22.77/8.19	WORST_CASE(Omega(n^1), O(n^1))
22.77/8.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.77/8.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.77/8.20	
22.77/8.20	
22.77/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.77/8.20	
22.77/8.20	(0) CpxTRS
22.77/8.20	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
22.77/8.20	(2) CpxTRS
22.77/8.20	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
22.77/8.20	    (4) CpxTRS
22.77/8.20	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 51 ms]
22.77/8.20	    (6) BOUNDS(1, n^1)
22.77/8.20	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
22.77/8.20	(8) TRS for Loop Detection
22.77/8.20	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
22.77/8.20	    (10) BEST
22.77/8.20	        (11) proven lower bound
22.77/8.20	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
22.77/8.20	            (13) BOUNDS(n^1, INF)
22.77/8.20	        (14) TRS for Loop Detection
22.77/8.20	
22.77/8.20	
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(0)
22.77/8.20	Obligation:
22.77/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.77/8.20	
22.77/8.20	
22.77/8.20	The TRS R consists of the following rules:
22.77/8.20	
22.77/8.20	   active(f(X)) -> mark(if(X, c, f(true)))
22.77/8.20	   active(if(true, X, Y)) -> mark(X)
22.77/8.20	   active(if(false, X, Y)) -> mark(Y)
22.77/8.20	   active(f(X)) -> f(active(X))
22.77/8.20	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
22.77/8.20	   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
22.77/8.20	   f(mark(X)) -> mark(f(X))
22.77/8.20	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
22.77/8.20	   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
22.77/8.20	   proper(f(X)) -> f(proper(X))
22.77/8.20	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
22.77/8.20	   proper(c) -> ok(c)
22.77/8.20	   proper(true) -> ok(true)
22.77/8.20	   proper(false) -> ok(false)
22.77/8.20	   f(ok(X)) -> ok(f(X))
22.77/8.20	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
22.77/8.20	   top(mark(X)) -> top(proper(X))
22.77/8.20	   top(ok(X)) -> top(active(X))
22.77/8.20	
22.77/8.20	S is empty.
22.77/8.20	Rewrite Strategy: FULL
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
22.77/8.20	The following defined symbols can occur below the 0th argument of top: proper, active
22.77/8.20	The following defined symbols can occur below the 0th argument of proper: proper, active
22.77/8.20	The following defined symbols can occur below the 0th argument of active: proper, active
22.77/8.20	
22.77/8.20	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
22.77/8.20	active(f(X)) -> mark(if(X, c, f(true)))
22.77/8.20	active(if(true, X, Y)) -> mark(X)
22.77/8.20	active(if(false, X, Y)) -> mark(Y)
22.77/8.20	active(f(X)) -> f(active(X))
22.77/8.20	active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
22.77/8.20	active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
22.77/8.20	proper(f(X)) -> f(proper(X))
22.77/8.20	proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
22.77/8.20	
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(2)
22.77/8.20	Obligation:
22.77/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
22.77/8.20	
22.77/8.20	
22.77/8.20	The TRS R consists of the following rules:
22.77/8.20	
22.77/8.20	   f(mark(X)) -> mark(f(X))
22.77/8.20	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
22.77/8.20	   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
22.77/8.20	   proper(c) -> ok(c)
22.77/8.20	   proper(true) -> ok(true)
22.77/8.20	   proper(false) -> ok(false)
22.77/8.20	   f(ok(X)) -> ok(f(X))
22.77/8.20	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
22.77/8.20	   top(mark(X)) -> top(proper(X))
22.77/8.20	   top(ok(X)) -> top(active(X))
22.77/8.20	
22.77/8.20	S is empty.
