5.14/2.07	WORST_CASE(NON_POLY, ?)
5.14/2.07	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.14/2.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.14/2.07	
5.14/2.07	
5.14/2.07	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.14/2.07	
5.14/2.07	(0) CpxTRS
5.14/2.07	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.14/2.07	(2) TRS for Loop Detection
5.14/2.07	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.14/2.07	(4) BEST
5.14/2.07	    (5) proven lower bound
5.14/2.07	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
5.14/2.07	        (7) BOUNDS(n^1, INF)
5.14/2.07	    (8) TRS for Loop Detection
5.14/2.07	        (9) DecreasingLoopProof [FINISHED, 409 ms]
5.14/2.07	        (10) BOUNDS(EXP, INF)
5.14/2.07	
5.14/2.07	
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(0)
5.14/2.07	Obligation:
5.14/2.07	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.14/2.07	
5.14/2.07	
5.14/2.07	The TRS R consists of the following rules:
5.14/2.07	
5.14/2.07	   minus(0, y) -> 0
5.14/2.07	   minus(s(x), 0) -> s(x)
5.14/2.07	   minus(s(x), s(y)) -> minus(x, y)
5.14/2.07	   le(0, y) -> true
5.14/2.07	   le(s(x), 0) -> false
5.14/2.07	   le(s(x), s(y)) -> le(x, y)
5.14/2.07	   if(true, x, y) -> x
5.14/2.07	   if(false, x, y) -> y
5.14/2.07	   perfectp(0) -> false
5.14/2.07	   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
5.14/2.07	   f(0, y, 0, u) -> true
5.14/2.07	   f(0, y, s(z), u) -> false
5.14/2.07	   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
5.14/2.07	   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))
5.14/2.07	
5.14/2.07	S is empty.
5.14/2.07	Rewrite Strategy: FULL
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.14/2.07	Transformed a relative TRS into a decreasing-loop problem.
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(2)
5.14/2.07	Obligation:
5.14/2.07	Analyzing the following TRS for decreasing loops:
5.14/2.07	
5.14/2.07	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.14/2.07	
5.14/2.07	
5.14/2.07	The TRS R consists of the following rules:
5.14/2.07	
5.14/2.07	   minus(0, y) -> 0
5.14/2.07	   minus(s(x), 0) -> s(x)
5.14/2.07	   minus(s(x), s(y)) -> minus(x, y)
5.14/2.07	   le(0, y) -> true
5.14/2.07	   le(s(x), 0) -> false
5.14/2.07	   le(s(x), s(y)) -> le(x, y)
5.14/2.07	   if(true, x, y) -> x
5.14/2.07	   if(false, x, y) -> y
5.14/2.07	   perfectp(0) -> false
5.14/2.07	   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
5.14/2.07	   f(0, y, 0, u) -> true
5.14/2.07	   f(0, y, s(z), u) -> false
5.14/2.07	   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
5.14/2.07	   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))
5.14/2.07	
5.14/2.07	S is empty.
5.14/2.07	Rewrite Strategy: FULL
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(3) DecreasingLoopProof (LOWER BOUND(ID))
5.14/2.07	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.14/2.07	
5.14/2.07	The rewrite sequence
5.14/2.07	
5.14/2.07	le(s(x), s(y)) ->^+ le(x, y)
5.14/2.07	
5.14/2.07	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
5.14/2.07	
5.14/2.07	The pumping substitution is [x / s(x), y / s(y)].
5.14/2.07	
5.14/2.07	The result substitution is [ ].
