Certificate for #3139 ⟨a, b | aabbbaabaab=1⟩

Completion settings:

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

Axiom: aabbbaabaab=1.

Referenced by [4].

Axiom: bb=c.

Defines rule #5.

Referenced by [4], [5], [7], [13], [22].

Axiom: aab=d.

Referenced by [4], [7].

Overlap of [1] aabbbaabaab=1 with [3] aab=d:

Critical pair: dbbaabaab=1.

Reduce LHS:

Referenced by [6], [8].

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

Critical pair: bc=cb.

Defines rule #3.

Referenced by [10], [23].

Overlap of [4] dcdd=1 with [4] dcdd=1:

Critical pair: dcd=cdd.

Referenced by [8], [9].

Overlap of [3] aab=d with [2] bb=c:

Critical pair: aac=db.

Referenced by [11], [13].

Overlap of [4] dcdd=1 with [6] dcd=cdd:

Critical pair: cddd=1.

Defines rule #2.

Referenced by [9], [10], [11], [12], [13], [14], [19], [21], [22], [24].

Overlap of [6] dcd=cdd with [6] dcd=cdd:

Critical pair: dccdd=cddcd.

Reduce RHS:

Referenced by [12].

Overlap of [5] bc=cb with [8] cddd=1:

Critical pair: b=cbddd.

Flip LHS and RHS.

Referenced by [13].

Overlap of [7] aac=db with [8] cddd=1:

Critical pair: aa=dbddd.

Referenced by [13], [17].

Overlap of [9] dccdd=c with [8] cddd=1:

Critical pair: dc=cd.

Defines rule #1.

Referenced by [13], [21], [22], [23].

Overlap of [7] aac=db with [10] cbddd=b:

Critical pair: aab=dbbddd.

Reduce LHS:

Reduce RHS:

Referenced by [14], [15], [16], [20].

Overlap of [8] cddd=1 with [13] dbdddb=d:

Critical pair: cddd=bdddb.

Reduce LHS:

Flip LHS and RHS.

Referenced by [16].

Overlap of [13] dbdddb=d with [13] dbdddb=d:

Critical pair: dbddd=ddddb.

Referenced by [17].

Overlap of [14] bdddb=1 with [13] dbdddb=d:

Critical pair: bddd=dddb.

Defines rule #4.

Referenced by [20].

Simplify [11] aa=dbddd.

Reduce RHS:

Defines rule #8.

Referenced by [18].

Overlap of [17] aa=ddddb with [17] aa=ddddb:

Critical pair: addddb=ddddba.

Flip LHS and RHS.

Referenced by [19], [20].

Overlap of [8] cddd=1 with [18] ddddba=addddb:

Critical pair: caddddb=dba.

Flip LHS and RHS.

Referenced by [21].

Overlap of [16] bddd=dddb with [18] ddddba=addddb:

Critical pair: bddaddddb=dddbdddba.

Reduce RHS:

Referenced by [22].

Overlap of [8] cddd=1 with [19] dba=caddddb:

Critical pair: cddcaddddb=ba.

Reduce LHS:

Flip LHS and RHS.

Defines rule #6.

Overlap of [20] bddaddddb=ddda with [2] bb=c:

Critical pair: bddaddddc=dddab.

Reduce LHS:

Referenced by [23].

Overlap of [22] bddad=dddab with [12] dc=cd:

Critical pair: bddacd=dddabc.

Reduce RHS:

Referenced by [24].

Overlap of [23] bddacd=dddacb with [8] cddd=1:

Critical pair: bdda=dddacbdd.

Defines rule #7.