4.31/1.95	WORST_CASE(NON_POLY, ?)
4.31/1.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.31/1.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.31/1.95	
4.31/1.95	
4.31/1.95	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.31/1.95	
4.31/1.95	(0) CpxTRS
4.31/1.95	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.31/1.95	(2) TRS for Loop Detection
4.31/1.95	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.31/1.95	(4) BEST
4.31/1.95	    (5) proven lower bound
4.31/1.95	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
4.31/1.95	        (7) BOUNDS(n^1, INF)
4.31/1.95	    (8) TRS for Loop Detection
4.31/1.95	        (9) DecreasingLoopProof [FINISHED, 142 ms]
4.31/1.95	        (10) BOUNDS(EXP, INF)
4.31/1.95	
4.31/1.95	
4.31/1.95	----------------------------------------
4.31/1.95	
4.31/1.95	(0)
4.31/1.95	Obligation:
4.31/1.95	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.31/1.95	
4.31/1.95	
4.31/1.95	The TRS R consists of the following rules:
4.31/1.95	
4.31/1.95	   U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2))
4.31/1.95	   U12(tt, V2) -> U13(isNat(activate(V2)))
4.31/1.95	   U13(tt) -> tt
4.31/1.95	   U21(tt, V1) -> U22(isNat(activate(V1)))
4.31/1.95	   U22(tt) -> tt
4.31/1.95	   U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2))
4.31/1.95	   U32(tt, V2) -> U33(isNat(activate(V2)))
4.31/1.95	   U33(tt) -> tt
4.31/1.95	   U41(tt, N) -> activate(N)
4.31/1.95	   U51(tt, M, N) -> s(plus(activate(N), activate(M)))
4.31/1.95	   U61(tt) -> 0
4.31/1.95	   U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
4.31/1.95	   and(tt, X) -> activate(X)
4.31/1.95	   isNat(n__0) -> tt
4.31/1.95	   isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
4.31/1.95	   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
4.31/1.95	   isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
4.31/1.95	   isNatKind(n__0) -> tt
4.31/1.95	   isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
4.31/1.95	   isNatKind(n__s(V1)) -> isNatKind(activate(V1))
4.31/1.95	   isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
4.31/1.95	   plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N)
4.31/1.95	   plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
4.31/1.96	   x(N, 0) -> U61(and(isNat(N), n__isNatKind(N)))
4.31/1.96	   x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
4.31/1.96	   0 -> n__0
4.31/1.96	   plus(X1, X2) -> n__plus(X1, X2)
4.31/1.96	   isNatKind(X) -> n__isNatKind(X)
4.31/1.96	   s(X) -> n__s(X)
4.31/1.96	   x(X1, X2) -> n__x(X1, X2)
4.31/1.96	   and(X1, X2) -> n__and(X1, X2)
4.31/1.96	   isNat(X) -> n__isNat(X)
4.31/1.96	   activate(n__0) -> 0
4.31/1.96	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
4.31/1.96	   activate(n__isNatKind(X)) -> isNatKind(X)
4.31/1.96	   activate(n__s(X)) -> s(activate(X))
4.31/1.96	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
4.31/1.96	   activate(n__and(X1, X2)) -> and(activate(X1), X2)
4.31/1.96	   activate(n__isNat(X)) -> isNat(X)
4.31/1.96	   activate(X) -> X
4.31/1.96	
4.31/1.96	S is empty.
4.31/1.96	Rewrite Strategy: FULL
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.31/1.96	Transformed a relative TRS into a decreasing-loop problem.
