3.31/1.67	WORST_CASE(NON_POLY, ?)
3.31/1.68	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.31/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.31/1.68	
3.31/1.68	
3.31/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.31/1.68	
3.31/1.68	(0) CpxTRS
3.31/1.68	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.31/1.68	(2) TRS for Loop Detection
3.31/1.68	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.31/1.68	(4) BEST
3.31/1.68	    (5) proven lower bound
3.31/1.68	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.31/1.68	        (7) BOUNDS(n^1, INF)
3.31/1.68	    (8) TRS for Loop Detection
3.31/1.68	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.31/1.68	        (10) BOUNDS(INF, INF)
3.31/1.68	
3.31/1.68	
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(0)
3.31/1.68	Obligation:
3.31/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.31/1.68	
3.31/1.68	
3.31/1.68	The TRS R consists of the following rules:
3.31/1.68	
3.31/1.68	   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
3.31/1.68	   add(0, X) -> X
3.31/1.68	   add(s(X), Y) -> s(add(X, Y))
3.31/1.68	   prod(0, X) -> 0
3.31/1.68	   prod(s(X), Y) -> add(Y, prod(X, Y))
3.31/1.68	   if(true, X, Y) -> X
3.31/1.68	   if(false, X, Y) -> Y
3.31/1.68	   zero(0) -> true
3.31/1.68	   zero(s(X)) -> false
3.31/1.68	   p(s(X)) -> X
3.31/1.68	
3.31/1.68	S is empty.
3.31/1.68	Rewrite Strategy: INNERMOST
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.31/1.68	Transformed a relative TRS into a decreasing-loop problem.
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(2)
3.31/1.68	Obligation:
3.31/1.68	Analyzing the following TRS for decreasing loops:
3.31/1.68	
3.31/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.31/1.68	
3.31/1.68	
3.31/1.68	The TRS R consists of the following rules:
3.31/1.68	
3.31/1.68	   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
3.31/1.68	   add(0, X) -> X
3.31/1.68	   add(s(X), Y) -> s(add(X, Y))
3.31/1.68	   prod(0, X) -> 0
3.31/1.68	   prod(s(X), Y) -> add(Y, prod(X, Y))
3.31/1.68	   if(true, X, Y) -> X
3.31/1.68	   if(false, X, Y) -> Y
3.31/1.68	   zero(0) -> true
3.31/1.68	   zero(s(X)) -> false
3.31/1.68	   p(s(X)) -> X
3.31/1.68	
3.31/1.68	S is empty.
3.31/1.68	Rewrite Strategy: INNERMOST
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.31/1.68	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.31/1.68	
3.31/1.68	The rewrite sequence
3.31/1.68	
3.31/1.68	add(s(X), Y) ->^+ s(add(X, Y))
3.31/1.68	
3.31/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.31/1.68	
3.31/1.68	The pumping substitution is [X / s(X)].
3.31/1.68	
3.31/1.68	The result substitution is [ ].
3.31/1.68	
3.31/1.68	
3.31/1.68	
3.31/1.68	
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(4)
3.31/1.68	Complex Obligation (BEST)
3.31/1.68	
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(5)
3.31/1.68	Obligation:
3.31/1.68	Proved the lower bound n^1 for the following obligation:
3.31/1.68	
3.31/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.31/1.68	
3.31/1.68	
3.31/1.68	The TRS R consists of the following rules:
3.31/1.68	
3.31/1.68	   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
3.31/1.68	   add(0, X) -> X
3.31/1.68	   add(s(X), Y) -> s(add(X, Y))
3.31/1.68	   prod(0, X) -> 0
3.31/1.68	   prod(s(X), Y) -> add(Y, prod(X, Y))
3.31/1.68	   if(true, X, Y) -> X
3.31/1.68	   if(false, X, Y) -> Y
3.31/1.68	   zero(0) -> true
3.31/1.68	   zero(s(X)) -> false
3.31/1.68	   p(s(X)) -> X
3.31/1.68	
3.31/1.68	S is empty.
3.31/1.68	Rewrite Strategy: INNERMOST
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(6) LowerBoundPropagationProof (FINISHED)
3.31/1.68	Propagated lower bound.
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(7)
3.31/1.68	BOUNDS(n^1, INF)
3.31/1.68	
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(8)
3.31/1.68	Obligation:
3.31/1.68	Analyzing the following TRS for decreasing loops:
3.31/1.68	
3.31/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.31/1.68	
3.31/1.68	
3.31/1.68	The TRS R consists of the following rules:
3.31/1.68	
3.31/1.68	   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
3.31/1.68	   add(0, X) -> X
3.31/1.68	   add(s(X), Y) -> s(add(X, Y))
3.31/1.68	   prod(0, X) -> 0
3.31/1.68	   prod(s(X), Y) -> add(Y, prod(X, Y))
3.31/1.68	   if(true, X, Y) -> X
3.31/1.68	   if(false, X, Y) -> Y
3.31/1.68	   zero(0) -> true
3.31/1.68	   zero(s(X)) -> false
3.31/1.68	   p(s(X)) -> X
3.31/1.68	
3.31/1.68	S is empty.
3.31/1.68	Rewrite Strategy: INNERMOST
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(9) InfiniteLowerBoundProof (FINISHED)
3.31/1.68	The following loop proves infinite runtime complexity:
3.31/1.68	
3.31/1.68	The rewrite sequence
3.31/1.68	
3.31/1.68	fact(X) ->^+ if(zero(X), s(0), prod(X, fact(p(X))))
3.31/1.68	
3.31/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
3.31/1.68	
3.31/1.68	The pumping substitution is [ ].
3.31/1.68	
3.31/1.68	The result substitution is [X / p(X)].
3.31/1.68	
3.31/1.68	
3.31/1.68	
3.31/1.68	
3.31/1.68	----------------------------------------
3.31/1.68	
3.31/1.68	(10)
3.31/1.68	BOUNDS(INF, INF)
3.31/1.71	EOF