3.33/1.64	WORST_CASE(NON_POLY, ?)
3.59/1.65	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.59/1.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.59/1.65	
3.59/1.65	
3.59/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.65	
3.59/1.65	(0) CpxTRS
3.59/1.65	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.59/1.65	(2) TRS for Loop Detection
3.59/1.65	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.59/1.65	(4) BEST
3.59/1.65	    (5) proven lower bound
3.59/1.65	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.59/1.65	        (7) BOUNDS(n^1, INF)
3.59/1.65	    (8) TRS for Loop Detection
3.59/1.65	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.59/1.65	        (10) BOUNDS(EXP, INF)
3.59/1.65	
3.59/1.65	
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(0)
3.59/1.65	Obligation:
3.59/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.65	
3.59/1.65	
3.59/1.65	The TRS R consists of the following rules:
3.59/1.65	
3.59/1.65	   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
3.59/1.65	   @(Nil, ys) -> ys
3.59/1.65	   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs')))
3.59/1.65	   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil)
3.59/1.65	   binom(Nil, k) -> Cons(Nil, Nil)
3.59/1.65	   goal(x, y) -> binom(x, y)
3.59/1.65	
3.59/1.65	S is empty.
3.59/1.65	Rewrite Strategy: INNERMOST
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.59/1.65	Transformed a relative TRS into a decreasing-loop problem.
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(2)
3.59/1.65	Obligation:
3.59/1.65	Analyzing the following TRS for decreasing loops:
3.59/1.65	
3.59/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.65	
3.59/1.65	
3.59/1.65	The TRS R consists of the following rules:
3.59/1.65	
3.59/1.65	   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
3.59/1.65	   @(Nil, ys) -> ys
3.59/1.65	   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs')))
3.59/1.65	   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil)
3.59/1.65	   binom(Nil, k) -> Cons(Nil, Nil)
3.59/1.65	   goal(x, y) -> binom(x, y)
3.59/1.65	
3.59/1.65	S is empty.
3.59/1.65	Rewrite Strategy: INNERMOST
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.59/1.65	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.59/1.65	
3.59/1.65	The rewrite sequence
3.59/1.65	
3.59/1.65	@(Cons(x, xs), ys) ->^+ Cons(x, @(xs, ys))
3.59/1.65	
3.59/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.59/1.65	
3.59/1.65	The pumping substitution is [xs / Cons(x, xs)].
3.59/1.65	
3.59/1.65	The result substitution is [ ].
3.59/1.65	
3.59/1.65	
3.59/1.65	
3.59/1.65	
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(4)
3.59/1.65	Complex Obligation (BEST)
3.59/1.65	
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(5)
3.59/1.65	Obligation:
3.59/1.65	Proved the lower bound n^1 for the following obligation:
3.59/1.65	
3.59/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.65	
3.59/1.65	
3.59/1.65	The TRS R consists of the following rules:
3.59/1.65	
3.59/1.65	   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
3.59/1.65	   @(Nil, ys) -> ys
3.59/1.65	   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs')))
3.59/1.65	   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil)
3.59/1.65	   binom(Nil, k) -> Cons(Nil, Nil)
3.59/1.65	   goal(x, y) -> binom(x, y)
3.59/1.65	
3.59/1.65	S is empty.
3.59/1.65	Rewrite Strategy: INNERMOST
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(6) LowerBoundPropagationProof (FINISHED)
3.59/1.65	Propagated lower bound.
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(7)
3.59/1.65	BOUNDS(n^1, INF)
3.59/1.65	
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(8)
3.59/1.65	Obligation:
3.59/1.65	Analyzing the following TRS for decreasing loops:
3.59/1.65	
3.59/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.65	
3.59/1.65	
3.59/1.65	The TRS R consists of the following rules:
3.59/1.65	
3.59/1.65	   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
3.59/1.65	   @(Nil, ys) -> ys
3.59/1.65	   binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs')))
3.59/1.65	   binom(Cons(x, xs), Nil) -> Cons(Nil, Nil)
3.59/1.65	   binom(Nil, k) -> Cons(Nil, Nil)
3.59/1.65	   goal(x, y) -> binom(x, y)
3.59/1.65	
3.59/1.65	S is empty.
3.59/1.65	Rewrite Strategy: INNERMOST
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(9) DecreasingLoopProof (FINISHED)
3.59/1.65	The following loop(s) give(s) rise to the lower bound EXP:
3.59/1.65	
3.59/1.65	The rewrite sequence
3.59/1.65	
3.59/1.65	binom(Cons(x, xs), Cons(x', xs')) ->^+ @(binom(xs, xs'), binom(xs, Cons(x', xs')))
3.59/1.65	
3.59/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.59/1.65	
3.59/1.65	The pumping substitution is [xs / Cons(x, xs), xs' / Cons(x', xs')].
3.59/1.65	
3.59/1.65	The result substitution is [ ].
3.59/1.65	
3.59/1.65	
3.59/1.65	
3.59/1.65	The rewrite sequence
3.59/1.65	
3.59/1.65	binom(Cons(x, xs), Cons(x', xs')) ->^+ @(binom(xs, xs'), binom(xs, Cons(x', xs')))
3.59/1.65	
3.59/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.59/1.65	
3.59/1.65	The pumping substitution is [xs / Cons(x, xs)].
3.59/1.65	
3.59/1.65	The result substitution is [ ].
3.59/1.65	
3.59/1.65	
3.59/1.65	
3.59/1.65	
3.59/1.65	----------------------------------------
3.59/1.65	
3.59/1.65	(10)
3.59/1.65	BOUNDS(EXP, INF)
3.63/1.69	EOF