23.33/7.54	WORST_CASE(Omega(n^1), O(n^1))
23.33/7.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.33/7.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.33/7.55	
23.33/7.55	
23.33/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.33/7.55	
23.33/7.55	(0) CpxTRS
23.33/7.55	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
23.33/7.55	(2) CpxTRS
23.33/7.55	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
23.33/7.55	    (4) CpxTRS
23.33/7.55	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 156 ms]
23.33/7.55	    (6) BOUNDS(1, n^1)
23.33/7.55	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
23.33/7.55	(8) TRS for Loop Detection
23.33/7.55	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
23.33/7.55	    (10) BEST
23.33/7.55	        (11) proven lower bound
23.33/7.55	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
23.33/7.55	            (13) BOUNDS(n^1, INF)
23.33/7.55	        (14) TRS for Loop Detection
23.33/7.55	
23.33/7.55	
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(0)
23.33/7.55	Obligation:
23.33/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.33/7.55	
23.33/7.55	
23.33/7.55	The TRS R consists of the following rules:
23.33/7.55	
23.33/7.55	   active(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
23.33/7.55	   active(g(b)) -> mark(c)
23.33/7.55	   active(b) -> mark(c)
23.33/7.55	   active(g(X)) -> g(active(X))
23.33/7.55	   g(mark(X)) -> mark(g(X))
23.33/7.55	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
23.33/7.55	   proper(g(X)) -> g(proper(X))
23.33/7.55	   proper(b) -> ok(b)
23.33/7.55	   proper(c) -> ok(c)
23.33/7.55	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
23.33/7.55	   g(ok(X)) -> ok(g(X))
23.33/7.55	   top(mark(X)) -> top(proper(X))
23.33/7.55	   top(ok(X)) -> top(active(X))
23.33/7.55	
23.33/7.55	S is empty.
23.33/7.55	Rewrite Strategy: FULL
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
23.33/7.55	The following defined symbols can occur below the 0th argument of top: proper, active
23.33/7.55	The following defined symbols can occur below the 0th argument of proper: proper, active
23.33/7.55	The following defined symbols can occur below the 0th argument of active: proper, active
23.33/7.55	
23.33/7.55	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
23.33/7.55	active(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
23.33/7.55	active(g(b)) -> mark(c)
23.33/7.55	active(g(X)) -> g(active(X))
23.33/7.55	proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
23.33/7.55	proper(g(X)) -> g(proper(X))
23.33/7.55	
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(2)
23.33/7.55	Obligation:
23.33/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
23.33/7.55	
23.33/7.55	
23.33/7.55	The TRS R consists of the following rules:
23.33/7.55	
23.33/7.55	   active(b) -> mark(c)
23.33/7.55	   g(mark(X)) -> mark(g(X))
23.33/7.55	   proper(b) -> ok(b)
23.33/7.55	   proper(c) -> ok(c)
23.33/7.55	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
23.33/7.55	   g(ok(X)) -> ok(g(X))
23.33/7.55	   top(mark(X)) -> top(proper(X))
23.33/7.55	   top(ok(X)) -> top(active(X))
23.33/7.55	
23.33/7.55	S is empty.
23.33/7.55	Rewrite Strategy: FULL
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(3) RelTrsToTrsProof (UPPER BOUND(ID))
23.33/7.55	transformed relative TRS to TRS
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(4)
23.33/7.55	Obligation:
23.33/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
23.33/7.55	
23.33/7.55	
23.33/7.55	The TRS R consists of the following rules:
23.33/7.55	
23.33/7.55	   active(b) -> mark(c)
23.33/7.55	   g(mark(X)) -> mark(g(X))
23.33/7.55	   proper(b) -> ok(b)
23.33/7.55	   proper(c) -> ok(c)
23.33/7.55	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
23.33/7.55	   g(ok(X)) -> ok(g(X))
23.33/7.55	   top(mark(X)) -> top(proper(X))
23.33/7.55	   top(ok(X)) -> top(active(X))
23.33/7.55	
23.33/7.55	S is empty.
23.33/7.55	Rewrite Strategy: FULL
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(5) CpxTrsMatchBoundsTAProof (FINISHED)
23.33/7.55	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. 
23.33/7.55	
23.33/7.55	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
23.33/7.55	final states : [1, 2, 3, 4, 5]
23.33/7.55	transitions: 
23.33/7.55	b0() -> 0
23.33/7.55	mark0(0) -> 0
23.33/7.55	c0() -> 0
23.33/7.55	ok0(0) -> 0
23.33/7.55	active0(0) -> 1
23.33/7.55	g0(0) -> 2
23.33/7.55	proper0(0) -> 3
23.33/7.55	f0(0, 0, 0) -> 4
23.33/7.55	top0(0) -> 5
23.33/7.55	c1() -> 6
23.33/7.55	mark1(6) -> 1
23.33/7.55	g1(0) -> 7
23.33/7.55	mark1(7) -> 2
23.33/7.55	b1() -> 8
23.33/7.55	ok1(8) -> 3
23.33/7.55	c1() -> 9
23.33/7.55	ok1(9) -> 3
23.33/7.55	f1(0, 0, 0) -> 10
23.33/7.55	ok1(10) -> 4
23.33/7.55	g1(0) -> 11
23.33/7.55	ok1(11) -> 2
23.33/7.55	proper1(0) -> 12
23.33/7.55	top1(12) -> 5
23.33/7.55	active1(0) -> 13
23.33/7.55	top1(13) -> 5
23.33/7.55	mark1(6) -> 13
23.33/7.55	mark1(7) -> 7
23.33/7.55	mark1(7) -> 11
23.33/7.55	ok1(8) -> 12
23.33/7.55	ok1(9) -> 12
23.33/7.55	ok1(10) -> 10
23.33/7.55	ok1(11) -> 7
23.33/7.55	ok1(11) -> 11
23.33/7.55	proper2(6) -> 14
23.33/7.55	top2(14) -> 5
23.33/7.55	active2(8) -> 15
23.33/7.55	top2(15) -> 5
23.33/7.55	active2(9) -> 15
23.33/7.55	c2() -> 16
23.33/7.55	mark2(16) -> 15
23.33/7.55	c2() -> 17
23.33/7.55	ok2(17) -> 14
23.33/7.55	proper3(16) -> 18
23.33/7.55	top3(18) -> 5
23.33/7.55	active3(17) -> 19
23.33/7.55	top3(19) -> 5
23.33/7.55	c3() -> 20
23.33/7.55	ok3(20) -> 18
23.33/7.55	active4(20) -> 21
23.33/7.55	top4(21) -> 5
23.33/7.55	
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(6)
23.33/7.55	BOUNDS(1, n^1)
23.33/7.55	
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
23.33/7.55	Transformed a relative TRS into a decreasing-loop problem.