5.14/2.07	
5.14/2.07	
5.14/2.07	
5.14/2.07	
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(4)
5.14/2.07	Complex Obligation (BEST)
5.14/2.07	
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(5)
5.14/2.07	Obligation:
5.14/2.07	Proved the lower bound n^1 for the following obligation:
5.14/2.07	
5.14/2.07	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.14/2.07	
5.14/2.07	
5.14/2.07	The TRS R consists of the following rules:
5.14/2.07	
5.14/2.07	   minus(0, y) -> 0
5.14/2.07	   minus(s(x), 0) -> s(x)
5.14/2.07	   minus(s(x), s(y)) -> minus(x, y)
5.14/2.07	   le(0, y) -> true
5.14/2.07	   le(s(x), 0) -> false
5.14/2.07	   le(s(x), s(y)) -> le(x, y)
5.14/2.07	   if(true, x, y) -> x
5.14/2.07	   if(false, x, y) -> y
5.14/2.07	   perfectp(0) -> false
5.14/2.07	   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
5.14/2.07	   f(0, y, 0, u) -> true
5.14/2.07	   f(0, y, s(z), u) -> false
5.14/2.07	   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
5.14/2.07	   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))
5.14/2.07	
5.14/2.07	S is empty.
5.14/2.07	Rewrite Strategy: FULL
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(6) LowerBoundPropagationProof (FINISHED)
5.14/2.07	Propagated lower bound.
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(7)
5.14/2.07	BOUNDS(n^1, INF)
5.14/2.07	
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(8)
5.14/2.07	Obligation:
5.14/2.07	Analyzing the following TRS for decreasing loops:
5.14/2.07	
5.14/2.07	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
5.14/2.07	
5.14/2.07	
5.14/2.07	The TRS R consists of the following rules:
5.14/2.07	
5.14/2.07	   minus(0, y) -> 0
5.14/2.07	   minus(s(x), 0) -> s(x)
5.14/2.07	   minus(s(x), s(y)) -> minus(x, y)
5.14/2.07	   le(0, y) -> true
5.14/2.07	   le(s(x), 0) -> false
5.14/2.07	   le(s(x), s(y)) -> le(x, y)
5.14/2.07	   if(true, x, y) -> x
5.14/2.07	   if(false, x, y) -> y
5.14/2.07	   perfectp(0) -> false
5.14/2.07	   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
5.14/2.07	   f(0, y, 0, u) -> true
5.14/2.07	   f(0, y, s(z), u) -> false
5.14/2.07	   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
5.14/2.07	   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))
5.14/2.07	
5.14/2.07	S is empty.
5.14/2.07	Rewrite Strategy: FULL
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(9) DecreasingLoopProof (FINISHED)
5.14/2.07	The following loop(s) give(s) rise to the lower bound EXP:
5.14/2.07	
5.14/2.07	The rewrite sequence
5.14/2.07	
5.14/2.07	f(s(s(x1_0)), s(y), z, s(0)) ->^+ if(le(s(x1_0), y), f(s(s(x1_0)), minus(y, s(x1_0)), z, s(0)), if(le(x1_0, 0), f(x1_0, s(0), minus(z, s(x1_0)), s(0)), f(x1_0, s(0), z, s(0))))
5.14/2.07	
5.14/2.07	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
5.14/2.07	
5.14/2.07	The pumping substitution is [x1_0 / s(s(x1_0))].
5.14/2.07	
5.14/2.07	The result substitution is [y / 0, z / minus(z, s(x1_0))].
5.14/2.07	
5.14/2.07	
5.14/2.07	
5.14/2.07	The rewrite sequence
5.14/2.07	
5.14/2.07	f(s(s(x1_0)), s(y), z, s(0)) ->^+ if(le(s(x1_0), y), f(s(s(x1_0)), minus(y, s(x1_0)), z, s(0)), if(le(x1_0, 0), f(x1_0, s(0), minus(z, s(x1_0)), s(0)), f(x1_0, s(0), z, s(0))))
5.14/2.07	
5.14/2.07	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,2].
5.14/2.07	
5.14/2.07	The pumping substitution is [x1_0 / s(s(x1_0))].
5.14/2.07	
5.14/2.07	The result substitution is [y / 0].
5.14/2.07	
5.14/2.07	
5.14/2.07	
5.14/2.07	
5.14/2.07	----------------------------------------
5.14/2.07	
5.14/2.07	(10)
5.14/2.07	BOUNDS(EXP, INF)
5.38/2.10	EOF