Certificate for #1723 ⟨a, b, c | aab=ca, abc=1⟩

Completion settings:

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

Axiom: aab=ca.

Referenced by [4], [14].

Axiom: abc=1.

Referenced by [4], [5], [7], [9].

Axiom: ac=d.

Referenced by [4], [6], [8], [10].

Overlap of [1] aab=ca with [2] abc=1:

Critical pair: a=cac.

Reduce RHS:

Defines rule #11.

Referenced by [5], [6], [7], [8], [9], [10], [14], [15].

Overlap of [2] abc=1 with [4] a=cd:

Critical pair: cdbc=1.

Referenced by [9], [10], [11], [16].

Overlap of [3] ac=d with [4] a=cd:

Critical pair: cdc=d.

Defines rule #1.

Referenced by [7], [8], [12], [15].

Overlap of [2] abc=1 with [6] cdc=d:

Critical pair: abd=dc.

Reduce LHS:

Defines rule #7.

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

Overlap of [3] ac=d with [6] cdc=d:

Critical pair: ad=ddc.

Reduce LHS:

Flip LHS and RHS.

Defines rule #5.

Overlap of [2] abc=1 with [5] cdbc=1:

Critical pair: ab=dbc.

Reduce LHS:

Flip LHS and RHS.

Defines rule #2.

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

Overlap of [3] ac=d with [5] cdbc=1:

Critical pair: a=ddbc.

Reduce LHS:

Reduce RHS:

Flip LHS and RHS.

Defines rule #10.

Overlap of [9] dbc=cdb with [5] cdbc=1:

Critical pair: db=cdbdbc.

Reduce RHS:

Flip LHS and RHS.

Defines rule #8.

Overlap of [9] dbc=cdb with [6] cdc=d:

Critical pair: dbd=cdbdc.

Reduce RHS:

Flip LHS and RHS.

Defines rule #3.

Referenced by [13].

Overlap of [7] cdbd=dc with [12] dcc=dbd:

Critical pair: cdbdbd=dccc.

Reduce LHS:

Reduce RHS:

Defines rule #9.

Simplify [1] aab=ca.

Reduce RHS:

Referenced by [15].

Overlap of [14] aab=ccd with [4] a=cd:

Critical pair: cdab=ccd.

Reduce LHS:

Defines rule #4.

Overlap of [5] cdbc=1 with [9] dbc=cdb:

Critical pair: ccdb=1.

Defines rule #6.