2.99/1.60	WORST_CASE(NON_POLY, ?)
2.99/1.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.99/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.99/1.61	
2.99/1.61	
2.99/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.61	
2.99/1.61	(0) CpxTRS
2.99/1.61	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
2.99/1.61	(2) TRS for Loop Detection
2.99/1.61	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
2.99/1.61	(4) BEST
2.99/1.61	    (5) proven lower bound
2.99/1.61	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
2.99/1.61	        (7) BOUNDS(n^1, INF)
2.99/1.61	    (8) TRS for Loop Detection
2.99/1.61	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
2.99/1.61	        (10) BOUNDS(INF, INF)
2.99/1.61	
2.99/1.61	
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(0)
2.99/1.61	Obligation:
2.99/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.61	
2.99/1.61	
2.99/1.61	The TRS R consists of the following rules:
2.99/1.61	
2.99/1.61	   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
2.99/1.61	   add(0, X) -> X
2.99/1.61	   add(s(X), Y) -> s(add(X, Y))
2.99/1.61	   prod(0, X) -> 0
2.99/1.61	   prod(s(X), Y) -> add(Y, prod(X, Y))
2.99/1.61	   if(true, X, Y) -> X
2.99/1.61	   if(false, X, Y) -> Y
2.99/1.61	   zero(0) -> true
2.99/1.61	   zero(s(X)) -> false
2.99/1.61	   p(s(X)) -> X
2.99/1.61	
2.99/1.61	S is empty.
2.99/1.61	Rewrite Strategy: FULL
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
2.99/1.61	Transformed a relative TRS into a decreasing-loop problem.
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(2)
2.99/1.61	Obligation:
2.99/1.61	Analyzing the following TRS for decreasing loops:
2.99/1.61	
2.99/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.61	
2.99/1.61	
2.99/1.61	The TRS R consists of the following rules:
2.99/1.61	
2.99/1.61	   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
2.99/1.61	   add(0, X) -> X
2.99/1.61	   add(s(X), Y) -> s(add(X, Y))
2.99/1.61	   prod(0, X) -> 0
2.99/1.61	   prod(s(X), Y) -> add(Y, prod(X, Y))
2.99/1.61	   if(true, X, Y) -> X
2.99/1.61	   if(false, X, Y) -> Y
2.99/1.61	   zero(0) -> true
2.99/1.61	   zero(s(X)) -> false
2.99/1.61	   p(s(X)) -> X
2.99/1.61	
2.99/1.61	S is empty.
2.99/1.61	Rewrite Strategy: FULL
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(3) DecreasingLoopProof (LOWER BOUND(ID))
2.99/1.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
2.99/1.61	
2.99/1.61	The rewrite sequence
2.99/1.61	
2.99/1.61	add(s(X), Y) ->^+ s(add(X, Y))
2.99/1.61	
2.99/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
2.99/1.61	
2.99/1.61	The pumping substitution is [X / s(X)].
2.99/1.61	
2.99/1.61	The result substitution is [ ].
2.99/1.61	
2.99/1.61	
2.99/1.61	
2.99/1.61	
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(4)
2.99/1.61	Complex Obligation (BEST)
2.99/1.61	
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(5)
2.99/1.61	Obligation:
2.99/1.61	Proved the lower bound n^1 for the following obligation:
2.99/1.61	
2.99/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.61	
2.99/1.61	
2.99/1.61	The TRS R consists of the following rules:
2.99/1.61	
2.99/1.61	   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
2.99/1.61	   add(0, X) -> X
2.99/1.61	   add(s(X), Y) -> s(add(X, Y))
2.99/1.61	   prod(0, X) -> 0
2.99/1.61	   prod(s(X), Y) -> add(Y, prod(X, Y))
2.99/1.61	   if(true, X, Y) -> X
2.99/1.61	   if(false, X, Y) -> Y
2.99/1.61	   zero(0) -> true
2.99/1.61	   zero(s(X)) -> false
2.99/1.61	   p(s(X)) -> X
2.99/1.61	
2.99/1.61	S is empty.
2.99/1.61	Rewrite Strategy: FULL
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(6) LowerBoundPropagationProof (FINISHED)
2.99/1.61	Propagated lower bound.
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(7)
2.99/1.61	BOUNDS(n^1, INF)
2.99/1.61	
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(8)
2.99/1.61	Obligation:
2.99/1.61	Analyzing the following TRS for decreasing loops:
2.99/1.61	
2.99/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.61	
2.99/1.61	
2.99/1.61	The TRS R consists of the following rules:
2.99/1.61	
2.99/1.61	   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
2.99/1.61	   add(0, X) -> X
2.99/1.61	   add(s(X), Y) -> s(add(X, Y))
2.99/1.61	   prod(0, X) -> 0
2.99/1.61	   prod(s(X), Y) -> add(Y, prod(X, Y))
2.99/1.61	   if(true, X, Y) -> X
2.99/1.61	   if(false, X, Y) -> Y
2.99/1.61	   zero(0) -> true
2.99/1.61	   zero(s(X)) -> false
2.99/1.61	   p(s(X)) -> X
2.99/1.61	
2.99/1.61	S is empty.
2.99/1.61	Rewrite Strategy: FULL
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(9) InfiniteLowerBoundProof (FINISHED)
2.99/1.61	The following loop proves infinite runtime complexity:
2.99/1.61	
2.99/1.61	The rewrite sequence
2.99/1.61	
2.99/1.61	fact(X) ->^+ if(zero(X), s(0), prod(X, fact(p(X))))
2.99/1.61	
2.99/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
2.99/1.61	
2.99/1.61	The pumping substitution is [ ].
2.99/1.61	
2.99/1.61	The result substitution is [X / p(X)].
2.99/1.61	
2.99/1.61	
2.99/1.61	
2.99/1.61	
2.99/1.61	----------------------------------------
2.99/1.61	
2.99/1.61	(10)
2.99/1.61	BOUNDS(INF, INF)
3.30/1.65	EOF