Certificate for #1324 ⟨a, b | aaaabbaaba=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. aaaaa=c
  2. bbaab=d

Axiom: aaaabbaaba=1.

Referenced by [4].

Axiom: aaaaa=c.

Defines rule #5.

Referenced by [5], [6], [7], [10], [15], [17], [19], [24], [26], [28], [32].

Axiom: bbaab=d.

Defines rule #15.

Referenced by [4], [11], [26].

Overlap of [1] aaaabbaaba=1 with [3] bbaab=d:

Critical pair: aaaada=1.

Referenced by [6], [7], [8], [9], [10], [12].

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

Critical pair: ac=ca.

Defines rule #3.

Referenced by [25], [29].

Overlap of [2] aaaaa=c with [4] aaaada=1:

Critical pair: a=cda.

Flip LHS and RHS.

Referenced by [9], [10].

Overlap of [2] aaaaa=c with [4] aaaada=1:

Critical pair: aa=cada.

Flip LHS and RHS.

Referenced by [10].

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

Critical pair: aaaad=aaada.

Referenced by [9], [12], [19].

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

Critical pair: cd=aaaada.

Reduce RHS:

Flip LHS and RHS.

Referenced by [12], [13].

Overlap of [7] cada=aa with [4] aaaada=1:

Critical pair: cad=aaaaada.

Reduce RHS:

Referenced by [18].

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

Critical pair: bbaad=dbaab.

Flip LHS and RHS.

Referenced by [20].

Overlap of [4] aaaada=1 with [8] aaaad=aaada:

Critical pair: aaadaa=1.

Reduce LHS:

Defines rule #1.

Referenced by [13], [15], [19], [21], [26], [30].

Simplify [9] aaadaa=cd.

Reduce RHS:

Referenced by [14], [16].

Overlap of [13] aaadaa=1 with [13] aaadaa=1:

Critical pair: aaad=adaa.

Referenced by [15], [16], [19].

Overlap of [2] aaaaa=c with [14] aaad=adaa:

Critical pair: aaadaa=cd.

Reduce LHS:

Reduce RHS:

Referenced by [16], [17].

Overlap of [13] aaadaa=1 with [14] aaad=adaa:

Critical pair: aaadaadaa=aad.

Reduce LHS:

Flip LHS and RHS.

Referenced by [17], [20], [22].

Overlap of [16] aad=daa with [15] adaaaa=1:

Critical pair: a=daaaaaa.

Reduce RHS:

Flip LHS and RHS.

Referenced by [18].

Overlap of [17] dca=a with [10] cad=a:

Critical pair: da=ad.

Flip LHS and RHS.

Defines rule #4.

Referenced by [19], [23], [31].

Overlap of [2] aaaaa=c with [18] ad=da:

Critical pair: aaaada=cd.

Reduce LHS:

Reduce RHS:

Defines rule #2.

Referenced by [24], [31], [32].

Simplify [11] dbaab=bbaad.

Reduce RHS:

Defines rule #7.

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

Overlap of [12] cd=1 with [20] dbaab=bbdaa:

Critical pair: cbbdaa=baab.

Referenced by [24].

Overlap of [16] aad=daa with [20] dbaab=bbdaa:

Critical pair: aabbdaa=daabaab.

Flip LHS and RHS.

Defines rule #12.

Referenced by [31].

Overlap of [18] ad=da with [20] dbaab=bbdaa:

Critical pair: abbdaa=dabaab.

Flip LHS and RHS.

Defines rule #9.

Overlap of [21] cbbdaa=baab with [2] aaaaa=c:

Critical pair: cbbdc=baabaaa.

Reduce LHS:

Defines rule #6.

Referenced by [25], [26].

Overlap of [5] ac=ca with [24] cbb=baabaaa:

Critical pair: abaabaaa=cabb.

Flip LHS and RHS.

Defines rule #8.

Referenced by [29].

Overlap of [24] cbb=baabaaa with [3] bbaab=d:

Critical pair: cd=baabaaaaab.

Reduce LHS:

Reduce RHS:

Flip LHS and RHS.

Referenced by [27].

Overlap of [26] baabcb=1 with [26] baabcb=1:

Critical pair: baabc=aabcb.

Flip LHS and RHS.

Defines rule #10.

Referenced by [28].

Overlap of [2] aaaaa=c with [27] aabcb=baabc:

Critical pair: aaabaabc=cbcb.

Referenced by [30].

Overlap of [5] ac=ca with [25] cabb=abaabaaa:

Critical pair: aabaabaaa=caabb.

Flip LHS and RHS.

Defines rule #11.

Overlap of [28] aaabaabc=cbcb with [12] cd=1:

Critical pair: aaabaab=cbcbd.

Defines rule #14.

Referenced by [31].

Overlap of [18] ad=da with [22] daabaab=aabbdaa:

Critical pair: aaabbdaa=daaabaab.

Reduce RHS:

Referenced by [32].

Overlap of [31] aaabbdaa=bcbd with [2] aaaaa=c:

Critical pair: aaabbdc=bcbdaaa.

Reduce LHS:

Defines rule #13.