316.56/291.53	WORST_CASE(Omega(n^1), ?)
316.56/291.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
316.56/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
316.56/291.53	
316.56/291.53	
316.56/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.56/291.53	
316.56/291.53	(0) CpxTRS
316.56/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
316.56/291.53	(2) TRS for Loop Detection
316.56/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
316.56/291.53	(4) BEST
316.56/291.53	    (5) proven lower bound
316.56/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
316.56/291.53	        (7) BOUNDS(n^1, INF)
316.56/291.53	    (8) TRS for Loop Detection
316.56/291.53	
316.56/291.53	
316.56/291.53	----------------------------------------
316.56/291.53	
316.56/291.53	(0)
316.56/291.53	Obligation:
316.56/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.56/291.53	
316.56/291.53	
316.56/291.53	The TRS R consists of the following rules:
316.56/291.53	
316.56/291.53	   eq(0, 0) -> true
316.56/291.53	   eq(0, s(y)) -> false
316.56/291.53	   eq(s(x), 0) -> false
316.56/291.53	   eq(s(x), s(y)) -> eq(x, y)
316.56/291.53	   lt(0, s(y)) -> true
316.56/291.53	   lt(x, 0) -> false
316.56/291.53	   lt(s(x), s(y)) -> lt(x, y)
316.56/291.53	   bin2s(nil) -> 0
316.56/291.53	   bin2s(cons(x, xs)) -> bin2ss(x, xs)
316.56/291.53	   bin2ss(x, nil) -> x
316.56/291.53	   bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs)
316.56/291.53	   bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs)
316.56/291.53	   half(0) -> 0
316.56/291.53	   half(s(0)) -> 0
316.56/291.53	   half(s(s(x))) -> s(half(x))
316.56/291.53	   log(0) -> 0
316.56/291.53	   log(s(0)) -> 0
316.56/291.53	   log(s(s(x))) -> s(log(half(s(s(x)))))
316.56/291.53	   more(nil) -> nil
316.56/291.53	   more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
316.56/291.53	   s2bin(x) -> s2bin1(x, 0, cons(nil, nil))
316.56/291.53	   s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists)
316.56/291.53	   if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists))
316.56/291.53	   if1(false, x, y, lists) -> s2bin2(x, lists)
316.56/291.53	   s2bin2(x, nil) -> bug_list_not
316.56/291.53	   s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys)
316.56/291.53	   if2(true, x, xs, ys) -> xs
316.56/291.53	   if2(false, x, xs, ys) -> s2bin2(x, ys)
316.56/291.53	
316.56/291.53	S is empty.
316.56/291.53	Rewrite Strategy: FULL
316.56/291.53	----------------------------------------
316.56/291.53	
316.56/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
316.56/291.53	Transformed a relative TRS into a decreasing-loop problem.
316.56/291.53	----------------------------------------
316.56/291.53	
316.56/291.53	(2)
316.56/291.53	Obligation:
316.56/291.53	Analyzing the following TRS for decreasing loops:
316.56/291.53	
316.56/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.56/291.54	
316.56/291.54	
316.56/291.54	The TRS R consists of the following rules:
316.56/291.54	
316.56/291.54	   eq(0, 0) -> true
316.56/291.54	   eq(0, s(y)) -> false
316.56/291.54	   eq(s(x), 0) -> false
316.56/291.54	   eq(s(x), s(y)) -> eq(x, y)
316.56/291.54	   lt(0, s(y)) -> true
316.56/291.54	   lt(x, 0) -> false
316.56/291.54	   lt(s(x), s(y)) -> lt(x, y)
316.56/291.54	   bin2s(nil) -> 0
316.56/291.54	   bin2s(cons(x, xs)) -> bin2ss(x, xs)
316.56/291.54	   bin2ss(x, nil) -> x
316.56/291.54	   bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs)
316.56/291.54	   bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs)
316.56/291.54	   half(0) -> 0
316.56/291.54	   half(s(0)) -> 0
316.56/291.54	   half(s(s(x))) -> s(half(x))
316.56/291.54	   log(0) -> 0
316.56/291.54	   log(s(0)) -> 0
316.56/291.54	   log(s(s(x))) -> s(log(half(s(s(x)))))
316.56/291.54	   more(nil) -> nil
316.56/291.54	   more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
316.56/291.54	   s2bin(x) -> s2bin1(x, 0, cons(nil, nil))
316.56/291.54	   s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists)
316.56/291.54	   if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists))
316.56/291.54	   if1(false, x, y, lists) -> s2bin2(x, lists)
316.56/291.54	   s2bin2(x, nil) -> bug_list_not
316.56/291.54	   s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys)
316.56/291.54	   if2(true, x, xs, ys) -> xs
316.56/291.54	   if2(false, x, xs, ys) -> s2bin2(x, ys)
316.56/291.54	
316.56/291.54	S is empty.
316.56/291.54	Rewrite Strategy: FULL
316.56/291.54	----------------------------------------
316.56/291.54	
316.56/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
316.56/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
316.56/291.54	
316.56/291.54	The rewrite sequence
316.56/291.54	
316.56/291.54	bin2ss(x, cons(0, xs)) ->^+ bin2ss(double(x), xs)
316.56/291.54	
316.56/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
316.56/291.54	
316.56/291.54	The pumping substitution is [xs / cons(0, xs)].
