311.30/291.52	WORST_CASE(Omega(n^1), ?)
311.30/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
311.30/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
311.30/291.52	
311.30/291.52	
311.30/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.30/291.52	
311.30/291.52	(0) CpxTRS
311.30/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
311.30/291.52	(2) TRS for Loop Detection
311.30/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
311.30/291.52	(4) BEST
311.30/291.52	    (5) proven lower bound
311.30/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
311.30/291.52	        (7) BOUNDS(n^1, INF)
311.30/291.52	    (8) TRS for Loop Detection
311.30/291.52	
311.30/291.52	
311.30/291.52	----------------------------------------
311.30/291.52	
311.30/291.52	(0)
311.30/291.52	Obligation:
311.30/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.30/291.52	
311.30/291.52	
311.30/291.52	The TRS R consists of the following rules:
311.30/291.52	
311.30/291.52	   times(x, y) -> sum(generate(x, y))
311.30/291.52	   generate(x, y) -> gen(x, y, 0)
311.30/291.52	   gen(x, y, z) -> if(ge(z, x), x, y, z)
311.30/291.52	   if(true, x, y, z) -> nil
311.30/291.52	   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
311.30/291.52	   sum(xs) -> sum2(xs, 0)
311.30/291.52	   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
311.30/291.52	   ifsum(true, b, xs, y) -> y
311.30/291.52	   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
311.30/291.52	   ifsum2(true, xs, y) -> sum2(tail(xs), y)
311.30/291.52	   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
311.30/291.52	   isNil(nil) -> true
311.30/291.52	   isNil(cons(x, xs)) -> false
311.30/291.52	   tail(nil) -> nil
311.30/291.52	   tail(cons(x, xs)) -> xs
311.30/291.52	   head(cons(x, xs)) -> x
311.30/291.52	   head(nil) -> error
311.30/291.52	   isZero(0) -> true
311.30/291.52	   isZero(s(0)) -> false
311.30/291.52	   isZero(s(s(x))) -> isZero(s(x))
311.30/291.52	   p(0) -> s(s(0))
311.30/291.52	   p(s(0)) -> 0
311.30/291.52	   p(s(s(x))) -> s(p(s(x)))
311.30/291.52	   ge(x, 0) -> true
311.30/291.52	   ge(0, s(y)) -> false
311.30/291.52	   ge(s(x), s(y)) -> ge(x, y)
311.30/291.52	   a -> c
311.30/291.52	   a -> d
311.30/291.52	
311.30/291.52	S is empty.
311.30/291.52	Rewrite Strategy: FULL
311.30/291.52	----------------------------------------
311.30/291.52	
311.30/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
311.30/291.52	Transformed a relative TRS into a decreasing-loop problem.
311.30/291.52	----------------------------------------
311.30/291.52	
311.30/291.52	(2)
311.30/291.52	Obligation:
311.30/291.52	Analyzing the following TRS for decreasing loops:
311.30/291.52	
311.30/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.30/291.52	
311.30/291.52	
311.30/291.52	The TRS R consists of the following rules:
311.30/291.52	
311.30/291.52	   times(x, y) -> sum(generate(x, y))
311.30/291.53	   generate(x, y) -> gen(x, y, 0)
311.30/291.53	   gen(x, y, z) -> if(ge(z, x), x, y, z)
311.30/291.53	   if(true, x, y, z) -> nil
311.30/291.53	   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
311.30/291.53	   sum(xs) -> sum2(xs, 0)
311.30/291.53	   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
311.30/291.53	   ifsum(true, b, xs, y) -> y
311.30/291.53	   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
311.30/291.53	   ifsum2(true, xs, y) -> sum2(tail(xs), y)
311.30/291.53	   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
311.30/291.53	   isNil(nil) -> true
311.30/291.53	   isNil(cons(x, xs)) -> false
311.30/291.53	   tail(nil) -> nil
311.30/291.53	   tail(cons(x, xs)) -> xs
311.30/291.53	   head(cons(x, xs)) -> x
311.30/291.53	   head(nil) -> error
311.30/291.53	   isZero(0) -> true
311.30/291.53	   isZero(s(0)) -> false
311.30/291.53	   isZero(s(s(x))) -> isZero(s(x))
311.30/291.53	   p(0) -> s(s(0))
311.30/291.53	   p(s(0)) -> 0
311.30/291.53	   p(s(s(x))) -> s(p(s(x)))
311.30/291.53	   ge(x, 0) -> true
311.30/291.53	   ge(0, s(y)) -> false
311.30/291.53	   ge(s(x), s(y)) -> ge(x, y)
311.30/291.53	   a -> c
311.30/291.53	   a -> d
311.30/291.53	
311.30/291.53	S is empty.
311.30/291.53	Rewrite Strategy: FULL
311.30/291.53	----------------------------------------
311.30/291.53	
311.30/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
311.30/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
311.30/291.53	
311.30/291.53	The rewrite sequence
311.30/291.53	
311.30/291.53	p(s(s(x))) ->^+ s(p(s(x)))
311.30/291.53	
311.30/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
311.30/291.53	
311.30/291.53	The pumping substitution is [x / s(x)].
