315.71/291.48	WORST_CASE(Omega(n^1), ?)
315.71/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
315.71/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
315.71/291.49	
315.71/291.49	
315.71/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.71/291.49	
315.71/291.49	(0) CpxTRS
315.71/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
315.71/291.49	(2) TRS for Loop Detection
315.71/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
315.71/291.49	(4) BEST
315.71/291.49	    (5) proven lower bound
315.71/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
315.71/291.49	        (7) BOUNDS(n^1, INF)
315.71/291.49	    (8) TRS for Loop Detection
315.71/291.49	
315.71/291.49	
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(0)
315.71/291.49	Obligation:
315.71/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.71/291.49	
315.71/291.49	
315.71/291.49	The TRS R consists of the following rules:
315.71/291.49	
315.71/291.49	   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
315.71/291.49	   plus(x, 0) -> x
315.71/291.49	   plus(x, s(y)) -> s(plus(x, y))
315.71/291.49	   length(nil) -> 0
315.71/291.49	   length(cons(x, y)) -> s(length(y))
315.71/291.49	   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
315.71/291.49	   ge(x, 0) -> true
315.71/291.49	   ge(0, s(x)) -> false
315.71/291.49	   ge(s(x), s(y)) -> ge(x, y)
315.71/291.49	   if(true, c, l, ys, zs) -> nil
315.71/291.49	   if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
315.71/291.49	   greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
315.71/291.49	   smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
315.71/291.49	   helpc(true, ys, zs) -> ys
315.71/291.49	   helpc(false, ys, zs) -> zs
315.71/291.49	   helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))
315.71/291.49	
315.71/291.49	S is empty.
315.71/291.49	Rewrite Strategy: FULL
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
315.71/291.49	Transformed a relative TRS into a decreasing-loop problem.
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(2)
315.71/291.49	Obligation:
315.71/291.49	Analyzing the following TRS for decreasing loops:
315.71/291.49	
315.71/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.71/291.49	
315.71/291.49	
315.71/291.49	The TRS R consists of the following rules:
315.71/291.49	
315.71/291.49	   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
315.71/291.49	   plus(x, 0) -> x
315.71/291.49	   plus(x, s(y)) -> s(plus(x, y))
315.71/291.49	   length(nil) -> 0
315.71/291.49	   length(cons(x, y)) -> s(length(y))
315.71/291.49	   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
315.71/291.49	   ge(x, 0) -> true
315.71/291.49	   ge(0, s(x)) -> false
315.71/291.49	   ge(s(x), s(y)) -> ge(x, y)
315.71/291.49	   if(true, c, l, ys, zs) -> nil
315.71/291.49	   if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
315.71/291.49	   greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
315.71/291.49	   smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
315.71/291.49	   helpc(true, ys, zs) -> ys
315.71/291.49	   helpc(false, ys, zs) -> zs
315.71/291.49	   helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))
315.71/291.49	
315.71/291.49	S is empty.
315.71/291.49	Rewrite Strategy: FULL
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
315.71/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
315.71/291.49	
315.71/291.49	The rewrite sequence
315.71/291.49	
315.71/291.49	plus(x, s(y)) ->^+ s(plus(x, y))
315.71/291.49	
315.71/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
315.71/291.49	
315.71/291.49	The pumping substitution is [y / s(y)].
315.71/291.49	
315.71/291.49	The result substitution is [ ].
315.71/291.49	
315.71/291.49	
315.71/291.49	
315.71/291.49	
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(4)
315.71/291.49	Complex Obligation (BEST)
315.71/291.49	
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(5)
315.71/291.49	Obligation:
315.71/291.49	Proved the lower bound n^1 for the following obligation:
315.71/291.49	
315.71/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.71/291.49	
315.71/291.49	
315.71/291.49	The TRS R consists of the following rules:
315.71/291.49	
315.71/291.49	   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
315.71/291.49	   plus(x, 0) -> x
315.71/291.49	   plus(x, s(y)) -> s(plus(x, y))
315.71/291.49	   length(nil) -> 0
315.71/291.49	   length(cons(x, y)) -> s(length(y))
315.71/291.49	   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
315.71/291.49	   ge(x, 0) -> true
315.71/291.49	   ge(0, s(x)) -> false
315.71/291.49	   ge(s(x), s(y)) -> ge(x, y)
315.71/291.49	   if(true, c, l, ys, zs) -> nil
315.71/291.49	   if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
315.71/291.49	   greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
315.71/291.49	   smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
315.71/291.49	   helpc(true, ys, zs) -> ys
315.71/291.49	   helpc(false, ys, zs) -> zs
315.71/291.49	   helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))
315.71/291.49	
315.71/291.49	S is empty.
315.71/291.49	Rewrite Strategy: FULL
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(6) LowerBoundPropagationProof (FINISHED)
315.71/291.49	Propagated lower bound.
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(7)
315.71/291.49	BOUNDS(n^1, INF)
315.71/291.49	
315.71/291.49	----------------------------------------
315.71/291.49	
315.71/291.49	(8)
315.71/291.49	Obligation:
315.71/291.49	Analyzing the following TRS for decreasing loops:
315.71/291.49	
315.71/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.71/291.49	
315.71/291.49	
315.71/291.49	The TRS R consists of the following rules:
315.71/291.49	
315.71/291.49	   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
315.71/291.49	   plus(x, 0) -> x
315.71/291.49	   plus(x, s(y)) -> s(plus(x, y))
315.71/291.49	   length(nil) -> 0
315.71/291.49	   length(cons(x, y)) -> s(length(y))
315.71/291.49	   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
315.71/291.49	   ge(x, 0) -> true
315.71/291.49	   ge(0, s(x)) -> false
315.71/291.49	   ge(s(x), s(y)) -> ge(x, y)
315.71/291.49	   if(true, c, l, ys, zs) -> nil
315.71/291.49	   if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
315.71/291.49	   greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
315.71/291.49	   smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
315.71/291.49	   helpc(true, ys, zs) -> ys
315.71/291.49	   helpc(false, ys, zs) -> zs
315.71/291.49	   helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))
315.71/291.49	
315.71/291.49	S is empty.
315.71/291.49	Rewrite Strategy: FULL
315.71/291.52	EOF