3.64/2.76	NO
3.64/2.76	We consider the system theBenchmark.
3.64/2.76	
3.64/2.76	  Alphabet:
3.64/2.76	
3.64/2.76	    cons : [d * e] --> e 
3.64/2.76	    f : [b * b * b] --> a 
3.64/2.76	    false : [] --> c 
3.64/2.76	    filter : [d -> c * e] --> e 
3.64/2.76	    filter2 : [c * d -> c * d * e] --> e 
3.64/2.76	    g : [b * b] --> b 
3.64/2.76	    map : [d -> d * e] --> e 
3.64/2.76	    nil : [] --> e 
3.64/2.76	    true : [] --> c 
3.64/2.76	
3.64/2.76	  Rules:
3.64/2.76	
3.64/2.76	    f(g(x, y), x, z) => f(z, z, z) 
3.64/2.76	    g(x, y) => x 
3.64/2.76	    g(x, y) => y 
3.64/2.76	    map(h, nil) => nil 
3.64/2.76	    map(h, cons(x, y)) => cons(h x, map(h, y)) 
3.64/2.76	    filter(h, nil) => nil 
3.64/2.76	    filter(h, cons(x, y)) => filter2(h x, h, x, y) 
3.64/2.76	    filter2(true, h, x, y) => cons(x, filter(h, y)) 
3.64/2.76	    filter2(false, h, x, y) => filter(h, y) 
3.64/2.76	
3.64/2.76	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
3.64/2.76	
3.64/2.76	This system is non-terminating, as demonstrated by our external first-order non-termination checker:
3.64/2.76	
3.64/2.76	 || The following well-typed term is terminating: f(g(%X', %Y), g(%X', %Y), g(%X', %Y))
3.64/2.76	 || 
3.64/2.76	 || proof of resources/system.trs
3.64/2.76	 || # AProVE Commit ID: 500ec9b2e2a919720cb177ef26031cb0220e008e fuhs 20130603
3.64/2.76	 || 
3.64/2.76	 || 
3.64/2.76	 || Termination w.r.t. Q of the given QTRS could be disproven:
3.64/2.76	 || 
3.64/2.76	 || (0) QTRS
3.64/2.76	 || (1) NonTerminationProof [EQUIVALENT, 0 ms]
3.64/2.76	 || (2) NO
3.64/2.76	 || 
3.64/2.76	 || 
3.64/2.76	 || ----------------------------------------
3.64/2.76	 || 
3.64/2.76	 || (0)
3.64/2.76	 || Obligation:
3.64/2.76	 || Q restricted rewrite system:
3.64/2.76	 || The TRS R consists of the following rules:
3.64/2.76	 || 
3.64/2.76	 ||    f(g(%X, %Y), %X, %Z) -> f(%Z, %Z, %Z)
3.64/2.76	 ||    g(%X, %Y) -> %X
3.64/2.76	 ||    g(%X, %Y) -> %Y
3.64/2.76	 || 
3.64/2.76	 || Q is empty.
3.64/2.76	 || 
3.64/2.76	 || ----------------------------------------
3.64/2.76	 || 
3.64/2.76	 || (1) NonTerminationProof (EQUIVALENT)
3.64/2.76	 || The following loops were found:
3.64/2.76	 || 
3.64/2.76	 || ---------- Loop: ----------
3.64/2.76	 || 
3.64/2.76	 || f(g(%X', %Y), g(%X', %Y), g(%X', %Y)) -> f(g(%X', %Y), %X', g(%X', %Y)) with rule g(%X'', %Y') -> %X'' at position [1] and matcher [%X'' / %X', %Y' / %Y]
3.64/2.76	 || 
3.64/2.76	 || f(g(%X', %Y), %X', g(%X', %Y)) -> f(g(%X', %Y), g(%X', %Y), g(%X', %Y)) with rule f(g(%X, %Y'), %X, %Z) -> f(%Z, %Z, %Z) at position [] and matcher [%X / %X', %Y' / %Y, %Z / g(%X', %Y)]
3.64/2.76	 || 
3.64/2.76	 || Now an instance of the first term with Matcher [ ] occurs in the last term at position [].
3.64/2.76	 || 
3.64/2.76	 || Context: []
3.64/2.76	 || 
3.64/2.76	 || 
3.64/2.76	 || ----------------------------------------
3.64/2.76	 || 
3.64/2.76	 || (2)
3.64/2.76	 || NO
3.64/2.76	 || 
3.64/2.76	EOF