NO
(ignored inputs)COMMENT Cops #647 - #721: generated ground TRSs; evenly distributed in the UNR/UNC/NFP/CR hierarchy submitted by: Bertram Felgenhauer
Rewrite Rules:
   [ a -> a,
     a -> h(a,b),
     b -> h(b,f(f(b))),
     b -> b,
     h(b,h(f(a),a)) -> a ]
Apply Direct Methods...
Inner CPs:
   [ h(b,h(f(a),a)) = a,
     h(b,h(f(a),a)) = a,
     h(b,h(f(h(a,b)),a)) = a,
     h(b,h(f(a),h(a,b))) = a,
     h(h(b,f(f(b))),h(f(a),a)) = a,
     h(b,h(f(a),a)) = a ]
Outer CPs:
   [ a = h(a,b),
     h(b,f(f(b))) = b ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
inner CP cond (upside-parallel)
innter CP Cond (outside)
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ h(a,b) = a,
     a = h(b,h(f(a),a)),
     a = h(a,b),
     a = h(b,h(f(h(a,b)),a)),
     a = h(b,h(f(a),h(a,b))),
     b = h(b,f(f(b))),
     a = h(h(b,f(f(b))),h(f(a),a)),
     h(b,f(f(b))) = b,
     h(h(b,f(f(b))),h(f(a),a)) = a,
     h(h(b,f(f(b))),h(f(a),h(a,b))) = a,
     h(h(b,f(f(b))),h(f(h(a,b)),a)) = a,
     h(h(b,f(f(b))),h(f(h(a,b)),h(a,b))) = a,
     h(b,h(f(a),a)) = a,
     h(b,h(f(a),h(a,b))) = a,
     h(b,h(f(h(a,b)),a)) = a,
     h(b,h(f(h(a,b)),h(a,b))) = a ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <h(b,h(f(a),a)), a> by Rules <0, 4> preceded by [(h,2),(h,1),(f,1)]
 joinable by a reduction of rules <[([],4)], []>
Critical Pair <h(b,h(f(a),a)), a> by Rules <0, 4> preceded by [(h,2),(h,2)]
 joinable by a reduction of rules <[([],4)], []>
Critical Pair <h(b,h(f(h(a,b)),a)), a> by Rules <1, 4> preceded by [(h,2),(h,1),(f,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 10 rules by 3 steps unfolding
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (success)
<F,Q,Q_f,Delta> where
   F = {(a,0),(b,0),(f,1),(h,2)}
   Q = {q_0,q_1,q_{a},q_{b},q_{f(a)},q_{f(b)},q_{f(f(b))},q_{h(a,b)},q_{h(f(a),h(a,b))}}
   Qf = {q_0,q_1}
   Delta = 
   [ a -> q_1,
     a -> q_{a},
     b -> q_{b},
     f(q_{a}) -> q_{f(a)},
     f(q_{b}) -> q_{f(b)},
     f(q_{f(b)}) -> q_{f(f(b))},
     h(q_1,q_{b}) -> q_1,
     h(q_1,q_{b}) -> q_{a},
     h(q_{a},q_{b}) -> q_{h(a,b)},
     h(q_{b},q_{f(f(b))}) -> q_{b},
     h(q_{b},q_{h(f(a),h(a,b))}) -> q_0,
     h(q_{f(a)},q_{h(a,b)}) -> q_{h(f(a),h(a,b))} ]
 (failure)
check by Interpretation(mod2) (failure)
check by Descendants-Approximation, check by Ordering(poly)Witness for Non-Confluence: <h(b,h(f(a),a)) <- h(b,h(f(a),a)) -> a>
Direct Methods: not CR

Combined result: not CR
672.trs: Success(not CR)
(403 msec.)