19.60/6.58	WORST_CASE(Omega(n^1), O(n^1))
19.60/6.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.60/6.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.60/6.59	
19.60/6.59	
19.60/6.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.60/6.59	
19.60/6.59	(0) CpxTRS
19.60/6.59	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
19.60/6.59	(2) CpxTRS
19.60/6.59	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
19.60/6.59	    (4) CpxTRS
19.60/6.59	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 8 ms]
19.60/6.59	    (6) BOUNDS(1, n^1)
19.60/6.59	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
19.60/6.59	(8) TRS for Loop Detection
19.60/6.59	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
19.60/6.59	    (10) BEST
19.60/6.59	        (11) proven lower bound
19.60/6.59	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
19.60/6.59	            (13) BOUNDS(n^1, INF)
19.60/6.59	        (14) TRS for Loop Detection
19.60/6.59	
19.60/6.59	
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(0)
19.60/6.59	Obligation:
19.60/6.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.60/6.59	
19.60/6.59	
19.60/6.59	The TRS R consists of the following rules:
19.60/6.59	
19.60/6.59	   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
19.60/6.59	   active(f(X1, X2)) -> f(active(X1), X2)
19.60/6.59	   active(g(X)) -> g(active(X))
19.60/6.59	   f(mark(X1), X2) -> mark(f(X1, X2))
19.60/6.59	   g(mark(X)) -> mark(g(X))
19.60/6.59	   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
19.60/6.59	   proper(g(X)) -> g(proper(X))
19.60/6.59	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
19.60/6.59	   g(ok(X)) -> ok(g(X))
19.60/6.59	   top(mark(X)) -> top(proper(X))
19.60/6.59	   top(ok(X)) -> top(active(X))
19.60/6.59	
19.60/6.59	S is empty.
19.60/6.59	Rewrite Strategy: FULL
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
19.60/6.59	The following defined symbols can occur below the 0th argument of top: proper, active
19.60/6.59	The following defined symbols can occur below the 0th argument of proper: proper, active
19.60/6.59	The following defined symbols can occur below the 0th argument of active: proper, active
19.60/6.59	
19.60/6.59	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
19.60/6.59	active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
19.60/6.59	active(f(X1, X2)) -> f(active(X1), X2)
19.60/6.59	active(g(X)) -> g(active(X))
19.60/6.59	proper(f(X1, X2)) -> f(proper(X1), proper(X2))
19.60/6.59	proper(g(X)) -> g(proper(X))
19.60/6.59	
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(2)
19.60/6.59	Obligation:
19.60/6.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
19.60/6.59	
19.60/6.59	
19.60/6.59	The TRS R consists of the following rules:
19.60/6.59	
19.60/6.59	   f(mark(X1), X2) -> mark(f(X1, X2))
19.60/6.59	   g(mark(X)) -> mark(g(X))
19.60/6.59	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
19.60/6.59	   g(ok(X)) -> ok(g(X))
19.60/6.59	   top(mark(X)) -> top(proper(X))
19.60/6.59	   top(ok(X)) -> top(active(X))
19.60/6.59	
19.60/6.59	S is empty.
19.60/6.59	Rewrite Strategy: FULL
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(3) RelTrsToTrsProof (UPPER BOUND(ID))
19.60/6.59	transformed relative TRS to TRS
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(4)
19.60/6.59	Obligation:
19.60/6.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
19.60/6.59	
19.60/6.59	
19.60/6.59	The TRS R consists of the following rules:
19.60/6.59	
19.60/6.59	   f(mark(X1), X2) -> mark(f(X1, X2))
19.60/6.59	   g(mark(X)) -> mark(g(X))
19.60/6.59	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
19.60/6.59	   g(ok(X)) -> ok(g(X))
19.60/6.59	   top(mark(X)) -> top(proper(X))
19.60/6.59	   top(ok(X)) -> top(active(X))
19.60/6.59	
19.60/6.59	S is empty.
19.60/6.59	Rewrite Strategy: FULL
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(5) CpxTrsMatchBoundsTAProof (FINISHED)
19.60/6.59	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. 
19.60/6.59	
19.60/6.59	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
19.60/6.59	final states : [1, 2, 3]
19.60/6.59	transitions: 
19.60/6.59	mark0(0) -> 0
19.60/6.59	ok0(0) -> 0
19.60/6.59	proper0(0) -> 0
19.60/6.59	active0(0) -> 0
19.60/6.59	f0(0, 0) -> 1
19.60/6.59	g0(0) -> 2
19.60/6.59	top0(0) -> 3
19.60/6.59	f1(0, 0) -> 4
19.60/6.59	mark1(4) -> 1
19.60/6.59	g1(0) -> 5
19.60/6.59	mark1(5) -> 2
19.60/6.59	f1(0, 0) -> 6
19.60/6.59	ok1(6) -> 1
19.60/6.59	g1(0) -> 7
19.60/6.59	ok1(7) -> 2
19.60/6.59	proper1(0) -> 8
19.60/6.59	top1(8) -> 3
19.60/6.59	active1(0) -> 9
19.60/6.59	top1(9) -> 3
19.60/6.59	mark1(4) -> 4
19.60/6.59	mark1(4) -> 6
19.60/6.59	mark1(5) -> 5
19.60/6.59	mark1(5) -> 7
19.60/6.59	ok1(6) -> 4
19.60/6.59	ok1(6) -> 6
19.60/6.59	ok1(7) -> 5
19.60/6.59	ok1(7) -> 7
19.60/6.59	
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(6)
19.60/6.59	BOUNDS(1, n^1)
19.60/6.59	
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
19.60/6.59	Transformed a relative TRS into a decreasing-loop problem.
