1122.71/292.35	WORST_CASE(Omega(n^1), ?)
1133.40/295.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1133.40/295.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1133.40/295.05	
1133.40/295.05	(0) CpxTRS
1133.40/295.05	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1133.40/295.05	(2) TRS for Loop Detection
1133.40/295.05	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1133.40/295.05	(4) BEST
1133.40/295.05	    (5) proven lower bound
1133.40/295.05	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1133.40/295.05	        (7) BOUNDS(n^1, INF)
1133.40/295.05	    (8) TRS for Loop Detection
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(0)
1133.40/295.05	Obligation:
1133.40/295.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	The TRS R consists of the following rules:
1133.40/295.05	
1133.40/295.05	   a__U11(tt, V2) -> a__U12(a__isNat(V2))
1133.40/295.05	   a__U12(tt) -> tt
1133.40/295.05	   a__U21(tt) -> tt
1133.40/295.05	   a__U31(tt, N) -> mark(N)
1133.40/295.05	   a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N)
1133.40/295.05	   a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1133.40/295.05	   a__isNat(0) -> tt
1133.40/295.05	   a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2)
1133.40/295.05	   a__isNat(s(V1)) -> a__U21(a__isNat(V1))
1133.40/295.05	   a__plus(N, 0) -> a__U31(a__isNat(N), N)
1133.40/295.05	   a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N)
1133.40/295.05	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
1133.40/295.05	   mark(U12(X)) -> a__U12(mark(X))
1133.40/295.05	   mark(isNat(X)) -> a__isNat(X)
1133.40/295.05	   mark(U21(X)) -> a__U21(mark(X))
1133.40/295.05	   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
1133.40/295.05	   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
1133.40/295.05	   mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
1133.40/295.05	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1133.40/295.05	   mark(tt) -> tt
1133.40/295.05	   mark(s(X)) -> s(mark(X))
1133.40/295.05	   mark(0) -> 0
1133.40/295.05	   a__U11(X1, X2) -> U11(X1, X2)
1133.40/295.05	   a__U12(X) -> U12(X)
1133.40/295.05	   a__isNat(X) -> isNat(X)
1133.40/295.05	   a__U21(X) -> U21(X)
1133.40/295.05	   a__U31(X1, X2) -> U31(X1, X2)
1133.40/295.05	   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
1133.40/295.05	   a__U42(X1, X2, X3) -> U42(X1, X2, X3)
1133.40/295.05	   a__plus(X1, X2) -> plus(X1, X2)
1133.40/295.05	
1133.40/295.05	S is empty.
1133.40/295.05	Rewrite Strategy: INNERMOST
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1133.40/295.05	Transformed a relative TRS into a decreasing-loop problem.
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(2)
1133.40/295.05	Obligation:
1133.40/295.05	Analyzing the following TRS for decreasing loops:
1133.40/295.05	
1133.40/295.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	The TRS R consists of the following rules:
1133.40/295.05	
1133.40/295.05	   a__U11(tt, V2) -> a__U12(a__isNat(V2))
1133.40/295.05	   a__U12(tt) -> tt
1133.40/295.05	   a__U21(tt) -> tt
1133.40/295.05	   a__U31(tt, N) -> mark(N)
1133.40/295.05	   a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N)
1133.40/295.05	   a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1133.40/295.05	   a__isNat(0) -> tt
1133.40/295.05	   a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2)
1133.40/295.05	   a__isNat(s(V1)) -> a__U21(a__isNat(V1))
1133.40/295.05	   a__plus(N, 0) -> a__U31(a__isNat(N), N)
1133.40/295.05	   a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N)
1133.40/295.05	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
1133.40/295.05	   mark(U12(X)) -> a__U12(mark(X))
1133.40/295.05	   mark(isNat(X)) -> a__isNat(X)
1133.40/295.05	   mark(U21(X)) -> a__U21(mark(X))
1133.40/295.05	   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
1133.40/295.05	   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
1133.40/295.05	   mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
1133.40/295.05	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1133.40/295.05	   mark(tt) -> tt
1133.40/295.05	   mark(s(X)) -> s(mark(X))
1133.40/295.05	   mark(0) -> 0
1133.40/295.05	   a__U11(X1, X2) -> U11(X1, X2)
1133.40/295.05	   a__U12(X) -> U12(X)
1133.40/295.05	   a__isNat(X) -> isNat(X)
1133.40/295.05	   a__U21(X) -> U21(X)
1133.40/295.05	   a__U31(X1, X2) -> U31(X1, X2)
1133.40/295.05	   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
1133.40/295.05	   a__U42(X1, X2, X3) -> U42(X1, X2, X3)
1133.40/295.05	   a__plus(X1, X2) -> plus(X1, X2)
1133.40/295.05	
1133.40/295.05	S is empty.
1133.40/295.05	Rewrite Strategy: INNERMOST
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(3) DecreasingLoopProof (LOWER BOUND(ID))
1133.40/295.05	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1133.40/295.05	
1133.40/295.05	The rewrite sequence
1133.40/295.05	
1133.40/295.05	mark(U12(X)) ->^+ a__U12(mark(X))
1133.40/295.05	
1133.40/295.05	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1133.40/295.05	
1133.40/295.05	The pumping substitution is [X / U12(X)].
