Certificate for #709 ⟨a, b | abaabbbba=1⟩

Completion settings:

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

Axiom: abaabbbba=1.

Referenced by [4].

Axiom: aa=c.

Defines rule #8.

Referenced by [4], [5], [8], [10], [11], [14].

Axiom: cbcb=d.

Referenced by [6], [7], [8], [17], [22].

Overlap of [1] abaabbbba=1 with [2] aa=c:

Critical pair: abcbbbba=1.

Referenced by [8], [9], [11], [14].

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

Critical pair: ac=ca.

Defines rule #6.

Referenced by [7].

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

Critical pair: cbd=dcb.

Flip LHS and RHS.

Referenced by [18].

Overlap of [5] ac=ca with [3] cbcb=d:

Critical pair: ad=cabcb.

Flip LHS and RHS.

Referenced by [13].

Overlap of [2] aa=c with [4] abcbbbba=1:

Critical pair: a=cbcbbbba.

Reduce RHS:

Flip LHS and RHS.

Referenced by [10], [11], [13], [14].

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

Critical pair: abcbbbb=bcbbbba.

Referenced by [11].

Overlap of [8] dbbba=a with [2] aa=c:

Critical pair: dbbbc=aa.

Reduce RHS:

Referenced by [12].

Overlap of [8] dbbba=a with [4] abcbbbba=1:

Critical pair: dbbb=abcbbbba.

Reduce RHS:

Flip LHS and RHS.

Referenced by [12], [13], [14], [15].

Overlap of [10] dbbbc=c with [11] bcbbbbc=dbbb:

Critical pair: dbbdbbb=cbbbbc.

Flip LHS and RHS.

Referenced by [16].

Overlap of [11] bcbbbbc=dbbb with [7] cabcb=ad:

Critical pair: bcbbbbad=dbbbabcb.

Reduce RHS:

Flip LHS and RHS.

Referenced by [14], [29].

Overlap of [4] abcbbbba=1 with [13] abcb=bcbbbbad:

Critical pair: bcbbbbadbbba=1.

Reduce LHS:

Referenced by [15], [16].

Simplify [11] bcbbbbc=dbbb.

Reduce RHS:

Referenced by [16].

Overlap of [15] bcbbbbc=1 with [12] cbbbbc=dbbdbbb:

Critical pair: bdbbdbbb=1.

Reduce LHS:

Referenced by [17], [18], [19], [20], [24].

Overlap of [3] cbcb=d with [16] bdbb=1:

Critical pair: cbc=ddbb.

Referenced by [21].

Overlap of [6] dcb=cbd with [16] bdbb=1:

Critical pair: dc=cbddbb.

Referenced by [23].

Overlap of [16] bdbb=1 with [16] bdbb=1:

Critical pair: bdb=dbb.

Flip LHS and RHS.

Referenced by [20].

Overlap of [19] dbb=bdb with [16] bdbb=1:

Critical pair: db=bdbdbb.

Reduce RHS:

Defines rule #1.

Referenced by [21], [22], [23], [24], [26], [27], [28], [29].

Simplify [17] cbc=ddbb.

Reduce RHS:

Defines rule #5.

Referenced by [22].

Overlap of [3] cbcb=d with [21] cbc=bbdd:

Critical pair: bbddb=d.

Reduce LHS:

Referenced by [23].

Simplify [18] dc=cbddbb.

Reduce RHS:

Defines rule #3.

Referenced by [25].

Overlap of [16] bdbb=1 with [20] db=bd:

Critical pair: bbdb=1.

Reduce LHS:

Defines rule #2.

Referenced by [25], [28], [29].

Overlap of [24] bbbd=1 with [23] dc=cd:

Critical pair: bbbcd=c.

Referenced by [26].

Overlap of [25] bbbcd=c with [20] db=bd:

Critical pair: bbbcbd=cb.

Referenced by [27].

Overlap of [26] bbbcbd=cb with [20] db=bd:

Critical pair: bbbcbbd=cbb.

Referenced by [28].

Overlap of [27] bbbcbbd=cbb with [20] db=bd:

Critical pair: bbbcbbbd=cbbb.

Reduce LHS:

Defines rule #4.

Overlap of [13] abcb=bcbbbbad with [24] bbbd=1:

Critical pair: abc=bcbbbbadbbd.

Reduce RHS:

Defines rule #7.