4.23/1.85	WORST_CASE(NON_POLY, ?)
4.23/1.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.23/1.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.23/1.86	
4.23/1.86	
4.23/1.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.23/1.86	
4.23/1.86	(0) CpxTRS
4.23/1.86	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.23/1.86	(2) TRS for Loop Detection
4.23/1.86	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.23/1.86	(4) BEST
4.23/1.86	    (5) proven lower bound
4.23/1.86	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
4.23/1.86	        (7) BOUNDS(n^1, INF)
4.23/1.86	    (8) TRS for Loop Detection
4.23/1.86	        (9) DecreasingLoopProof [FINISHED, 154 ms]
4.23/1.86	        (10) BOUNDS(EXP, INF)
4.23/1.86	
4.23/1.86	
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(0)
4.23/1.86	Obligation:
4.23/1.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.23/1.86	
4.23/1.86	
4.23/1.86	The TRS R consists of the following rules:
4.23/1.86	
4.23/1.86	   U11(tt, N) -> activate(N)
4.23/1.86	   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
4.23/1.86	   and(tt, X) -> activate(X)
4.23/1.86	   isNat(n__0) -> tt
4.23/1.86	   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
4.23/1.86	   isNat(n__s(V1)) -> isNat(activate(V1))
4.23/1.86	   plus(N, 0) -> U11(isNat(N), N)
4.23/1.86	   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
4.23/1.86	   0 -> n__0
4.23/1.86	   plus(X1, X2) -> n__plus(X1, X2)
4.23/1.86	   isNat(X) -> n__isNat(X)
4.23/1.86	   s(X) -> n__s(X)
4.23/1.86	   activate(n__0) -> 0
4.23/1.86	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
4.23/1.86	   activate(n__isNat(X)) -> isNat(X)
4.23/1.86	   activate(n__s(X)) -> s(activate(X))
4.23/1.86	   activate(X) -> X
4.23/1.86	
4.23/1.86	S is empty.
4.23/1.86	Rewrite Strategy: FULL
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.23/1.86	Transformed a relative TRS into a decreasing-loop problem.
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(2)
4.23/1.86	Obligation:
4.23/1.86	Analyzing the following TRS for decreasing loops:
4.23/1.86	
4.23/1.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.23/1.86	
4.23/1.86	
4.23/1.86	The TRS R consists of the following rules:
4.23/1.86	
4.23/1.86	   U11(tt, N) -> activate(N)
4.23/1.86	   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
4.23/1.86	   and(tt, X) -> activate(X)
4.23/1.86	   isNat(n__0) -> tt
4.23/1.86	   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
4.23/1.86	   isNat(n__s(V1)) -> isNat(activate(V1))
4.23/1.86	   plus(N, 0) -> U11(isNat(N), N)
4.23/1.86	   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
4.23/1.86	   0 -> n__0
4.23/1.86	   plus(X1, X2) -> n__plus(X1, X2)
4.23/1.86	   isNat(X) -> n__isNat(X)
4.23/1.86	   s(X) -> n__s(X)
4.23/1.86	   activate(n__0) -> 0
4.23/1.86	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
4.23/1.86	   activate(n__isNat(X)) -> isNat(X)
4.23/1.86	   activate(n__s(X)) -> s(activate(X))
4.23/1.86	   activate(X) -> X
4.23/1.86	
4.23/1.86	S is empty.
4.23/1.86	Rewrite Strategy: FULL
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(3) DecreasingLoopProof (LOWER BOUND(ID))
4.23/1.86	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.23/1.86	
4.23/1.86	The rewrite sequence
4.23/1.86	
4.23/1.86	activate(n__s(X)) ->^+ s(activate(X))
4.23/1.86	
4.23/1.86	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.23/1.86	
4.23/1.86	The pumping substitution is [X / n__s(X)].
4.23/1.86	
4.23/1.86	The result substitution is [ ].
