(ignored inputs)COMMENT handcrafted "UNC & ~NFP" submitted by: Franziska Rapp
Rewrite Rules:
   [ if(true,a,?x) -> a,
     if(true,b,?x) -> b,
     if(true,g(a),?x) -> g(a),
     if(true,g(b),?x) -> g(b),
     if(false,?x,a) -> a,
     if(false,?x,b) -> b,
     if(false,?x,g(a)) -> g(a),
     if(false,?x,g(b)) -> g(b),
     g(a) -> g(g(a)),
     g(b) -> a,
     f(a,b) -> b,
     f(g(g(a)),?x) -> b ]
Apply Direct Methods...
Inner CPs:
   [ if(true,g(g(a)),?x_2) = g(a),
     if(true,a,?x_3) = g(b),
     if(false,?x_6,g(g(a))) = g(a),
     if(false,?x_7,a) = g(b),
     f(g(g(g(a))),?x_8) = b ]
Outer CPs:
   [  ]
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:
   [ if(true,g(g(a)),?x) = g(a),
     if(true,a,?x) = g(b),
     if(false,?x,g(g(a))) = g(a),
     if(false,?x,a) = g(b),
     g(a) = if(true,g(g(a)),?x_2),
     g(a) = if(false,?x_6,g(g(a))),
     b = f(g(g(g(a))),?x_8),
     g(b) = if(true,a,?x_3),
     g(b) = if(false,?x_7,a),
     f(g(g(g(a))),?x) = b ]
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 <if(true,g(g(a)),?x_2), g(a)> by Rules <8, 2> preceded by [(if,2)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 15 rules by 3 steps unfolding
obtain 43 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,2),(g,1),(if,3),(c_1,0),(true,0),(false,0)}
   Q = {q_0,q_1,q_{c_1},q_{g(g(a))},q_{a},q_{g(a)},q_{g(g(g(a)))}}
   Qf = {q_0,q_1}
   Delta = 
   [ a -> q_{a},
     b -> q_1,
     f(q_{g(g(g(a)))},q_{c_1}) -> q_0,
     g(q_{g(g(a))}) -> q_{g(g(g(a)))},
     g(q_{a}) -> q_{g(a)},
     g(q_{g(a)}) -> q_{g(g(a))},
     g(q_{g(a)}) -> q_{g(a)},
     c_1 -> q_{c_1} ]
Witness for Non-Confluence: <f(g(g(g(a))),c_1), b>
Direct Methods: not CR

Combined result: not CR
741.trs: Success(not CR)
NO
(97 msec.)