5.30/2.10	WORST_CASE(NON_POLY, ?)
5.30/2.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.30/2.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.30/2.12	
5.30/2.12	
5.30/2.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.30/2.12	
5.30/2.12	(0) CpxTRS
5.30/2.12	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.30/2.12	(2) TRS for Loop Detection
5.30/2.12	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.30/2.12	(4) BEST
5.30/2.12	    (5) proven lower bound
5.30/2.12	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
5.30/2.12	        (7) BOUNDS(n^1, INF)
5.30/2.12	    (8) TRS for Loop Detection
5.30/2.12	        (9) DecreasingLoopProof [FINISHED, 460 ms]
5.30/2.12	        (10) BOUNDS(EXP, INF)
5.30/2.12	
5.30/2.12	
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(0)
5.30/2.12	Obligation:
5.30/2.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.30/2.12	
5.30/2.12	
5.30/2.12	The TRS R consists of the following rules:
5.30/2.12	
5.30/2.12	   U11(tt, V2) -> U12(isNat(activate(V2)))
5.30/2.12	   U12(tt) -> tt
5.30/2.12	   U21(tt) -> tt
5.30/2.12	   U31(tt, V2) -> U32(isNat(activate(V2)))
5.30/2.12	   U32(tt) -> tt
5.30/2.12	   U41(tt, N) -> activate(N)
5.30/2.12	   U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N))
5.30/2.12	   U52(tt, M, N) -> s(plus(activate(N), activate(M)))
5.30/2.12	   U61(tt) -> 0
5.30/2.12	   U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N))
5.30/2.12	   U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
5.30/2.12	   isNat(n__0) -> tt
5.30/2.12	   isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
5.30/2.12	   isNat(n__s(V1)) -> U21(isNat(activate(V1)))
5.30/2.12	   isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2))
5.30/2.12	   plus(N, 0) -> U41(isNat(N), N)
5.30/2.12	   plus(N, s(M)) -> U51(isNat(M), M, N)
5.30/2.12	   x(N, 0) -> U61(isNat(N))
5.30/2.12	   x(N, s(M)) -> U71(isNat(M), M, N)
5.30/2.12	   0 -> n__0
5.30/2.12	   plus(X1, X2) -> n__plus(X1, X2)
5.30/2.12	   s(X) -> n__s(X)
5.30/2.12	   x(X1, X2) -> n__x(X1, X2)
5.30/2.12	   activate(n__0) -> 0
5.30/2.12	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
5.30/2.12	   activate(n__s(X)) -> s(activate(X))
5.30/2.12	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
5.30/2.12	   activate(X) -> X
5.30/2.12	
5.30/2.12	S is empty.
5.30/2.12	Rewrite Strategy: FULL
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.30/2.12	Transformed a relative TRS into a decreasing-loop problem.
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(2)
5.30/2.12	Obligation:
5.30/2.12	Analyzing the following TRS for decreasing loops:
5.30/2.12	
5.30/2.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.30/2.12	
5.30/2.12	
5.30/2.12	The TRS R consists of the following rules:
5.30/2.12	
5.30/2.12	   U11(tt, V2) -> U12(isNat(activate(V2)))
5.30/2.12	   U12(tt) -> tt
5.30/2.12	   U21(tt) -> tt
5.30/2.12	   U31(tt, V2) -> U32(isNat(activate(V2)))
5.30/2.12	   U32(tt) -> tt
5.30/2.12	   U41(tt, N) -> activate(N)
5.30/2.12	   U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N))
5.30/2.12	   U52(tt, M, N) -> s(plus(activate(N), activate(M)))
5.30/2.12	   U61(tt) -> 0
5.30/2.12	   U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N))
5.30/2.12	   U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
5.30/2.12	   isNat(n__0) -> tt
5.30/2.12	   isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
5.30/2.12	   isNat(n__s(V1)) -> U21(isNat(activate(V1)))
5.30/2.12	   isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2))
5.30/2.12	   plus(N, 0) -> U41(isNat(N), N)
5.30/2.12	   plus(N, s(M)) -> U51(isNat(M), M, N)
5.30/2.12	   x(N, 0) -> U61(isNat(N))
5.30/2.12	   x(N, s(M)) -> U71(isNat(M), M, N)
5.30/2.12	   0 -> n__0
5.30/2.12	   plus(X1, X2) -> n__plus(X1, X2)
5.30/2.12	   s(X) -> n__s(X)
5.30/2.12	   x(X1, X2) -> n__x(X1, X2)
5.30/2.12	   activate(n__0) -> 0
5.30/2.12	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
5.30/2.12	   activate(n__s(X)) -> s(activate(X))
5.30/2.12	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
5.30/2.12	   activate(X) -> X
5.30/2.12	
5.30/2.12	S is empty.
5.30/2.12	Rewrite Strategy: FULL
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(3) DecreasingLoopProof (LOWER BOUND(ID))
5.30/2.12	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.30/2.12	
5.30/2.12	The rewrite sequence
5.30/2.12	
5.30/2.12	activate(n__s(X)) ->^+ s(activate(X))
5.30/2.12	
5.30/2.12	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
5.30/2.12	
5.30/2.12	The pumping substitution is [X / n__s(X)].
5.30/2.12	
5.30/2.12	The result substitution is [ ].
