3.15/1.56	WORST_CASE(NON_POLY, ?)
3.15/1.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.15/1.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.15/1.57	
3.15/1.57	
3.15/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.15/1.57	
3.15/1.57	(0) CpxTRS
3.15/1.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.15/1.57	(2) TRS for Loop Detection
3.15/1.57	(3) DecreasingLoopProof [FINISHED, 0 ms]
3.15/1.57	(4) BOUNDS(EXP, INF)
3.15/1.57	
3.15/1.57	
3.15/1.57	----------------------------------------
3.15/1.57	
3.15/1.57	(0)
3.15/1.57	Obligation:
3.15/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.15/1.57	
3.15/1.57	
3.15/1.57	The TRS R consists of the following rules:
3.15/1.57	
3.15/1.57	   lt(0, s(x)) -> true
3.15/1.57	   lt(x, 0) -> false
3.15/1.57	   lt(s(x), s(y)) -> lt(x, y)
3.15/1.57	   fibo(0) -> fib(0)
3.15/1.57	   fibo(s(0)) -> fib(s(0))
3.15/1.57	   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
3.15/1.57	   fib(0) -> s(0)
3.15/1.57	   fib(s(0)) -> s(0)
3.15/1.57	   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0)
3.15/1.57	   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
3.15/1.57	   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
3.15/1.57	   sum(x, 0) -> x
3.15/1.57	   sum(x, s(y)) -> s(sum(x, y))
3.15/1.57	
3.15/1.57	S is empty.
3.15/1.57	Rewrite Strategy: INNERMOST
3.15/1.57	----------------------------------------
3.15/1.57	
3.15/1.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.15/1.57	Transformed a relative TRS into a decreasing-loop problem.
3.15/1.57	----------------------------------------
3.15/1.57	
3.15/1.57	(2)
3.15/1.57	Obligation:
3.15/1.57	Analyzing the following TRS for decreasing loops:
3.15/1.57	
3.15/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.15/1.57	
3.15/1.57	
3.15/1.57	The TRS R consists of the following rules:
3.15/1.57	
3.15/1.57	   lt(0, s(x)) -> true
3.15/1.57	   lt(x, 0) -> false
3.15/1.57	   lt(s(x), s(y)) -> lt(x, y)
3.15/1.57	   fibo(0) -> fib(0)
3.15/1.57	   fibo(s(0)) -> fib(s(0))
3.15/1.57	   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
3.15/1.57	   fib(0) -> s(0)
3.15/1.57	   fib(s(0)) -> s(0)
3.15/1.57	   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0)
3.15/1.57	   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
3.15/1.57	   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
3.15/1.57	   sum(x, 0) -> x
3.15/1.57	   sum(x, s(y)) -> s(sum(x, y))
3.15/1.57	
3.15/1.57	S is empty.
3.15/1.57	Rewrite Strategy: INNERMOST
3.15/1.57	----------------------------------------
3.15/1.57	
3.15/1.57	(3) DecreasingLoopProof (FINISHED)
3.15/1.57	The following loop(s) give(s) rise to the lower bound EXP:
3.15/1.57	
3.15/1.57	The rewrite sequence
3.15/1.57	
3.15/1.57	fibo(s(s(x))) ->^+ sum(fibo(s(x)), fibo(x))
3.15/1.57	
3.15/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.15/1.57	
3.15/1.57	The pumping substitution is [x / s(x)].
3.15/1.57	
3.15/1.57	The result substitution is [ ].
3.15/1.57	
3.15/1.57	
3.15/1.57	
3.15/1.57	The rewrite sequence
3.15/1.57	
3.15/1.57	fibo(s(s(x))) ->^+ sum(fibo(s(x)), fibo(x))
3.15/1.57	
3.15/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.15/1.57	
3.15/1.57	The pumping substitution is [x / s(s(x))].
3.15/1.57	
3.15/1.57	The result substitution is [ ].
3.15/1.57	
3.15/1.57	
3.15/1.57	
3.15/1.57	
3.15/1.57	----------------------------------------
3.15/1.57	
3.15/1.57	(4)
3.15/1.57	BOUNDS(EXP, INF)
3.15/1.60	EOF