22.77/8.20	Rewrite Strategy: FULL
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(3) RelTrsToTrsProof (UPPER BOUND(ID))
22.77/8.20	transformed relative TRS to TRS
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(4)
22.77/8.20	Obligation:
22.77/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
22.77/8.20	
22.77/8.20	
22.77/8.20	The TRS R consists of the following rules:
22.77/8.20	
22.77/8.20	   f(mark(X)) -> mark(f(X))
22.77/8.20	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
22.77/8.20	   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
22.77/8.20	   proper(c) -> ok(c)
22.77/8.20	   proper(true) -> ok(true)
22.77/8.20	   proper(false) -> ok(false)
22.77/8.20	   f(ok(X)) -> ok(f(X))
22.77/8.20	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
22.77/8.20	   top(mark(X)) -> top(proper(X))
22.77/8.20	   top(ok(X)) -> top(active(X))
22.77/8.20	
22.77/8.20	S is empty.
22.77/8.20	Rewrite Strategy: FULL
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(5) CpxTrsMatchBoundsTAProof (FINISHED)
22.77/8.20	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. 
22.77/8.20	
22.77/8.20	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
22.77/8.20	final states : [1, 2, 3, 4]
22.77/8.20	transitions: 
22.77/8.20	mark0(0) -> 0
22.77/8.20	c0() -> 0
22.77/8.20	ok0(0) -> 0
22.77/8.20	true0() -> 0
22.77/8.20	false0() -> 0
22.77/8.20	active0(0) -> 0
22.77/8.20	f0(0) -> 1
22.77/8.20	if0(0, 0, 0) -> 2
22.77/8.20	proper0(0) -> 3
22.77/8.20	top0(0) -> 4
22.77/8.20	f1(0) -> 5
22.77/8.20	mark1(5) -> 1
22.77/8.20	if1(0, 0, 0) -> 6
22.77/8.20	mark1(6) -> 2
22.77/8.20	c1() -> 7
22.77/8.20	ok1(7) -> 3
22.77/8.20	true1() -> 8
22.77/8.20	ok1(8) -> 3
22.77/8.20	false1() -> 9
22.77/8.20	ok1(9) -> 3
22.77/8.20	f1(0) -> 10
22.77/8.20	ok1(10) -> 1
22.77/8.20	if1(0, 0, 0) -> 11
22.77/8.20	ok1(11) -> 2
22.77/8.20	proper1(0) -> 12
22.77/8.20	top1(12) -> 4
22.77/8.20	active1(0) -> 13
22.77/8.20	top1(13) -> 4
22.77/8.20	mark1(5) -> 5
22.77/8.20	mark1(5) -> 10
22.77/8.20	mark1(6) -> 6
22.77/8.20	mark1(6) -> 11
22.77/8.20	ok1(7) -> 12
22.77/8.20	ok1(8) -> 12
22.77/8.20	ok1(9) -> 12
22.77/8.20	ok1(10) -> 5
22.77/8.20	ok1(10) -> 10
22.77/8.20	ok1(11) -> 6
22.77/8.20	ok1(11) -> 11
22.77/8.20	active2(7) -> 14
22.77/8.20	top2(14) -> 4
22.77/8.20	active2(8) -> 14
22.77/8.20	active2(9) -> 14
22.77/8.20	
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(6)
22.77/8.20	BOUNDS(1, n^1)
22.77/8.20	
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
22.77/8.20	Transformed a relative TRS into a decreasing-loop problem.
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(8)
22.77/8.20	Obligation:
22.77/8.20	Analyzing the following TRS for decreasing loops:
22.77/8.20	
22.77/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.77/8.20	
22.77/8.20	
22.77/8.20	The TRS R consists of the following rules:
22.77/8.20	
22.77/8.20	   active(f(X)) -> mark(if(X, c, f(true)))
22.77/8.20	   active(if(true, X, Y)) -> mark(X)
22.77/8.20	   active(if(false, X, Y)) -> mark(Y)
22.77/8.20	   active(f(X)) -> f(active(X))
22.77/8.20	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
22.77/8.20	   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
22.77/8.20	   f(mark(X)) -> mark(f(X))
22.77/8.20	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
22.77/8.20	   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
22.77/8.20	   proper(f(X)) -> f(proper(X))
22.77/8.20	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
22.77/8.20	   proper(c) -> ok(c)
22.77/8.20	   proper(true) -> ok(true)
22.77/8.20	   proper(false) -> ok(false)
22.77/8.20	   f(ok(X)) -> ok(f(X))
22.77/8.20	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
22.77/8.20	   top(mark(X)) -> top(proper(X))
22.77/8.20	   top(ok(X)) -> top(active(X))
22.77/8.20	
22.77/8.20	S is empty.
