314.11/291.50	WORST_CASE(Omega(n^1), ?)
314.11/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
314.11/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
314.11/291.50	
314.11/291.50	
314.11/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.11/291.50	
314.11/291.50	(0) CpxTRS
314.11/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
314.11/291.50	(2) TRS for Loop Detection
314.11/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
314.11/291.50	(4) BEST
314.11/291.50	    (5) proven lower bound
314.11/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
314.11/291.50	        (7) BOUNDS(n^1, INF)
314.11/291.50	    (8) TRS for Loop Detection
314.11/291.50	
314.11/291.50	
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(0)
314.11/291.50	Obligation:
314.11/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.11/291.50	
314.11/291.50	
314.11/291.50	The TRS R consists of the following rules:
314.11/291.50	
314.11/291.50	   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
314.11/291.50	   plus(x, 0) -> x
314.11/291.50	   plus(x, s(y)) -> s(plus(x, y))
314.11/291.50	   length(nil) -> 0
314.11/291.50	   length(cons(x, y)) -> s(length(y))
314.11/291.50	   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
314.11/291.50	   ge(x, 0) -> true
314.11/291.50	   ge(0, s(x)) -> false
314.11/291.50	   ge(s(x), s(y)) -> ge(x, y)
314.11/291.50	   if(true, c, l, ys, zs) -> nil
314.11/291.50	   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
314.11/291.50	   take(0, cons(x, xs), ys) -> x
314.11/291.50	   take(0, nil, cons(y, ys)) -> y
314.11/291.50	   take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
314.11/291.50	   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
314.11/291.50	   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
314.11/291.50	
314.11/291.50	S is empty.
314.11/291.50	Rewrite Strategy: FULL
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
314.11/291.50	Transformed a relative TRS into a decreasing-loop problem.
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(2)
314.11/291.50	Obligation:
314.11/291.50	Analyzing the following TRS for decreasing loops:
314.11/291.50	
314.11/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.11/291.50	
314.11/291.50	
314.11/291.50	The TRS R consists of the following rules:
314.11/291.50	
314.11/291.50	   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
314.11/291.50	   plus(x, 0) -> x
314.11/291.50	   plus(x, s(y)) -> s(plus(x, y))
314.11/291.50	   length(nil) -> 0
314.11/291.50	   length(cons(x, y)) -> s(length(y))
314.11/291.50	   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
314.11/291.50	   ge(x, 0) -> true
314.11/291.50	   ge(0, s(x)) -> false
314.11/291.50	   ge(s(x), s(y)) -> ge(x, y)
314.11/291.50	   if(true, c, l, ys, zs) -> nil
314.11/291.50	   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
314.11/291.50	   take(0, cons(x, xs), ys) -> x
314.11/291.50	   take(0, nil, cons(y, ys)) -> y
314.11/291.50	   take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
314.11/291.50	   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
314.11/291.50	   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
314.11/291.50	
314.11/291.50	S is empty.
314.11/291.50	Rewrite Strategy: FULL
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
314.11/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
314.11/291.50	
314.11/291.50	The rewrite sequence
314.11/291.50	
314.11/291.50	plus(x, s(y)) ->^+ s(plus(x, y))
314.11/291.50	
314.11/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
314.11/291.50	
314.11/291.50	The pumping substitution is [y / s(y)].
314.11/291.50	
314.11/291.50	The result substitution is [ ].
314.11/291.50	
314.11/291.50	
314.11/291.50	
314.11/291.50	
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(4)
314.11/291.50	Complex Obligation (BEST)
314.11/291.50	
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(5)
314.11/291.50	Obligation:
314.11/291.50	Proved the lower bound n^1 for the following obligation:
314.11/291.50	
314.11/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.11/291.50	
314.11/291.50	
314.11/291.50	The TRS R consists of the following rules:
314.11/291.50	
314.11/291.50	   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
314.11/291.50	   plus(x, 0) -> x
314.11/291.50	   plus(x, s(y)) -> s(plus(x, y))
314.11/291.50	   length(nil) -> 0
314.11/291.50	   length(cons(x, y)) -> s(length(y))
314.11/291.50	   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
314.11/291.50	   ge(x, 0) -> true
314.11/291.50	   ge(0, s(x)) -> false
314.11/291.50	   ge(s(x), s(y)) -> ge(x, y)
314.11/291.50	   if(true, c, l, ys, zs) -> nil
314.11/291.50	   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
314.11/291.50	   take(0, cons(x, xs), ys) -> x
314.11/291.50	   take(0, nil, cons(y, ys)) -> y
314.11/291.50	   take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
314.11/291.50	   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
314.11/291.50	   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
314.11/291.50	
314.11/291.50	S is empty.
314.11/291.50	Rewrite Strategy: FULL
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(6) LowerBoundPropagationProof (FINISHED)
314.11/291.50	Propagated lower bound.
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(7)
314.11/291.50	BOUNDS(n^1, INF)
314.11/291.50	
314.11/291.50	----------------------------------------
314.11/291.50	
314.11/291.50	(8)
314.11/291.50	Obligation:
314.11/291.50	Analyzing the following TRS for decreasing loops:
314.11/291.50	
314.11/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.11/291.50	
314.11/291.50	
314.11/291.50	The TRS R consists of the following rules:
314.11/291.50	
314.11/291.50	   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
314.11/291.50	   plus(x, 0) -> x
314.11/291.50	   plus(x, s(y)) -> s(plus(x, y))
314.11/291.50	   length(nil) -> 0
314.11/291.50	   length(cons(x, y)) -> s(length(y))
314.11/291.50	   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
314.11/291.50	   ge(x, 0) -> true
314.11/291.50	   ge(0, s(x)) -> false
314.11/291.50	   ge(s(x), s(y)) -> ge(x, y)
314.11/291.50	   if(true, c, l, ys, zs) -> nil
314.11/291.50	   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
314.11/291.50	   take(0, cons(x, xs), ys) -> x
314.11/291.50	   take(0, nil, cons(y, ys)) -> y
314.11/291.50	   take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
314.11/291.50	   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
314.11/291.50	   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))
314.11/291.50	
314.11/291.50	S is empty.
314.11/291.50	Rewrite Strategy: FULL
314.11/291.53	EOF