Certificate for #1851 ⟨a, b, c | aba=bb, acb=1⟩

Completion settings:

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

Axiom: aba=bb.

Referenced by [4].

Axiom: acb=1.

Defines rule #9.

Referenced by [5], [7], [12], [13].

Axiom: ab=d.

Defines rule #2.

Referenced by [4], [6], [8], [14], [16], [18].

Overlap of [1] aba=bb with [3] ab=d:

Critical pair: da=bb.

Defines rule #7.

Referenced by [5], [6], [10], [12], [15].

Overlap of [4] da=bb with [2] acb=1:

Critical pair: d=bbcb.

Flip LHS and RHS.

Defines rule #5.

Referenced by [7], [8], [9], [16], [18].

Overlap of [4] da=bb with [3] ab=d:

Critical pair: dd=bbb.

Defines rule #6.

Referenced by [10], [11].

Overlap of [2] acb=1 with [5] bbcb=d:

Critical pair: acd=bcb.

Defines rule #15.

Referenced by [12].

Overlap of [3] ab=d with [5] bbcb=d:

Critical pair: ad=dbcb.

Defines rule #8.

Referenced by [18].

Overlap of [5] bbcb=d with [5] bbcb=d:

Critical pair: bbcd=dbcb.

Defines rule #13.

Overlap of [6] dd=bbb with [4] da=bb:

Critical pair: dbb=bbba.

Flip LHS and RHS.

Defines rule #3.

Referenced by [17].

Overlap of [6] dd=bbb with [6] dd=bbb:

Critical pair: dbbb=bbbd.

Flip LHS and RHS.

Defines rule #1.

Overlap of [7] acd=bcb with [4] da=bb:

Critical pair: acbb=bcba.

Reduce LHS:

Flip LHS and RHS.

Referenced by [13].

Overlap of [2] acb=1 with [12] bcba=b:

Critical pair: acb=cba.

Reduce LHS:

Flip LHS and RHS.

Defines rule #14.

Referenced by [14].

Overlap of [13] cba=1 with [3] ab=d:

Critical pair: cbd=b.

Defines rule #11.

Referenced by [15].

Overlap of [14] cbd=b with [4] da=bb:

Critical pair: cbbb=ba.

Defines rule #4.

Referenced by [16], [17].

Overlap of [15] cbbb=ba with [5] bbcb=d:

Critical pair: cbbd=babcb.

Reduce RHS:

Defines rule #12.

Overlap of [15] cbbb=ba with [10] bbba=dbb:

Critical pair: cdbb=baa.

Defines rule #10.

Referenced by [18].

Overlap of [17] cdbb=baa with [5] bbcb=d:

Critical pair: cdbd=baabcb.

Reduce RHS:

Defines rule #16.