Certificate for #12429 ⟨a, b | aaba=bb, bbbb=b⟩

Completion settings:

• Reduction order:
  • Right-to-left recursive path with deg(a) = 0; deg(b) = 1

Axiom: aaba=bb.

Flip LHS and RHS.

Referenced by [2], [3], [6], [7], [8], [9], [10], [17], [18], [25].

Axiom: bbbb=b.

Reduce LHS:

Referenced by [4], [6], [7], [8], [9], [10], [12], [16], [18], [20].

Overlap of [1] bb=aaba with [1] bb=aaba:

Critical pair: baaba=aabab.

Referenced by [4], [5], [7], [9], [13], [21].

Overlap of [2] aabaaaba=b with [3] baaba=aabab:

Critical pair: aabaaaaabab=baba.

Referenced by [6], [11], [14].

Overlap of [3] baaba=aabab with [3] baaba=aabab:

Critical pair: baaaabab=aabababa.

Flip LHS and RHS.

Referenced by [22].

Overlap of [4] aabaaaaabab=baba with [1] bb=aaba:

Critical pair: aabaaaaabaaaba=babab.

Reduce LHS:

Flip LHS and RHS.

Referenced by [7], [8], [9], [10], [15].

Overlap of [1] bb=aaba with [6] babab=aabaaab:

Critical pair: baabaaab=aabaabab.

Reduce LHS:

Reduce RHS:

Referenced by [11], [13].

Overlap of [2] aabaaaba=b with [6] babab=aabaaab:

Critical pair: aabaaaaabaaab=bbab.

Reduce RHS:

Referenced by [14].

Overlap of [3] baaba=aabab with [6] babab=aabaaab:

Critical pair: baaaabaaab=aababbab.

Reduce RHS:

Referenced by [19].

Overlap of [6] babab=aabaaab with [6] babab=aabaaab:

Critical pair: baaabaaab=aabaaabab.

Reduce RHS:

Referenced by [12], [13], [14], [15], [16].

Overlap of [4] aabaaaaabab=baba with [7] aababaab=aab:

Critical pair: aabaaaaab=babaaab.

Flip LHS and RHS.

Referenced by [15].

Overlap of [2] aabaaaba=b with [10] baaabaaab=aaba:

Critical pair: aaaaba=baab.

Flip LHS and RHS.

Referenced by [14], [17], [18], [19], [22].

Overlap of [3] baaba=aabab with [10] baaabaaab=aaba:

Critical pair: baaaaba=aababaabaaab.

Reduce RHS:

Referenced by [14], [19].

Overlap of [4] aabaaaaabab=baba with [10] baaabaaab=aaba:

Critical pair: aabaaaaabaaaba=babaaaabaaab.

Reduce LHS:

Reduce RHS:

Flip LHS and RHS.

Referenced by [19].

Overlap of [6] babab=aabaaab with [10] baaabaaab=aaba:

Critical pair: babaaaba=aabaaabaaabaaab.

Reduce LHS:

Reduce RHS:

Referenced by [18].

Overlap of [10] baaabaaab=aaba with [2] aabaaaba=b:

Critical pair: bab=aabaa.

Referenced by [17], [18], [19], [20], [21], [23].

Overlap of [1] bb=aaba with [16] bab=aabaa:

Critical pair: baabaa=aabaab.

Reduce LHS:

Reduce RHS:

Referenced by [18], [19], [20], [21], [23].

Overlap of [2] aabaaaba=b with [16] bab=aabaa:

Critical pair: aabaaaaabaa=bb.

Reduce LHS:

Reduce RHS:

Referenced by [20].

Simplify [9] baaaabaaab=bab.

Reduce LHS:

Reduce RHS:

Flip LHS and RHS.

Referenced by [20], [21].

Overlap of [2] aabaaaba=b with [19] aabaa=aaaaaaaaaaaaba:

Critical pair: aaaaaaaaaaaabaaba=b.

Reduce LHS:

Referenced by [21], [22], [23], [24], [26].

Overlap of [3] baaba=aabab with [19] aabaa=aaaaaaaaaaaaba:

Critical pair: baaaaaaaaaaaaba=aababa.

Reduce LHS:

Reduce RHS:

Referenced by [22].

Simplify [5] aabababa=baaaabab.

Reduce RHS:

Referenced by [23].

Overlap of [22] aabababa=aab with [16] bab=aabaa:

Critical pair: aaaabaaaba=aab.

Reduce LHS:

Referenced by [24].

Overlap of [20] aaaaaaaaba=b with [23] baa=aab:

Critical pair: aaaaaaaaaab=ba.

Flip LHS and RHS.

Defines rule #2.

Referenced by [25], [26].

Simplify [1] bb=aaba.

Reduce RHS:

Defines rule #3.

Overlap of [20] aaaaaaaaba=b with [24] ba=aaaaaaaaaab:

Critical pair: aaaaaaaaaaaaaaaaaab=b.

Defines rule #1.