3.63/1.71	WORST_CASE(Omega(n^1), O(n^1))
3.63/1.72	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.63/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.63/1.72	
3.63/1.72	
3.63/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.63/1.72	
3.63/1.72	(0) CpxTRS
3.63/1.72	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.63/1.72	(2) CpxTRS
3.63/1.72	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 32 ms]
3.63/1.72	    (4) BOUNDS(1, n^1)
3.63/1.72	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.63/1.72	(6) TRS for Loop Detection
3.63/1.72	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.63/1.72	    (8) BEST
3.63/1.72	        (9) proven lower bound
3.63/1.72	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.63/1.72	            (11) BOUNDS(n^1, INF)
3.63/1.72	        (12) TRS for Loop Detection
3.63/1.72	
3.63/1.72	
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(0)
3.63/1.72	Obligation:
3.63/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.63/1.72	
3.63/1.72	
3.63/1.72	The TRS R consists of the following rules:
3.63/1.72	
3.63/1.72	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.63/1.72	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.63/1.72	
3.63/1.72	S is empty.
3.63/1.72	Rewrite Strategy: INNERMOST
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.63/1.72	transformed relative TRS to TRS
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(2)
3.63/1.72	Obligation:
3.63/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.63/1.72	
3.63/1.72	
3.63/1.72	The TRS R consists of the following rules:
3.63/1.72	
3.63/1.72	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.63/1.72	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.63/1.72	
3.63/1.72	S is empty.
3.63/1.72	Rewrite Strategy: INNERMOST
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.63/1.72	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. 
3.63/1.72	
3.63/1.72	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.63/1.72	final states : [1, 2]
3.63/1.72	transitions: 
3.63/1.72	c0(0, 0) -> 0
3.63/1.72	s0(0) -> 0
3.63/1.72	f0(0) -> 1
3.63/1.72	g0(0) -> 2
3.63/1.72	s1(0) -> 4
3.63/1.72	c1(4, 0) -> 3
3.63/1.72	f1(3) -> 1
3.63/1.72	s1(0) -> 6
3.63/1.72	c1(0, 6) -> 5
3.63/1.72	f1(5) -> 2
3.63/1.72	s1(4) -> 4
3.63/1.72	s2(0) -> 8
3.63/1.72	c2(8, 0) -> 7
3.63/1.72	f2(7) -> 2
3.63/1.72	s1(8) -> 4
3.63/1.72	f1(3) -> 2
3.63/1.72	
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(4)
3.63/1.72	BOUNDS(1, n^1)
3.63/1.72	
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.63/1.72	Transformed a relative TRS into a decreasing-loop problem.
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(6)
3.63/1.72	Obligation:
3.63/1.72	Analyzing the following TRS for decreasing loops:
3.63/1.72	
3.63/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.63/1.72	
3.63/1.72	
3.63/1.72	The TRS R consists of the following rules:
3.63/1.72	
3.63/1.72	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.63/1.72	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.63/1.72	
3.63/1.72	S is empty.
3.63/1.72	Rewrite Strategy: INNERMOST
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.63/1.72	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.63/1.72	
3.63/1.72	The rewrite sequence
3.63/1.72	
3.63/1.72	f(c(X, s(Y))) ->^+ f(c(s(X), Y))
3.63/1.72	
3.63/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.63/1.72	
3.63/1.72	The pumping substitution is [Y / s(Y)].
3.63/1.72	
3.63/1.72	The result substitution is [X / s(X)].
3.63/1.72	
3.63/1.72	
3.63/1.72	
3.63/1.72	
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(8)
3.63/1.72	Complex Obligation (BEST)
3.63/1.72	
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(9)
3.63/1.72	Obligation:
3.63/1.72	Proved the lower bound n^1 for the following obligation:
3.63/1.72	
3.63/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.63/1.72	
3.63/1.72	
3.63/1.72	The TRS R consists of the following rules:
3.63/1.72	
3.63/1.72	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.63/1.72	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.63/1.72	
3.63/1.72	S is empty.
3.63/1.72	Rewrite Strategy: INNERMOST
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(10) LowerBoundPropagationProof (FINISHED)
3.63/1.72	Propagated lower bound.
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(11)
3.63/1.72	BOUNDS(n^1, INF)
3.63/1.72	
3.63/1.72	----------------------------------------
3.63/1.72	
3.63/1.72	(12)
3.63/1.72	Obligation:
3.63/1.72	Analyzing the following TRS for decreasing loops:
3.63/1.72	
3.63/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.63/1.72	
3.63/1.72	
3.63/1.72	The TRS R consists of the following rules:
3.63/1.72	
3.63/1.72	   f(c(X, s(Y))) -> f(c(s(X), Y))
3.63/1.72	   g(c(s(X), Y)) -> f(c(X, s(Y)))
3.63/1.72	
3.63/1.72	S is empty.
3.63/1.72	Rewrite Strategy: INNERMOST
3.69/2.34	EOF