7.43/2.73	WORST_CASE(Omega(n^1), O(n^1))
7.81/2.82	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
7.81/2.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.81/2.82	
7.81/2.82	
7.81/2.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.81/2.82	
7.81/2.82	(0) CpxTRS
7.81/2.82	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
7.81/2.82	(2) CpxTRS
7.81/2.82	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
7.81/2.82	    (4) CpxTRS
7.81/2.82	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 16 ms]
7.81/2.82	    (6) BOUNDS(1, n^1)
7.81/2.82	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
7.81/2.82	(8) TRS for Loop Detection
7.81/2.82	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
7.81/2.82	    (10) BEST
7.81/2.82	        (11) proven lower bound
7.81/2.82	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
7.81/2.82	            (13) BOUNDS(n^1, INF)
7.81/2.82	        (14) TRS for Loop Detection
7.81/2.82	
7.81/2.82	
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(0)
7.81/2.82	Obligation:
7.81/2.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.81/2.82	
7.81/2.82	
7.81/2.82	The TRS R consists of the following rules:
7.81/2.82	
7.81/2.82	   double(0) -> 0
7.81/2.82	   double(s(x)) -> s(s(double(x)))
7.81/2.82	   half(0) -> 0
7.81/2.82	   half(s(0)) -> 0
7.81/2.82	   half(s(s(x))) -> s(half(x))
7.81/2.82	   -(x, 0) -> x
7.81/2.82	   -(s(x), s(y)) -> -(x, y)
7.81/2.82	   if(0, y, z) -> y
7.81/2.82	   if(s(x), y, z) -> z
7.81/2.82	   half(double(x)) -> x
7.81/2.82	
7.81/2.82	S is empty.
7.81/2.82	Rewrite Strategy: FULL
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
7.81/2.82	The TRS does not nest defined symbols.
7.81/2.82	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
7.81/2.82	half(double(x)) -> x
7.81/2.82	
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(2)
7.81/2.82	Obligation:
7.81/2.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
7.81/2.82	
7.81/2.82	
7.81/2.82	The TRS R consists of the following rules:
7.81/2.82	
7.81/2.82	   double(0) -> 0
7.81/2.82	   double(s(x)) -> s(s(double(x)))
7.81/2.82	   half(0) -> 0
7.81/2.82	   half(s(0)) -> 0
7.81/2.82	   half(s(s(x))) -> s(half(x))
7.81/2.82	   -(x, 0) -> x
7.81/2.82	   -(s(x), s(y)) -> -(x, y)
7.81/2.82	   if(0, y, z) -> y
7.81/2.82	   if(s(x), y, z) -> z
7.81/2.82	
7.81/2.82	S is empty.
7.81/2.82	Rewrite Strategy: FULL
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(3) RelTrsToTrsProof (UPPER BOUND(ID))
7.81/2.82	transformed relative TRS to TRS
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(4)
7.81/2.82	Obligation:
7.81/2.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
7.81/2.82	
7.81/2.82	
7.81/2.82	The TRS R consists of the following rules:
7.81/2.82	
7.81/2.82	   double(0) -> 0
7.81/2.82	   double(s(x)) -> s(s(double(x)))
7.81/2.82	   half(0) -> 0
7.81/2.82	   half(s(0)) -> 0
7.81/2.82	   half(s(s(x))) -> s(half(x))
7.81/2.82	   -(x, 0) -> x
7.81/2.82	   -(s(x), s(y)) -> -(x, y)
7.81/2.82	   if(0, y, z) -> y
7.81/2.82	   if(s(x), y, z) -> z
7.81/2.82	
7.81/2.82	S is empty.
7.81/2.82	Rewrite Strategy: FULL
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(5) CpxTrsMatchBoundsTAProof (FINISHED)
7.81/2.82	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 
7.81/2.82	
7.81/2.82	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
7.81/2.82	final states : [1, 2, 3, 4]
7.81/2.82	transitions: 
7.81/2.82	00() -> 0
7.81/2.82	s0(0) -> 0
7.81/2.82	double0(0) -> 1
7.81/2.82	half0(0) -> 2
7.81/2.82	-0(0, 0) -> 3
7.81/2.82	if0(0, 0, 0) -> 4
7.81/2.82	01() -> 1
7.81/2.82	double1(0) -> 6
7.81/2.82	s1(6) -> 5
7.81/2.82	s1(5) -> 1
7.81/2.82	01() -> 2
7.81/2.82	half1(0) -> 7
7.81/2.82	s1(7) -> 2
7.81/2.82	-1(0, 0) -> 3
7.81/2.82	01() -> 6
7.81/2.82	s1(5) -> 6
7.81/2.82	01() -> 7
7.81/2.82	s1(7) -> 7
7.81/2.82	0 -> 3
7.81/2.82	0 -> 4
7.81/2.82	
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(6)
7.81/2.82	BOUNDS(1, n^1)
7.81/2.82	
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
7.81/2.82	Transformed a relative TRS into a decreasing-loop problem.