4.23/1.86	
4.23/1.86	
4.23/1.86	
4.23/1.86	
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(4)
4.23/1.86	Complex Obligation (BEST)
4.23/1.86	
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(5)
4.23/1.86	Obligation:
4.23/1.86	Proved the lower bound n^1 for the following obligation:
4.23/1.86	
4.23/1.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.23/1.86	
4.23/1.86	
4.23/1.86	The TRS R consists of the following rules:
4.23/1.86	
4.23/1.86	   U11(tt, N) -> activate(N)
4.23/1.86	   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
4.23/1.86	   and(tt, X) -> activate(X)
4.23/1.86	   isNat(n__0) -> tt
4.23/1.86	   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
4.23/1.86	   isNat(n__s(V1)) -> isNat(activate(V1))
4.23/1.86	   plus(N, 0) -> U11(isNat(N), N)
4.23/1.86	   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
4.23/1.86	   0 -> n__0
4.23/1.86	   plus(X1, X2) -> n__plus(X1, X2)
4.23/1.86	   isNat(X) -> n__isNat(X)
4.23/1.86	   s(X) -> n__s(X)
4.23/1.86	   activate(n__0) -> 0
4.23/1.86	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
4.23/1.86	   activate(n__isNat(X)) -> isNat(X)
4.23/1.86	   activate(n__s(X)) -> s(activate(X))
4.23/1.86	   activate(X) -> X
4.23/1.86	
4.23/1.86	S is empty.
4.23/1.86	Rewrite Strategy: FULL
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(6) LowerBoundPropagationProof (FINISHED)
4.23/1.86	Propagated lower bound.
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(7)
4.23/1.86	BOUNDS(n^1, INF)
4.23/1.86	
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(8)
4.23/1.86	Obligation:
4.23/1.86	Analyzing the following TRS for decreasing loops:
4.23/1.86	
4.23/1.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.23/1.86	
4.23/1.86	
4.23/1.86	The TRS R consists of the following rules:
4.23/1.86	
4.23/1.86	   U11(tt, N) -> activate(N)
4.23/1.86	   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
4.23/1.86	   and(tt, X) -> activate(X)
4.23/1.86	   isNat(n__0) -> tt
4.23/1.86	   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
4.23/1.86	   isNat(n__s(V1)) -> isNat(activate(V1))
4.23/1.86	   plus(N, 0) -> U11(isNat(N), N)
4.23/1.86	   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
4.23/1.86	   0 -> n__0
4.23/1.86	   plus(X1, X2) -> n__plus(X1, X2)
4.23/1.86	   isNat(X) -> n__isNat(X)
4.23/1.86	   s(X) -> n__s(X)
4.23/1.86	   activate(n__0) -> 0
4.23/1.86	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
4.23/1.86	   activate(n__isNat(X)) -> isNat(X)
4.23/1.86	   activate(n__s(X)) -> s(activate(X))
4.23/1.86	   activate(X) -> X
4.23/1.86	
4.23/1.86	S is empty.
4.23/1.86	Rewrite Strategy: FULL
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(9) DecreasingLoopProof (FINISHED)
4.23/1.86	The following loop(s) give(s) rise to the lower bound EXP:
4.23/1.86	
4.23/1.86	The rewrite sequence
4.23/1.86	
4.23/1.86	activate(n__plus(X1, n__s(X1_0))) ->^+ U21(and(isNat(activate(X1_0)), n__isNat(activate(X1))), activate(X1_0), activate(X1))
4.23/1.86	
4.23/1.86	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
4.23/1.86	
4.23/1.86	The pumping substitution is [X1_0 / n__plus(X1, n__s(X1_0))].
4.23/1.86	
4.23/1.86	The result substitution is [ ].
4.23/1.86	
4.23/1.86	
4.23/1.86	
4.23/1.86	The rewrite sequence
4.23/1.86	
4.23/1.86	activate(n__plus(X1, n__s(X1_0))) ->^+ U21(and(isNat(activate(X1_0)), n__isNat(activate(X1))), activate(X1_0), activate(X1))
4.23/1.86	
4.23/1.86	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
4.23/1.86	
4.23/1.86	The pumping substitution is [X1_0 / n__plus(X1, n__s(X1_0))].
4.23/1.86	
4.23/1.86	The result substitution is [ ].
4.23/1.86	
4.23/1.86	
4.23/1.86	
4.23/1.86	
4.23/1.86	----------------------------------------
4.23/1.86	
4.23/1.86	(10)
4.23/1.86	BOUNDS(EXP, INF)
4.38/1.90	EOF