Certificate for #5819 ⟨a, b, c | ab=a, aca=bc⟩

Completion settings:

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

Axiom: ab=a.

Defines rule #1.

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

Axiom: aca=bc.

Flip LHS and RHS.

Defines rule #5.

Referenced by [4], [8].

Axiom: acc=d.

Defines rule #10.

Referenced by [6], [7], [10], [13].

Overlap of [1] ab=a with [2] bc=aca:

Critical pair: aaca=ac.

Defines rule #6.

Referenced by [5], [6], [7], [11], [13].

Overlap of [4] aaca=ac with [1] ab=a:

Critical pair: aaca=acb.

Reduce LHS:

Flip LHS and RHS.

Defines rule #4.

Referenced by [7], [12].

Overlap of [4] aaca=ac with [4] aaca=ac:

Critical pair: aacac=acaca.

Reduce LHS:

Flip LHS and RHS.

Defines rule #11.

Referenced by [13], [14], [15].

Overlap of [4] aaca=ac with [5] acb=ac:

Critical pair: aacac=accb.

Reduce LHS:

Reduce RHS:

Flip LHS and RHS.

Defines rule #2.

Referenced by [8].

Overlap of [7] db=d with [2] bc=aca:

Critical pair: daca=dc.

Defines rule #8.

Referenced by [9], [10], [11], [12], [15], [17], [18].

Overlap of [8] daca=dc with [1] ab=a:

Critical pair: daca=dcb.

Reduce LHS:

Flip LHS and RHS.

Defines rule #7.

Overlap of [8] daca=dc with [3] acc=d:

Critical pair: dacd=dccc.

Flip LHS and RHS.

Referenced by [16].

Overlap of [8] daca=dc with [4] aaca=ac:

Critical pair: dacac=dcaca.

Reduce LHS:

Flip LHS and RHS.

Defines rule #15.

Referenced by [18].

Overlap of [8] daca=dc with [5] acb=ac:

Critical pair: dacac=dccb.

Reduce LHS:

Flip LHS and RHS.

Defines rule #14.

Overlap of [4] aaca=ac with [6] acaca=d:

Critical pair: ad=acca.

Reduce RHS:

Defines rule #3.

Referenced by [17].

Overlap of [6] acaca=d with [6] acaca=d:

Critical pair: acd=dca.

Defines rule #9.

Referenced by [16], [17], [18].

Overlap of [8] daca=dc with [6] acaca=d:

Critical pair: dd=dcca.

Flip LHS and RHS.

Defines rule #13.

Simplify [10] dccc=dacd.

Reduce RHS:

Defines rule #17.

Overlap of [8] daca=dc with [13] ad=da:

Critical pair: dacda=dcd.

Reduce LHS:

Flip LHS and RHS.

Defines rule #12.

Overlap of [8] daca=dc with [14] acd=dca:

Critical pair: dacdca=dccd.

Reduce LHS:

Flip LHS and RHS.

Defines rule #16.