21.30/7.33	WORST_CASE(Omega(n^1), O(n^1))
21.30/7.33	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
21.30/7.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.30/7.33	
21.30/7.33	
21.30/7.33	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.30/7.33	
21.30/7.33	(0) CpxTRS
21.30/7.33	(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
21.30/7.33	(2) CpxTRS
21.30/7.33	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
21.30/7.33	    (4) CpxTRS
21.30/7.33	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
21.30/7.33	    (6) BOUNDS(1, n^1)
21.30/7.33	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
21.30/7.33	(8) TRS for Loop Detection
21.30/7.33	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
21.30/7.33	    (10) BEST
21.30/7.33	        (11) proven lower bound
21.30/7.33	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
21.30/7.33	            (13) BOUNDS(n^1, INF)
21.30/7.33	        (14) TRS for Loop Detection
21.30/7.33	
21.30/7.33	
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(0)
21.30/7.33	Obligation:
21.30/7.33	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.30/7.33	
21.30/7.33	
21.30/7.33	The TRS R consists of the following rules:
21.30/7.33	
21.30/7.33	   f(c(s(x), y)) -> f(c(x, s(y)))
21.30/7.33	   g(c(x, s(y))) -> g(c(s(x), y))
21.30/7.33	   g(s(f(x))) -> g(f(x))
21.30/7.33	
21.30/7.33	S is empty.
21.30/7.33	Rewrite Strategy: FULL
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(1) DependencyGraphProof (UPPER BOUND(ID))
21.30/7.33	The following rules are not reachable from basic terms in the dependency graph and can be removed:
21.30/7.33	
21.30/7.33	g(s(f(x))) -> g(f(x))
21.30/7.33	
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(2)
21.30/7.33	Obligation:
21.30/7.33	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
21.30/7.33	
21.30/7.33	
21.30/7.33	The TRS R consists of the following rules:
21.30/7.33	
21.30/7.33	   f(c(s(x), y)) -> f(c(x, s(y)))
21.30/7.33	   g(c(x, s(y))) -> g(c(s(x), y))
21.30/7.33	
21.30/7.33	S is empty.
21.30/7.33	Rewrite Strategy: FULL
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(3) RelTrsToTrsProof (UPPER BOUND(ID))
21.30/7.33	transformed relative TRS to TRS
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(4)
21.30/7.33	Obligation:
21.30/7.33	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
21.30/7.33	
21.30/7.33	
21.30/7.33	The TRS R consists of the following rules:
21.30/7.33	
21.30/7.33	   f(c(s(x), y)) -> f(c(x, s(y)))
21.30/7.33	   g(c(x, s(y))) -> g(c(s(x), y))
21.30/7.33	
21.30/7.33	S is empty.
21.30/7.33	Rewrite Strategy: FULL
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(5) CpxTrsMatchBoundsTAProof (FINISHED)
21.30/7.33	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. 
21.30/7.33	
21.30/7.33	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
21.30/7.33	final states : [1, 2]
21.30/7.33	transitions: 
21.30/7.33	c0(0, 0) -> 0
21.30/7.33	s0(0) -> 0
21.30/7.33	f0(0) -> 1
21.30/7.33	g0(0) -> 2
21.30/7.33	s1(0) -> 4
21.30/7.33	c1(0, 4) -> 3
21.30/7.33	f1(3) -> 1
21.30/7.33	s1(0) -> 6
21.30/7.33	c1(6, 0) -> 5
21.30/7.33	g1(5) -> 2
21.30/7.33	s1(4) -> 4
21.30/7.33	s1(6) -> 6
21.30/7.33	
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(6)
21.30/7.33	BOUNDS(1, n^1)
21.30/7.33	
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
21.30/7.33	Transformed a relative TRS into a decreasing-loop problem.
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(8)
21.30/7.33	Obligation:
21.30/7.33	Analyzing the following TRS for decreasing loops:
21.30/7.33	
21.30/7.33	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.30/7.33	
21.30/7.33	
21.30/7.33	The TRS R consists of the following rules:
21.30/7.33	
21.30/7.33	   f(c(s(x), y)) -> f(c(x, s(y)))
21.30/7.33	   g(c(x, s(y))) -> g(c(s(x), y))
21.30/7.33	   g(s(f(x))) -> g(f(x))
21.30/7.33	
21.30/7.33	S is empty.
21.30/7.33	Rewrite Strategy: FULL
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(9) DecreasingLoopProof (LOWER BOUND(ID))
21.30/7.33	The following loop(s) give(s) rise to the lower bound Omega(n^1):
21.30/7.33	
21.30/7.33	The rewrite sequence
21.30/7.33	
21.30/7.33	g(c(x, s(y))) ->^+ g(c(s(x), y))
21.30/7.33	
21.30/7.33	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
21.30/7.33	
21.30/7.33	The pumping substitution is [y / s(y)].
21.30/7.33	
21.30/7.33	The result substitution is [x / s(x)].
21.30/7.33	
21.30/7.33	
21.30/7.33	
21.30/7.33	
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(10)
21.30/7.33	Complex Obligation (BEST)
21.30/7.33	
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(11)
21.30/7.33	Obligation:
21.30/7.33	Proved the lower bound n^1 for the following obligation:
21.30/7.33	
21.30/7.33	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.30/7.33	
21.30/7.33	
21.30/7.33	The TRS R consists of the following rules:
21.30/7.33	
21.30/7.33	   f(c(s(x), y)) -> f(c(x, s(y)))
21.30/7.33	   g(c(x, s(y))) -> g(c(s(x), y))
21.30/7.33	   g(s(f(x))) -> g(f(x))
21.30/7.33	
21.30/7.33	S is empty.
21.30/7.33	Rewrite Strategy: FULL
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(12) LowerBoundPropagationProof (FINISHED)
21.30/7.33	Propagated lower bound.
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(13)
21.30/7.33	BOUNDS(n^1, INF)
21.30/7.33	
21.30/7.33	----------------------------------------
21.30/7.33	
21.30/7.33	(14)
21.30/7.33	Obligation:
21.30/7.33	Analyzing the following TRS for decreasing loops:
21.30/7.33	
21.30/7.33	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.30/7.33	
21.30/7.33	
21.30/7.33	The TRS R consists of the following rules:
21.30/7.33	
21.30/7.33	   f(c(s(x), y)) -> f(c(x, s(y)))
21.30/7.33	   g(c(x, s(y))) -> g(c(s(x), y))
21.30/7.33	   g(s(f(x))) -> g(f(x))
21.30/7.33	
21.30/7.33	S is empty.
21.30/7.33	Rewrite Strategy: FULL
21.43/11.95	EOF