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(8)
19.60/6.59	Obligation:
19.60/6.59	Analyzing the following TRS for decreasing loops:
19.60/6.59	
19.60/6.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.60/6.59	
19.60/6.59	
19.60/6.59	The TRS R consists of the following rules:
19.60/6.59	
19.60/6.59	   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
19.60/6.59	   active(f(X1, X2)) -> f(active(X1), X2)
19.60/6.59	   active(g(X)) -> g(active(X))
19.60/6.59	   f(mark(X1), X2) -> mark(f(X1, X2))
19.60/6.59	   g(mark(X)) -> mark(g(X))
19.60/6.59	   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
19.60/6.59	   proper(g(X)) -> g(proper(X))
19.60/6.59	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
19.60/6.59	   g(ok(X)) -> ok(g(X))
19.60/6.59	   top(mark(X)) -> top(proper(X))
19.60/6.59	   top(ok(X)) -> top(active(X))
19.60/6.59	
19.60/6.59	S is empty.
19.60/6.59	Rewrite Strategy: FULL
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(9) DecreasingLoopProof (LOWER BOUND(ID))
19.60/6.59	The following loop(s) give(s) rise to the lower bound Omega(n^1):
19.60/6.59	
19.60/6.59	The rewrite sequence
19.60/6.59	
19.60/6.59	f(mark(X1), X2) ->^+ mark(f(X1, X2))
19.60/6.59	
19.60/6.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
19.60/6.59	
19.60/6.59	The pumping substitution is [X1 / mark(X1)].
19.60/6.59	
19.60/6.59	The result substitution is [ ].
19.60/6.59	
19.60/6.59	
19.60/6.59	
19.60/6.59	
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(10)
19.60/6.59	Complex Obligation (BEST)
19.60/6.59	
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(11)
19.60/6.59	Obligation:
19.60/6.59	Proved the lower bound n^1 for the following obligation:
19.60/6.59	
19.60/6.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.60/6.59	
19.60/6.59	
19.60/6.59	The TRS R consists of the following rules:
19.60/6.59	
19.60/6.59	   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
19.60/6.59	   active(f(X1, X2)) -> f(active(X1), X2)
19.60/6.59	   active(g(X)) -> g(active(X))
19.60/6.59	   f(mark(X1), X2) -> mark(f(X1, X2))
19.60/6.59	   g(mark(X)) -> mark(g(X))
19.60/6.59	   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
19.60/6.59	   proper(g(X)) -> g(proper(X))
19.60/6.59	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
19.60/6.59	   g(ok(X)) -> ok(g(X))
19.60/6.59	   top(mark(X)) -> top(proper(X))
19.60/6.59	   top(ok(X)) -> top(active(X))
19.60/6.59	
19.60/6.59	S is empty.
19.60/6.59	Rewrite Strategy: FULL
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(12) LowerBoundPropagationProof (FINISHED)
19.60/6.59	Propagated lower bound.
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(13)
19.60/6.59	BOUNDS(n^1, INF)
19.60/6.59	
19.60/6.59	----------------------------------------
19.60/6.59	
19.60/6.59	(14)
19.60/6.59	Obligation:
19.60/6.59	Analyzing the following TRS for decreasing loops:
19.60/6.59	
19.60/6.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.60/6.59	
19.60/6.59	
19.60/6.59	The TRS R consists of the following rules:
19.60/6.59	
19.60/6.59	   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
19.60/6.59	   active(f(X1, X2)) -> f(active(X1), X2)
19.60/6.59	   active(g(X)) -> g(active(X))
19.60/6.59	   f(mark(X1), X2) -> mark(f(X1, X2))
19.60/6.59	   g(mark(X)) -> mark(g(X))
19.60/6.59	   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
19.60/6.59	   proper(g(X)) -> g(proper(X))
19.60/6.59	   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
19.60/6.59	   g(ok(X)) -> ok(g(X))
19.60/6.59	   top(mark(X)) -> top(proper(X))
19.60/6.59	   top(ok(X)) -> top(active(X))
19.60/6.59	
19.60/6.59	S is empty.
19.60/6.59	Rewrite Strategy: FULL
19.70/6.62	EOF