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(2)
4.31/1.96	Obligation:
4.31/1.96	Analyzing the following TRS for decreasing loops:
4.31/1.96	
4.31/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.31/1.96	
4.31/1.96	
4.31/1.96	The TRS R consists of the following rules:
4.31/1.96	
4.31/1.96	   U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2))
4.31/1.96	   U12(tt, V2) -> U13(isNat(activate(V2)))
4.31/1.96	   U13(tt) -> tt
4.31/1.96	   U21(tt, V1) -> U22(isNat(activate(V1)))
4.31/1.96	   U22(tt) -> tt
4.31/1.96	   U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2))
4.31/1.96	   U32(tt, V2) -> U33(isNat(activate(V2)))
4.31/1.96	   U33(tt) -> tt
4.31/1.96	   U41(tt, N) -> activate(N)
4.31/1.96	   U51(tt, M, N) -> s(plus(activate(N), activate(M)))
4.31/1.96	   U61(tt) -> 0
4.31/1.96	   U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
4.31/1.96	   and(tt, X) -> activate(X)
4.31/1.96	   isNat(n__0) -> tt
4.31/1.96	   isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
4.31/1.96	   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
4.31/1.96	   isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
4.31/1.96	   isNatKind(n__0) -> tt
4.31/1.96	   isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
4.31/1.96	   isNatKind(n__s(V1)) -> isNatKind(activate(V1))
4.31/1.96	   isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
4.31/1.96	   plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N)
4.31/1.96	   plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
4.31/1.96	   x(N, 0) -> U61(and(isNat(N), n__isNatKind(N)))
4.31/1.96	   x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
4.31/1.96	   0 -> n__0
4.31/1.96	   plus(X1, X2) -> n__plus(X1, X2)
4.31/1.96	   isNatKind(X) -> n__isNatKind(X)
4.31/1.96	   s(X) -> n__s(X)
4.31/1.96	   x(X1, X2) -> n__x(X1, X2)
4.31/1.96	   and(X1, X2) -> n__and(X1, X2)
4.31/1.96	   isNat(X) -> n__isNat(X)
4.31/1.96	   activate(n__0) -> 0
4.31/1.96	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
4.31/1.96	   activate(n__isNatKind(X)) -> isNatKind(X)
4.31/1.96	   activate(n__s(X)) -> s(activate(X))
4.31/1.96	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
4.31/1.96	   activate(n__and(X1, X2)) -> and(activate(X1), X2)
4.31/1.96	   activate(n__isNat(X)) -> isNat(X)
4.31/1.96	   activate(X) -> X
4.31/1.96	
4.31/1.96	S is empty.
4.31/1.96	Rewrite Strategy: FULL
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(3) DecreasingLoopProof (LOWER BOUND(ID))
4.31/1.96	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.31/1.96	
4.31/1.96	The rewrite sequence
4.31/1.96	
4.31/1.96	activate(n__s(X)) ->^+ s(activate(X))
4.31/1.96	
4.31/1.96	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.31/1.96	
4.31/1.96	The pumping substitution is [X / n__s(X)].
4.31/1.96	
4.31/1.96	The result substitution is [ ].
4.31/1.96	
4.31/1.96	
4.31/1.96	
4.31/1.96	
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(4)
4.31/1.96	Complex Obligation (BEST)
4.31/1.96	
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(5)
4.31/1.96	Obligation:
4.31/1.96	Proved the lower bound n^1 for the following obligation:
4.31/1.96	
4.31/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.31/1.96	
4.31/1.96	
4.31/1.96	The TRS R consists of the following rules:
4.31/1.96	
4.31/1.96	   U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2))
4.31/1.96	   U12(tt, V2) -> U13(isNat(activate(V2)))
4.31/1.96	   U13(tt) -> tt
4.31/1.96	   U21(tt, V1) -> U22(isNat(activate(V1)))
4.31/1.96	   U22(tt) -> tt
4.31/1.96	   U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2))
4.31/1.96	   U32(tt, V2) -> U33(isNat(activate(V2)))
4.31/1.96	   U33(tt) -> tt
4.31/1.96	   U41(tt, N) -> activate(N)
4.31/1.96	   U51(tt, M, N) -> s(plus(activate(N), activate(M)))
4.31/1.96	   U61(tt) -> 0
4.31/1.96	   U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
4.31/1.96	   and(tt, X) -> activate(X)
4.31/1.96	   isNat(n__0) -> tt
4.31/1.96	   isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
4.31/1.96	   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
4.31/1.96	   isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
4.31/1.96	   isNatKind(n__0) -> tt
4.31/1.96	   isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
4.31/1.96	   isNatKind(n__s(V1)) -> isNatKind(activate(V1))
4.31/1.96	   isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
4.31/1.96	   plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N)
4.31/1.96	   plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
4.31/1.96	   x(N, 0) -> U61(and(isNat(N), n__isNatKind(N)))
4.31/1.96	   x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
4.31/1.96	   0 -> n__0
4.31/1.96	   plus(X1, X2) -> n__plus(X1, X2)
4.31/1.96	   isNatKind(X) -> n__isNatKind(X)
4.31/1.96	   s(X) -> n__s(X)
4.31/1.96	   x(X1, X2) -> n__x(X1, X2)
4.31/1.96	   and(X1, X2) -> n__and(X1, X2)
4.31/1.96	   isNat(X) -> n__isNat(X)
4.31/1.96	   activate(n__0) -> 0
4.31/1.96	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
4.31/1.96	   activate(n__isNatKind(X)) -> isNatKind(X)
4.31/1.96	   activate(n__s(X)) -> s(activate(X))
4.31/1.96	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
4.31/1.96	   activate(n__and(X1, X2)) -> and(activate(X1), X2)
4.31/1.96	   activate(n__isNat(X)) -> isNat(X)
4.31/1.96	   activate(X) -> X
4.31/1.96	
4.31/1.96	S is empty.
