305.11/291.53	WORST_CASE(Omega(n^1), ?)
305.28/291.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
305.28/291.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
305.28/291.58	
305.28/291.58	
305.28/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.28/291.58	
305.28/291.58	(0) CpxTRS
305.28/291.58	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
305.28/291.58	(2) TRS for Loop Detection
305.28/291.58	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
305.28/291.58	(4) BEST
305.28/291.58	    (5) proven lower bound
305.28/291.58	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
305.28/291.58	        (7) BOUNDS(n^1, INF)
305.28/291.58	    (8) TRS for Loop Detection
305.28/291.58	
305.28/291.58	
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(0)
305.28/291.58	Obligation:
305.28/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.28/291.58	
305.28/291.58	
305.28/291.58	The TRS R consists of the following rules:
305.28/291.58	
305.28/291.58	   half(0) -> 0
305.28/291.58	   half(s(0)) -> 0
305.28/291.58	   half(s(s(x))) -> s(half(x))
305.28/291.58	   le(0, y) -> true
305.28/291.58	   le(s(x), 0) -> false
305.28/291.58	   le(s(x), s(y)) -> le(x, y)
305.28/291.58	   inc(s(x)) -> s(inc(x))
305.28/291.58	   inc(0) -> s(0)
305.28/291.58	   logarithm(x) -> logIter(x, 0)
305.28/291.58	   logIter(x, y) -> if(le(s(0), x), le(s(s(0)), x), half(x), inc(y))
305.28/291.58	   if(false, b, x, y) -> logZeroError
305.28/291.58	   if(true, false, x, s(y)) -> y
305.28/291.58	   if(true, true, x, y) -> logIter(x, y)
305.28/291.58	   f -> g
305.28/291.58	   f -> h
305.28/291.58	
305.28/291.58	S is empty.
305.28/291.58	Rewrite Strategy: FULL
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
305.28/291.58	Transformed a relative TRS into a decreasing-loop problem.
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(2)
305.28/291.58	Obligation:
305.28/291.58	Analyzing the following TRS for decreasing loops:
305.28/291.58	
305.28/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.28/291.58	
305.28/291.58	
305.28/291.58	The TRS R consists of the following rules:
305.28/291.58	
305.28/291.58	   half(0) -> 0
305.28/291.58	   half(s(0)) -> 0
305.28/291.58	   half(s(s(x))) -> s(half(x))
305.28/291.58	   le(0, y) -> true
305.28/291.58	   le(s(x), 0) -> false
305.28/291.58	   le(s(x), s(y)) -> le(x, y)
305.28/291.58	   inc(s(x)) -> s(inc(x))
305.28/291.58	   inc(0) -> s(0)
305.28/291.58	   logarithm(x) -> logIter(x, 0)
305.28/291.58	   logIter(x, y) -> if(le(s(0), x), le(s(s(0)), x), half(x), inc(y))
305.28/291.58	   if(false, b, x, y) -> logZeroError
305.28/291.58	   if(true, false, x, s(y)) -> y
305.28/291.58	   if(true, true, x, y) -> logIter(x, y)
305.28/291.58	   f -> g
305.28/291.58	   f -> h
305.28/291.58	
305.28/291.58	S is empty.
305.28/291.58	Rewrite Strategy: FULL
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(3) DecreasingLoopProof (LOWER BOUND(ID))
305.28/291.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
305.28/291.58	
305.28/291.58	The rewrite sequence
305.28/291.58	
305.28/291.58	le(s(x), s(y)) ->^+ le(x, y)
305.28/291.58	
305.28/291.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
305.28/291.58	
305.28/291.58	The pumping substitution is [x / s(x), y / s(y)].
305.28/291.58	
305.28/291.58	The result substitution is [ ].
305.28/291.58	
305.28/291.58	
305.28/291.58	
305.28/291.58	
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(4)
305.28/291.58	Complex Obligation (BEST)
305.28/291.58	
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(5)
305.28/291.58	Obligation:
305.28/291.58	Proved the lower bound n^1 for the following obligation:
305.28/291.58	
305.28/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.28/291.58	
305.28/291.58	
305.28/291.58	The TRS R consists of the following rules:
305.28/291.58	
305.28/291.58	   half(0) -> 0
305.28/291.58	   half(s(0)) -> 0
305.28/291.58	   half(s(s(x))) -> s(half(x))
305.28/291.58	   le(0, y) -> true
305.28/291.58	   le(s(x), 0) -> false
305.28/291.58	   le(s(x), s(y)) -> le(x, y)
305.28/291.58	   inc(s(x)) -> s(inc(x))
305.28/291.58	   inc(0) -> s(0)
305.28/291.58	   logarithm(x) -> logIter(x, 0)
305.28/291.58	   logIter(x, y) -> if(le(s(0), x), le(s(s(0)), x), half(x), inc(y))
305.28/291.58	   if(false, b, x, y) -> logZeroError
305.28/291.58	   if(true, false, x, s(y)) -> y
305.28/291.58	   if(true, true, x, y) -> logIter(x, y)
305.28/291.58	   f -> g
305.28/291.58	   f -> h
305.28/291.58	
305.28/291.58	S is empty.
305.28/291.58	Rewrite Strategy: FULL
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(6) LowerBoundPropagationProof (FINISHED)
305.28/291.58	Propagated lower bound.
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(7)
305.28/291.58	BOUNDS(n^1, INF)
305.28/291.58	
305.28/291.58	----------------------------------------
305.28/291.58	
305.28/291.58	(8)
305.28/291.58	Obligation:
305.28/291.58	Analyzing the following TRS for decreasing loops:
305.28/291.58	
305.28/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.28/291.58	
305.28/291.58	
305.28/291.58	The TRS R consists of the following rules:
305.28/291.58	
305.28/291.58	   half(0) -> 0
305.28/291.58	   half(s(0)) -> 0
305.28/291.58	   half(s(s(x))) -> s(half(x))
305.28/291.58	   le(0, y) -> true
305.28/291.58	   le(s(x), 0) -> false
305.28/291.58	   le(s(x), s(y)) -> le(x, y)
305.28/291.58	   inc(s(x)) -> s(inc(x))
305.28/291.58	   inc(0) -> s(0)
305.28/291.58	   logarithm(x) -> logIter(x, 0)
305.28/291.58	   logIter(x, y) -> if(le(s(0), x), le(s(s(0)), x), half(x), inc(y))
305.28/291.58	   if(false, b, x, y) -> logZeroError
305.28/291.58	   if(true, false, x, s(y)) -> y
305.28/291.58	   if(true, true, x, y) -> logIter(x, y)
305.28/291.58	   f -> g
305.28/291.58	   f -> h
305.28/291.58	
305.28/291.58	S is empty.
305.28/291.58	Rewrite Strategy: FULL
305.28/291.62	EOF