5.46/2.16	WORST_CASE(NON_POLY, ?)
5.46/2.18	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.46/2.18	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.46/2.18	
5.46/2.18	
5.46/2.18	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.46/2.18	
5.46/2.18	(0) CpxTRS
5.46/2.18	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.46/2.18	(2) TRS for Loop Detection
5.46/2.18	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.46/2.18	(4) BEST
5.46/2.18	    (5) proven lower bound
5.46/2.18	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
5.46/2.18	        (7) BOUNDS(n^1, INF)
5.46/2.18	    (8) TRS for Loop Detection
5.46/2.18	        (9) DecreasingLoopProof [FINISHED, 286 ms]
5.46/2.18	        (10) BOUNDS(EXP, INF)
5.46/2.18	
5.46/2.18	
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(0)
5.46/2.18	Obligation:
5.46/2.18	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.46/2.18	
5.46/2.18	
5.46/2.18	The TRS R consists of the following rules:
5.46/2.18	
5.46/2.18	   U11(tt, N) -> activate(N)
5.46/2.18	   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
5.46/2.18	   U31(tt) -> 0
5.46/2.18	   U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
5.46/2.18	   and(tt, X) -> activate(X)
5.46/2.18	   isNat(n__0) -> tt
5.46/2.18	   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
5.46/2.18	   isNat(n__s(V1)) -> isNat(activate(V1))
5.46/2.18	   isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
5.46/2.18	   plus(N, 0) -> U11(isNat(N), N)
5.46/2.18	   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
5.46/2.18	   x(N, 0) -> U31(isNat(N))
5.46/2.18	   x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N)
5.46/2.18	   0 -> n__0
5.46/2.18	   plus(X1, X2) -> n__plus(X1, X2)
5.46/2.18	   isNat(X) -> n__isNat(X)
5.46/2.18	   s(X) -> n__s(X)
5.46/2.18	   x(X1, X2) -> n__x(X1, X2)
5.46/2.18	   activate(n__0) -> 0
5.46/2.18	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
5.46/2.18	   activate(n__isNat(X)) -> isNat(X)
5.46/2.18	   activate(n__s(X)) -> s(activate(X))
5.46/2.18	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
5.46/2.18	   activate(X) -> X
5.46/2.18	
5.46/2.18	S is empty.
5.46/2.18	Rewrite Strategy: FULL
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.46/2.18	Transformed a relative TRS into a decreasing-loop problem.
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(2)
5.46/2.18	Obligation:
5.46/2.18	Analyzing the following TRS for decreasing loops:
5.46/2.18	
5.46/2.18	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.46/2.18	
5.46/2.18	
5.46/2.18	The TRS R consists of the following rules:
5.46/2.18	
5.46/2.18	   U11(tt, N) -> activate(N)
5.46/2.18	   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
5.46/2.18	   U31(tt) -> 0
5.46/2.18	   U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
5.46/2.18	   and(tt, X) -> activate(X)
5.46/2.18	   isNat(n__0) -> tt
5.46/2.18	   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
5.46/2.18	   isNat(n__s(V1)) -> isNat(activate(V1))
5.46/2.18	   isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
5.46/2.18	   plus(N, 0) -> U11(isNat(N), N)
5.46/2.18	   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
5.46/2.18	   x(N, 0) -> U31(isNat(N))
5.46/2.18	   x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N)
5.46/2.18	   0 -> n__0
5.46/2.18	   plus(X1, X2) -> n__plus(X1, X2)
5.46/2.18	   isNat(X) -> n__isNat(X)
5.46/2.18	   s(X) -> n__s(X)
5.46/2.18	   x(X1, X2) -> n__x(X1, X2)
5.46/2.18	   activate(n__0) -> 0
5.46/2.18	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
5.46/2.18	   activate(n__isNat(X)) -> isNat(X)
5.46/2.18	   activate(n__s(X)) -> s(activate(X))
5.46/2.18	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
5.46/2.18	   activate(X) -> X
5.46/2.18	
5.46/2.18	S is empty.
5.46/2.18	Rewrite Strategy: FULL
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(3) DecreasingLoopProof (LOWER BOUND(ID))
5.46/2.18	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.46/2.18	
5.46/2.18	The rewrite sequence
5.46/2.18	
5.46/2.18	activate(n__s(X)) ->^+ s(activate(X))
5.46/2.18	
5.46/2.18	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
5.46/2.18	
5.46/2.18	The pumping substitution is [X / n__s(X)].
5.46/2.18	
5.46/2.18	The result substitution is [ ].