5.30/2.12	
5.30/2.12	
5.30/2.12	
5.30/2.12	
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(4)
5.30/2.12	Complex Obligation (BEST)
5.30/2.12	
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(5)
5.30/2.12	Obligation:
5.30/2.12	Proved the lower bound n^1 for the following obligation:
5.30/2.12	
5.30/2.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.30/2.12	
5.30/2.12	
5.30/2.12	The TRS R consists of the following rules:
5.30/2.12	
5.30/2.12	   U11(tt, V2) -> U12(isNat(activate(V2)))
5.30/2.12	   U12(tt) -> tt
5.30/2.12	   U21(tt) -> tt
5.30/2.12	   U31(tt, V2) -> U32(isNat(activate(V2)))
5.30/2.12	   U32(tt) -> tt
5.30/2.12	   U41(tt, N) -> activate(N)
5.30/2.12	   U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N))
5.30/2.12	   U52(tt, M, N) -> s(plus(activate(N), activate(M)))
5.30/2.12	   U61(tt) -> 0
5.30/2.12	   U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N))
5.30/2.12	   U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
5.30/2.12	   isNat(n__0) -> tt
5.30/2.12	   isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
5.30/2.12	   isNat(n__s(V1)) -> U21(isNat(activate(V1)))
5.30/2.12	   isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2))
5.30/2.12	   plus(N, 0) -> U41(isNat(N), N)
5.30/2.12	   plus(N, s(M)) -> U51(isNat(M), M, N)
5.30/2.12	   x(N, 0) -> U61(isNat(N))
5.30/2.12	   x(N, s(M)) -> U71(isNat(M), M, N)
5.30/2.12	   0 -> n__0
5.30/2.12	   plus(X1, X2) -> n__plus(X1, X2)
5.30/2.12	   s(X) -> n__s(X)
5.30/2.12	   x(X1, X2) -> n__x(X1, X2)
5.30/2.12	   activate(n__0) -> 0
5.30/2.12	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
5.30/2.12	   activate(n__s(X)) -> s(activate(X))
5.30/2.12	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
5.30/2.12	   activate(X) -> X
5.30/2.12	
5.30/2.12	S is empty.
5.30/2.12	Rewrite Strategy: FULL
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(6) LowerBoundPropagationProof (FINISHED)
5.30/2.12	Propagated lower bound.
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(7)
5.30/2.12	BOUNDS(n^1, INF)
5.30/2.12	
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(8)
5.30/2.12	Obligation:
5.30/2.12	Analyzing the following TRS for decreasing loops:
5.30/2.12	
5.30/2.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.30/2.12	
5.30/2.12	
5.30/2.12	The TRS R consists of the following rules:
5.30/2.12	
5.30/2.12	   U11(tt, V2) -> U12(isNat(activate(V2)))
5.30/2.12	   U12(tt) -> tt
5.30/2.12	   U21(tt) -> tt
5.30/2.12	   U31(tt, V2) -> U32(isNat(activate(V2)))
5.30/2.12	   U32(tt) -> tt
5.30/2.12	   U41(tt, N) -> activate(N)
5.30/2.12	   U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N))
5.30/2.12	   U52(tt, M, N) -> s(plus(activate(N), activate(M)))
5.30/2.12	   U61(tt) -> 0
5.30/2.12	   U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N))
5.30/2.12	   U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
5.30/2.12	   isNat(n__0) -> tt
5.30/2.12	   isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
5.30/2.12	   isNat(n__s(V1)) -> U21(isNat(activate(V1)))
5.30/2.12	   isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2))
5.30/2.12	   plus(N, 0) -> U41(isNat(N), N)
5.30/2.12	   plus(N, s(M)) -> U51(isNat(M), M, N)
5.30/2.12	   x(N, 0) -> U61(isNat(N))
5.30/2.12	   x(N, s(M)) -> U71(isNat(M), M, N)
5.30/2.12	   0 -> n__0
5.30/2.12	   plus(X1, X2) -> n__plus(X1, X2)
5.30/2.12	   s(X) -> n__s(X)
5.30/2.12	   x(X1, X2) -> n__x(X1, X2)
5.30/2.12	   activate(n__0) -> 0
5.30/2.12	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
5.30/2.12	   activate(n__s(X)) -> s(activate(X))
5.30/2.12	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
5.30/2.12	   activate(X) -> X
5.30/2.12	
5.30/2.12	S is empty.
5.30/2.12	Rewrite Strategy: FULL
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(9) DecreasingLoopProof (FINISHED)
5.30/2.12	The following loop(s) give(s) rise to the lower bound EXP:
5.30/2.12	
5.30/2.12	The rewrite sequence
5.30/2.12	
5.30/2.12	activate(n__x(X1, n__s(X1_0))) ->^+ U71(isNat(activate(X1_0)), activate(X1_0), activate(X1))
5.30/2.12	
5.30/2.12	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
5.30/2.12	
5.30/2.12	The pumping substitution is [X1_0 / n__x(X1, n__s(X1_0))].
5.30/2.12	
5.30/2.12	The result substitution is [ ].
5.30/2.12	
5.30/2.12	
5.30/2.12	
5.30/2.12	The rewrite sequence
5.30/2.12	
5.30/2.12	activate(n__x(X1, n__s(X1_0))) ->^+ U71(isNat(activate(X1_0)), activate(X1_0), activate(X1))
5.30/2.12	
5.30/2.12	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
5.30/2.12	
5.30/2.12	The pumping substitution is [X1_0 / n__x(X1, n__s(X1_0))].
5.30/2.12	
5.30/2.12	The result substitution is [ ].
5.30/2.12	
5.30/2.12	
5.30/2.12	
5.30/2.12	
5.30/2.12	----------------------------------------
5.30/2.12	
5.30/2.12	(10)
5.30/2.12	BOUNDS(EXP, INF)
5.66/2.15	EOF