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(8)
7.81/2.82	Obligation:
7.81/2.82	Analyzing the following TRS for decreasing loops:
7.81/2.82	
7.81/2.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.81/2.82	
7.81/2.82	
7.81/2.82	The TRS R consists of the following rules:
7.81/2.82	
7.81/2.82	   double(0) -> 0
7.81/2.82	   double(s(x)) -> s(s(double(x)))
7.81/2.82	   half(0) -> 0
7.81/2.82	   half(s(0)) -> 0
7.81/2.82	   half(s(s(x))) -> s(half(x))
7.81/2.82	   -(x, 0) -> x
7.81/2.82	   -(s(x), s(y)) -> -(x, y)
7.81/2.82	   if(0, y, z) -> y
7.81/2.82	   if(s(x), y, z) -> z
7.81/2.82	   half(double(x)) -> x
7.81/2.82	
7.81/2.82	S is empty.
7.81/2.82	Rewrite Strategy: FULL
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(9) DecreasingLoopProof (LOWER BOUND(ID))
7.81/2.82	The following loop(s) give(s) rise to the lower bound Omega(n^1):
7.81/2.82	
7.81/2.82	The rewrite sequence
7.81/2.82	
7.81/2.82	-(s(x), s(y)) ->^+ -(x, y)
7.81/2.82	
7.81/2.82	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
7.81/2.82	
7.81/2.82	The pumping substitution is [x / s(x), y / s(y)].
7.81/2.82	
7.81/2.82	The result substitution is [ ].
7.81/2.82	
7.81/2.82	
7.81/2.82	
7.81/2.82	
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(10)
7.81/2.82	Complex Obligation (BEST)
7.81/2.82	
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(11)
7.81/2.82	Obligation:
7.81/2.82	Proved the lower bound n^1 for the following obligation:
7.81/2.82	
7.81/2.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.81/2.82	
7.81/2.82	
7.81/2.82	The TRS R consists of the following rules:
7.81/2.82	
7.81/2.82	   double(0) -> 0
7.81/2.82	   double(s(x)) -> s(s(double(x)))
7.81/2.82	   half(0) -> 0
7.81/2.82	   half(s(0)) -> 0
7.81/2.82	   half(s(s(x))) -> s(half(x))
7.81/2.82	   -(x, 0) -> x
7.81/2.82	   -(s(x), s(y)) -> -(x, y)
7.81/2.82	   if(0, y, z) -> y
7.81/2.82	   if(s(x), y, z) -> z
7.81/2.82	   half(double(x)) -> x
7.81/2.82	
7.81/2.82	S is empty.
7.81/2.82	Rewrite Strategy: FULL
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(12) LowerBoundPropagationProof (FINISHED)
7.81/2.82	Propagated lower bound.
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(13)
7.81/2.82	BOUNDS(n^1, INF)
7.81/2.82	
7.81/2.82	----------------------------------------
7.81/2.82	
7.81/2.82	(14)
7.81/2.82	Obligation:
7.81/2.82	Analyzing the following TRS for decreasing loops:
7.81/2.82	
7.81/2.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.81/2.82	
7.81/2.82	
7.81/2.82	The TRS R consists of the following rules:
7.81/2.82	
7.81/2.82	   double(0) -> 0
7.81/2.82	   double(s(x)) -> s(s(double(x)))
7.81/2.82	   half(0) -> 0
7.81/2.82	   half(s(0)) -> 0
7.81/2.82	   half(s(s(x))) -> s(half(x))
7.81/2.82	   -(x, 0) -> x
7.81/2.82	   -(s(x), s(y)) -> -(x, y)
7.81/2.82	   if(0, y, z) -> y
7.81/2.82	   if(s(x), y, z) -> z
7.81/2.82	   half(double(x)) -> x
7.81/2.82	
7.81/2.82	S is empty.
7.81/2.82	Rewrite Strategy: FULL
8.06/2.93	EOF