Certificate for #331 ⟨a, b | abababba=1⟩

Completion settings:

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

Axiom: abababba=1.

Referenced by [4].

Axiom: ba=c.

Defines rule #7.

Referenced by [4], [5], [7], [11], [22].

Axiom: bca=d.

Referenced by [6].

Overlap of [1] abababba=1 with [2] ba=c:

Critical pair: acbabba=1.

Reduce LHS:

Referenced by [5], [6], [8], [9], [10], [12].

Overlap of [2] ba=c with [4] accbc=1:

Critical pair: b=cccbc.

Flip LHS and RHS.

Referenced by [13], [15].

Overlap of [4] accbc=1 with [3] bca=d:

Critical pair: accd=a.

Referenced by [7].

Overlap of [2] ba=c with [6] accd=a:

Critical pair: ba=cccd.

Reduce LHS:

Flip LHS and RHS.

Referenced by [8], [14].

Overlap of [4] accbc=1 with [7] cccd=c:

Critical pair: accbc=ccd.

Reduce LHS:

Flip LHS and RHS.

Referenced by [9], [14].

Overlap of [4] accbc=1 with [8] ccd=1:

Critical pair: accb=cd.

Referenced by [10], [11], [12], [16].

Overlap of [4] accbc=1 with [9] accb=cd:

Critical pair: cdc=1.

Referenced by [12], [13], [14].

Overlap of [9] accb=cd with [2] ba=c:

Critical pair: accc=cda.

Referenced by [17].

Overlap of [4] accbc=1 with [10] cdc=1:

Critical pair: accb=dc.

Reduce LHS:

Defines rule #1.

Referenced by [14], [16], [17], [18], [19], [20], [24], [26].

Overlap of [5] cccbc=b with [10] cdc=1:

Critical pair: cccb=bdc.

Flip LHS and RHS.

Referenced by [20], [24], [25].

Overlap of [10] cdc=1 with [7] cccd=c:

Critical pair: cdc=ccd.

Reduce LHS:

Reduce RHS:

Defines rule #2.

Referenced by [15], [18], [19], [23], [25], [26].

Overlap of [14] dcc=1 with [5] cccbc=b:

Critical pair: db=cbc.

Flip LHS and RHS.

Referenced by [18], [19], [20], [21].

Simplify [9] accb=cd.

Reduce RHS:

Defines rule #8.

Referenced by [18].

Simplify [11] accc=cda.

Reduce RHS:

Defines rule #4.

Overlap of [16] accb=dc with [15] cbc=db:

Critical pair: acdb=dcc.

Reduce LHS:

Reduce RHS:

Referenced by [21].

Overlap of [14] dcc=1 with [15] cbc=db:

Critical pair: dcdb=bc.

Reduce LHS:

Flip LHS and RHS.

Defines rule #5.

Referenced by [26].

Overlap of [15] cbc=db with [12] cd=dc:

Critical pair: cbdc=dbd.

Reduce LHS:

Flip LHS and RHS.

Referenced by [25].

Overlap of [18] adcb=1 with [15] cbc=db:

Critical pair: addb=c.

Referenced by [22].

Overlap of [21] addb=c with [2] ba=c:

Critical pair: addc=ca.

Referenced by [23].

Overlap of [22] addc=ca with [14] dcc=1:

Critical pair: ad=cac.

Defines rule #3.

Overlap of [13] bdc=cccb with [12] cd=dc:

Critical pair: bddc=cccbd.

Referenced by [25], [26].

Overlap of [20] dbd=ccccb with [24] bddc=cccbd:

Critical pair: dcccbd=ccccbdc.

Reduce LHS:

Reduce RHS:

Referenced by [26].

Overlap of [24] bddc=cccbd with [14] dcc=1:

Critical pair: bd=cccbdc.

Reduce RHS:

Defines rule #6.