308.71/291.55	WORST_CASE(Omega(n^1), ?)
308.71/291.56	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
308.71/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
308.71/291.56	
308.71/291.56	
308.71/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.71/291.56	
308.71/291.56	(0) CpxTRS
308.71/291.56	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
308.71/291.56	(2) TRS for Loop Detection
308.71/291.56	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
308.71/291.56	(4) BEST
308.71/291.56	    (5) proven lower bound
308.71/291.56	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
308.71/291.56	        (7) BOUNDS(n^1, INF)
308.71/291.56	    (8) TRS for Loop Detection
308.71/291.56	
308.71/291.56	
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(0)
308.71/291.56	Obligation:
308.71/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.71/291.56	
308.71/291.56	
308.71/291.56	The TRS R consists of the following rules:
308.71/291.56	
308.71/291.56	   le(0, y) -> true
308.71/291.56	   le(s(x), 0) -> false
308.71/291.56	   le(s(x), s(y)) -> le(x, y)
308.71/291.56	   inc(0) -> 0
308.71/291.56	   inc(s(x)) -> s(inc(x))
308.71/291.56	   minus(0, y) -> 0
308.71/291.56	   minus(x, 0) -> x
308.71/291.56	   minus(s(x), s(y)) -> minus(x, y)
308.71/291.56	   quot(0, s(y)) -> 0
308.71/291.56	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
308.71/291.56	   log(x) -> log2(x, 0)
308.71/291.56	   log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y))
308.71/291.56	   if(true, b, x, y) -> log_undefined
308.71/291.56	   if(false, b, x, y) -> if2(b, x, y)
308.71/291.56	   if2(true, x, s(y)) -> y
308.71/291.56	   if2(false, x, y) -> log2(quot(x, s(s(0))), y)
308.71/291.56	
308.71/291.56	S is empty.
308.71/291.56	Rewrite Strategy: FULL
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
308.71/291.56	Transformed a relative TRS into a decreasing-loop problem.
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(2)
308.71/291.56	Obligation:
308.71/291.56	Analyzing the following TRS for decreasing loops:
308.71/291.56	
308.71/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.71/291.56	
308.71/291.56	
308.71/291.56	The TRS R consists of the following rules:
308.71/291.56	
308.71/291.56	   le(0, y) -> true
308.71/291.56	   le(s(x), 0) -> false
308.71/291.56	   le(s(x), s(y)) -> le(x, y)
308.71/291.56	   inc(0) -> 0
308.71/291.56	   inc(s(x)) -> s(inc(x))
308.71/291.56	   minus(0, y) -> 0
308.71/291.56	   minus(x, 0) -> x
308.71/291.56	   minus(s(x), s(y)) -> minus(x, y)
308.71/291.56	   quot(0, s(y)) -> 0
308.71/291.56	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
308.71/291.56	   log(x) -> log2(x, 0)
308.71/291.56	   log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y))
308.71/291.56	   if(true, b, x, y) -> log_undefined
308.71/291.56	   if(false, b, x, y) -> if2(b, x, y)
308.71/291.56	   if2(true, x, s(y)) -> y
308.71/291.56	   if2(false, x, y) -> log2(quot(x, s(s(0))), y)
308.71/291.56	
308.71/291.56	S is empty.
308.71/291.56	Rewrite Strategy: FULL
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(3) DecreasingLoopProof (LOWER BOUND(ID))
308.71/291.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
308.71/291.56	
308.71/291.56	The rewrite sequence
308.71/291.56	
308.71/291.56	le(s(x), s(y)) ->^+ le(x, y)
308.71/291.56	
308.71/291.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
308.71/291.56	
308.71/291.56	The pumping substitution is [x / s(x), y / s(y)].
308.71/291.56	
308.71/291.56	The result substitution is [ ].
308.71/291.56	
308.71/291.56	
308.71/291.56	
308.71/291.56	
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(4)
308.71/291.56	Complex Obligation (BEST)
308.71/291.56	
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(5)
308.71/291.56	Obligation:
308.71/291.56	Proved the lower bound n^1 for the following obligation:
308.71/291.56	
308.71/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.71/291.56	
308.71/291.56	
308.71/291.56	The TRS R consists of the following rules:
308.71/291.56	
308.71/291.56	   le(0, y) -> true
308.71/291.56	   le(s(x), 0) -> false
308.71/291.56	   le(s(x), s(y)) -> le(x, y)
308.71/291.56	   inc(0) -> 0
308.71/291.56	   inc(s(x)) -> s(inc(x))
308.71/291.56	   minus(0, y) -> 0
308.71/291.56	   minus(x, 0) -> x
308.71/291.56	   minus(s(x), s(y)) -> minus(x, y)
308.71/291.56	   quot(0, s(y)) -> 0
308.71/291.56	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
308.71/291.56	   log(x) -> log2(x, 0)
308.71/291.56	   log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y))
308.71/291.56	   if(true, b, x, y) -> log_undefined
308.71/291.56	   if(false, b, x, y) -> if2(b, x, y)
308.71/291.56	   if2(true, x, s(y)) -> y
308.71/291.56	   if2(false, x, y) -> log2(quot(x, s(s(0))), y)
308.71/291.56	
308.71/291.56	S is empty.
308.71/291.56	Rewrite Strategy: FULL
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(6) LowerBoundPropagationProof (FINISHED)
308.71/291.56	Propagated lower bound.
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(7)
308.71/291.56	BOUNDS(n^1, INF)
308.71/291.56	
308.71/291.56	----------------------------------------
308.71/291.56	
308.71/291.56	(8)
308.71/291.56	Obligation:
308.71/291.56	Analyzing the following TRS for decreasing loops:
308.71/291.56	
308.71/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.71/291.56	
308.71/291.56	
308.71/291.56	The TRS R consists of the following rules:
308.71/291.56	
308.71/291.56	   le(0, y) -> true
308.71/291.56	   le(s(x), 0) -> false
308.71/291.56	   le(s(x), s(y)) -> le(x, y)
308.71/291.56	   inc(0) -> 0
308.71/291.56	   inc(s(x)) -> s(inc(x))
308.71/291.56	   minus(0, y) -> 0
308.71/291.56	   minus(x, 0) -> x
308.71/291.56	   minus(s(x), s(y)) -> minus(x, y)
308.71/291.56	   quot(0, s(y)) -> 0
308.71/291.56	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
308.71/291.56	   log(x) -> log2(x, 0)
308.71/291.56	   log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y))
308.71/291.56	   if(true, b, x, y) -> log_undefined
308.71/291.56	   if(false, b, x, y) -> if2(b, x, y)
308.71/291.56	   if2(true, x, s(y)) -> y
308.71/291.56	   if2(false, x, y) -> log2(quot(x, s(s(0))), y)
308.71/291.56	
308.71/291.56	S is empty.
308.71/291.56	Rewrite Strategy: FULL
308.80/291.60	EOF