Certificate for #1332 ⟨a, b | aaaabbbaba=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. bbbab=d

Axiom: aaaabbbaba=1.

Referenced by [4].

Axiom: aaaaa=c.

Defines rule #5.

Referenced by [5], [6], [7], [10], [15], [17], [19], [24], [26], [29], [37].

Axiom: bbbab=d.

Defines rule #18.

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

Overlap of [1] aaaabbbaba=1 with [3] bbbab=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], [32], [35].

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

Critical pair: bbbad=dbbab.

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], [34].

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], [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], [20], [23], [33], [36].

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

Critical pair: aaaada=cd.

Reduce LHS:

Reduce RHS:

Defines rule #2.

Referenced by [24], [36], [37].

Simplify [11] dbbab=bbbad.

Reduce RHS:

Defines rule #9.

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

Overlap of [12] cd=1 with [20] dbbab=bbbda:

Critical pair: cbbbda=bbab.

Referenced by [24].

Overlap of [16] aad=daa with [20] dbbab=bbbda:

Critical pair: aabbbda=daabbab.

Flip LHS and RHS.

Defines rule #13.

Referenced by [33].

Overlap of [18] ad=da with [20] dbbab=bbbda:

Critical pair: abbbda=dabbab.

Flip LHS and RHS.

Defines rule #11.

Overlap of [21] cbbbda=bbab with [2] aaaaa=c:

Critical pair: cbbbdc=bbabaaaa.

Reduce LHS:

Defines rule #8.

Referenced by [25], [26].

Overlap of [5] ac=ca with [24] cbbb=bbabaaaa:

Critical pair: abbabaaaa=cabbb.

Flip LHS and RHS.

Defines rule #10.

Referenced by [32].

Overlap of [24] cbbb=bbabaaaa with [3] bbbab=d:

Critical pair: cd=bbabaaaaab.

Reduce LHS:

Reduce RHS:

Flip LHS and RHS.

Referenced by [27], [28].

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

Critical pair: bbabc=babcb.

Flip LHS and RHS.

Referenced by [28], [31].

Overlap of [26] bbabcb=1 with [27] babcb=bbabc:

Critical pair: bbabcbbabc=abcb.

Reduce LHS:

Flip LHS and RHS.

Defines rule #6.

Referenced by [29].

Overlap of [2] aaaaa=c with [28] abcb=babc:

Critical pair: aaaababc=cbcb.

Referenced by [30], [31].

Overlap of [29] aaaababc=cbcb with [12] cd=1:

Critical pair: aaaabab=cbcbd.

Defines rule #7.

Overlap of [29] aaaababc=cbcb with [27] babcb=bbabc:

Critical pair: aaaabbabc=cbcbb.

Referenced by [34].

Overlap of [5] ac=ca with [25] cabbb=abbabaaaa:

Critical pair: aabbabaaaa=caabbb.

Flip LHS and RHS.

Defines rule #12.

Referenced by [35].

Overlap of [18] ad=da with [22] daabbab=aabbbda:

Critical pair: aaabbbda=daaabbab.

Flip LHS and RHS.

Defines rule #15.

Referenced by [36].

Overlap of [31] aaaabbabc=cbcbb with [12] cd=1:

Critical pair: aaaabbab=cbcbbd.

Defines rule #17.

Referenced by [36].

Overlap of [5] ac=ca with [32] caabbb=aabbabaaaa:

Critical pair: aaabbabaaaa=caaabbb.

Flip LHS and RHS.

Defines rule #14.

Overlap of [18] ad=da with [33] daaabbab=aaabbbda:

Critical pair: aaaabbbda=daaaabbab.

Reduce RHS:

Referenced by [37].

Overlap of [36] aaaabbbda=bcbbd with [2] aaaaa=c:

Critical pair: aaaabbbdc=bcbbdaaaa.

Reduce LHS:

Defines rule #16.