305.19/291.52	WORST_CASE(Omega(n^1), ?)
305.19/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
305.19/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
305.19/291.52	
305.19/291.52	
305.19/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.19/291.52	
305.19/291.52	(0) CpxTRS
305.19/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
305.19/291.52	(2) TRS for Loop Detection
305.19/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
305.19/291.52	(4) BEST
305.19/291.52	    (5) proven lower bound
305.19/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
305.19/291.52	        (7) BOUNDS(n^1, INF)
305.19/291.52	    (8) TRS for Loop Detection
305.19/291.52	
305.19/291.52	
305.19/291.52	----------------------------------------
305.19/291.52	
305.19/291.52	(0)
305.19/291.52	Obligation:
305.19/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.19/291.52	
305.19/291.52	
305.19/291.52	The TRS R consists of the following rules:
305.19/291.52	
305.19/291.52	   length(nil) -> 0
305.19/291.52	   length(cons(x, l)) -> s(length(l))
305.19/291.52	   lt(x, 0) -> false
305.19/291.52	   lt(0, s(y)) -> true
305.19/291.52	   lt(s(x), s(y)) -> lt(x, y)
305.19/291.52	   head(cons(x, l)) -> x
305.19/291.52	   head(nil) -> undefined
305.19/291.52	   tail(nil) -> nil
305.19/291.52	   tail(cons(x, l)) -> l
305.19/291.52	   reverse(l) -> rev(0, l, nil, l)
305.19/291.52	   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
305.19/291.53	   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
305.19/291.53	   if(false, x, l, accu, orig) -> accu
305.19/291.53	
305.19/291.53	S is empty.
305.19/291.53	Rewrite Strategy: FULL
305.19/291.53	----------------------------------------
305.19/291.53	
305.19/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
305.19/291.53	Transformed a relative TRS into a decreasing-loop problem.
305.19/291.53	----------------------------------------
305.19/291.53	
305.19/291.53	(2)
305.19/291.53	Obligation:
305.19/291.53	Analyzing the following TRS for decreasing loops:
305.19/291.53	
305.19/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.19/291.53	
305.19/291.53	
305.19/291.53	The TRS R consists of the following rules:
305.19/291.53	
305.19/291.53	   length(nil) -> 0
305.19/291.53	   length(cons(x, l)) -> s(length(l))
305.19/291.53	   lt(x, 0) -> false
305.19/291.53	   lt(0, s(y)) -> true
305.19/291.53	   lt(s(x), s(y)) -> lt(x, y)
305.19/291.53	   head(cons(x, l)) -> x
305.19/291.53	   head(nil) -> undefined
305.19/291.53	   tail(nil) -> nil
305.19/291.53	   tail(cons(x, l)) -> l
305.19/291.53	   reverse(l) -> rev(0, l, nil, l)
305.19/291.53	   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
305.19/291.53	   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
305.19/291.53	   if(false, x, l, accu, orig) -> accu
305.19/291.53	
305.19/291.53	S is empty.
305.19/291.53	Rewrite Strategy: FULL
305.19/291.53	----------------------------------------
305.19/291.53	
305.19/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
305.19/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
305.19/291.53	
305.19/291.53	The rewrite sequence
305.19/291.53	
305.19/291.53	lt(s(x), s(y)) ->^+ lt(x, y)
305.19/291.53	
305.19/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
305.19/291.53	
305.19/291.53	The pumping substitution is [x / s(x), y / s(y)].
305.19/291.53	
305.19/291.53	The result substitution is [ ].
305.19/291.53	
305.19/291.53	
305.19/291.53	
305.19/291.53	
305.19/291.53	----------------------------------------
305.19/291.53	
305.19/291.53	(4)
305.19/291.53	Complex Obligation (BEST)
305.19/291.53	
305.19/291.53	----------------------------------------
305.19/291.53	
305.19/291.53	(5)
305.19/291.53	Obligation:
305.19/291.53	Proved the lower bound n^1 for the following obligation:
305.19/291.53	
305.19/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.19/291.53	
305.19/291.53	
305.19/291.53	The TRS R consists of the following rules:
305.19/291.53	
305.19/291.53	   length(nil) -> 0
305.19/291.53	   length(cons(x, l)) -> s(length(l))
305.19/291.53	   lt(x, 0) -> false
305.19/291.53	   lt(0, s(y)) -> true
305.19/291.53	   lt(s(x), s(y)) -> lt(x, y)
305.19/291.53	   head(cons(x, l)) -> x
305.19/291.53	   head(nil) -> undefined
305.19/291.53	   tail(nil) -> nil
305.19/291.53	   tail(cons(x, l)) -> l
305.19/291.53	   reverse(l) -> rev(0, l, nil, l)
305.19/291.53	   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
305.19/291.53	   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
305.19/291.53	   if(false, x, l, accu, orig) -> accu
305.19/291.53	
305.19/291.53	S is empty.
305.19/291.53	Rewrite Strategy: FULL
305.19/291.53	----------------------------------------
305.19/291.53	
305.19/291.53	(6) LowerBoundPropagationProof (FINISHED)
305.19/291.53	Propagated lower bound.
305.19/291.53	----------------------------------------
305.19/291.53	
305.19/291.53	(7)
305.19/291.53	BOUNDS(n^1, INF)
305.19/291.53	
305.19/291.53	----------------------------------------
305.19/291.53	
305.19/291.53	(8)
305.19/291.53	Obligation:
305.19/291.53	Analyzing the following TRS for decreasing loops:
305.19/291.53	
305.19/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.19/291.53	
305.19/291.53	
305.19/291.53	The TRS R consists of the following rules:
305.19/291.53	
305.19/291.53	   length(nil) -> 0
305.19/291.53	   length(cons(x, l)) -> s(length(l))
305.19/291.53	   lt(x, 0) -> false
305.19/291.53	   lt(0, s(y)) -> true
305.19/291.53	   lt(s(x), s(y)) -> lt(x, y)
305.19/291.53	   head(cons(x, l)) -> x
305.19/291.53	   head(nil) -> undefined
305.19/291.53	   tail(nil) -> nil
305.19/291.53	   tail(cons(x, l)) -> l
305.19/291.53	   reverse(l) -> rev(0, l, nil, l)
305.19/291.53	   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
305.19/291.53	   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
305.19/291.53	   if(false, x, l, accu, orig) -> accu
305.19/291.53	
305.19/291.53	S is empty.
305.19/291.53	Rewrite Strategy: FULL
305.19/291.55	EOF