Certificate for #1378 ⟨a, b | aaabbbaaab=1⟩

Completion settings:

• Auxiliary generators:
  1. c = aaa
  2. d = bcb
• Reduction order:
  • Right-to-left recursive path with deg(d) = 0; deg(c) = deg(b) = deg(a) = 1, c < b < a

Axiom: aaabbbaaab=1.

Referenced by [4].

Axiom: aaa=c.

Defines rule #8.

Referenced by [4], [9].

Axiom: bcb=d.

Referenced by [4], [5], [6], [12], [14], [15].

Overlap of [1] aaabbbaaab=1 with [2] aaa=c:

Critical pair: cbbbaaab=1.

Reduce LHS:

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

Overlap of [3] bcb=d with [3] bcb=d:

Critical pair: bcd=dcb.

Flip LHS and RHS.

Referenced by [10], [11].

Overlap of [3] bcb=d with [4] cbbd=1:

Critical pair: b=dbd.

Flip LHS and RHS.

Defines rule #2.

Referenced by [7], [8], [11], [13], [16], [19], [24], [26], [27].

Overlap of [4] cbbd=1 with [6] dbd=b:

Critical pair: cbbb=bd.

Referenced by [10].

Overlap of [6] dbd=b with [6] dbd=b:

Critical pair: dbb=bbd.

Flip LHS and RHS.

Defines rule #4.

Referenced by [19], [20], [25].

Overlap of [2] aaa=c with [2] aaa=c:

Critical pair: ac=ca.

Flip LHS and RHS.

Defines rule #5.

Referenced by [17].

Overlap of [4] cbbd=1 with [5] dcb=bcd:

Critical pair: cbbbcd=cb.

Reduce LHS:

Flip LHS and RHS.

Referenced by [11], [18].

Overlap of [4] cbbd=1 with [10] cb=bdcd:

Critical pair: bdcdbd=1.

Reduce LHS:

Referenced by [12], [13], [14].

Overlap of [3] bcb=d with [11] bbcd=1:

Critical pair: bc=dbcd.

Flip LHS and RHS.

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

Overlap of [6] dbd=b with [12] dbcd=bc:

Critical pair: dbbc=bbcd.

Reduce RHS:

Defines rule #6.

Referenced by [16], [17], [19], [21], [25], [26].

Overlap of [11] bbcd=1 with [12] dbcd=bc:

Critical pair: bbcbc=bcd.

Reduce LHS:

Flip LHS and RHS.

Referenced by [19].

Overlap of [12] dbcd=bc with [12] dbcd=bc:

Critical pair: dbcbc=bcbcd.

Reduce LHS:

Reduce RHS:

Flip LHS and RHS.

Referenced by [18], [23].

Overlap of [6] dbd=b with [13] dbbc=1:

Critical pair: db=bbbc.

Flip LHS and RHS.

Defines rule #9.

Referenced by [26].

Overlap of [13] dbbc=1 with [9] ca=ac:

Critical pair: dbbac=a.

Referenced by [22].

Simplify [10] cb=bdcd.

Reduce RHS:

Defines rule #3.

Referenced by [19], [24], [25].

Overlap of [18] cb=bddc with [8] bbd=dbb:

Critical pair: cdbb=bddcbd.

Reduce RHS:

Referenced by [20].

Overlap of [19] cdbb=1 with [8] bbd=dbb:

Critical pair: cddbb=d.

Referenced by [21].

Overlap of [20] cddbb=d with [13] dbbc=1:

Critical pair: cd=dc.

Defines rule #1.

Referenced by [22].

Overlap of [17] dbbac=a with [21] cd=dc:

Critical pair: dbbadc=ad.

Referenced by [23].

Overlap of [22] dbbadc=ad with [15] dcd=ddc:

Critical pair: dbbaddc=add.

Referenced by [24].

Overlap of [23] dbbaddc=add with [18] cb=bddc:

Critical pair: dbbaddbddc=addb.

Reduce LHS:

Referenced by [25].

Overlap of [24] dbbabc=addb with [18] cb=bddc:

Critical pair: dbbabbddc=addbb.

Reduce LHS:

Referenced by [26].

Overlap of [25] dbbad=addbb with [13] dbbc=1:

Critical pair: dbba=addbbbbc.

Reduce RHS:

Defines rule #7.

Referenced by [27].

Overlap of [6] dbd=b with [26] dbba=adbb:

Critical pair: dbadbb=bbba.

Flip LHS and RHS.

Defines rule #10.