NO
(ignored inputs)COMMENT doi:10.4230/LIPIcs.RTA.2012.165 [37] Example 3.1 ( R )
Rewrite Rules:
   [ a -> b,
     a -> f(a),
     b -> f(f(b)) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ b = f(a) ]
Overlay, check Innermost Termination...
unknown Innermost 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) = b,
     b = f(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(a), b> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[([(f,1)],1),([(f,1),(f,1)],0)], [([],2)]>
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)}
   Q = {q_0,q_1,q_{b},q_{f(b)}}
   Qf = {q_0,q_1}
   Delta = 
   [ b -> q_1,
     b -> q_{b},
     f(q_{b}) -> q_0,
     f(q_{b}) -> q_{f(b)},
     f(q_{f(b)}) -> q_1,
     f(q_{f(b)}) -> q_{b} ]
 (failure)
check by Interpretation(mod2) (success)
Polynomial Interpretation:
  b:= (1)
  f:= (1)+(1)*x1
 [b]-[f(f(b))] = (~2) = 0 (mod 2)
 [f(f(b))]-[b] = (2) = 0 (mod 2)
 [f(b)]-[b] = (1) = 1 (mod 2)
 (success)
Witness for Non-Confluence: <f(b) <- a -> b>
Direct Methods: not CR

Combined result: not CR
212.trs: Success(not CR)
(20 msec.)