5.30/2.13	WORST_CASE(NON_POLY, ?)
5.41/2.14	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.41/2.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.41/2.14	
5.41/2.14	
5.41/2.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.41/2.14	
5.41/2.14	(0) CpxTRS
5.41/2.14	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.41/2.14	(2) TRS for Loop Detection
5.41/2.14	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.41/2.14	(4) BEST
5.41/2.14	    (5) proven lower bound
5.41/2.14	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
5.41/2.14	        (7) BOUNDS(n^1, INF)
5.41/2.14	    (8) TRS for Loop Detection
5.41/2.14	        (9) DecreasingLoopProof [FINISHED, 309 ms]
5.41/2.14	        (10) BOUNDS(EXP, INF)
5.41/2.14	
5.41/2.14	
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(0)
5.41/2.14	Obligation:
5.41/2.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.41/2.14	
5.41/2.14	
5.41/2.14	The TRS R consists of the following rules:
5.41/2.14	
5.41/2.14	   a__U11(tt, N) -> mark(N)
5.41/2.14	   a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M)))
5.41/2.14	   a__U31(tt) -> 0
5.41/2.14	   a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
5.41/2.14	   a__and(tt, X) -> mark(X)
5.41/2.14	   a__isNat(0) -> tt
5.41/2.14	   a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
5.41/2.14	   a__isNat(s(V1)) -> a__isNat(V1)
5.41/2.14	   a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
5.41/2.14	   a__plus(N, 0) -> a__U11(a__isNat(N), N)
5.41/2.14	   a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N)
5.41/2.14	   a__x(N, 0) -> a__U31(a__isNat(N))
5.41/2.14	   a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N)
5.41/2.14	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
5.41/2.14	   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
5.41/2.14	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
5.41/2.14	   mark(U31(X)) -> a__U31(mark(X))
5.41/2.14	   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
5.41/2.14	   mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
5.41/2.14	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
5.41/2.14	   mark(isNat(X)) -> a__isNat(X)
5.41/2.14	   mark(tt) -> tt
5.41/2.14	   mark(s(X)) -> s(mark(X))
5.41/2.14	   mark(0) -> 0
5.41/2.14	   a__U11(X1, X2) -> U11(X1, X2)
5.41/2.14	   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
5.41/2.14	   a__plus(X1, X2) -> plus(X1, X2)
5.41/2.14	   a__U31(X) -> U31(X)
5.41/2.14	   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
5.41/2.14	   a__x(X1, X2) -> x(X1, X2)
5.41/2.14	   a__and(X1, X2) -> and(X1, X2)
5.41/2.14	   a__isNat(X) -> isNat(X)
5.41/2.14	
5.41/2.14	S is empty.
5.41/2.14	Rewrite Strategy: INNERMOST
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.41/2.14	Transformed a relative TRS into a decreasing-loop problem.
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(2)
5.41/2.14	Obligation:
5.41/2.14	Analyzing the following TRS for decreasing loops:
5.41/2.14	
5.41/2.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.41/2.14	
5.41/2.14	
5.41/2.14	The TRS R consists of the following rules:
5.41/2.14	
5.41/2.14	   a__U11(tt, N) -> mark(N)
5.41/2.14	   a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M)))
5.41/2.14	   a__U31(tt) -> 0
5.41/2.14	   a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
5.41/2.14	   a__and(tt, X) -> mark(X)
5.41/2.14	   a__isNat(0) -> tt
5.41/2.14	   a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
5.41/2.14	   a__isNat(s(V1)) -> a__isNat(V1)
5.41/2.14	   a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
5.41/2.14	   a__plus(N, 0) -> a__U11(a__isNat(N), N)
5.41/2.14	   a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N)
5.41/2.14	   a__x(N, 0) -> a__U31(a__isNat(N))
5.41/2.14	   a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N)
5.41/2.14	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
5.41/2.14	   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
5.41/2.14	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
5.41/2.14	   mark(U31(X)) -> a__U31(mark(X))
5.41/2.14	   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
5.41/2.14	   mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
5.41/2.14	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
5.41/2.14	   mark(isNat(X)) -> a__isNat(X)
5.41/2.14	   mark(tt) -> tt
5.41/2.14	   mark(s(X)) -> s(mark(X))
5.41/2.14	   mark(0) -> 0
5.41/2.14	   a__U11(X1, X2) -> U11(X1, X2)
5.41/2.14	   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
5.41/2.14	   a__plus(X1, X2) -> plus(X1, X2)
5.41/2.14	   a__U31(X) -> U31(X)
5.41/2.14	   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
5.41/2.14	   a__x(X1, X2) -> x(X1, X2)
5.41/2.14	   a__and(X1, X2) -> and(X1, X2)
5.41/2.14	   a__isNat(X) -> isNat(X)
5.41/2.14	
5.41/2.14	S is empty.
5.41/2.14	Rewrite Strategy: INNERMOST
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(3) DecreasingLoopProof (LOWER BOUND(ID))
5.41/2.14	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.41/2.14	
5.41/2.14	The rewrite sequence
5.41/2.14	
5.41/2.14	mark(U41(X1, X2, X3)) ->^+ a__U41(mark(X1), X2, X3)
5.41/2.14	
5.41/2.14	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
5.41/2.14	
5.41/2.14	The pumping substitution is [X1 / U41(X1, X2, X3)].
5.41/2.14	
5.41/2.14	The result substitution is [ ].
