4.29/2.22	NO
4.29/2.22	We consider the system theBenchmark.
4.29/2.22	
4.29/2.22	  Alphabet:
4.29/2.22	
4.29/2.22	    0 : [] --> b 
4.29/2.22	    1 : [] --> b 
4.29/2.22	    cons : [e * f] --> f 
4.29/2.22	    f : [b * b * b * c] --> a 
4.29/2.22	    false : [] --> d 
4.29/2.22	    filter : [e -> d * f] --> f 
4.29/2.22	    filter2 : [d * e -> d * e * f] --> f 
4.29/2.22	    g : [b * b] --> b 
4.29/2.22	    h : [b] --> c 
4.29/2.22	    map : [e -> e * f] --> f 
4.29/2.22	    nil : [] --> f 
4.29/2.22	    true : [] --> d 
4.29/2.22	
4.29/2.22	  Rules:
4.29/2.22	
4.29/2.22	    f(0, 1, g(x, y), z) => f(g(x, y), g(x, y), g(x, y), h(x)) 
4.29/2.22	    g(0, 1) => 0 
4.29/2.22	    g(0, 1) => 1 
4.29/2.22	    h(g(x, y)) => h(x) 
4.29/2.22	    map(i, nil) => nil 
4.29/2.22	    map(i, cons(x, y)) => cons(i x, map(i, y)) 
4.29/2.22	    filter(i, nil) => nil 
4.29/2.22	    filter(i, cons(x, y)) => filter2(i x, i, x, y) 
4.29/2.22	    filter2(true, i, x, y) => cons(x, filter(i, y)) 
4.29/2.22	    filter2(false, i, x, y) => filter(i, y) 
4.29/2.22	
4.29/2.22	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
4.29/2.22	
4.29/2.22	This system is non-terminating, as demonstrated by our external first-order non-termination checker:
4.29/2.22	
4.29/2.22	 || The following well-typed term is terminating: f(g(0, 1), g(0, 1), g(0, 1), h(0))
4.29/2.22	 || 
4.29/2.22	 || proof of resources/system.trs
4.29/2.22	 || # AProVE Commit ID: 500ec9b2e2a919720cb177ef26031cb0220e008e fuhs 20130603
4.29/2.22	 || 
4.29/2.22	 || 
4.29/2.22	 || Termination w.r.t. Q of the given QTRS could be disproven:
4.29/2.22	 || 
4.29/2.22	 || (0) QTRS
4.29/2.22	 || (1) NonTerminationProof [EQUIVALENT, 0 ms]
4.29/2.22	 || (2) NO
4.29/2.22	 || 
4.29/2.22	 || 
4.29/2.22	 || ----------------------------------------
4.29/2.22	 || 
4.29/2.22	 || (0)
4.29/2.22	 || Obligation:
4.29/2.22	 || Q restricted rewrite system:
4.29/2.22	 || The TRS R consists of the following rules:
4.29/2.22	 || 
4.29/2.22	 ||    f(0, 1, g(%X, %Y), %Z) -> f(g(%X, %Y), g(%X, %Y), g(%X, %Y), h(%X))
4.29/2.22	 ||    g(0, 1) -> 0
4.29/2.22	 ||    g(0, 1) -> 1
4.29/2.22	 ||    h(g(%X, %Y)) -> h(%X)
4.29/2.22	 || 
4.29/2.22	 || Q is empty.
4.29/2.22	 || 
4.29/2.22	 || ----------------------------------------
4.29/2.22	 || 
4.29/2.22	 || (1) NonTerminationProof (EQUIVALENT)
4.29/2.22	 || The following loops were found:
4.29/2.22	 || 
4.29/2.22	 || ---------- Loop: ----------
4.29/2.22	 || 
4.29/2.22	 || f(g(0, 1), g(0, 1), g(0, 1), h(0)) -> f(g(0, 1), 1, g(0, 1), h(0)) with rule g(0, 1) -> 1 at position [1] and matcher [ ]
4.29/2.22	 || 
4.29/2.22	 || f(g(0, 1), 1, g(0, 1), h(0)) -> f(0, 1, g(0, 1), h(0)) with rule g(0, 1) -> 0 at position [0] and matcher [ ]
4.29/2.22	 || 
4.29/2.22	 || f(0, 1, g(0, 1), h(0)) -> f(g(0, 1), g(0, 1), g(0, 1), h(0)) with rule f(0, 1, g(%X, %Y), %Z) -> f(g(%X, %Y), g(%X, %Y), g(%X, %Y), h(%X)) at position [] and matcher [%X / 0, %Y / 1, %Z / h(0)]
4.29/2.22	 || 
4.29/2.22	 || Now an instance of the first term with Matcher [ ] occurs in the last term at position [].
4.29/2.22	 || 
4.29/2.22	 || Context: []
4.29/2.22	 || 
4.29/2.22	 || 
4.29/2.22	 || ----------------------------------------
4.29/2.22	 || 
4.29/2.22	 || (2)
4.29/2.22	 || NO
4.29/2.22	 || 
4.29/2.22	EOF