3.98/1.78	WORST_CASE(Omega(n^1), O(n^1))
3.98/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.98/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.98/1.79	
3.98/1.79	
3.98/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.98/1.79	
3.98/1.79	(0) CpxTRS
3.98/1.79	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.98/1.79	(2) CpxTRS
3.98/1.79	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 83 ms]
3.98/1.79	    (4) BOUNDS(1, n^1)
3.98/1.79	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.98/1.79	(6) TRS for Loop Detection
3.98/1.79	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.98/1.79	    (8) BEST
3.98/1.79	        (9) proven lower bound
3.98/1.79	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.98/1.79	            (11) BOUNDS(n^1, INF)
3.98/1.79	        (12) TRS for Loop Detection
3.98/1.79	
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(0)
3.98/1.79	Obligation:
3.98/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.98/1.79	
3.98/1.79	
3.98/1.79	The TRS R consists of the following rules:
3.98/1.79	
3.98/1.79	   f(X) -> if(X, c, n__f(n__true))
3.98/1.79	   if(true, X, Y) -> X
3.98/1.79	   if(false, X, Y) -> activate(Y)
3.98/1.79	   f(X) -> n__f(X)
3.98/1.79	   true -> n__true
3.98/1.79	   activate(n__f(X)) -> f(activate(X))
3.98/1.79	   activate(n__true) -> true
3.98/1.79	   activate(X) -> X
3.98/1.79	
3.98/1.79	S is empty.
3.98/1.79	Rewrite Strategy: FULL
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.98/1.79	transformed relative TRS to TRS
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(2)
3.98/1.79	Obligation:
3.98/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.98/1.79	
3.98/1.79	
3.98/1.79	The TRS R consists of the following rules:
3.98/1.79	
3.98/1.79	   f(X) -> if(X, c, n__f(n__true))
3.98/1.79	   if(true, X, Y) -> X
3.98/1.79	   if(false, X, Y) -> activate(Y)
3.98/1.79	   f(X) -> n__f(X)
3.98/1.79	   true -> n__true
3.98/1.79	   activate(n__f(X)) -> f(activate(X))
3.98/1.79	   activate(n__true) -> true
3.98/1.79	   activate(X) -> X
3.98/1.79	
3.98/1.79	S is empty.
3.98/1.79	Rewrite Strategy: FULL
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.98/1.79	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. 
3.98/1.79	
3.98/1.79	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.98/1.79	final states : [1, 2, 3, 4]
3.98/1.79	transitions: 
3.98/1.79	c0() -> 0
3.98/1.79	n__f0(0) -> 0
3.98/1.79	n__true0() -> 0
3.98/1.79	false0() -> 0
3.98/1.79	f0(0) -> 1
3.98/1.79	if0(0, 0, 0) -> 2
3.98/1.79	true0() -> 3
3.98/1.79	activate0(0) -> 4
3.98/1.79	c1() -> 5
3.98/1.79	n__true1() -> 7
3.98/1.79	n__f1(7) -> 6
3.98/1.79	if1(0, 5, 6) -> 1
3.98/1.79	activate1(0) -> 2
3.98/1.79	n__f1(0) -> 1
3.98/1.79	n__true1() -> 3
3.98/1.79	activate1(0) -> 8
3.98/1.79	f1(8) -> 4
3.98/1.79	true1() -> 4
3.98/1.79	c2() -> 9
3.98/1.79	n__true2() -> 11
3.98/1.79	n__f2(11) -> 10
3.98/1.79	if2(8, 9, 10) -> 4
3.98/1.79	activate1(6) -> 1
3.98/1.79	n__f2(8) -> 4
3.98/1.79	n__true2() -> 4
3.98/1.79	f1(8) -> 2
3.98/1.79	f1(8) -> 8
3.98/1.79	true1() -> 2
3.98/1.79	true1() -> 8
3.98/1.79	if2(8, 9, 10) -> 2
3.98/1.79	if2(8, 9, 10) -> 8
3.98/1.79	n__f2(8) -> 2
3.98/1.79	n__f2(8) -> 8
3.98/1.79	n__true2() -> 2
3.98/1.79	n__true2() -> 8
3.98/1.79	activate2(7) -> 12
3.98/1.79	f2(12) -> 1
3.98/1.79	activate1(10) -> 4
3.98/1.79	activate1(10) -> 2
3.98/1.79	activate1(10) -> 8
3.98/1.79	activate2(11) -> 12
3.98/1.79	f2(12) -> 4
3.98/1.79	c3() -> 13
3.98/1.79	n__true3() -> 15
3.98/1.79	n__f3(15) -> 14
3.98/1.79	if3(12, 13, 14) -> 1
3.98/1.79	n__f3(12) -> 1
3.98/1.79	true2() -> 12
3.98/1.79	f2(12) -> 2
3.98/1.79	f2(12) -> 8
3.98/1.79	if3(12, 13, 14) -> 4
3.98/1.79	n__f3(12) -> 4
3.98/1.79	true3() -> 12
3.98/1.79	n__true3() -> 12
3.98/1.79	if3(12, 13, 14) -> 2
3.98/1.79	if3(12, 13, 14) -> 8
3.98/1.79	n__f3(12) -> 2
3.98/1.79	n__f3(12) -> 8
3.98/1.79	n__true4() -> 12
3.98/1.79	0 -> 4
3.98/1.79	0 -> 2
3.98/1.79	0 -> 8
3.98/1.79	6 -> 1
3.98/1.79	9 -> 4
3.98/1.79	9 -> 2
3.98/1.79	9 -> 8
3.98/1.79	10 -> 4
3.98/1.79	10 -> 2
3.98/1.79	10 -> 8
3.98/1.79	7 -> 12
3.98/1.79	11 -> 12
3.98/1.79	13 -> 1
3.98/1.79	13 -> 4
3.98/1.79	13 -> 2
3.98/1.79	13 -> 8
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(4)
3.98/1.79	BOUNDS(1, n^1)
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.98/1.79	Transformed a relative TRS into a decreasing-loop problem.