5.41/2.14	
5.41/2.14	
5.41/2.14	
5.41/2.14	
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(4)
5.41/2.14	Complex Obligation (BEST)
5.41/2.14	
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(5)
5.41/2.14	Obligation:
5.41/2.14	Proved the lower bound n^1 for the following obligation:
5.41/2.14	
5.41/2.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.41/2.14	
5.41/2.14	
5.41/2.14	The TRS R consists of the following rules:
5.41/2.14	
5.41/2.14	   a__U11(tt, N) -> mark(N)
5.41/2.14	   a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M)))
5.41/2.14	   a__U31(tt) -> 0
5.41/2.14	   a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
5.41/2.14	   a__and(tt, X) -> mark(X)
5.41/2.14	   a__isNat(0) -> tt
5.41/2.14	   a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
5.41/2.14	   a__isNat(s(V1)) -> a__isNat(V1)
5.41/2.14	   a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
5.41/2.14	   a__plus(N, 0) -> a__U11(a__isNat(N), N)
5.41/2.14	   a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N)
5.41/2.14	   a__x(N, 0) -> a__U31(a__isNat(N))
5.41/2.14	   a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N)
5.41/2.14	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
5.41/2.14	   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
5.41/2.14	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
5.41/2.14	   mark(U31(X)) -> a__U31(mark(X))
5.41/2.14	   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
5.41/2.14	   mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
5.41/2.14	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
5.41/2.14	   mark(isNat(X)) -> a__isNat(X)
5.41/2.14	   mark(tt) -> tt
5.41/2.14	   mark(s(X)) -> s(mark(X))
5.41/2.14	   mark(0) -> 0
5.41/2.14	   a__U11(X1, X2) -> U11(X1, X2)
5.41/2.14	   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
5.41/2.14	   a__plus(X1, X2) -> plus(X1, X2)
5.41/2.14	   a__U31(X) -> U31(X)
5.41/2.14	   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
5.41/2.14	   a__x(X1, X2) -> x(X1, X2)
5.41/2.14	   a__and(X1, X2) -> and(X1, X2)
5.41/2.14	   a__isNat(X) -> isNat(X)
5.41/2.14	
5.41/2.14	S is empty.
5.41/2.14	Rewrite Strategy: INNERMOST
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(6) LowerBoundPropagationProof (FINISHED)
5.41/2.14	Propagated lower bound.
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(7)
5.41/2.14	BOUNDS(n^1, INF)
5.41/2.14	
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(8)
5.41/2.14	Obligation:
5.41/2.14	Analyzing the following TRS for decreasing loops:
5.41/2.14	
5.41/2.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.41/2.14	
5.41/2.14	
5.41/2.14	The TRS R consists of the following rules:
5.41/2.14	
5.41/2.14	   a__U11(tt, N) -> mark(N)
5.41/2.14	   a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M)))
5.41/2.14	   a__U31(tt) -> 0
5.41/2.14	   a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
5.41/2.14	   a__and(tt, X) -> mark(X)
5.41/2.14	   a__isNat(0) -> tt
5.41/2.14	   a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
5.41/2.14	   a__isNat(s(V1)) -> a__isNat(V1)
5.41/2.14	   a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
5.41/2.14	   a__plus(N, 0) -> a__U11(a__isNat(N), N)
5.41/2.14	   a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N)
5.41/2.14	   a__x(N, 0) -> a__U31(a__isNat(N))
5.41/2.14	   a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N)
5.41/2.14	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
5.41/2.14	   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
5.41/2.14	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
5.41/2.14	   mark(U31(X)) -> a__U31(mark(X))
5.41/2.14	   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
5.41/2.14	   mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
5.41/2.14	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
5.41/2.14	   mark(isNat(X)) -> a__isNat(X)
5.41/2.14	   mark(tt) -> tt
5.41/2.14	   mark(s(X)) -> s(mark(X))
5.41/2.14	   mark(0) -> 0
5.41/2.14	   a__U11(X1, X2) -> U11(X1, X2)
5.41/2.14	   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
5.41/2.14	   a__plus(X1, X2) -> plus(X1, X2)
5.41/2.14	   a__U31(X) -> U31(X)
5.41/2.14	   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
5.41/2.14	   a__x(X1, X2) -> x(X1, X2)
5.41/2.14	   a__and(X1, X2) -> and(X1, X2)
5.41/2.14	   a__isNat(X) -> isNat(X)
5.41/2.14	
5.41/2.14	S is empty.
5.41/2.14	Rewrite Strategy: INNERMOST
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(9) DecreasingLoopProof (FINISHED)
5.41/2.14	The following loop(s) give(s) rise to the lower bound EXP:
5.41/2.14	
5.41/2.14	The rewrite sequence
5.41/2.14	
5.41/2.14	mark(U41(tt, X2, X3)) ->^+ a__plus(a__x(mark(X3), mark(X2)), mark(X3))
5.41/2.14	
5.41/2.14	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
5.41/2.14	
5.41/2.14	The pumping substitution is [X3 / U41(tt, X2, X3)].
5.41/2.14	
5.41/2.14	The result substitution is [ ].
5.41/2.14	
5.41/2.14	
5.41/2.14	
5.41/2.14	The rewrite sequence
5.41/2.14	
5.41/2.14	mark(U41(tt, X2, X3)) ->^+ a__plus(a__x(mark(X3), mark(X2)), mark(X3))
5.41/2.14	
5.41/2.14	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
5.41/2.14	
5.41/2.14	The pumping substitution is [X3 / U41(tt, X2, X3)].
5.41/2.14	
5.41/2.14	The result substitution is [ ].
5.41/2.14	
5.41/2.14	
5.41/2.14	
5.41/2.14	
5.41/2.14	----------------------------------------
5.41/2.14	
5.41/2.14	(10)
5.41/2.14	BOUNDS(EXP, INF)
5.41/2.17	EOF