3.78/1.73	WORST_CASE(NON_POLY, ?)
3.78/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.78/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.78/1.73	
3.78/1.73	
3.78/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.73	
3.78/1.73	(0) CpxTRS
3.78/1.73	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.78/1.73	(2) TRS for Loop Detection
3.78/1.73	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.78/1.73	(4) BEST
3.78/1.73	    (5) proven lower bound
3.78/1.73	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.78/1.73	        (7) BOUNDS(n^1, INF)
3.78/1.73	    (8) TRS for Loop Detection
3.78/1.73	        (9) DecreasingLoopProof [FINISHED, 19 ms]
3.78/1.73	        (10) BOUNDS(EXP, INF)
3.78/1.73	
3.78/1.73	
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(0)
3.78/1.73	Obligation:
3.78/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.73	
3.78/1.73	
3.78/1.73	The TRS R consists of the following rules:
3.78/1.73	
3.78/1.73	   a__p(0) -> 0
3.78/1.73	   a__p(s(X)) -> mark(X)
3.78/1.73	   a__leq(0, Y) -> true
3.78/1.73	   a__leq(s(X), 0) -> false
3.78/1.73	   a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
3.78/1.73	   a__if(true, X, Y) -> mark(X)
3.78/1.73	   a__if(false, X, Y) -> mark(Y)
3.78/1.73	   a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
3.78/1.73	   mark(p(X)) -> a__p(mark(X))
3.78/1.73	   mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
3.78/1.73	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
3.78/1.73	   mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
3.78/1.73	   mark(0) -> 0
3.78/1.73	   mark(s(X)) -> s(mark(X))
3.78/1.73	   mark(true) -> true
3.78/1.73	   mark(false) -> false
3.78/1.73	   a__p(X) -> p(X)
3.78/1.73	   a__leq(X1, X2) -> leq(X1, X2)
3.78/1.73	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.78/1.73	   a__diff(X1, X2) -> diff(X1, X2)
3.78/1.73	
3.78/1.73	S is empty.
3.78/1.73	Rewrite Strategy: FULL
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.78/1.73	Transformed a relative TRS into a decreasing-loop problem.
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(2)
3.78/1.73	Obligation:
3.78/1.73	Analyzing the following TRS for decreasing loops:
3.78/1.73	
3.78/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.73	
3.78/1.73	
3.78/1.73	The TRS R consists of the following rules:
3.78/1.73	
3.78/1.73	   a__p(0) -> 0
3.78/1.73	   a__p(s(X)) -> mark(X)
3.78/1.73	   a__leq(0, Y) -> true
3.78/1.73	   a__leq(s(X), 0) -> false
3.78/1.73	   a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
3.78/1.73	   a__if(true, X, Y) -> mark(X)
3.78/1.73	   a__if(false, X, Y) -> mark(Y)
3.78/1.73	   a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
3.78/1.73	   mark(p(X)) -> a__p(mark(X))
3.78/1.73	   mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
3.78/1.73	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
3.78/1.73	   mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
3.78/1.73	   mark(0) -> 0
3.78/1.73	   mark(s(X)) -> s(mark(X))
3.78/1.73	   mark(true) -> true
3.78/1.73	   mark(false) -> false
3.78/1.73	   a__p(X) -> p(X)
3.78/1.73	   a__leq(X1, X2) -> leq(X1, X2)
3.78/1.73	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.78/1.73	   a__diff(X1, X2) -> diff(X1, X2)
3.78/1.73	
3.78/1.73	S is empty.
3.78/1.73	Rewrite Strategy: FULL
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.78/1.73	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.78/1.73	
3.78/1.73	The rewrite sequence
3.78/1.73	
3.78/1.73	mark(diff(X1, X2)) ->^+ a__diff(mark(X1), mark(X2))
3.78/1.73	
3.78/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.78/1.73	
3.78/1.73	The pumping substitution is [X1 / diff(X1, X2)].
3.78/1.73	
3.78/1.73	The result substitution is [ ].