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(6)
3.98/1.79	Obligation:
3.98/1.79	Analyzing the following TRS for decreasing loops:
3.98/1.79	
3.98/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.98/1.79	
3.98/1.79	
3.98/1.79	The TRS R consists of the following rules:
3.98/1.79	
3.98/1.79	   f(X) -> if(X, c, n__f(n__true))
3.98/1.79	   if(true, X, Y) -> X
3.98/1.79	   if(false, X, Y) -> activate(Y)
3.98/1.79	   f(X) -> n__f(X)
3.98/1.79	   true -> n__true
3.98/1.79	   activate(n__f(X)) -> f(activate(X))
3.98/1.79	   activate(n__true) -> true
3.98/1.79	   activate(X) -> X
3.98/1.79	
3.98/1.79	S is empty.
3.98/1.79	Rewrite Strategy: FULL
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.98/1.79	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.98/1.79	
3.98/1.79	The rewrite sequence
3.98/1.79	
3.98/1.79	activate(n__f(X)) ->^+ f(activate(X))
3.98/1.79	
3.98/1.79	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.98/1.79	
3.98/1.79	The pumping substitution is [X / n__f(X)].
3.98/1.79	
3.98/1.79	The result substitution is [ ].
3.98/1.79	
3.98/1.79	
3.98/1.79	
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(8)
3.98/1.79	Complex Obligation (BEST)
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(9)
3.98/1.79	Obligation:
3.98/1.79	Proved the lower bound n^1 for the following obligation:
3.98/1.79	
3.98/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.98/1.79	
3.98/1.79	
3.98/1.79	The TRS R consists of the following rules:
3.98/1.79	
3.98/1.79	   f(X) -> if(X, c, n__f(n__true))
3.98/1.79	   if(true, X, Y) -> X
3.98/1.79	   if(false, X, Y) -> activate(Y)
3.98/1.79	   f(X) -> n__f(X)
3.98/1.79	   true -> n__true
3.98/1.79	   activate(n__f(X)) -> f(activate(X))
3.98/1.79	   activate(n__true) -> true
3.98/1.79	   activate(X) -> X
3.98/1.79	
3.98/1.79	S is empty.
3.98/1.79	Rewrite Strategy: FULL
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(10) LowerBoundPropagationProof (FINISHED)
3.98/1.79	Propagated lower bound.
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(11)
3.98/1.79	BOUNDS(n^1, INF)
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(12)
3.98/1.79	Obligation:
3.98/1.79	Analyzing the following TRS for decreasing loops:
3.98/1.79	
3.98/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.98/1.79	
3.98/1.79	
3.98/1.79	The TRS R consists of the following rules:
3.98/1.79	
3.98/1.79	   f(X) -> if(X, c, n__f(n__true))
3.98/1.79	   if(true, X, Y) -> X
3.98/1.79	   if(false, X, Y) -> activate(Y)
3.98/1.79	   f(X) -> n__f(X)
3.98/1.79	   true -> n__true
3.98/1.79	   activate(n__f(X)) -> f(activate(X))
3.98/1.79	   activate(n__true) -> true
3.98/1.79	   activate(X) -> X
3.98/1.79	
3.98/1.79	S is empty.
3.98/1.79	Rewrite Strategy: FULL
4.06/1.83	EOF