841.19/291.59	WORST_CASE(Omega(n^1), ?)
841.19/291.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
841.19/291.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
841.19/291.60	
841.19/291.60	
841.19/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
841.19/291.60	
841.19/291.60	(0) CpxTRS
841.19/291.60	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
841.19/291.60	(2) TRS for Loop Detection
841.19/291.60	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
841.19/291.60	(4) BEST
841.19/291.60	    (5) proven lower bound
841.19/291.60	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
841.19/291.60	        (7) BOUNDS(n^1, INF)
841.19/291.60	    (8) TRS for Loop Detection
841.19/291.60	
841.19/291.60	
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(0)
841.19/291.60	Obligation:
841.19/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
841.19/291.60	
841.19/291.60	
841.19/291.60	The TRS R consists of the following rules:
841.19/291.60	
841.19/291.60	   -(x, 0) -> x
841.19/291.60	   -(s(x), s(y)) -> -(x, y)
841.19/291.60	   *(x, 0) -> 0
841.19/291.60	   *(x, s(y)) -> +(*(x, y), x)
841.19/291.60	   if(true, x, y) -> x
841.19/291.60	   if(false, x, y) -> y
841.19/291.60	   odd(0) -> false
841.19/291.60	   odd(s(0)) -> true
841.19/291.60	   odd(s(s(x))) -> odd(x)
841.19/291.60	   half(0) -> 0
841.19/291.60	   half(s(0)) -> 0
841.19/291.60	   half(s(s(x))) -> s(half(x))
841.19/291.60	   if(true, x, y) -> true
841.19/291.60	   if(false, x, y) -> false
841.19/291.60	   pow(x, y) -> f(x, y, s(0))
841.19/291.60	   f(x, 0, z) -> z
841.19/291.60	   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
841.19/291.60	
841.19/291.60	S is empty.
841.19/291.60	Rewrite Strategy: INNERMOST
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
841.19/291.60	Transformed a relative TRS into a decreasing-loop problem.
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(2)
841.19/291.60	Obligation:
841.19/291.60	Analyzing the following TRS for decreasing loops:
841.19/291.60	
841.19/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
841.19/291.60	
841.19/291.60	
841.19/291.60	The TRS R consists of the following rules:
841.19/291.60	
841.19/291.60	   -(x, 0) -> x
841.19/291.60	   -(s(x), s(y)) -> -(x, y)
841.19/291.60	   *(x, 0) -> 0
841.19/291.60	   *(x, s(y)) -> +(*(x, y), x)
841.19/291.60	   if(true, x, y) -> x
841.19/291.60	   if(false, x, y) -> y
841.19/291.60	   odd(0) -> false
841.19/291.60	   odd(s(0)) -> true
841.19/291.60	   odd(s(s(x))) -> odd(x)
841.19/291.60	   half(0) -> 0
841.19/291.60	   half(s(0)) -> 0
841.19/291.60	   half(s(s(x))) -> s(half(x))
841.19/291.60	   if(true, x, y) -> true
841.19/291.60	   if(false, x, y) -> false
841.19/291.60	   pow(x, y) -> f(x, y, s(0))
841.19/291.60	   f(x, 0, z) -> z
841.19/291.60	   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
841.19/291.60	
841.19/291.60	S is empty.
841.19/291.60	Rewrite Strategy: INNERMOST
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(3) DecreasingLoopProof (LOWER BOUND(ID))
841.19/291.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
841.19/291.60	
841.19/291.60	The rewrite sequence
841.19/291.60	
841.19/291.60	-(s(x), s(y)) ->^+ -(x, y)
841.19/291.60	
841.19/291.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
841.19/291.60	
841.19/291.60	The pumping substitution is [x / s(x), y / s(y)].
841.19/291.60	
841.19/291.60	The result substitution is [ ].
841.19/291.60	
841.19/291.60	
841.19/291.60	
841.19/291.60	
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(4)
841.19/291.60	Complex Obligation (BEST)
841.19/291.60	
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(5)
841.19/291.60	Obligation:
841.19/291.60	Proved the lower bound n^1 for the following obligation:
841.19/291.60	
841.19/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
841.19/291.60	
841.19/291.60	
841.19/291.60	The TRS R consists of the following rules:
841.19/291.60	
841.19/291.60	   -(x, 0) -> x
841.19/291.60	   -(s(x), s(y)) -> -(x, y)
841.19/291.60	   *(x, 0) -> 0
841.19/291.60	   *(x, s(y)) -> +(*(x, y), x)
841.19/291.60	   if(true, x, y) -> x
841.19/291.60	   if(false, x, y) -> y
841.19/291.60	   odd(0) -> false
841.19/291.60	   odd(s(0)) -> true
841.19/291.60	   odd(s(s(x))) -> odd(x)
841.19/291.60	   half(0) -> 0
841.19/291.60	   half(s(0)) -> 0
841.19/291.60	   half(s(s(x))) -> s(half(x))
841.19/291.60	   if(true, x, y) -> true
841.19/291.60	   if(false, x, y) -> false
841.19/291.60	   pow(x, y) -> f(x, y, s(0))
841.19/291.60	   f(x, 0, z) -> z
841.19/291.60	   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
841.19/291.60	
841.19/291.60	S is empty.
841.19/291.60	Rewrite Strategy: INNERMOST
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(6) LowerBoundPropagationProof (FINISHED)
841.19/291.60	Propagated lower bound.
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(7)
841.19/291.60	BOUNDS(n^1, INF)
841.19/291.60	
841.19/291.60	----------------------------------------
841.19/291.60	
841.19/291.60	(8)
841.19/291.60	Obligation:
841.19/291.60	Analyzing the following TRS for decreasing loops:
841.19/291.60	
841.19/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
841.19/291.60	
841.19/291.60	
841.19/291.60	The TRS R consists of the following rules:
841.19/291.60	
841.19/291.60	   -(x, 0) -> x
841.19/291.60	   -(s(x), s(y)) -> -(x, y)
841.19/291.60	   *(x, 0) -> 0
841.19/291.60	   *(x, s(y)) -> +(*(x, y), x)
841.19/291.60	   if(true, x, y) -> x
841.19/291.60	   if(false, x, y) -> y
841.19/291.60	   odd(0) -> false
841.19/291.60	   odd(s(0)) -> true
841.19/291.60	   odd(s(s(x))) -> odd(x)
841.19/291.60	   half(0) -> 0
841.19/291.60	   half(s(0)) -> 0
841.19/291.60	   half(s(s(x))) -> s(half(x))
841.19/291.60	   if(true, x, y) -> true
841.19/291.60	   if(false, x, y) -> false
841.19/291.60	   pow(x, y) -> f(x, y, s(0))
841.19/291.60	   f(x, 0, z) -> z
841.19/291.60	   f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
841.19/291.60	
841.19/291.60	S is empty.
841.19/291.60	Rewrite Strategy: INNERMOST
841.28/291.68	EOF