1133.40/295.05	
1133.40/295.05	The result substitution is [ ].
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(4)
1133.40/295.05	Complex Obligation (BEST)
1133.40/295.05	
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(5)
1133.40/295.05	Obligation:
1133.40/295.05	Proved the lower bound n^1 for the following obligation:
1133.40/295.05	
1133.40/295.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	The TRS R consists of the following rules:
1133.40/295.05	
1133.40/295.05	   a__U11(tt, V2) -> a__U12(a__isNat(V2))
1133.40/295.05	   a__U12(tt) -> tt
1133.40/295.05	   a__U21(tt) -> tt
1133.40/295.05	   a__U31(tt, N) -> mark(N)
1133.40/295.05	   a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N)
1133.40/295.05	   a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1133.40/295.05	   a__isNat(0) -> tt
1133.40/295.05	   a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2)
1133.40/295.05	   a__isNat(s(V1)) -> a__U21(a__isNat(V1))
1133.40/295.05	   a__plus(N, 0) -> a__U31(a__isNat(N), N)
1133.40/295.05	   a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N)
1133.40/295.05	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
1133.40/295.05	   mark(U12(X)) -> a__U12(mark(X))
1133.40/295.05	   mark(isNat(X)) -> a__isNat(X)
1133.40/295.05	   mark(U21(X)) -> a__U21(mark(X))
1133.40/295.05	   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
1133.40/295.05	   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
1133.40/295.05	   mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
1133.40/295.05	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1133.40/295.05	   mark(tt) -> tt
1133.40/295.05	   mark(s(X)) -> s(mark(X))
1133.40/295.05	   mark(0) -> 0
1133.40/295.05	   a__U11(X1, X2) -> U11(X1, X2)
1133.40/295.05	   a__U12(X) -> U12(X)
1133.40/295.05	   a__isNat(X) -> isNat(X)
1133.40/295.05	   a__U21(X) -> U21(X)
1133.40/295.05	   a__U31(X1, X2) -> U31(X1, X2)
1133.40/295.05	   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
1133.40/295.05	   a__U42(X1, X2, X3) -> U42(X1, X2, X3)
1133.40/295.05	   a__plus(X1, X2) -> plus(X1, X2)
1133.40/295.05	
1133.40/295.05	S is empty.
1133.40/295.05	Rewrite Strategy: INNERMOST
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(6) LowerBoundPropagationProof (FINISHED)
1133.40/295.05	Propagated lower bound.
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(7)
1133.40/295.05	BOUNDS(n^1, INF)
1133.40/295.05	
1133.40/295.05	----------------------------------------
1133.40/295.05	
1133.40/295.05	(8)
1133.40/295.05	Obligation:
1133.40/295.05	Analyzing the following TRS for decreasing loops:
1133.40/295.05	
1133.40/295.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1133.40/295.05	
1133.40/295.05	
1133.40/295.05	The TRS R consists of the following rules:
1133.40/295.05	
1133.40/295.05	   a__U11(tt, V2) -> a__U12(a__isNat(V2))
1133.40/295.05	   a__U12(tt) -> tt
1133.40/295.05	   a__U21(tt) -> tt
1133.40/295.05	   a__U31(tt, N) -> mark(N)
1133.40/295.05	   a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N)
1133.40/295.05	   a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M)))
1133.40/295.05	   a__isNat(0) -> tt
1133.40/295.05	   a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2)
1133.40/295.05	   a__isNat(s(V1)) -> a__U21(a__isNat(V1))
1133.40/295.05	   a__plus(N, 0) -> a__U31(a__isNat(N), N)
1133.40/295.05	   a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N)
1133.40/295.05	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
1133.40/295.05	   mark(U12(X)) -> a__U12(mark(X))
1133.40/295.05	   mark(isNat(X)) -> a__isNat(X)
1133.40/295.05	   mark(U21(X)) -> a__U21(mark(X))
1133.40/295.05	   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
1133.40/295.05	   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
1133.40/295.05	   mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
1133.40/295.05	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
1133.40/295.05	   mark(tt) -> tt
1133.40/295.05	   mark(s(X)) -> s(mark(X))
1133.40/295.05	   mark(0) -> 0
1133.40/295.05	   a__U11(X1, X2) -> U11(X1, X2)
1133.40/295.05	   a__U12(X) -> U12(X)
1133.40/295.05	   a__isNat(X) -> isNat(X)
1133.40/295.05	   a__U21(X) -> U21(X)
1133.40/295.05	   a__U31(X1, X2) -> U31(X1, X2)
1133.40/295.05	   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
1133.40/295.05	   a__U42(X1, X2, X3) -> U42(X1, X2, X3)
1133.40/295.05	   a__plus(X1, X2) -> plus(X1, X2)
1133.40/295.05	
1133.40/295.05	S is empty.
1133.40/295.05	Rewrite Strategy: INNERMOST
1133.51/295.12	EOF