301.23/291.51	WORST_CASE(Omega(n^1), ?)
301.23/291.56	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
301.23/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
301.23/291.56	
301.23/291.56	
301.23/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.23/291.56	
301.23/291.56	(0) CpxTRS
301.23/291.56	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
301.23/291.56	(2) TRS for Loop Detection
301.23/291.56	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
301.23/291.56	(4) BEST
301.23/291.56	    (5) proven lower bound
301.23/291.56	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
301.23/291.56	        (7) BOUNDS(n^1, INF)
301.23/291.56	    (8) TRS for Loop Detection
301.23/291.56	
301.23/291.56	
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(0)
301.23/291.56	Obligation:
301.23/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.23/291.56	
301.23/291.56	
301.23/291.56	The TRS R consists of the following rules:
301.23/291.56	
301.23/291.56	   minus(0, y) -> 0
301.23/291.56	   minus(s(x), y) -> if(gt(s(x), y), x, y)
301.23/291.56	   if(true, x, y) -> s(minus(x, y))
301.23/291.56	   if(false, x, y) -> 0
301.23/291.56	   mod(x, 0) -> 0
301.23/291.56	   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
301.23/291.56	   if1(true, x, y) -> x
301.23/291.56	   if1(false, x, y) -> mod(minus(x, y), y)
301.23/291.56	   gt(0, y) -> false
301.23/291.56	   gt(s(x), 0) -> true
301.23/291.56	   gt(s(x), s(y)) -> gt(x, y)
301.23/291.56	   lt(x, 0) -> false
301.23/291.56	   lt(0, s(x)) -> true
301.23/291.56	   lt(s(x), s(y)) -> lt(x, y)
301.23/291.56	
301.23/291.56	S is empty.
301.23/291.56	Rewrite Strategy: FULL
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
301.23/291.56	Transformed a relative TRS into a decreasing-loop problem.
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(2)
301.23/291.56	Obligation:
301.23/291.56	Analyzing the following TRS for decreasing loops:
301.23/291.56	
301.23/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.23/291.56	
301.23/291.56	
301.23/291.56	The TRS R consists of the following rules:
301.23/291.56	
301.23/291.56	   minus(0, y) -> 0
301.23/291.56	   minus(s(x), y) -> if(gt(s(x), y), x, y)
301.23/291.56	   if(true, x, y) -> s(minus(x, y))
301.23/291.56	   if(false, x, y) -> 0
301.23/291.56	   mod(x, 0) -> 0
301.23/291.56	   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
301.23/291.56	   if1(true, x, y) -> x
301.23/291.56	   if1(false, x, y) -> mod(minus(x, y), y)
301.23/291.56	   gt(0, y) -> false
301.23/291.56	   gt(s(x), 0) -> true
301.23/291.56	   gt(s(x), s(y)) -> gt(x, y)
301.23/291.56	   lt(x, 0) -> false
301.23/291.56	   lt(0, s(x)) -> true
301.23/291.56	   lt(s(x), s(y)) -> lt(x, y)
301.23/291.56	
301.23/291.56	S is empty.
301.23/291.56	Rewrite Strategy: FULL
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(3) DecreasingLoopProof (LOWER BOUND(ID))
301.23/291.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
301.23/291.56	
301.23/291.56	The rewrite sequence
301.23/291.56	
301.23/291.56	lt(s(x), s(y)) ->^+ lt(x, y)
301.23/291.56	
301.23/291.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
301.23/291.56	
301.23/291.56	The pumping substitution is [x / s(x), y / s(y)].
301.23/291.56	
301.23/291.56	The result substitution is [ ].
301.23/291.56	
301.23/291.56	
301.23/291.56	
301.23/291.56	
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(4)
301.23/291.56	Complex Obligation (BEST)
301.23/291.56	
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(5)
301.23/291.56	Obligation:
301.23/291.56	Proved the lower bound n^1 for the following obligation:
301.23/291.56	
301.23/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.23/291.56	
301.23/291.56	
301.23/291.56	The TRS R consists of the following rules:
301.23/291.56	
301.23/291.56	   minus(0, y) -> 0
301.23/291.56	   minus(s(x), y) -> if(gt(s(x), y), x, y)
301.23/291.56	   if(true, x, y) -> s(minus(x, y))
301.23/291.56	   if(false, x, y) -> 0
301.23/291.56	   mod(x, 0) -> 0
301.23/291.56	   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
301.23/291.56	   if1(true, x, y) -> x
301.23/291.56	   if1(false, x, y) -> mod(minus(x, y), y)
301.23/291.56	   gt(0, y) -> false
301.23/291.56	   gt(s(x), 0) -> true
301.23/291.56	   gt(s(x), s(y)) -> gt(x, y)
301.23/291.56	   lt(x, 0) -> false
301.23/291.56	   lt(0, s(x)) -> true
301.23/291.56	   lt(s(x), s(y)) -> lt(x, y)
301.23/291.56	
301.23/291.56	S is empty.
301.23/291.56	Rewrite Strategy: FULL
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(6) LowerBoundPropagationProof (FINISHED)
301.23/291.56	Propagated lower bound.
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(7)
301.23/291.56	BOUNDS(n^1, INF)
301.23/291.56	
301.23/291.56	----------------------------------------
301.23/291.56	
301.23/291.56	(8)
301.23/291.56	Obligation:
301.23/291.56	Analyzing the following TRS for decreasing loops:
301.23/291.56	
301.23/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.23/291.56	
301.23/291.56	
301.23/291.56	The TRS R consists of the following rules:
301.23/291.56	
301.23/291.56	   minus(0, y) -> 0
301.23/291.56	   minus(s(x), y) -> if(gt(s(x), y), x, y)
301.23/291.56	   if(true, x, y) -> s(minus(x, y))
301.23/291.56	   if(false, x, y) -> 0
301.23/291.56	   mod(x, 0) -> 0
301.23/291.56	   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
301.23/291.56	   if1(true, x, y) -> x
301.23/291.56	   if1(false, x, y) -> mod(minus(x, y), y)
301.23/291.56	   gt(0, y) -> false
301.23/291.56	   gt(s(x), 0) -> true
301.23/291.56	   gt(s(x), s(y)) -> gt(x, y)
301.23/291.56	   lt(x, 0) -> false
301.23/291.56	   lt(0, s(x)) -> true
301.23/291.56	   lt(s(x), s(y)) -> lt(x, y)
301.23/291.56	
301.23/291.56	S is empty.
301.23/291.56	Rewrite Strategy: FULL
301.36/291.58	EOF