313.29/291.48	WORST_CASE(Omega(n^1), ?)
313.29/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
313.29/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
313.29/291.49	
313.29/291.49	
313.29/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.29/291.49	
313.29/291.49	(0) CpxTRS
313.29/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
313.29/291.49	(2) TRS for Loop Detection
313.29/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
313.29/291.49	(4) BEST
313.29/291.49	    (5) proven lower bound
313.29/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
313.29/291.49	        (7) BOUNDS(n^1, INF)
313.29/291.49	    (8) TRS for Loop Detection
313.29/291.49	
313.29/291.49	
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(0)
313.29/291.49	Obligation:
313.29/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.29/291.49	
313.29/291.49	
313.29/291.49	The TRS R consists of the following rules:
313.29/291.49	
313.29/291.49	   plus(0, y) -> y
313.29/291.49	   plus(s(x), y) -> s(plus(x, y))
313.29/291.49	   lt(0, s(y)) -> true
313.29/291.49	   lt(x, 0) -> false
313.29/291.49	   lt(s(x), s(y)) -> lt(x, y)
313.29/291.49	   fib(x) -> fibiter(x, 0, 0, s(0))
313.29/291.49	   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
313.29/291.49	   if(false, b, c, x, y) -> x
313.29/291.49	   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
313.29/291.49	
313.29/291.49	S is empty.
313.29/291.49	Rewrite Strategy: FULL
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
313.29/291.49	Transformed a relative TRS into a decreasing-loop problem.
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(2)
313.29/291.49	Obligation:
313.29/291.49	Analyzing the following TRS for decreasing loops:
313.29/291.49	
313.29/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.29/291.49	
313.29/291.49	
313.29/291.49	The TRS R consists of the following rules:
313.29/291.49	
313.29/291.49	   plus(0, y) -> y
313.29/291.49	   plus(s(x), y) -> s(plus(x, y))
313.29/291.49	   lt(0, s(y)) -> true
313.29/291.49	   lt(x, 0) -> false
313.29/291.49	   lt(s(x), s(y)) -> lt(x, y)
313.29/291.49	   fib(x) -> fibiter(x, 0, 0, s(0))
313.29/291.49	   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
313.29/291.49	   if(false, b, c, x, y) -> x
313.29/291.49	   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
313.29/291.49	
313.29/291.49	S is empty.
313.29/291.49	Rewrite Strategy: FULL
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
313.29/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
313.29/291.49	
313.29/291.49	The rewrite sequence
313.29/291.49	
313.29/291.49	lt(s(x), s(y)) ->^+ lt(x, y)
313.29/291.49	
313.29/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
313.29/291.49	
313.29/291.49	The pumping substitution is [x / s(x), y / s(y)].
313.29/291.49	
313.29/291.49	The result substitution is [ ].
313.29/291.49	
313.29/291.49	
313.29/291.49	
313.29/291.49	
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(4)
313.29/291.49	Complex Obligation (BEST)
313.29/291.49	
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(5)
313.29/291.49	Obligation:
313.29/291.49	Proved the lower bound n^1 for the following obligation:
313.29/291.49	
313.29/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.29/291.49	
313.29/291.49	
313.29/291.49	The TRS R consists of the following rules:
313.29/291.49	
313.29/291.49	   plus(0, y) -> y
313.29/291.49	   plus(s(x), y) -> s(plus(x, y))
313.29/291.49	   lt(0, s(y)) -> true
313.29/291.49	   lt(x, 0) -> false
313.29/291.49	   lt(s(x), s(y)) -> lt(x, y)
313.29/291.49	   fib(x) -> fibiter(x, 0, 0, s(0))
313.29/291.49	   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
313.29/291.49	   if(false, b, c, x, y) -> x
313.29/291.49	   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
313.29/291.49	
313.29/291.49	S is empty.
313.29/291.49	Rewrite Strategy: FULL
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(6) LowerBoundPropagationProof (FINISHED)
313.29/291.49	Propagated lower bound.
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(7)
313.29/291.49	BOUNDS(n^1, INF)
313.29/291.49	
313.29/291.49	----------------------------------------
313.29/291.49	
313.29/291.49	(8)
313.29/291.49	Obligation:
313.29/291.49	Analyzing the following TRS for decreasing loops:
313.29/291.49	
313.29/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.29/291.49	
313.29/291.49	
313.29/291.49	The TRS R consists of the following rules:
313.29/291.49	
313.29/291.49	   plus(0, y) -> y
313.29/291.49	   plus(s(x), y) -> s(plus(x, y))
313.29/291.49	   lt(0, s(y)) -> true
313.29/291.49	   lt(x, 0) -> false
313.29/291.49	   lt(s(x), s(y)) -> lt(x, y)
313.29/291.49	   fib(x) -> fibiter(x, 0, 0, s(0))
313.29/291.49	   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
313.29/291.49	   if(false, b, c, x, y) -> x
313.29/291.49	   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
313.29/291.49	
313.29/291.49	S is empty.
313.29/291.49	Rewrite Strategy: FULL
313.33/291.52	EOF