Certificate for #3813 ⟨a, b | abbabaaaab=b⟩

Completion settings:

• Reduction order:
  • Left-to-right recursive path with deg(a) = deg(c) = 0, a < c; deg(d) = deg(b) = 1, d < b
• Auxiliary generators:
  1. ab=c
  2. bbc=d

Axiom: abbabaaaab=b.

Referenced by [4].

Axiom: ab=c.

Referenced by [4], [5], [9].

Axiom: bbc=d.

Referenced by [5], [10].

Overlap of [1] abbabaaaab=b with [2] ab=c:

Critical pair: cbabaaaab=b.

Reduce LHS:

Referenced by [7].

Overlap of [2] ab=c with [3] bbc=d:

Critical pair: ad=cbc.

Flip LHS and RHS.

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

Overlap of [5] cbc=ad with [5] cbc=ad:

Critical pair: cbad=adbc.

Referenced by [8].

Simplify [4] cbcaaac=b.

Reduce LHS:

Flip LHS and RHS.

Defines rule #7.

Referenced by [8], [9], [10], [11].

Simplify [6] cbad=adbc.

Reduce LHS:

Reduce RHS:

Referenced by [14], [20].

Overlap of [2] ab=c with [7] b=adaaac:

Critical pair: aadaaac=c.

Defines rule #5.

Referenced by [12], [15], [17], [20], [21].

Overlap of [3] bbc=d with [7] b=adaaac:

Critical pair: adaaacbc=d.

Reduce LHS:

Defines rule #12.

Referenced by [12], [13], [14], [16], [18], [19].

Overlap of [5] cbc=ad with [7] b=adaaac:

Critical pair: cadaaacc=ad.

Defines rule #6.

Referenced by [14].

Overlap of [10] adaaaad=d with [9] aadaaac=c:

Critical pair: adaac=daaac.

Defines rule #3.

Overlap of [10] adaaaad=d with [10] adaaaad=d:

Critical pair: adaaad=daaaad.

Defines rule #11.

Referenced by [15], [16].

Overlap of [8] cadaaacad=adadaaacc with [11] cadaaacc=ad:

Critical pair: cadaaaad=adadaaaccaaacc.

Reduce LHS:

Flip LHS and RHS.

Referenced by [21].

Overlap of [13] adaaad=daaaad with [9] aadaaac=c:

Critical pair: adac=daaaadaaac.

Reduce RHS:

Defines rule #2.

Overlap of [13] adaaad=daaaad with [10] adaaaad=d:

Critical pair: adaad=daaaadaaaad.

Reduce RHS:

Defines rule #10.

Referenced by [17], [18].

Overlap of [16] adaad=daaad with [9] aadaaac=c:

Critical pair: adc=daaadaaac.

Reduce RHS:

Defines rule #1.

Overlap of [16] adaad=daaad with [10] adaaaad=d:

Critical pair: adad=daaadaaaad.

Reduce RHS:

Defines rule #9.

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

Overlap of [18] adad=daad with [10] adaaaad=d:

Critical pair: add=daadaaaad.

Reduce RHS:

Defines rule #8.

Simplify [8] cadaaacad=adadaaacc.

Reduce RHS:

Defines rule #13.

Overlap of [14] adadaaaccaaacc=cd with [18] adad=daad:

Critical pair: daadaaaccaaacc=cd.

Reduce LHS:

Flip LHS and RHS.

Defines rule #4.