3.24/1.66	WORST_CASE(Omega(n^1), O(n^1))
3.24/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.24/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.24/1.67	
3.24/1.67	
3.24/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.24/1.67	
3.24/1.67	(0) CpxTRS
3.24/1.67	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.24/1.67	(2) CpxTRS
3.24/1.67	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 34 ms]
3.24/1.67	    (4) BOUNDS(1, n^1)
3.24/1.67	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.24/1.67	(6) TRS for Loop Detection
3.24/1.67	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.24/1.67	    (8) BEST
3.24/1.67	        (9) proven lower bound
3.24/1.67	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.24/1.67	            (11) BOUNDS(n^1, INF)
3.24/1.67	        (12) TRS for Loop Detection
3.24/1.67	
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(0)
3.24/1.67	Obligation:
3.24/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.24/1.67	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: FULL
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.24/1.67	transformed relative TRS to TRS
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(2)
3.24/1.67	Obligation:
3.24/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.24/1.67	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: FULL
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.24/1.67	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 2. 
3.24/1.67	
3.24/1.67	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.24/1.67	final states : [1, 2]
3.24/1.67	transitions: 
3.24/1.67	c0(0, 0) -> 0
3.24/1.67	s0(0) -> 0
3.24/1.67	f0(0) -> 1
3.24/1.67	g0(0) -> 2
3.24/1.67	s1(0) -> 4
3.24/1.67	c1(4, 0) -> 3
3.24/1.67	f1(3) -> 1
3.24/1.67	s1(0) -> 6
3.24/1.67	c1(0, 6) -> 5
3.24/1.67	f1(5) -> 2
3.24/1.67	s1(4) -> 4
3.24/1.67	s2(0) -> 8
3.24/1.67	c2(8, 0) -> 7
3.24/1.67	f2(7) -> 2
3.24/1.67	s1(8) -> 4
3.24/1.67	f1(3) -> 2
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(4)
3.24/1.67	BOUNDS(1, n^1)
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.24/1.67	Transformed a relative TRS into a decreasing-loop problem.
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(6)
3.24/1.67	Obligation:
3.24/1.67	Analyzing the following TRS for decreasing loops:
3.24/1.67	
3.24/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.24/1.67	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: FULL
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.24/1.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.24/1.67	
3.24/1.67	The rewrite sequence
3.24/1.67	
3.24/1.67	f(c(X, s(Y))) ->^+ f(c(s(X), Y))
3.24/1.67	
3.24/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.24/1.67	
3.24/1.67	The pumping substitution is [Y / s(Y)].
3.24/1.67	
3.24/1.67	The result substitution is [X / s(X)].
3.24/1.67	
3.24/1.67	
3.24/1.67	
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(8)
3.24/1.67	Complex Obligation (BEST)
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(9)
3.24/1.67	Obligation:
3.24/1.67	Proved the lower bound n^1 for the following obligation:
3.24/1.67	
3.24/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.24/1.67	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: FULL
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(10) LowerBoundPropagationProof (FINISHED)
3.24/1.67	Propagated lower bound.
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(11)
3.24/1.67	BOUNDS(n^1, INF)
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(12)
3.24/1.67	Obligation:
3.24/1.67	Analyzing the following TRS for decreasing loops:
3.24/1.67	
3.24/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.24/1.67	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: FULL
3.46/1.70	EOF