5.46/2.18	
5.46/2.18	
5.46/2.18	
5.46/2.18	
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(4)
5.46/2.18	Complex Obligation (BEST)
5.46/2.18	
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(5)
5.46/2.18	Obligation:
5.46/2.18	Proved the lower bound n^1 for the following obligation:
5.46/2.18	
5.46/2.18	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.46/2.18	
5.46/2.18	
5.46/2.18	The TRS R consists of the following rules:
5.46/2.18	
5.46/2.18	   U11(tt, N) -> activate(N)
5.46/2.18	   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
5.46/2.18	   U31(tt) -> 0
5.46/2.18	   U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
5.46/2.18	   and(tt, X) -> activate(X)
5.46/2.18	   isNat(n__0) -> tt
5.46/2.18	   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
5.46/2.18	   isNat(n__s(V1)) -> isNat(activate(V1))
5.46/2.18	   isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
5.46/2.18	   plus(N, 0) -> U11(isNat(N), N)
5.46/2.18	   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
5.46/2.18	   x(N, 0) -> U31(isNat(N))
5.46/2.18	   x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N)
5.46/2.18	   0 -> n__0
5.46/2.18	   plus(X1, X2) -> n__plus(X1, X2)
5.46/2.18	   isNat(X) -> n__isNat(X)
5.46/2.18	   s(X) -> n__s(X)
5.46/2.18	   x(X1, X2) -> n__x(X1, X2)
5.46/2.18	   activate(n__0) -> 0
5.46/2.18	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
5.46/2.18	   activate(n__isNat(X)) -> isNat(X)
5.46/2.18	   activate(n__s(X)) -> s(activate(X))
5.46/2.18	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
5.46/2.18	   activate(X) -> X
5.46/2.18	
5.46/2.18	S is empty.
5.46/2.18	Rewrite Strategy: FULL
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(6) LowerBoundPropagationProof (FINISHED)
5.46/2.18	Propagated lower bound.
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(7)
5.46/2.18	BOUNDS(n^1, INF)
5.46/2.18	
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(8)
5.46/2.18	Obligation:
5.46/2.18	Analyzing the following TRS for decreasing loops:
5.46/2.18	
5.46/2.18	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.46/2.18	
5.46/2.18	
5.46/2.18	The TRS R consists of the following rules:
5.46/2.18	
5.46/2.18	   U11(tt, N) -> activate(N)
5.46/2.18	   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
5.46/2.18	   U31(tt) -> 0
5.46/2.18	   U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
5.46/2.18	   and(tt, X) -> activate(X)
5.46/2.18	   isNat(n__0) -> tt
5.46/2.18	   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
5.46/2.18	   isNat(n__s(V1)) -> isNat(activate(V1))
5.46/2.18	   isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
5.46/2.18	   plus(N, 0) -> U11(isNat(N), N)
5.46/2.18	   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
5.46/2.18	   x(N, 0) -> U31(isNat(N))
5.46/2.18	   x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N)
5.46/2.18	   0 -> n__0
5.46/2.18	   plus(X1, X2) -> n__plus(X1, X2)
5.46/2.18	   isNat(X) -> n__isNat(X)
5.46/2.18	   s(X) -> n__s(X)
5.46/2.18	   x(X1, X2) -> n__x(X1, X2)
5.46/2.18	   activate(n__0) -> 0
5.46/2.18	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
5.46/2.18	   activate(n__isNat(X)) -> isNat(X)
5.46/2.18	   activate(n__s(X)) -> s(activate(X))
5.46/2.18	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
5.46/2.18	   activate(X) -> X
5.46/2.18	
5.46/2.18	S is empty.
5.46/2.18	Rewrite Strategy: FULL
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(9) DecreasingLoopProof (FINISHED)
5.46/2.18	The following loop(s) give(s) rise to the lower bound EXP:
5.46/2.18	
5.46/2.18	The rewrite sequence
5.46/2.18	
5.46/2.18	activate(n__x(X1, n__s(X1_0))) ->^+ U41(and(isNat(activate(X1_0)), n__isNat(activate(X1))), activate(X1_0), activate(X1))
5.46/2.18	
5.46/2.18	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
5.46/2.18	
5.46/2.18	The pumping substitution is [X1_0 / n__x(X1, n__s(X1_0))].
5.46/2.18	
5.46/2.18	The result substitution is [ ].
5.46/2.18	
5.46/2.18	
5.46/2.18	
5.46/2.18	The rewrite sequence
5.46/2.18	
5.46/2.18	activate(n__x(X1, n__s(X1_0))) ->^+ U41(and(isNat(activate(X1_0)), n__isNat(activate(X1))), activate(X1_0), activate(X1))
5.46/2.18	
5.46/2.18	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
5.46/2.18	
5.46/2.18	The pumping substitution is [X1_0 / n__x(X1, n__s(X1_0))].
5.46/2.18	
5.46/2.18	The result substitution is [ ].
5.46/2.18	
5.46/2.18	
5.46/2.18	
5.46/2.18	
5.46/2.18	----------------------------------------
5.46/2.18	
5.46/2.18	(10)
5.46/2.18	BOUNDS(EXP, INF)
5.56/2.21	EOF