3.78/1.73	
3.78/1.73	
3.78/1.73	
3.78/1.73	
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(4)
3.78/1.73	Complex Obligation (BEST)
3.78/1.73	
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(5)
3.78/1.73	Obligation:
3.78/1.73	Proved the lower bound n^1 for the following obligation:
3.78/1.73	
3.78/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.73	
3.78/1.73	
3.78/1.73	The TRS R consists of the following rules:
3.78/1.73	
3.78/1.73	   a__p(0) -> 0
3.78/1.73	   a__p(s(X)) -> mark(X)
3.78/1.73	   a__leq(0, Y) -> true
3.78/1.73	   a__leq(s(X), 0) -> false
3.78/1.73	   a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
3.78/1.73	   a__if(true, X, Y) -> mark(X)
3.78/1.73	   a__if(false, X, Y) -> mark(Y)
3.78/1.73	   a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
3.78/1.73	   mark(p(X)) -> a__p(mark(X))
3.78/1.73	   mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
3.78/1.73	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
3.78/1.73	   mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
3.78/1.73	   mark(0) -> 0
3.78/1.73	   mark(s(X)) -> s(mark(X))
3.78/1.73	   mark(true) -> true
3.78/1.73	   mark(false) -> false
3.78/1.73	   a__p(X) -> p(X)
3.78/1.73	   a__leq(X1, X2) -> leq(X1, X2)
3.78/1.73	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.78/1.73	   a__diff(X1, X2) -> diff(X1, X2)
3.78/1.73	
3.78/1.73	S is empty.
3.78/1.73	Rewrite Strategy: FULL
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(6) LowerBoundPropagationProof (FINISHED)
3.78/1.73	Propagated lower bound.
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(7)
3.78/1.73	BOUNDS(n^1, INF)
3.78/1.73	
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(8)
3.78/1.73	Obligation:
3.78/1.73	Analyzing the following TRS for decreasing loops:
3.78/1.73	
3.78/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.73	
3.78/1.73	
3.78/1.73	The TRS R consists of the following rules:
3.78/1.73	
3.78/1.73	   a__p(0) -> 0
3.78/1.73	   a__p(s(X)) -> mark(X)
3.78/1.73	   a__leq(0, Y) -> true
3.78/1.73	   a__leq(s(X), 0) -> false
3.78/1.73	   a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
3.78/1.73	   a__if(true, X, Y) -> mark(X)
3.78/1.73	   a__if(false, X, Y) -> mark(Y)
3.78/1.73	   a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
3.78/1.73	   mark(p(X)) -> a__p(mark(X))
3.78/1.73	   mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
3.78/1.73	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
3.78/1.73	   mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
3.78/1.73	   mark(0) -> 0
3.78/1.73	   mark(s(X)) -> s(mark(X))
3.78/1.73	   mark(true) -> true
3.78/1.73	   mark(false) -> false
3.78/1.73	   a__p(X) -> p(X)
3.78/1.73	   a__leq(X1, X2) -> leq(X1, X2)
3.78/1.73	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.78/1.73	   a__diff(X1, X2) -> diff(X1, X2)
3.78/1.73	
3.78/1.73	S is empty.
3.78/1.73	Rewrite Strategy: FULL
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(9) DecreasingLoopProof (FINISHED)
3.78/1.73	The following loop(s) give(s) rise to the lower bound EXP:
3.78/1.73	
3.78/1.73	The rewrite sequence
3.78/1.73	
3.78/1.73	mark(diff(X1, X2)) ->^+ a__if(a__leq(mark(mark(X1)), mark(mark(X2))), 0, s(diff(p(mark(X1)), mark(X2))))
3.78/1.73	
3.78/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
3.78/1.73	
3.78/1.73	The pumping substitution is [X1 / diff(X1, X2)].
3.78/1.73	
3.78/1.73	The result substitution is [ ].
3.78/1.73	
3.78/1.73	
3.78/1.73	
3.78/1.73	The rewrite sequence
3.78/1.73	
3.78/1.73	mark(diff(X1, X2)) ->^+ a__if(a__leq(mark(mark(X1)), mark(mark(X2))), 0, s(diff(p(mark(X1)), mark(X2))))
3.78/1.73	
3.78/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,0,0,0].
3.78/1.73	
3.78/1.73	The pumping substitution is [X1 / diff(X1, X2)].
3.78/1.73	
3.78/1.73	The result substitution is [ ].
3.78/1.73	
3.78/1.73	
3.78/1.73	
3.78/1.73	
3.78/1.73	----------------------------------------
3.78/1.73	
3.78/1.73	(10)
3.78/1.73	BOUNDS(EXP, INF)
3.91/1.75	EOF