311.30/291.53	
311.30/291.53	The result substitution is [ ].
311.30/291.53	
311.30/291.53	
311.30/291.53	
311.30/291.53	
311.30/291.53	----------------------------------------
311.30/291.53	
311.30/291.53	(4)
311.30/291.53	Complex Obligation (BEST)
311.30/291.53	
311.30/291.53	----------------------------------------
311.30/291.53	
311.30/291.53	(5)
311.30/291.53	Obligation:
311.30/291.53	Proved the lower bound n^1 for the following obligation:
311.30/291.53	
311.30/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.30/291.53	
311.30/291.53	
311.30/291.53	The TRS R consists of the following rules:
311.30/291.53	
311.30/291.53	   times(x, y) -> sum(generate(x, y))
311.30/291.53	   generate(x, y) -> gen(x, y, 0)
311.30/291.53	   gen(x, y, z) -> if(ge(z, x), x, y, z)
311.30/291.53	   if(true, x, y, z) -> nil
311.30/291.53	   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
311.30/291.53	   sum(xs) -> sum2(xs, 0)
311.30/291.53	   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
311.30/291.53	   ifsum(true, b, xs, y) -> y
311.30/291.53	   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
311.30/291.53	   ifsum2(true, xs, y) -> sum2(tail(xs), y)
311.30/291.53	   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
311.30/291.53	   isNil(nil) -> true
311.30/291.53	   isNil(cons(x, xs)) -> false
311.30/291.53	   tail(nil) -> nil
311.30/291.53	   tail(cons(x, xs)) -> xs
311.30/291.53	   head(cons(x, xs)) -> x
311.30/291.53	   head(nil) -> error
311.30/291.53	   isZero(0) -> true
311.30/291.53	   isZero(s(0)) -> false
311.30/291.53	   isZero(s(s(x))) -> isZero(s(x))
311.30/291.53	   p(0) -> s(s(0))
311.30/291.53	   p(s(0)) -> 0
311.30/291.53	   p(s(s(x))) -> s(p(s(x)))
311.30/291.53	   ge(x, 0) -> true
311.30/291.53	   ge(0, s(y)) -> false
311.30/291.53	   ge(s(x), s(y)) -> ge(x, y)
311.30/291.53	   a -> c
311.30/291.53	   a -> d
311.30/291.53	
311.30/291.53	S is empty.
311.30/291.53	Rewrite Strategy: FULL
311.30/291.53	----------------------------------------
311.30/291.53	
311.30/291.53	(6) LowerBoundPropagationProof (FINISHED)
311.30/291.53	Propagated lower bound.
311.30/291.53	----------------------------------------
311.30/291.53	
311.30/291.53	(7)
311.30/291.53	BOUNDS(n^1, INF)
311.30/291.53	
311.30/291.53	----------------------------------------
311.30/291.53	
311.30/291.53	(8)
311.30/291.53	Obligation:
311.30/291.53	Analyzing the following TRS for decreasing loops:
311.30/291.53	
311.30/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.30/291.53	
311.30/291.53	
311.30/291.53	The TRS R consists of the following rules:
311.30/291.53	
311.30/291.53	   times(x, y) -> sum(generate(x, y))
311.30/291.53	   generate(x, y) -> gen(x, y, 0)
311.30/291.53	   gen(x, y, z) -> if(ge(z, x), x, y, z)
311.30/291.53	   if(true, x, y, z) -> nil
311.30/291.53	   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
311.30/291.53	   sum(xs) -> sum2(xs, 0)
311.30/291.53	   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
311.30/291.53	   ifsum(true, b, xs, y) -> y
311.30/291.53	   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
311.30/291.53	   ifsum2(true, xs, y) -> sum2(tail(xs), y)
311.30/291.53	   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
311.30/291.53	   isNil(nil) -> true
311.30/291.53	   isNil(cons(x, xs)) -> false
311.30/291.53	   tail(nil) -> nil
311.30/291.53	   tail(cons(x, xs)) -> xs
311.30/291.53	   head(cons(x, xs)) -> x
311.30/291.53	   head(nil) -> error
311.30/291.53	   isZero(0) -> true
311.30/291.53	   isZero(s(0)) -> false
311.30/291.53	   isZero(s(s(x))) -> isZero(s(x))
311.30/291.53	   p(0) -> s(s(0))
311.30/291.53	   p(s(0)) -> 0
311.30/291.53	   p(s(s(x))) -> s(p(s(x)))
311.30/291.53	   ge(x, 0) -> true
311.30/291.53	   ge(0, s(y)) -> false
311.30/291.53	   ge(s(x), s(y)) -> ge(x, y)
311.30/291.53	   a -> c
311.30/291.53	   a -> d
311.30/291.53	
311.30/291.53	S is empty.
311.30/291.53	Rewrite Strategy: FULL
311.30/291.55	EOF