1125.74/291.57	WORST_CASE(Omega(n^1), ?)
1125.87/291.62	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1125.87/291.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1125.87/291.62	
1125.87/291.62	(0) CpxTRS
1125.87/291.62	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1125.87/291.62	(2) TRS for Loop Detection
1125.87/291.62	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1125.87/291.62	(4) BEST
1125.87/291.62	    (5) proven lower bound
1125.87/291.62	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1125.87/291.62	        (7) BOUNDS(n^1, INF)
1125.87/291.62	    (8) TRS for Loop Detection
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(0)
1125.87/291.62	Obligation:
1125.87/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	The TRS R consists of the following rules:
1125.87/291.62	
1125.87/291.62	   1024 -> 1024_1(0)
1125.87/291.62	   1024_1(x) -> if(lt(x, 10), x)
1125.87/291.62	   if(true, x) -> double(1024_1(s(x)))
1125.87/291.62	   if(false, x) -> s(0)
1125.87/291.62	   lt(0, s(y)) -> true
1125.87/291.62	   lt(x, 0) -> false
1125.87/291.62	   lt(s(x), s(y)) -> lt(x, y)
1125.87/291.62	   double(0) -> 0
1125.87/291.62	   double(s(x)) -> s(s(double(x)))
1125.87/291.62	   10 -> double(s(double(s(s(0)))))
1125.87/291.62	
1125.87/291.62	S is empty.
1125.87/291.62	Rewrite Strategy: FULL
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1125.87/291.62	Transformed a relative TRS into a decreasing-loop problem.
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(2)
1125.87/291.62	Obligation:
1125.87/291.62	Analyzing the following TRS for decreasing loops:
1125.87/291.62	
1125.87/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	The TRS R consists of the following rules:
1125.87/291.62	
1125.87/291.62	   1024 -> 1024_1(0)
1125.87/291.62	   1024_1(x) -> if(lt(x, 10), x)
1125.87/291.62	   if(true, x) -> double(1024_1(s(x)))
1125.87/291.62	   if(false, x) -> s(0)
1125.87/291.62	   lt(0, s(y)) -> true
1125.87/291.62	   lt(x, 0) -> false
1125.87/291.62	   lt(s(x), s(y)) -> lt(x, y)
1125.87/291.62	   double(0) -> 0
1125.87/291.62	   double(s(x)) -> s(s(double(x)))
1125.87/291.62	   10 -> double(s(double(s(s(0)))))
1125.87/291.62	
1125.87/291.62	S is empty.
1125.87/291.62	Rewrite Strategy: FULL
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(3) DecreasingLoopProof (LOWER BOUND(ID))
1125.87/291.62	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1125.87/291.62	
1125.87/291.62	The rewrite sequence
1125.87/291.62	
1125.87/291.62	lt(s(x), s(y)) ->^+ lt(x, y)
1125.87/291.62	
1125.87/291.62	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1125.87/291.62	
1125.87/291.62	The pumping substitution is [x / s(x), y / s(y)].
1125.87/291.62	
1125.87/291.62	The result substitution is [ ].
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(4)
1125.87/291.62	Complex Obligation (BEST)
1125.87/291.62	
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(5)
1125.87/291.62	Obligation:
1125.87/291.62	Proved the lower bound n^1 for the following obligation:
1125.87/291.62	
1125.87/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	The TRS R consists of the following rules:
1125.87/291.62	
1125.87/291.62	   1024 -> 1024_1(0)
1125.87/291.62	   1024_1(x) -> if(lt(x, 10), x)
1125.87/291.62	   if(true, x) -> double(1024_1(s(x)))
1125.87/291.62	   if(false, x) -> s(0)
1125.87/291.62	   lt(0, s(y)) -> true
1125.87/291.62	   lt(x, 0) -> false
1125.87/291.62	   lt(s(x), s(y)) -> lt(x, y)
1125.87/291.62	   double(0) -> 0
1125.87/291.62	   double(s(x)) -> s(s(double(x)))
1125.87/291.62	   10 -> double(s(double(s(s(0)))))
1125.87/291.62	
1125.87/291.62	S is empty.
1125.87/291.62	Rewrite Strategy: FULL
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(6) LowerBoundPropagationProof (FINISHED)
1125.87/291.62	Propagated lower bound.
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(7)
1125.87/291.62	BOUNDS(n^1, INF)
1125.87/291.62	
1125.87/291.62	----------------------------------------
1125.87/291.62	
1125.87/291.62	(8)
1125.87/291.62	Obligation:
1125.87/291.62	Analyzing the following TRS for decreasing loops:
1125.87/291.62	
1125.87/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1125.87/291.62	
1125.87/291.62	
1125.87/291.62	The TRS R consists of the following rules:
1125.87/291.62	
1125.87/291.62	   1024 -> 1024_1(0)
1125.87/291.62	   1024_1(x) -> if(lt(x, 10), x)
1125.87/291.62	   if(true, x) -> double(1024_1(s(x)))
1125.87/291.62	   if(false, x) -> s(0)
1125.87/291.62	   lt(0, s(y)) -> true
1125.87/291.62	   lt(x, 0) -> false
1125.87/291.62	   lt(s(x), s(y)) -> lt(x, y)
1125.87/291.62	   double(0) -> 0
1125.87/291.62	   double(s(x)) -> s(s(double(x)))
1125.87/291.62	   10 -> double(s(double(s(s(0)))))
1125.87/291.62	
1125.87/291.62	S is empty.
1125.87/291.62	Rewrite Strategy: FULL
1125.96/291.71	EOF