Certificate for #648 ⟨a, b | aaabbbaba=1⟩

Completion settings:

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

Axiom: aaabbbaba=1.

Referenced by [4].

Axiom: aaaa=c.

Defines rule #5.

Referenced by [5], [6], [13], [14], [19], [21], [24], [30].

Axiom: bbbab=d.

Defines rule #16.

Referenced by [4], [9], [21].

Overlap of [1] aaabbbaba=1 with [3] bbbab=d:

Critical pair: aaada=1.

Referenced by [6], [7], [8], [10].

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

Critical pair: ac=ca.

Defines rule #3.

Referenced by [20], [27].

Overlap of [2] aaaa=c with [4] aaada=1:

Critical pair: a=cda.

Flip LHS and RHS.

Referenced by [8].

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

Critical pair: aaad=aada.

Referenced by [8], [10], [14].

Overlap of [6] cda=a with [4] aaada=1:

Critical pair: cd=aaada.

Reduce RHS:

Flip LHS and RHS.

Referenced by [10], [11].

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

Critical pair: bbbad=dbbab.

Flip LHS and RHS.

Referenced by [15].

Overlap of [4] aaada=1 with [7] aaad=aada:

Critical pair: aadaa=1.

Reduce LHS:

Defines rule #1.

Referenced by [11], [13], [16], [21], [25], [28].

Simplify [8] aadaa=cd.

Reduce RHS:

Referenced by [12], [14].

Overlap of [11] aadaa=1 with [11] aadaa=1:

Critical pair: aad=daa.

Referenced by [13], [14], [17].

Overlap of [2] aaaa=c with [12] aad=daa:

Critical pair: aadaa=cd.

Reduce LHS:

Reduce RHS:

Defines rule #2.

Referenced by [14], [19], [29], [30].

Overlap of [11] aadaa=1 with [12] aad=daa:

Critical pair: aadadaa=ad.

Reduce LHS:

Flip LHS and RHS.

Defines rule #4.

Referenced by [15], [18], [29].

Simplify [9] dbbab=bbbad.

Reduce RHS:

Defines rule #9.

Referenced by [16], [17], [18].

Overlap of [10] cd=1 with [15] dbbab=bbbda:

Critical pair: cbbbda=bbab.

Referenced by [19].

Overlap of [12] aad=daa with [15] dbbab=bbbda:

Critical pair: aabbbda=daabbab.

Flip LHS and RHS.

Defines rule #13.

Referenced by [29].

Overlap of [14] ad=da with [15] dbbab=bbbda:

Critical pair: abbbda=dabbab.

Flip LHS and RHS.

Defines rule #11.

Overlap of [16] cbbbda=bbab with [2] aaaa=c:

Critical pair: cbbbdc=bbabaaa.

Reduce LHS:

Defines rule #8.

Referenced by [20], [21].

Overlap of [5] ac=ca with [19] cbbb=bbabaaa:

Critical pair: abbabaaa=cabbb.

Flip LHS and RHS.

Defines rule #10.

Referenced by [27].

Overlap of [19] cbbb=bbabaaa with [3] bbbab=d:

Critical pair: cd=bbabaaaab.

Reduce LHS:

Reduce RHS:

Flip LHS and RHS.

Referenced by [22], [23].

Overlap of [21] bbabcb=1 with [21] bbabcb=1:

Critical pair: bbabc=babcb.

Flip LHS and RHS.

Referenced by [23].

Overlap of [21] bbabcb=1 with [22] babcb=bbabc:

Critical pair: bbabcbbabc=abcb.

Reduce LHS:

Flip LHS and RHS.

Defines rule #6.

Referenced by [24], [26].

Overlap of [2] aaaa=c with [23] abcb=babc:

Critical pair: aaababc=cbcb.

Referenced by [25], [26].

Overlap of [24] aaababc=cbcb with [10] cd=1:

Critical pair: aaabab=cbcbd.

Defines rule #7.

Overlap of [24] aaababc=cbcb with [23] abcb=babc:

Critical pair: aaabbabc=cbcbb.

Referenced by [28].

Overlap of [5] ac=ca with [20] cabbb=abbabaaa:

Critical pair: aabbabaaa=caabbb.

Flip LHS and RHS.

Defines rule #12.

Overlap of [26] aaabbabc=cbcbb with [10] cd=1:

Critical pair: aaabbab=cbcbbd.

Defines rule #15.

Referenced by [29].

Overlap of [14] ad=da with [17] daabbab=aabbbda:

Critical pair: aaabbbda=daaabbab.

Reduce RHS:

Referenced by [30].

Overlap of [29] aaabbbda=bcbbd with [2] aaaa=c:

Critical pair: aaabbbdc=bcbbdaaa.

Reduce LHS:

Defines rule #14.