596.18/291.52	WORST_CASE(Omega(n^1), ?)
596.18/291.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
596.18/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
596.18/291.52	
596.18/291.52	
596.18/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
596.18/291.52	
596.18/291.52	(0) CpxTRS
596.18/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
596.18/291.52	(2) TRS for Loop Detection
596.18/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
596.18/291.52	(4) BEST
596.18/291.52	    (5) proven lower bound
596.18/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
596.18/291.52	        (7) BOUNDS(n^1, INF)
596.18/291.52	    (8) TRS for Loop Detection
596.18/291.52	
596.18/291.52	
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(0)
596.18/291.52	Obligation:
596.18/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
596.18/291.52	
596.18/291.52	
596.18/291.52	The TRS R consists of the following rules:
596.18/291.52	
596.18/291.52	   isEmpty(cons(x, xs)) -> false
596.18/291.52	   isEmpty(nil) -> true
596.18/291.52	   isZero(0) -> true
596.18/291.52	   isZero(s(x)) -> false
596.18/291.52	   head(cons(x, xs)) -> x
596.18/291.52	   tail(cons(x, xs)) -> xs
596.18/291.52	   tail(nil) -> nil
596.18/291.52	   append(nil, x) -> cons(x, nil)
596.18/291.52	   append(cons(y, ys), x) -> cons(y, append(ys, x))
596.18/291.52	   p(s(s(x))) -> s(p(s(x)))
596.18/291.52	   p(s(0)) -> 0
596.18/291.52	   p(0) -> 0
596.18/291.52	   inc(s(x)) -> s(inc(x))
596.18/291.52	   inc(0) -> s(0)
596.18/291.52	   addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
596.18/291.52	   if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
596.18/291.52	   if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
596.18/291.52	   if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
596.18/291.52	   if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
596.18/291.52	   if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
596.18/291.52	   addList(xs, ys) -> addLists(xs, ys, nil)
596.18/291.52	
596.18/291.52	S is empty.
596.18/291.52	Rewrite Strategy: FULL
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
596.18/291.52	Transformed a relative TRS into a decreasing-loop problem.
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(2)
596.18/291.52	Obligation:
596.18/291.52	Analyzing the following TRS for decreasing loops:
596.18/291.52	
596.18/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
596.18/291.52	
596.18/291.52	
596.18/291.52	The TRS R consists of the following rules:
596.18/291.52	
596.18/291.52	   isEmpty(cons(x, xs)) -> false
596.18/291.52	   isEmpty(nil) -> true
596.18/291.52	   isZero(0) -> true
596.18/291.52	   isZero(s(x)) -> false
596.18/291.52	   head(cons(x, xs)) -> x
596.18/291.52	   tail(cons(x, xs)) -> xs
596.18/291.52	   tail(nil) -> nil
596.18/291.52	   append(nil, x) -> cons(x, nil)
596.18/291.52	   append(cons(y, ys), x) -> cons(y, append(ys, x))
596.18/291.52	   p(s(s(x))) -> s(p(s(x)))
596.18/291.52	   p(s(0)) -> 0
596.18/291.52	   p(0) -> 0
596.18/291.52	   inc(s(x)) -> s(inc(x))
596.18/291.52	   inc(0) -> s(0)
596.18/291.52	   addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
596.18/291.52	   if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
596.18/291.52	   if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
596.18/291.52	   if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
596.18/291.52	   if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
596.18/291.52	   if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
596.18/291.52	   addList(xs, ys) -> addLists(xs, ys, nil)
596.18/291.52	
596.18/291.52	S is empty.
596.18/291.52	Rewrite Strategy: FULL
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
596.18/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
596.18/291.52	
596.18/291.52	The rewrite sequence
596.18/291.52	
596.18/291.52	append(cons(y, ys), x) ->^+ cons(y, append(ys, x))
596.18/291.52	
596.18/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
596.18/291.52	
596.18/291.52	The pumping substitution is [ys / cons(y, ys)].
596.18/291.52	
596.18/291.52	The result substitution is [ ].
596.18/291.52	
596.18/291.52	
596.18/291.52	
596.18/291.52	
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(4)
596.18/291.52	Complex Obligation (BEST)
596.18/291.52	
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(5)
596.18/291.52	Obligation:
596.18/291.52	Proved the lower bound n^1 for the following obligation:
596.18/291.52	
596.18/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
596.18/291.52	
596.18/291.52	
596.18/291.52	The TRS R consists of the following rules:
596.18/291.52	
596.18/291.52	   isEmpty(cons(x, xs)) -> false
596.18/291.52	   isEmpty(nil) -> true
596.18/291.52	   isZero(0) -> true
596.18/291.52	   isZero(s(x)) -> false
596.18/291.52	   head(cons(x, xs)) -> x
596.18/291.52	   tail(cons(x, xs)) -> xs
596.18/291.52	   tail(nil) -> nil
596.18/291.52	   append(nil, x) -> cons(x, nil)
596.18/291.52	   append(cons(y, ys), x) -> cons(y, append(ys, x))
596.18/291.52	   p(s(s(x))) -> s(p(s(x)))
596.18/291.52	   p(s(0)) -> 0
596.18/291.52	   p(0) -> 0
596.18/291.52	   inc(s(x)) -> s(inc(x))
596.18/291.52	   inc(0) -> s(0)
596.18/291.52	   addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
596.18/291.52	   if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
596.18/291.52	   if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
596.18/291.52	   if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
596.18/291.52	   if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
596.18/291.52	   if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
596.18/291.52	   addList(xs, ys) -> addLists(xs, ys, nil)
596.18/291.52	
596.18/291.52	S is empty.
596.18/291.52	Rewrite Strategy: FULL
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(6) LowerBoundPropagationProof (FINISHED)
596.18/291.52	Propagated lower bound.
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(7)
596.18/291.52	BOUNDS(n^1, INF)
596.18/291.52	
596.18/291.52	----------------------------------------
596.18/291.52	
596.18/291.52	(8)
596.18/291.52	Obligation:
596.18/291.52	Analyzing the following TRS for decreasing loops:
596.18/291.52	
596.18/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
596.18/291.52	
596.18/291.52	
596.18/291.52	The TRS R consists of the following rules:
596.18/291.52	
596.18/291.52	   isEmpty(cons(x, xs)) -> false
596.18/291.52	   isEmpty(nil) -> true
596.18/291.52	   isZero(0) -> true
596.18/291.52	   isZero(s(x)) -> false
596.18/291.52	   head(cons(x, xs)) -> x
596.18/291.52	   tail(cons(x, xs)) -> xs
596.18/291.52	   tail(nil) -> nil
596.18/291.52	   append(nil, x) -> cons(x, nil)
596.18/291.52	   append(cons(y, ys), x) -> cons(y, append(ys, x))
596.18/291.52	   p(s(s(x))) -> s(p(s(x)))
596.18/291.52	   p(s(0)) -> 0
596.18/291.52	   p(0) -> 0
596.18/291.52	   inc(s(x)) -> s(inc(x))
596.18/291.52	   inc(0) -> s(0)
596.18/291.52	   addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
596.18/291.52	   if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
596.18/291.52	   if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
596.18/291.52	   if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
596.18/291.52	   if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
596.18/291.52	   if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
596.18/291.52	   addList(xs, ys) -> addLists(xs, ys, nil)
596.18/291.52	
596.18/291.52	S is empty.
596.18/291.52	Rewrite Strategy: FULL
596.29/291.56	EOF