322.73/291.51	WORST_CASE(Omega(n^1), ?)
322.73/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
322.73/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
322.73/291.51	
322.73/291.51	
322.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
322.73/291.51	
322.73/291.51	(0) CpxTRS
322.73/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
322.73/291.51	(2) TRS for Loop Detection
322.73/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
322.73/291.51	(4) BEST
322.73/291.51	    (5) proven lower bound
322.73/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
322.73/291.51	        (7) BOUNDS(n^1, INF)
322.73/291.51	    (8) TRS for Loop Detection
322.73/291.51	
322.73/291.51	
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(0)
322.73/291.51	Obligation:
322.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
322.73/291.51	
322.73/291.51	
322.73/291.51	The TRS R consists of the following rules:
322.73/291.51	
322.73/291.51	   -(x, 0) -> x
322.73/291.51	   -(s(x), s(y)) -> -(x, y)
322.73/291.51	   *(x, 0) -> 0
322.73/291.51	   *(x, s(y)) -> +(*(x, y), x)
322.73/291.51	   if(true, x, y) -> x
322.73/291.51	   if(false, x, y) -> y
322.73/291.51	   odd(0) -> false
322.73/291.51	   odd(s(0)) -> true
322.73/291.51	   odd(s(s(x))) -> odd(x)
322.73/291.51	   half(0) -> 0
322.73/291.51	   half(s(0)) -> 0
322.73/291.51	   half(s(s(x))) -> s(half(x))
322.73/291.51	   if(true, x, y) -> true
322.73/291.51	   if(false, x, y) -> false
322.73/291.51	   pow(x, y) -> f(x, y, s(0))
322.73/291.51	   f(x, 0, z) -> z
322.73/291.51	   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
322.73/291.51	
322.73/291.51	S is empty.
322.73/291.51	Rewrite Strategy: FULL
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
322.73/291.51	Transformed a relative TRS into a decreasing-loop problem.
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(2)
322.73/291.51	Obligation:
322.73/291.51	Analyzing the following TRS for decreasing loops:
322.73/291.51	
322.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
322.73/291.51	
322.73/291.51	
322.73/291.51	The TRS R consists of the following rules:
322.73/291.51	
322.73/291.51	   -(x, 0) -> x
322.73/291.51	   -(s(x), s(y)) -> -(x, y)
322.73/291.51	   *(x, 0) -> 0
322.73/291.51	   *(x, s(y)) -> +(*(x, y), x)
322.73/291.51	   if(true, x, y) -> x
322.73/291.51	   if(false, x, y) -> y
322.73/291.51	   odd(0) -> false
322.73/291.51	   odd(s(0)) -> true
322.73/291.51	   odd(s(s(x))) -> odd(x)
322.73/291.51	   half(0) -> 0
322.73/291.51	   half(s(0)) -> 0
322.73/291.51	   half(s(s(x))) -> s(half(x))
322.73/291.51	   if(true, x, y) -> true
322.73/291.51	   if(false, x, y) -> false
322.73/291.51	   pow(x, y) -> f(x, y, s(0))
322.73/291.51	   f(x, 0, z) -> z
322.73/291.51	   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
322.73/291.51	
322.73/291.51	S is empty.
322.73/291.51	Rewrite Strategy: FULL
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
322.73/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
322.73/291.51	
322.73/291.51	The rewrite sequence
322.73/291.51	
322.73/291.51	-(s(x), s(y)) ->^+ -(x, y)
322.73/291.51	
322.73/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
322.73/291.51	
322.73/291.51	The pumping substitution is [x / s(x), y / s(y)].
322.73/291.51	
322.73/291.51	The result substitution is [ ].
322.73/291.51	
322.73/291.51	
322.73/291.51	
322.73/291.51	
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(4)
322.73/291.51	Complex Obligation (BEST)
322.73/291.51	
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(5)
322.73/291.51	Obligation:
322.73/291.51	Proved the lower bound n^1 for the following obligation:
322.73/291.51	
322.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
322.73/291.51	
322.73/291.51	
322.73/291.51	The TRS R consists of the following rules:
322.73/291.51	
322.73/291.51	   -(x, 0) -> x
322.73/291.51	   -(s(x), s(y)) -> -(x, y)
322.73/291.51	   *(x, 0) -> 0
322.73/291.51	   *(x, s(y)) -> +(*(x, y), x)
322.73/291.51	   if(true, x, y) -> x
322.73/291.51	   if(false, x, y) -> y
322.73/291.51	   odd(0) -> false
322.73/291.51	   odd(s(0)) -> true
322.73/291.51	   odd(s(s(x))) -> odd(x)
322.73/291.51	   half(0) -> 0
322.73/291.51	   half(s(0)) -> 0
322.73/291.51	   half(s(s(x))) -> s(half(x))
322.73/291.51	   if(true, x, y) -> true
322.73/291.51	   if(false, x, y) -> false
322.73/291.51	   pow(x, y) -> f(x, y, s(0))
322.73/291.51	   f(x, 0, z) -> z
322.73/291.51	   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
322.73/291.51	
322.73/291.51	S is empty.
322.73/291.51	Rewrite Strategy: FULL
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(6) LowerBoundPropagationProof (FINISHED)
322.73/291.51	Propagated lower bound.
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(7)
322.73/291.51	BOUNDS(n^1, INF)
322.73/291.51	
322.73/291.51	----------------------------------------
322.73/291.51	
322.73/291.51	(8)
322.73/291.51	Obligation:
322.73/291.51	Analyzing the following TRS for decreasing loops:
322.73/291.51	
322.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
322.73/291.51	
322.73/291.51	
322.73/291.51	The TRS R consists of the following rules:
322.73/291.51	
322.73/291.51	   -(x, 0) -> x
322.73/291.51	   -(s(x), s(y)) -> -(x, y)
322.73/291.51	   *(x, 0) -> 0
322.73/291.51	   *(x, s(y)) -> +(*(x, y), x)
322.73/291.51	   if(true, x, y) -> x
322.73/291.51	   if(false, x, y) -> y
322.73/291.51	   odd(0) -> false
322.73/291.51	   odd(s(0)) -> true
322.73/291.51	   odd(s(s(x))) -> odd(x)
322.73/291.51	   half(0) -> 0
322.73/291.51	   half(s(0)) -> 0
322.73/291.51	   half(s(s(x))) -> s(half(x))
322.73/291.51	   if(true, x, y) -> true
322.73/291.51	   if(false, x, y) -> false
322.73/291.51	   pow(x, y) -> f(x, y, s(0))
322.73/291.51	   f(x, 0, z) -> z
322.73/291.51	   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
322.73/291.51	
322.73/291.51	S is empty.
322.73/291.51	Rewrite Strategy: FULL
322.85/291.54	EOF