316.56/291.54	
316.56/291.54	The result substitution is [x / double(x)].
316.56/291.54	
316.56/291.54	
316.56/291.54	
316.56/291.54	
316.56/291.54	----------------------------------------
316.56/291.54	
316.56/291.54	(4)
316.56/291.54	Complex Obligation (BEST)
316.56/291.54	
316.56/291.54	----------------------------------------
316.56/291.54	
316.56/291.54	(5)
316.56/291.54	Obligation:
316.56/291.54	Proved the lower bound n^1 for the following obligation:
316.56/291.54	
316.56/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.56/291.54	
316.56/291.54	
316.56/291.54	The TRS R consists of the following rules:
316.56/291.54	
316.56/291.54	   eq(0, 0) -> true
316.56/291.54	   eq(0, s(y)) -> false
316.56/291.54	   eq(s(x), 0) -> false
316.56/291.54	   eq(s(x), s(y)) -> eq(x, y)
316.56/291.54	   lt(0, s(y)) -> true
316.56/291.54	   lt(x, 0) -> false
316.56/291.54	   lt(s(x), s(y)) -> lt(x, y)
316.56/291.54	   bin2s(nil) -> 0
316.56/291.54	   bin2s(cons(x, xs)) -> bin2ss(x, xs)
316.56/291.54	   bin2ss(x, nil) -> x
316.56/291.54	   bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs)
316.56/291.54	   bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs)
316.56/291.54	   half(0) -> 0
316.56/291.54	   half(s(0)) -> 0
316.56/291.54	   half(s(s(x))) -> s(half(x))
316.56/291.54	   log(0) -> 0
316.56/291.54	   log(s(0)) -> 0
316.56/291.54	   log(s(s(x))) -> s(log(half(s(s(x)))))
316.56/291.54	   more(nil) -> nil
316.56/291.54	   more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
316.56/291.54	   s2bin(x) -> s2bin1(x, 0, cons(nil, nil))
316.56/291.54	   s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists)
316.56/291.54	   if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists))
316.56/291.54	   if1(false, x, y, lists) -> s2bin2(x, lists)
316.56/291.54	   s2bin2(x, nil) -> bug_list_not
316.56/291.54	   s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys)
316.56/291.54	   if2(true, x, xs, ys) -> xs
316.56/291.54	   if2(false, x, xs, ys) -> s2bin2(x, ys)
316.56/291.54	
316.56/291.54	S is empty.
316.56/291.54	Rewrite Strategy: FULL
316.56/291.54	----------------------------------------
316.56/291.54	
316.56/291.54	(6) LowerBoundPropagationProof (FINISHED)
316.56/291.54	Propagated lower bound.
316.56/291.54	----------------------------------------
316.56/291.54	
316.56/291.54	(7)
316.56/291.54	BOUNDS(n^1, INF)
316.56/291.54	
316.56/291.54	----------------------------------------
316.56/291.54	
316.56/291.54	(8)
316.56/291.54	Obligation:
316.56/291.54	Analyzing the following TRS for decreasing loops:
316.56/291.54	
316.56/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.56/291.54	
316.56/291.54	
316.56/291.54	The TRS R consists of the following rules:
316.56/291.54	
316.56/291.54	   eq(0, 0) -> true
316.56/291.54	   eq(0, s(y)) -> false
316.56/291.54	   eq(s(x), 0) -> false
316.56/291.54	   eq(s(x), s(y)) -> eq(x, y)
316.56/291.54	   lt(0, s(y)) -> true
316.56/291.54	   lt(x, 0) -> false
316.56/291.54	   lt(s(x), s(y)) -> lt(x, y)
316.56/291.54	   bin2s(nil) -> 0
316.56/291.54	   bin2s(cons(x, xs)) -> bin2ss(x, xs)
316.56/291.54	   bin2ss(x, nil) -> x
316.56/291.54	   bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs)
316.56/291.54	   bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs)
316.56/291.54	   half(0) -> 0
316.56/291.54	   half(s(0)) -> 0
316.56/291.54	   half(s(s(x))) -> s(half(x))
316.56/291.54	   log(0) -> 0
316.56/291.54	   log(s(0)) -> 0
316.56/291.54	   log(s(s(x))) -> s(log(half(s(s(x)))))
316.56/291.54	   more(nil) -> nil
316.56/291.54	   more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
316.56/291.54	   s2bin(x) -> s2bin1(x, 0, cons(nil, nil))
316.56/291.54	   s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists)
316.56/291.54	   if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists))
316.56/291.54	   if1(false, x, y, lists) -> s2bin2(x, lists)
316.56/291.54	   s2bin2(x, nil) -> bug_list_not
316.56/291.54	   s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys)
316.56/291.54	   if2(true, x, xs, ys) -> xs
316.56/291.54	   if2(false, x, xs, ys) -> s2bin2(x, ys)
316.56/291.54	
316.56/291.54	S is empty.
316.56/291.54	Rewrite Strategy: FULL
316.59/291.56	EOF