Certificate for #417 ⟨a, b, c | abc=b, cba=1⟩

Completion settings:

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

Axiom: abc=b.

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

Axiom: cba=1.

Defines rule #8.

Referenced by [4], [5], [6], [8], [13], [16].

Axiom: ac=d.

Defines rule #7.

Referenced by [4], [5], [15].

Overlap of [2] cba=1 with [3] ac=d:

Critical pair: cbd=c.

Referenced by [7].

Overlap of [3] ac=d with [2] cba=1:

Critical pair: a=dba.

Flip LHS and RHS.

Referenced by [9].

Overlap of [1] abc=b with [2] cba=1:

Critical pair: ab=bba.

Flip LHS and RHS.

Defines rule #4.

Referenced by [11].

Overlap of [1] abc=b with [4] cbd=c:

Critical pair: abc=bbd.

Reduce LHS:

Flip LHS and RHS.

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

Overlap of [2] cba=1 with [1] abc=b:

Critical pair: cbb=bc.

Flip LHS and RHS.

Defines rule #5.

Overlap of [5] dba=a with [1] abc=b:

Critical pair: dbb=abc.

Reduce RHS:

Referenced by [10].

Overlap of [9] dbb=b with [7] bbd=b:

Critical pair: db=bd.

Referenced by [11], [17].

Overlap of [10] db=bd with [6] bba=ab:

Critical pair: dab=bdba.

Reduce RHS:

Referenced by [12], [14].

Overlap of [11] dab=ba with [7] bbd=b:

Critical pair: dab=babd.

Reduce LHS:

Flip LHS and RHS.

Referenced by [13].

Overlap of [2] cba=1 with [12] babd=ba:

Critical pair: cba=bd.

Reduce LHS:

Flip LHS and RHS.

Defines rule #1.

Referenced by [14], [17].

Overlap of [11] dab=ba with [13] bd=1:

Critical pair: da=bad.

Defines rule #3.

Referenced by [15].

Overlap of [14] da=bad with [3] ac=d:

Critical pair: dd=badc.

Flip LHS and RHS.

Referenced by [16].

Overlap of [2] cba=1 with [15] badc=dd:

Critical pair: cdd=dc.

Flip LHS and RHS.

Defines rule #6.

Simplify [10] db=bd.

Reduce RHS:

Defines rule #2.