22.77/8.20	Rewrite Strategy: FULL
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(9) DecreasingLoopProof (LOWER BOUND(ID))
22.77/8.20	The following loop(s) give(s) rise to the lower bound Omega(n^1):
22.77/8.20	
22.77/8.20	The rewrite sequence
22.77/8.20	
22.77/8.20	f(ok(X)) ->^+ ok(f(X))
22.77/8.20	
22.77/8.20	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
22.77/8.20	
22.77/8.20	The pumping substitution is [X / ok(X)].
22.77/8.20	
22.77/8.20	The result substitution is [ ].
22.77/8.20	
22.77/8.20	
22.77/8.20	
22.77/8.20	
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(10)
22.77/8.20	Complex Obligation (BEST)
22.77/8.20	
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(11)
22.77/8.20	Obligation:
22.77/8.20	Proved the lower bound n^1 for the following obligation:
22.77/8.20	
22.77/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.77/8.20	
22.77/8.20	
22.77/8.20	The TRS R consists of the following rules:
22.77/8.20	
22.77/8.20	   active(f(X)) -> mark(if(X, c, f(true)))
22.77/8.20	   active(if(true, X, Y)) -> mark(X)
22.77/8.20	   active(if(false, X, Y)) -> mark(Y)
22.77/8.20	   active(f(X)) -> f(active(X))
22.77/8.20	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
22.77/8.20	   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
22.77/8.20	   f(mark(X)) -> mark(f(X))
22.77/8.20	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
22.77/8.20	   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
22.77/8.20	   proper(f(X)) -> f(proper(X))
22.77/8.20	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
22.77/8.20	   proper(c) -> ok(c)
22.77/8.20	   proper(true) -> ok(true)
22.77/8.20	   proper(false) -> ok(false)
22.77/8.20	   f(ok(X)) -> ok(f(X))
22.77/8.20	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
22.77/8.20	   top(mark(X)) -> top(proper(X))
22.77/8.20	   top(ok(X)) -> top(active(X))
22.77/8.20	
22.77/8.20	S is empty.
22.77/8.20	Rewrite Strategy: FULL
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(12) LowerBoundPropagationProof (FINISHED)
22.77/8.20	Propagated lower bound.
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(13)
22.77/8.20	BOUNDS(n^1, INF)
22.77/8.20	
22.77/8.20	----------------------------------------
22.77/8.20	
22.77/8.20	(14)
22.77/8.20	Obligation:
22.77/8.20	Analyzing the following TRS for decreasing loops:
22.77/8.20	
22.77/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.77/8.20	
22.77/8.20	
22.77/8.20	The TRS R consists of the following rules:
22.77/8.20	
22.77/8.20	   active(f(X)) -> mark(if(X, c, f(true)))
22.77/8.20	   active(if(true, X, Y)) -> mark(X)
22.77/8.20	   active(if(false, X, Y)) -> mark(Y)
22.77/8.20	   active(f(X)) -> f(active(X))
22.77/8.20	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
22.77/8.20	   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
22.77/8.20	   f(mark(X)) -> mark(f(X))
22.77/8.20	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
22.77/8.20	   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
22.77/8.20	   proper(f(X)) -> f(proper(X))
22.77/8.20	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
22.77/8.20	   proper(c) -> ok(c)
22.77/8.20	   proper(true) -> ok(true)
22.77/8.20	   proper(false) -> ok(false)
22.77/8.20	   f(ok(X)) -> ok(f(X))
22.77/8.20	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
22.77/8.20	   top(mark(X)) -> top(proper(X))
22.77/8.20	   top(ok(X)) -> top(active(X))
22.77/8.20	
22.77/8.20	S is empty.
22.77/8.20	Rewrite Strategy: FULL
22.77/8.24	EOF