4.03/1.94	WORST_CASE(Omega(n^1), O(n^1))
4.03/1.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.03/1.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.03/1.96	
4.03/1.96	
4.03/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.96	
4.03/1.96	(0) CpxTRS
4.03/1.96	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
4.03/1.96	(2) CpxTRS
4.03/1.96	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 154 ms]
4.03/1.96	    (4) BOUNDS(1, n^1)
4.03/1.96	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.03/1.96	(6) TRS for Loop Detection
4.03/1.96	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.03/1.96	    (8) BEST
4.03/1.96	        (9) proven lower bound
4.03/1.96	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
4.03/1.96	            (11) BOUNDS(n^1, INF)
4.03/1.96	        (12) TRS for Loop Detection
4.03/1.96	
4.03/1.96	
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(0)
4.03/1.96	Obligation:
4.03/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.96	
4.03/1.96	
4.03/1.96	The TRS R consists of the following rules:
4.03/1.96	
4.03/1.96	   f(0) -> cons(0, n__f(n__s(n__0)))
4.03/1.96	   f(s(0)) -> f(p(s(0)))
4.03/1.96	   p(s(X)) -> X
4.03/1.96	   f(X) -> n__f(X)
4.03/1.96	   s(X) -> n__s(X)
4.03/1.96	   0 -> n__0
4.03/1.96	   activate(n__f(X)) -> f(activate(X))
4.03/1.96	   activate(n__s(X)) -> s(activate(X))
4.03/1.96	   activate(n__0) -> 0
4.03/1.96	   activate(X) -> X
4.03/1.96	
4.03/1.96	S is empty.
4.03/1.96	Rewrite Strategy: FULL
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(1) RelTrsToTrsProof (UPPER BOUND(ID))
4.03/1.96	transformed relative TRS to TRS
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(2)
4.03/1.96	Obligation:
4.03/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
4.03/1.96	
4.03/1.96	
4.03/1.96	The TRS R consists of the following rules:
4.03/1.96	
4.03/1.96	   f(0) -> cons(0, n__f(n__s(n__0)))
4.03/1.96	   f(s(0)) -> f(p(s(0)))
4.03/1.96	   p(s(X)) -> X
4.03/1.96	   f(X) -> n__f(X)
4.03/1.96	   s(X) -> n__s(X)
4.03/1.96	   0 -> n__0
4.03/1.96	   activate(n__f(X)) -> f(activate(X))
4.03/1.96	   activate(n__s(X)) -> s(activate(X))
4.03/1.96	   activate(n__0) -> 0
4.03/1.96	   activate(X) -> X
4.03/1.96	
4.03/1.96	S is empty.
4.03/1.96	Rewrite Strategy: FULL
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(3) CpxTrsMatchBoundsTAProof (FINISHED)
4.03/1.96	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 4. 
4.03/1.96	
4.03/1.96	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
4.03/1.96	final states : [1, 2, 3, 4, 5]
4.03/1.96	transitions: 
4.03/1.96	cons0(0, 0) -> 0
4.03/1.96	n__f0(0) -> 0
4.03/1.96	n__s0(0) -> 0
4.03/1.96	n__00() -> 0
4.03/1.96	f0(0) -> 1
4.03/1.96	p0(0) -> 2
4.03/1.96	s0(0) -> 3
4.03/1.96	00() -> 4
4.03/1.96	activate0(0) -> 5
4.03/1.96	n__f1(0) -> 1
4.03/1.96	n__s1(0) -> 3
4.03/1.96	n__01() -> 4
4.03/1.96	activate1(0) -> 6
4.03/1.96	f1(6) -> 5
4.03/1.96	activate1(0) -> 7
4.03/1.96	s1(7) -> 5
4.03/1.96	01() -> 5
4.03/1.96	n__f2(6) -> 5
4.03/1.96	n__s2(7) -> 5
4.03/1.96	n__02() -> 5
4.03/1.96	f1(6) -> 6
4.03/1.96	f1(6) -> 7
4.03/1.96	s1(7) -> 6
4.03/1.96	s1(7) -> 7
4.03/1.96	01() -> 6
4.03/1.96	01() -> 7
4.03/1.96	n__f2(6) -> 6
4.03/1.96	n__f2(6) -> 7
4.03/1.96	n__s2(7) -> 6
4.03/1.96	n__s2(7) -> 7
4.03/1.96	n__02() -> 6
4.03/1.96	n__02() -> 7
4.03/1.96	02() -> 8
4.03/1.96	n__02() -> 11
4.03/1.96	n__s2(11) -> 10
4.03/1.96	n__f2(10) -> 9
4.03/1.96	cons2(8, 9) -> 5
4.03/1.96	cons2(8, 9) -> 6
4.03/1.96	cons2(8, 9) -> 7
4.03/1.96	02() -> 14
4.03/1.96	s2(14) -> 13
4.03/1.96	p2(13) -> 12
4.03/1.96	f2(12) -> 5
4.03/1.96	f2(12) -> 6
4.03/1.96	f2(12) -> 7
4.03/1.96	n__f3(12) -> 5
4.03/1.96	n__f3(12) -> 6
4.03/1.96	n__f3(12) -> 7
4.03/1.96	n__s3(14) -> 13
4.03/1.96	n__03() -> 8
4.03/1.96	n__03() -> 14
4.03/1.96	03() -> 15
4.03/1.96	n__03() -> 18
4.03/1.96	n__s3(18) -> 17
4.03/1.96	n__f3(17) -> 16
4.03/1.96	cons3(15, 16) -> 5
4.03/1.96	cons3(15, 16) -> 6
4.03/1.96	cons3(15, 16) -> 7
4.03/1.96	n__04() -> 15
4.03/1.96	0 -> 5
4.03/1.96	0 -> 6
4.03/1.96	0 -> 7
4.03/1.96	14 -> 12
4.03/1.96	
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(4)
4.03/1.96	BOUNDS(1, n^1)
4.03/1.96	
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.03/1.96	Transformed a relative TRS into a decreasing-loop problem.