4.31/1.96	Rewrite Strategy: FULL
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(6) LowerBoundPropagationProof (FINISHED)
4.31/1.96	Propagated lower bound.
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(7)
4.31/1.96	BOUNDS(n^1, INF)
4.31/1.96	
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(8)
4.31/1.96	Obligation:
4.31/1.96	Analyzing the following TRS for decreasing loops:
4.31/1.96	
4.31/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.31/1.96	
4.31/1.96	
4.31/1.96	The TRS R consists of the following rules:
4.31/1.96	
4.31/1.96	   U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2))
4.31/1.96	   U12(tt, V2) -> U13(isNat(activate(V2)))
4.31/1.96	   U13(tt) -> tt
4.31/1.96	   U21(tt, V1) -> U22(isNat(activate(V1)))
4.31/1.96	   U22(tt) -> tt
4.31/1.96	   U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2))
4.31/1.96	   U32(tt, V2) -> U33(isNat(activate(V2)))
4.31/1.96	   U33(tt) -> tt
4.31/1.96	   U41(tt, N) -> activate(N)
4.31/1.96	   U51(tt, M, N) -> s(plus(activate(N), activate(M)))
4.31/1.96	   U61(tt) -> 0
4.31/1.96	   U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
4.31/1.96	   and(tt, X) -> activate(X)
4.31/1.96	   isNat(n__0) -> tt
4.31/1.96	   isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
4.31/1.96	   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
4.31/1.96	   isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
4.31/1.96	   isNatKind(n__0) -> tt
4.31/1.96	   isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
4.31/1.96	   isNatKind(n__s(V1)) -> isNatKind(activate(V1))
4.31/1.96	   isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
4.31/1.96	   plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N)
4.31/1.96	   plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
4.31/1.96	   x(N, 0) -> U61(and(isNat(N), n__isNatKind(N)))
4.31/1.96	   x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
4.31/1.96	   0 -> n__0
4.31/1.96	   plus(X1, X2) -> n__plus(X1, X2)
4.31/1.96	   isNatKind(X) -> n__isNatKind(X)
4.31/1.96	   s(X) -> n__s(X)
4.31/1.96	   x(X1, X2) -> n__x(X1, X2)
4.31/1.96	   and(X1, X2) -> n__and(X1, X2)
4.31/1.96	   isNat(X) -> n__isNat(X)
4.31/1.96	   activate(n__0) -> 0
4.31/1.96	   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
4.31/1.96	   activate(n__isNatKind(X)) -> isNatKind(X)
4.31/1.96	   activate(n__s(X)) -> s(activate(X))
4.31/1.96	   activate(n__x(X1, X2)) -> x(activate(X1), activate(X2))
4.31/1.96	   activate(n__and(X1, X2)) -> and(activate(X1), X2)
4.31/1.96	   activate(n__isNat(X)) -> isNat(X)
4.31/1.96	   activate(X) -> X
4.31/1.96	
4.31/1.96	S is empty.
4.31/1.96	Rewrite Strategy: FULL
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(9) DecreasingLoopProof (FINISHED)
4.31/1.96	The following loop(s) give(s) rise to the lower bound EXP:
4.31/1.96	
4.31/1.96	The rewrite sequence
4.31/1.96	
4.31/1.96	activate(n__isNat(n__x(V11_0, V22_0))) ->^+ U31(and(isNatKind(activate(V11_0)), n__isNatKind(activate(V22_0))), activate(V11_0), activate(V22_0))
4.31/1.96	
4.31/1.96	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
4.31/1.96	
4.31/1.96	The pumping substitution is [V11_0 / n__isNat(n__x(V11_0, V22_0))].
4.31/1.96	
4.31/1.96	The result substitution is [ ].
4.31/1.96	
4.31/1.96	
4.31/1.96	
4.31/1.96	The rewrite sequence
4.31/1.96	
4.31/1.96	activate(n__isNat(n__x(V11_0, V22_0))) ->^+ U31(and(isNatKind(activate(V11_0)), n__isNatKind(activate(V22_0))), activate(V11_0), activate(V22_0))
4.31/1.96	
4.31/1.96	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
4.31/1.96	
4.31/1.96	The pumping substitution is [V11_0 / n__isNat(n__x(V11_0, V22_0))].
4.31/1.96	
4.31/1.96	The result substitution is [ ].
4.31/1.96	
4.31/1.96	
4.31/1.96	
4.31/1.96	
4.31/1.96	----------------------------------------
4.31/1.96	
4.31/1.96	(10)
4.31/1.96	BOUNDS(EXP, INF)
4.44/2.01	EOF