4.82/2.16	NO
4.82/2.16	We consider the system theBenchmark.
4.82/2.16	
4.82/2.16	  Alphabet:
4.82/2.16	
4.82/2.16	    fapp : [o * o] --> o 
4.82/2.16	    lam : [o] --> o 
4.82/2.16	    subst : [o * o] --> o 
4.82/2.16	    v : [] --> o 
4.82/2.16	
4.82/2.16	  Rules:
4.82/2.16	
4.82/2.16	    fapp(lam(x), y) => subst(x, y) 
4.82/2.16	    subst(v, x) => x 
4.82/2.16	    subst(lam(x), y) => lam(x) 
4.82/2.16	    subst(fapp(x, y), z) => fapp(subst(x, z), subst(y, z)) 
4.82/2.16	
4.82/2.16	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
4.82/2.16	
4.82/2.16	This system is non-terminating, as demonstrated by our external first-order non-termination checker:
4.82/2.16	
4.82/2.16	 || The following well-typed term is terminating: subst(fapp(v, v), subst(v, lam(fapp(v, v))))
4.82/2.16	 || 
4.82/2.16	 || proof of resources/system.trs
4.82/2.16	 || # AProVE Commit ID: 500ec9b2e2a919720cb177ef26031cb0220e008e fuhs 20130603
4.82/2.16	 || 
4.82/2.16	 || 
4.82/2.16	 || Termination w.r.t. Q of the given QTRS could be disproven:
4.82/2.16	 || 
4.82/2.16	 || (0) QTRS
4.82/2.16	 || (1) NonTerminationProof [EQUIVALENT, 78 ms]
4.82/2.16	 || (2) NO
4.82/2.16	 || 
4.82/2.16	 || 
4.82/2.16	 || ----------------------------------------
4.82/2.16	 || 
4.82/2.16	 || (0)
4.82/2.16	 || Obligation:
4.82/2.16	 || Q restricted rewrite system:
4.82/2.16	 || The TRS R consists of the following rules:
4.82/2.16	 || 
4.82/2.16	 ||    fapp(lam(%X), %Y) -> subst(%X, %Y)
4.82/2.16	 ||    subst(v, %X) -> %X
4.82/2.16	 ||    subst(lam(%X), %Y) -> lam(%X)
4.82/2.16	 ||    subst(fapp(%X, %Y), %Z) -> fapp(subst(%X, %Z), subst(%Y, %Z))
4.82/2.16	 || 
4.82/2.16	 || Q is empty.
4.82/2.16	 || 
4.82/2.16	 || ----------------------------------------
4.82/2.16	 || 
4.82/2.16	 || (1) NonTerminationProof (EQUIVALENT)
4.82/2.16	 || The following loops were found:
4.82/2.16	 || 
4.82/2.16	 || ---------- Loop: ----------
4.82/2.16	 || 
4.82/2.16	 || subst(fapp(v, v), subst(v, lam(fapp(v, v)))) -> subst(fapp(v, v), lam(fapp(v, v))) with rule subst(v, %X') -> %X' at position [1] and matcher [%X' / lam(fapp(v, v))]
4.82/2.16	 || 
4.82/2.16	 || subst(fapp(v, v), lam(fapp(v, v))) -> fapp(subst(v, lam(fapp(v, v))), subst(v, lam(fapp(v, v)))) with rule subst(fapp(%X, %Y), %Z) -> fapp(subst(%X, %Z), subst(%Y, %Z)) at position [] and matcher [%X / v, %Y / v, %Z / lam(fapp(v, v))]
4.82/2.16	 || 
4.82/2.16	 || fapp(subst(v, lam(fapp(v, v))), subst(v, lam(fapp(v, v)))) -> fapp(lam(fapp(v, v)), subst(v, lam(fapp(v, v)))) with rule subst(v, %X') -> %X' at position [0] and matcher [%X' / lam(fapp(v, v))]
4.82/2.16	 || 
4.82/2.16	 || fapp(lam(fapp(v, v)), subst(v, lam(fapp(v, v)))) -> subst(fapp(v, v), subst(v, lam(fapp(v, v)))) with rule fapp(lam(%X), %Y) -> subst(%X, %Y) at position [] and matcher [%X / fapp(v, v), %Y / subst(v, lam(fapp(v, v)))]
4.82/2.16	 || 
4.82/2.16	 || Now an instance of the first term with Matcher [ ] occurs in the last term at position [].
4.82/2.16	 || 
4.82/2.16	 || Context: []
4.82/2.16	 || 
4.82/2.16	 || 
4.82/2.16	 || ----------------------------------------
4.82/2.16	 || 
4.82/2.16	 || (2)
4.82/2.16	 || NO
4.82/2.16	 || 
4.82/2.16	EOF