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(6)
4.03/1.96	Obligation:
4.03/1.96	Analyzing the following TRS for decreasing loops:
4.03/1.96	
4.03/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.96	
4.03/1.96	
4.03/1.96	The TRS R consists of the following rules:
4.03/1.96	
4.03/1.96	   f(0) -> cons(0, n__f(n__s(n__0)))
4.03/1.96	   f(s(0)) -> f(p(s(0)))
4.03/1.96	   p(s(X)) -> X
4.03/1.96	   f(X) -> n__f(X)
4.03/1.96	   s(X) -> n__s(X)
4.03/1.96	   0 -> n__0
4.03/1.96	   activate(n__f(X)) -> f(activate(X))
4.03/1.96	   activate(n__s(X)) -> s(activate(X))
4.03/1.96	   activate(n__0) -> 0
4.03/1.96	   activate(X) -> X
4.03/1.96	
4.03/1.96	S is empty.
4.03/1.96	Rewrite Strategy: FULL
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(7) DecreasingLoopProof (LOWER BOUND(ID))
4.03/1.96	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.03/1.96	
4.03/1.96	The rewrite sequence
4.03/1.96	
4.03/1.96	activate(n__f(X)) ->^+ f(activate(X))
4.03/1.96	
4.03/1.96	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.03/1.96	
4.03/1.96	The pumping substitution is [X / n__f(X)].
4.03/1.96	
4.03/1.96	The result substitution is [ ].
4.03/1.96	
4.03/1.96	
4.03/1.96	
4.03/1.96	
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(8)
4.03/1.96	Complex Obligation (BEST)
4.03/1.96	
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(9)
4.03/1.96	Obligation:
4.03/1.96	Proved the lower bound n^1 for the following obligation:
4.03/1.96	
4.03/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.96	
4.03/1.96	
4.03/1.96	The TRS R consists of the following rules:
4.03/1.96	
4.03/1.96	   f(0) -> cons(0, n__f(n__s(n__0)))
4.03/1.96	   f(s(0)) -> f(p(s(0)))
4.03/1.96	   p(s(X)) -> X
4.03/1.96	   f(X) -> n__f(X)
4.03/1.96	   s(X) -> n__s(X)
4.03/1.96	   0 -> n__0
4.03/1.96	   activate(n__f(X)) -> f(activate(X))
4.03/1.96	   activate(n__s(X)) -> s(activate(X))
4.03/1.96	   activate(n__0) -> 0
4.03/1.96	   activate(X) -> X
4.03/1.96	
4.03/1.96	S is empty.
4.03/1.96	Rewrite Strategy: FULL
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(10) LowerBoundPropagationProof (FINISHED)
4.03/1.96	Propagated lower bound.
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(11)
4.03/1.96	BOUNDS(n^1, INF)
4.03/1.96	
4.03/1.96	----------------------------------------
4.03/1.96	
4.03/1.96	(12)
4.03/1.96	Obligation:
4.03/1.96	Analyzing the following TRS for decreasing loops:
4.03/1.96	
4.03/1.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.96	
4.03/1.96	
4.03/1.96	The TRS R consists of the following rules:
4.03/1.96	
4.03/1.96	   f(0) -> cons(0, n__f(n__s(n__0)))
4.03/1.96	   f(s(0)) -> f(p(s(0)))
4.03/1.96	   p(s(X)) -> X
4.03/1.96	   f(X) -> n__f(X)
4.03/1.96	   s(X) -> n__s(X)
4.03/1.96	   0 -> n__0
4.03/1.96	   activate(n__f(X)) -> f(activate(X))
4.03/1.96	   activate(n__s(X)) -> s(activate(X))
4.03/1.96	   activate(n__0) -> 0
4.03/1.96	   activate(X) -> X
4.03/1.96	
4.03/1.96	S is empty.
4.03/1.96	Rewrite Strategy: FULL
4.37/2.99	EOF