1122.44/291.56	WORST_CASE(Omega(n^1), ?)
1122.56/291.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1122.56/291.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.56/291.60	
1122.56/291.60	(0) CpxTRS
1122.56/291.60	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1122.56/291.60	(2) TRS for Loop Detection
1122.56/291.60	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1122.56/291.60	(4) BEST
1122.56/291.60	    (5) proven lower bound
1122.56/291.60	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1122.56/291.60	        (7) BOUNDS(n^1, INF)
1122.56/291.60	    (8) TRS for Loop Detection
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(0)
1122.56/291.60	Obligation:
1122.56/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	The TRS R consists of the following rules:
1122.56/291.60	
1122.56/291.60	   1024 -> 1024_1(0)
1122.56/291.60	   1024_1(x) -> if(lt(x, 10), x)
1122.56/291.60	   if(true, x) -> double(1024_1(s(x)))
1122.56/291.60	   if(false, x) -> s(0)
1122.56/291.60	   lt(0, s(y)) -> true
1122.56/291.60	   lt(x, 0) -> false
1122.56/291.60	   lt(s(x), s(y)) -> lt(x, y)
1122.56/291.60	   double(0) -> 0
1122.56/291.60	   double(s(x)) -> s(s(double(x)))
1122.56/291.60	   10 -> double(s(double(s(s(0)))))
1122.56/291.60	
1122.56/291.60	S is empty.
1122.56/291.60	Rewrite Strategy: INNERMOST
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1122.56/291.60	Transformed a relative TRS into a decreasing-loop problem.
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(2)
1122.56/291.60	Obligation:
1122.56/291.60	Analyzing the following TRS for decreasing loops:
1122.56/291.60	
1122.56/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	The TRS R consists of the following rules:
1122.56/291.60	
1122.56/291.60	   1024 -> 1024_1(0)
1122.56/291.60	   1024_1(x) -> if(lt(x, 10), x)
1122.56/291.60	   if(true, x) -> double(1024_1(s(x)))
1122.56/291.60	   if(false, x) -> s(0)
1122.56/291.60	   lt(0, s(y)) -> true
1122.56/291.60	   lt(x, 0) -> false
1122.56/291.60	   lt(s(x), s(y)) -> lt(x, y)
1122.56/291.60	   double(0) -> 0
1122.56/291.60	   double(s(x)) -> s(s(double(x)))
1122.56/291.60	   10 -> double(s(double(s(s(0)))))
1122.56/291.60	
1122.56/291.60	S is empty.
1122.56/291.60	Rewrite Strategy: INNERMOST
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(3) DecreasingLoopProof (LOWER BOUND(ID))
1122.56/291.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1122.56/291.60	
1122.56/291.60	The rewrite sequence
1122.56/291.60	
1122.56/291.60	lt(s(x), s(y)) ->^+ lt(x, y)
1122.56/291.60	
1122.56/291.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1122.56/291.60	
1122.56/291.60	The pumping substitution is [x / s(x), y / s(y)].
1122.56/291.60	
1122.56/291.60	The result substitution is [ ].
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(4)
1122.56/291.60	Complex Obligation (BEST)
1122.56/291.60	
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(5)
1122.56/291.60	Obligation:
1122.56/291.60	Proved the lower bound n^1 for the following obligation:
1122.56/291.60	
1122.56/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	The TRS R consists of the following rules:
1122.56/291.60	
1122.56/291.60	   1024 -> 1024_1(0)
1122.56/291.60	   1024_1(x) -> if(lt(x, 10), x)
1122.56/291.60	   if(true, x) -> double(1024_1(s(x)))
1122.56/291.60	   if(false, x) -> s(0)
1122.56/291.60	   lt(0, s(y)) -> true
1122.56/291.60	   lt(x, 0) -> false
1122.56/291.60	   lt(s(x), s(y)) -> lt(x, y)
1122.56/291.60	   double(0) -> 0
1122.56/291.60	   double(s(x)) -> s(s(double(x)))
1122.56/291.60	   10 -> double(s(double(s(s(0)))))
1122.56/291.60	
1122.56/291.60	S is empty.
1122.56/291.60	Rewrite Strategy: INNERMOST
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(6) LowerBoundPropagationProof (FINISHED)
1122.56/291.60	Propagated lower bound.
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(7)
1122.56/291.60	BOUNDS(n^1, INF)
1122.56/291.60	
1122.56/291.60	----------------------------------------
1122.56/291.60	
1122.56/291.60	(8)
1122.56/291.60	Obligation:
1122.56/291.60	Analyzing the following TRS for decreasing loops:
1122.56/291.60	
1122.56/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.56/291.60	
1122.56/291.60	
1122.56/291.60	The TRS R consists of the following rules:
1122.56/291.60	
1122.56/291.60	   1024 -> 1024_1(0)
1122.56/291.60	   1024_1(x) -> if(lt(x, 10), x)
1122.56/291.60	   if(true, x) -> double(1024_1(s(x)))
1122.56/291.60	   if(false, x) -> s(0)
1122.56/291.60	   lt(0, s(y)) -> true
1122.56/291.60	   lt(x, 0) -> false
1122.56/291.60	   lt(s(x), s(y)) -> lt(x, y)
1122.56/291.60	   double(0) -> 0
1122.56/291.60	   double(s(x)) -> s(s(double(x)))
1122.56/291.60	   10 -> double(s(double(s(s(0)))))
1122.56/291.60	
1122.56/291.60	S is empty.
1122.56/291.60	Rewrite Strategy: INNERMOST
1122.84/291.71	EOF