NO
(ignored inputs)COMMENT [111] p. 29 ( GWCR & ~WCR & ~GCR )
Rewrite Rules:
   [ a -> b,
     a -> f(a,a),
     f(?x,a) -> a ]
Apply Direct Methods...
Inner CPs:
   [ f(?x,b) = a,
     f(?x,f(a,a)) = a ]
Outer CPs:
   [ b = f(a,a) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ f(a,a) = b,
     a = f(?x,b),
     b = f(a,a),
     a = f(?x,f(a,a)),
     f(?x,b) = a,
     f(?x,f(a,a)) = 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 <f(?x,b), a> by Rules <0, 2> preceded by [(f,2)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 8 rules by 3 steps unfolding
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (success)
 (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (success)
<F,Q,Q_f,Delta> where
   F = {(a,0),(b,0),(f,2)}
   Q = {q_0,q_1,q_{a},q_{b}}
   Qf = {q_0,q_1}
   Delta = 
   [ a -> q_{a},
     b -> q_1,
     b -> q_{a},
     b -> q_{b},
     f(q_{a},q_{a}) -> q_{a},
     f(q_{a},q_{b}) -> q_0 ]
 (failure)
check by Interpretation(mod2) (success)
Polynomial Interpretation:
  b:= (1)
  f:= 0
 [f(b,b)]-[b] = (~1) = 1 (mod 2)
 (success)
Witness for Non-Confluence: <f(b,b) <- a -> b>
Direct Methods: not CR

Combined result: not CR
507.trs: Success(not CR)
(27 msec.)