbbc.

Flip LHS and RHS.

Referenced by [15].

Overlap of [2] aaa=c with [14] adbbc=bab:

Critical pair: aabab=cdbbc.

Reduce RHS:

Flip LHS and RHS.

Defines rule #6.

Referenced by [16], [17].

Overlap of [3] babb=d with [15] bbc=aabab:

Critical pair: baaabab=dc.

Reduce LHS:

Reduce RHS:

Referenced by [18].

Overlap of [15] bbc=aabab with [5] ca=ac:

Critical pair: bbac=aababa.

Defines rule #9.

Referenced by [20].

Overlap of [16] bcbab=1 with [16] bcbab=1:

Critical pair: bcba=cbab.

Defines rule #8.

Referenced by [19].

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

Critical pair: bcbc=cbabaa.

Flip LHS and RHS.

Referenced by [21].

Overlap of [17] bbac=aababa with [5] ca=ac:

Critical pair: bbaac=aababaa.

Referenced by [22].

Overlap of [11] dc=1 with [19] cbabaa=bcbc:

Critical pair: dbcbc=babaa.

Flip LHS and RHS.

Defines rule #12.

Referenced by [22].

Simplify [20] bbaac=aababaa.

Reduce RHS:

Referenced by [23].

Overlap of [22] bbaac=aadbcbc with [8] cd=1:

Critical pair: bbaa=aadbcbcd.

Reduce RHS:

Defines rule #11.