3.74/1.97	NO
3.74/1.97	We consider the system theBenchmark.
3.74/1.97	
3.74/1.97	  Alphabet:
3.74/1.97	
3.74/1.97	    0 : [] --> b 
3.74/1.97	    1 : [] --> b 
3.74/1.97	    cons : [e * f] --> f 
3.74/1.97	    f : [b * c] --> a 
3.74/1.97	    false : [] --> d 
3.74/1.97	    filter : [e -> d * f] --> f 
3.74/1.97	    filter2 : [d * e -> d * e * f] --> f 
3.74/1.97	    g : [b] --> c 
3.74/1.97	    map : [e -> e * f] --> f 
3.74/1.97	    nil : [] --> f 
3.74/1.97	    true : [] --> d 
3.74/1.97	
3.74/1.97	  Rules:
3.74/1.97	
3.74/1.97	    f(x, g(x)) => f(1, g(x)) 
3.74/1.97	    g(1) => g(0) 
3.74/1.97	    map(h, nil) => nil 
3.74/1.97	    map(h, cons(x, y)) => cons(h x, map(h, y)) 
3.74/1.97	    filter(h, nil) => nil 
3.74/1.97	    filter(h, cons(x, y)) => filter2(h x, h, x, y) 
3.74/1.97	    filter2(true, h, x, y) => cons(x, filter(h, y)) 
3.74/1.97	    filter2(false, h, x, y) => filter(h, y) 
3.74/1.97	
3.74/1.97	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
3.74/1.97	
3.74/1.97	This system is non-terminating, as demonstrated by our external first-order non-termination checker:
3.74/1.97	
3.74/1.97	 || The following well-typed term is terminating: f(1, g(1))
3.74/1.97	 || 
3.74/1.97	 || proof of resources/system.trs
3.74/1.97	 || # AProVE Commit ID: 500ec9b2e2a919720cb177ef26031cb0220e008e fuhs 20130603
3.74/1.97	 || 
3.74/1.97	 || 
3.74/1.97	 || Termination w.r.t. Q of the given QTRS could be disproven:
3.74/1.97	 || 
3.74/1.97	 || (0) QTRS
3.74/1.97	 || (1) NonTerminationProof [EQUIVALENT, 0 ms]
3.74/1.97	 || (2) NO
3.74/1.97	 || 
3.74/1.97	 || 
3.74/1.97	 || ----------------------------------------
3.74/1.97	 || 
3.74/1.97	 || (0)
3.74/1.97	 || Obligation:
3.74/1.97	 || Q restricted rewrite system:
3.74/1.97	 || The TRS R consists of the following rules:
3.74/1.97	 || 
3.74/1.97	 ||    f(%X, g(%X)) -> f(1, g(%X))
3.74/1.97	 ||    g(1) -> g(0)
3.74/1.97	 || 
3.74/1.97	 || Q is empty.
3.74/1.97	 || 
3.74/1.97	 || ----------------------------------------
3.74/1.97	 || 
3.74/1.97	 || (1) NonTerminationProof (EQUIVALENT)
3.74/1.97	 || The following loops were found:
3.74/1.97	 || 
3.74/1.97	 || ---------- Loop: ----------
3.74/1.97	 || 
3.74/1.97	 || f(1, g(1)) -> f(1, g(1)) with rule f(%X, g(%X)) -> f(1, g(%X)) at position [] and matcher [%X / 1]
3.74/1.97	 || 
3.74/1.97	 || Now an instance of the first term with Matcher [ ] occurs in the last term at position [].
3.74/1.97	 || 
3.74/1.97	 || Context: []
3.74/1.97	 || 
3.74/1.97	 || 
3.74/1.97	 || ----------------------------------------
3.74/1.97	 || 
3.74/1.97	 || (2)
3.74/1.97	 || NO
3.74/1.97	 || 
3.74/1.97	EOF