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(8)
23.33/7.55	Obligation:
23.33/7.55	Analyzing the following TRS for decreasing loops:
23.33/7.55	
23.33/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.33/7.55	
23.33/7.55	
23.33/7.55	The TRS R consists of the following rules:
23.33/7.55	
23.33/7.55	   active(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
23.33/7.55	   active(g(b)) -> mark(c)
23.33/7.55	   active(b) -> mark(c)
23.33/7.55	   active(g(X)) -> g(active(X))
23.33/7.55	   g(mark(X)) -> mark(g(X))
23.33/7.55	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
23.33/7.55	   proper(g(X)) -> g(proper(X))
23.33/7.55	   proper(b) -> ok(b)
23.33/7.55	   proper(c) -> ok(c)
23.33/7.55	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
23.33/7.55	   g(ok(X)) -> ok(g(X))
23.33/7.55	   top(mark(X)) -> top(proper(X))
23.33/7.55	   top(ok(X)) -> top(active(X))
23.33/7.55	
23.33/7.55	S is empty.
23.33/7.55	Rewrite Strategy: FULL
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(9) DecreasingLoopProof (LOWER BOUND(ID))
23.33/7.55	The following loop(s) give(s) rise to the lower bound Omega(n^1):
23.33/7.55	
23.33/7.55	The rewrite sequence
23.33/7.55	
23.33/7.55	g(ok(X)) ->^+ ok(g(X))
23.33/7.55	
23.33/7.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
23.33/7.55	
23.33/7.55	The pumping substitution is [X / ok(X)].
23.33/7.55	
23.33/7.55	The result substitution is [ ].
23.33/7.55	
23.33/7.55	
23.33/7.55	
23.33/7.55	
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(10)
23.33/7.55	Complex Obligation (BEST)
23.33/7.55	
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(11)
23.33/7.55	Obligation:
23.33/7.55	Proved the lower bound n^1 for the following obligation:
23.33/7.55	
23.33/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.33/7.55	
23.33/7.55	
23.33/7.55	The TRS R consists of the following rules:
23.33/7.55	
23.33/7.55	   active(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
23.33/7.55	   active(g(b)) -> mark(c)
23.33/7.55	   active(b) -> mark(c)
23.33/7.55	   active(g(X)) -> g(active(X))
23.33/7.55	   g(mark(X)) -> mark(g(X))
23.33/7.55	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
23.33/7.55	   proper(g(X)) -> g(proper(X))
23.33/7.55	   proper(b) -> ok(b)
23.33/7.55	   proper(c) -> ok(c)
23.33/7.55	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
23.33/7.55	   g(ok(X)) -> ok(g(X))
23.33/7.55	   top(mark(X)) -> top(proper(X))
23.33/7.55	   top(ok(X)) -> top(active(X))
23.33/7.55	
23.33/7.55	S is empty.
23.33/7.55	Rewrite Strategy: FULL
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(12) LowerBoundPropagationProof (FINISHED)
23.33/7.55	Propagated lower bound.
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(13)
23.33/7.55	BOUNDS(n^1, INF)
23.33/7.55	
23.33/7.55	----------------------------------------
23.33/7.55	
23.33/7.55	(14)
23.33/7.55	Obligation:
23.33/7.55	Analyzing the following TRS for decreasing loops:
23.33/7.55	
23.33/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.33/7.55	
23.33/7.55	
23.33/7.55	The TRS R consists of the following rules:
23.33/7.55	
23.33/7.55	   active(f(X, g(X), Y)) -> mark(f(Y, Y, Y))
23.33/7.55	   active(g(b)) -> mark(c)
23.33/7.55	   active(b) -> mark(c)
23.33/7.55	   active(g(X)) -> g(active(X))
23.33/7.55	   g(mark(X)) -> mark(g(X))
23.33/7.55	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
23.33/7.55	   proper(g(X)) -> g(proper(X))
23.33/7.55	   proper(b) -> ok(b)
23.33/7.55	   proper(c) -> ok(c)
23.33/7.55	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
23.33/7.55	   g(ok(X)) -> ok(g(X))
23.33/7.55	   top(mark(X)) -> top(proper(X))
23.33/7.55	   top(ok(X)) -> top(active(X))
23.33/7.55	
23.33/7.55	S is empty.
23.33/7.55	Rewrite Strategy: FULL
23.54/7.58	EOF