NO
(ignored inputs)COMMENT [120] Example 3 submitted by: Aart Middeldorp
Rewrite Rules:
   [ g(?x) -> h(k(?x)),
     g(?x) -> ?x,
     h(k(?x)) -> f(?x),
     f(?x) -> ?x,
     k(c) -> c,
     f(c) -> g(c) ]
Apply Direct Methods...
Inner CPs:
   [ h(c) = f(c) ]
Outer CPs:
   [ h(k(?x)) = ?x,
     c = g(c) ]
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:
   [ ?x = h(k(?x)),
     h(k(?x)) = ?x,
     h(c) = f(c),
     g(c) = c,
     f(c) = h(c),
     c = g(c) ]
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(c), f(c)> by Rules <4, 2> preceded by [(h,1)]
 joinable by a reduction of rules <[], [([],5),([],0),([(h,1)],4)]>
Critical Pair <?x_1, h(k(?x_1))> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[], [([],2),([],3)]>
Critical Pair <g(c), c> by Rules <5, 3> preceded by []
 joinable by a reduction of rules <[([],1)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 12 rules by 3 steps unfolding
obtain 16 candidates for checking non-joinability
check by TCAP-Approximation (success)
 (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (success)
Witness for Non-Confluence: <h(c) <- g(c) -> c>
Direct Methods: not CR

Combined result: not CR
544.trs: Success(not CR)
(34 msec.)