3.10/1.52	WORST_CASE(NON_POLY, ?)
3.10/1.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.10/1.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.10/1.53	
3.10/1.53	
3.10/1.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.10/1.53	
3.10/1.53	(0) CpxTRS
3.10/1.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.10/1.53	(2) TRS for Loop Detection
3.10/1.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.10/1.53	(4) BEST
3.10/1.53	    (5) proven lower bound
3.10/1.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.10/1.53	        (7) BOUNDS(n^1, INF)
3.10/1.53	    (8) TRS for Loop Detection
3.10/1.53	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.10/1.53	        (10) BOUNDS(INF, INF)
3.10/1.53	
3.10/1.53	
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(0)
3.10/1.53	Obligation:
3.10/1.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.10/1.53	
3.10/1.53	
3.10/1.53	The TRS R consists of the following rules:
3.10/1.53	
3.10/1.53	   p(0) -> 0
3.10/1.53	   p(s(X)) -> X
3.10/1.53	   leq(0, Y) -> true
3.10/1.53	   leq(s(X), 0) -> false
3.10/1.53	   leq(s(X), s(Y)) -> leq(X, Y)
3.10/1.53	   if(true, X, Y) -> X
3.10/1.53	   if(false, X, Y) -> Y
3.10/1.53	   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))
3.10/1.53	
3.10/1.53	S is empty.
3.10/1.53	Rewrite Strategy: FULL
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.10/1.53	Transformed a relative TRS into a decreasing-loop problem.
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(2)
3.10/1.53	Obligation:
3.10/1.53	Analyzing the following TRS for decreasing loops:
3.10/1.53	
3.10/1.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.10/1.53	
3.10/1.53	
3.10/1.53	The TRS R consists of the following rules:
3.10/1.53	
3.10/1.53	   p(0) -> 0
3.10/1.53	   p(s(X)) -> X
3.10/1.53	   leq(0, Y) -> true
3.10/1.53	   leq(s(X), 0) -> false
3.10/1.53	   leq(s(X), s(Y)) -> leq(X, Y)
3.10/1.53	   if(true, X, Y) -> X
3.10/1.53	   if(false, X, Y) -> Y
3.10/1.53	   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))
3.10/1.53	
3.10/1.53	S is empty.
3.10/1.53	Rewrite Strategy: FULL
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.10/1.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.10/1.53	
3.10/1.53	The rewrite sequence
3.10/1.53	
3.10/1.53	leq(s(X), s(Y)) ->^+ leq(X, Y)
3.10/1.53	
3.10/1.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.10/1.53	
3.10/1.53	The pumping substitution is [X / s(X), Y / s(Y)].
3.10/1.53	
3.10/1.53	The result substitution is [ ].
3.10/1.53	
3.10/1.53	
3.10/1.53	
3.10/1.53	
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(4)
3.10/1.53	Complex Obligation (BEST)
3.10/1.53	
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(5)
3.10/1.53	Obligation:
3.10/1.53	Proved the lower bound n^1 for the following obligation:
3.10/1.53	
3.10/1.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.10/1.53	
3.10/1.53	
3.10/1.53	The TRS R consists of the following rules:
3.10/1.53	
3.10/1.53	   p(0) -> 0
3.10/1.53	   p(s(X)) -> X
3.10/1.53	   leq(0, Y) -> true
3.10/1.53	   leq(s(X), 0) -> false
3.10/1.53	   leq(s(X), s(Y)) -> leq(X, Y)
3.10/1.53	   if(true, X, Y) -> X
3.10/1.53	   if(false, X, Y) -> Y
3.10/1.53	   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))
3.10/1.53	
3.10/1.53	S is empty.
3.10/1.53	Rewrite Strategy: FULL
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(6) LowerBoundPropagationProof (FINISHED)
3.10/1.53	Propagated lower bound.
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(7)
3.10/1.53	BOUNDS(n^1, INF)
3.10/1.53	
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(8)
3.10/1.53	Obligation:
3.10/1.53	Analyzing the following TRS for decreasing loops:
3.10/1.53	
3.10/1.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.10/1.53	
3.10/1.53	
3.10/1.53	The TRS R consists of the following rules:
3.10/1.53	
3.10/1.53	   p(0) -> 0
3.10/1.53	   p(s(X)) -> X
3.10/1.53	   leq(0, Y) -> true
3.10/1.53	   leq(s(X), 0) -> false
3.10/1.53	   leq(s(X), s(Y)) -> leq(X, Y)
3.10/1.53	   if(true, X, Y) -> X
3.10/1.53	   if(false, X, Y) -> Y
3.10/1.53	   diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y)))
3.10/1.53	
3.10/1.53	S is empty.
3.10/1.53	Rewrite Strategy: FULL
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(9) InfiniteLowerBoundProof (FINISHED)
3.10/1.53	The following loop proves infinite runtime complexity:
3.10/1.53	
3.10/1.53	The rewrite sequence
3.10/1.53	
3.10/1.53	diff(X, Y) ->^+ if(leq(X, Y), 0, s(diff(p(X), Y)))
3.10/1.53	
3.10/1.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,0].
3.10/1.53	
3.10/1.53	The pumping substitution is [ ].
3.10/1.53	
3.10/1.53	The result substitution is [X / p(X)].
3.10/1.53	
3.10/1.53	
3.10/1.53	
3.10/1.53	
3.10/1.53	----------------------------------------
3.10/1.53	
3.10/1.53	(10)
3.10/1.53	BOUNDS(INF, INF)
3.10/1.55	EOF