3.12/1.60	WORST_CASE(NON_POLY, ?)
3.12/1.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.12/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.12/1.61	
3.12/1.61	
3.12/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.12/1.61	
3.12/1.61	(0) CpxTRS
3.12/1.61	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.12/1.61	(2) TRS for Loop Detection
3.12/1.61	(3) DecreasingLoopProof [FINISHED, 0 ms]
3.12/1.61	(4) BOUNDS(EXP, INF)
3.12/1.61	
3.12/1.61	
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(0)
3.12/1.61	Obligation:
3.12/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.12/1.61	
3.12/1.61	
3.12/1.61	The TRS R consists of the following rules:
3.12/1.61	
3.12/1.61	   fib(0) -> 0
3.12/1.61	   fib(s(0)) -> s(0)
3.12/1.61	   fib(s(s(x))) -> +(fib(s(x)), fib(x))
3.12/1.61	   +(x, 0) -> x
3.12/1.61	   +(x, s(y)) -> s(+(x, y))
3.12/1.61	
3.12/1.61	S is empty.
3.12/1.61	Rewrite Strategy: FULL
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.12/1.61	Transformed a relative TRS into a decreasing-loop problem.
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(2)
3.12/1.61	Obligation:
3.12/1.61	Analyzing the following TRS for decreasing loops:
3.12/1.61	
3.12/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.12/1.61	
3.12/1.61	
3.12/1.61	The TRS R consists of the following rules:
3.12/1.61	
3.12/1.61	   fib(0) -> 0
3.12/1.61	   fib(s(0)) -> s(0)
3.12/1.61	   fib(s(s(x))) -> +(fib(s(x)), fib(x))
3.12/1.61	   +(x, 0) -> x
3.12/1.61	   +(x, s(y)) -> s(+(x, y))
3.12/1.61	
3.12/1.61	S is empty.
3.12/1.61	Rewrite Strategy: FULL
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(3) DecreasingLoopProof (FINISHED)
3.12/1.61	The following loop(s) give(s) rise to the lower bound EXP:
3.12/1.61	
3.12/1.61	The rewrite sequence
3.12/1.61	
3.12/1.61	fib(s(s(x))) ->^+ +(fib(s(x)), fib(x))
3.12/1.61	
3.12/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.12/1.61	
3.12/1.61	The pumping substitution is [x / s(x)].
3.12/1.61	
3.12/1.61	The result substitution is [ ].
3.12/1.61	
3.12/1.61	
3.12/1.61	
3.12/1.61	The rewrite sequence
3.12/1.61	
3.12/1.61	fib(s(s(x))) ->^+ +(fib(s(x)), fib(x))
3.12/1.61	
3.12/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.12/1.61	
3.12/1.61	The pumping substitution is [x / s(s(x))].
3.12/1.61	
3.12/1.61	The result substitution is [ ].
3.12/1.61	
3.12/1.61	
3.12/1.61	
3.12/1.61	
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(4)
3.12/1.61	BOUNDS(EXP, INF)
3.12/1.64	EOF