Certificate for #312 ⟨a, b | aabbaaba=1⟩

Completion settings:

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

Axiom: aabbaaba=1.

Referenced by [4].

Axiom: aaa=c.

Defines rule #5.

Referenced by [5], [6], [10], [20], [23], [28], [32].

Axiom: bbaab=d.

Referenced by [4], [11], [14].

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

Critical pair: aada=1.

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

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

Critical pair: ac=ca.

Flip LHS and RHS.

Defines rule #3.

Referenced by [20], [31], [32].

Overlap of [2] aaa=c with [4] aada=1:

Critical pair: a=cda.

Flip LHS and RHS.

Referenced by [8].

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

Critical pair: aad=ada.

Flip LHS and RHS.

Referenced by [9], [10].

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

Critical pair: cd=aada.

Reduce RHS:

Defines rule #2.

Referenced by [15], [17], [22], [30], [31], [32].

Overlap of [4] aada=1 with [7] ada=aad:

Critical pair: aadaad=da.

Reduce LHS:

Flip LHS and RHS.

Defines rule #4.

Referenced by [10], [18], [24], [29], [30].

Overlap of [9] da=ad with [2] aaa=c:

Critical pair: dc=adaa.

Reduce RHS:

Defines rule #1.

Referenced by [13], [24], [25], [26], [29].

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

Critical pair: bbaad=dbaab.

Referenced by [12], [13].

Overlap of [11] bbaad=dbaab with [4] aada=1:

Critical pair: bb=dbaaba.

Flip LHS and RHS.

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

Overlap of [11] bbaad=dbaab with [10] dc=1:

Critical pair: bbaa=dbaabc.

Referenced by [14], [27].

Overlap of [3] bbaab=d with [13] bbaa=dbaabc:

Critical pair: dbaabcb=d.

Referenced by [17], [18].

Overlap of [8] cd=1 with [12] dbaaba=bb:

Critical pair: cbb=baaba.

Flip LHS and RHS.

Defines rule #9.

Referenced by [16].

Overlap of [12] dbaaba=bb with [15] baaba=cbb:

Critical pair: dbaacbb=bbaba.

Flip LHS and RHS.

Defines rule #14.

Overlap of [8] cd=1 with [14] dbaabcb=d:

Critical pair: cd=baabcb.

Reduce LHS:

Flip LHS and RHS.

Referenced by [18], [19], [21].

Overlap of [14] dbaabcb=d with [17] baabcb=1:

Critical pair: dbaabc=daabcb.

Reduce RHS:

Referenced by [27].

Overlap of [17] baabcb=1 with [17] baabcb=1:

Critical pair: baabc=aabcb.

Defines rule #6.

Referenced by [20], [21], [22].

Overlap of [12] dbaaba=bb with [19] baabc=aabcb:

Critical pair: dbaaaabcb=bbabc.

Reduce LHS:

Flip LHS and RHS.

Defines rule #12.

Referenced by [30].

Overlap of [17] baabcb=1 with [19] baabc=aabcb:

Critical pair: aabcbb=1.

Referenced by [23].

Overlap of [19] baabc=aabcb with [8] cd=1:

Critical pair: baab=aabcbd.

Flip LHS and RHS.

Referenced by [28].

Overlap of [2] aaa=c with [21] aabcbb=1:

Critical pair: a=cbcbb.

Flip LHS and RHS.

Referenced by [24].

Overlap of [10] dc=1 with [23] cbcbb=a:

Critical pair: da=bcbb.

Reduce LHS:

Flip LHS and RHS.

Defines rule #11.

Referenced by [25], [30], [31].

Overlap of [24] bcbb=ad with [24] bcbb=ad:

Critical pair: bcbad=adcbb.

Reduce RHS:

Referenced by [26].

Overlap of [25] bcbad=abb with [10] dc=1:

Critical pair: bcba=abbc.

Defines rule #8.

Simplify [13] bbaa=dbaabc.

Reduce RHS:

Defines rule #10.

Referenced by [32].

Overlap of [2] aaa=c with [22] aabcbd=baab:

Critical pair: abaab=cbcbd.

Flip LHS and RHS.

Referenced by [29].

Overlap of [10] dc=1 with [28] cbcbd=abaab:

Critical pair: dabaab=bcbd.

Reduce LHS:

Flip LHS and RHS.

Defines rule #7.

Overlap of [24] bcbb=ad with [20] bbabc=dbacbcb:

Critical pair: bcdbacbcb=adabc.

Reduce LHS:

Reduce RHS:

Referenced by [31].

Overlap of [30] bbacbcb=aadbc with [24] bcbb=ad:

Critical pair: bbacbcad=aadbccbb.

Reduce LHS:

Defines rule #15.

Referenced by [32].

Overlap of [31] bbacba=aadbccbb with [2] aaa=c:

Critical pair: bbacbc=aadbccbbaa.

Reduce RHS:

Defines rule #13.