3.97/1.80	WORST_CASE(Omega(n^1), O(n^1))
3.97/1.81	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.97/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.97/1.81	
3.97/1.81	
3.97/1.81	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.97/1.81	
3.97/1.81	(0) CpxTRS
3.97/1.81	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.97/1.81	(2) CpxTRS
3.97/1.81	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 97 ms]
3.97/1.81	    (4) BOUNDS(1, n^1)
3.97/1.81	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.97/1.81	(6) TRS for Loop Detection
3.97/1.81	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.97/1.81	    (8) BEST
3.97/1.81	        (9) proven lower bound
3.97/1.81	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.97/1.81	            (11) BOUNDS(n^1, INF)
3.97/1.81	        (12) TRS for Loop Detection
3.97/1.81	
3.97/1.81	
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(0)
3.97/1.81	Obligation:
3.97/1.81	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.97/1.81	
3.97/1.81	
3.97/1.81	The TRS R consists of the following rules:
3.97/1.81	
3.97/1.81	   a__zeros -> cons(0, zeros)
3.97/1.81	   a__tail(cons(X, XS)) -> mark(XS)
3.97/1.81	   mark(zeros) -> a__zeros
3.97/1.81	   mark(tail(X)) -> a__tail(mark(X))
3.97/1.81	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.97/1.81	   mark(0) -> 0
3.97/1.81	   a__zeros -> zeros
3.97/1.81	   a__tail(X) -> tail(X)
3.97/1.81	
3.97/1.81	S is empty.
3.97/1.81	Rewrite Strategy: INNERMOST
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.97/1.81	transformed relative TRS to TRS
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(2)
3.97/1.81	Obligation:
3.97/1.81	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.97/1.81	
3.97/1.81	
3.97/1.81	The TRS R consists of the following rules:
3.97/1.81	
3.97/1.81	   a__zeros -> cons(0, zeros)
3.97/1.81	   a__tail(cons(X, XS)) -> mark(XS)
3.97/1.81	   mark(zeros) -> a__zeros
3.97/1.81	   mark(tail(X)) -> a__tail(mark(X))
3.97/1.81	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.97/1.81	   mark(0) -> 0
3.97/1.81	   a__zeros -> zeros
3.97/1.81	   a__tail(X) -> tail(X)
3.97/1.81	
3.97/1.81	S is empty.
3.97/1.81	Rewrite Strategy: INNERMOST
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.97/1.81	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. 
3.97/1.81	
3.97/1.81	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.97/1.81	final states : [1, 2, 3]
3.97/1.81	transitions: 
3.97/1.81	cons0(0, 0) -> 0
3.97/1.81	00() -> 0
3.97/1.81	zeros0() -> 0
3.97/1.81	tail0(0) -> 0
3.97/1.81	a__zeros0() -> 1
3.97/1.81	a__tail0(0) -> 2
3.97/1.81	mark0(0) -> 3
3.97/1.81	01() -> 4
3.97/1.81	zeros1() -> 5
3.97/1.81	cons1(4, 5) -> 1
3.97/1.81	mark1(0) -> 2
3.97/1.81	a__zeros1() -> 3
3.97/1.81	mark1(0) -> 6
3.97/1.81	a__tail1(6) -> 3
3.97/1.81	mark1(0) -> 7
3.97/1.81	cons1(7, 0) -> 3
3.97/1.81	01() -> 3
3.97/1.81	zeros1() -> 1
3.97/1.81	tail1(0) -> 2
3.97/1.81	02() -> 8
3.97/1.81	zeros2() -> 9
3.97/1.81	cons2(8, 9) -> 3
3.97/1.81	a__zeros1() -> 2
3.97/1.81	a__zeros1() -> 6
3.97/1.81	a__zeros1() -> 7
3.97/1.81	a__tail1(6) -> 2
3.97/1.81	a__tail1(6) -> 6
3.97/1.81	a__tail1(6) -> 7
3.97/1.81	cons1(7, 0) -> 2
3.97/1.81	cons1(7, 0) -> 6
3.97/1.81	cons1(7, 0) -> 7
3.97/1.81	01() -> 2
3.97/1.81	01() -> 6
3.97/1.81	01() -> 7
3.97/1.81	zeros2() -> 3
3.97/1.81	tail2(6) -> 3
3.97/1.81	cons2(8, 9) -> 2
3.97/1.81	cons2(8, 9) -> 6
3.97/1.81	cons2(8, 9) -> 7
3.97/1.81	mark2(0) -> 2
3.97/1.81	mark2(0) -> 3
3.97/1.81	mark2(0) -> 6
3.97/1.81	mark2(0) -> 7
3.97/1.81	zeros2() -> 2
3.97/1.81	zeros2() -> 6
3.97/1.81	zeros2() -> 7
3.97/1.81	tail2(6) -> 2
3.97/1.81	tail2(6) -> 6
3.97/1.81	tail2(6) -> 7
3.97/1.81	mark2(9) -> 2
3.97/1.81	mark2(9) -> 3
3.97/1.81	mark2(9) -> 6
3.97/1.81	mark2(9) -> 7
3.97/1.81	a__zeros3() -> 2
3.97/1.81	a__zeros3() -> 3
3.97/1.81	a__zeros3() -> 6
3.97/1.81	a__zeros3() -> 7
3.97/1.81	04() -> 10
3.97/1.81	zeros4() -> 11
3.97/1.81	cons4(10, 11) -> 2
3.97/1.81	cons4(10, 11) -> 3
3.97/1.81	cons4(10, 11) -> 6
3.97/1.81	cons4(10, 11) -> 7
3.97/1.81	zeros4() -> 2
3.97/1.81	zeros4() -> 3
3.97/1.81	zeros4() -> 6
3.97/1.81	zeros4() -> 7
3.97/1.81	mark2(11) -> 2
3.97/1.81	mark2(11) -> 3
3.97/1.81	mark2(11) -> 6
3.97/1.81	mark2(11) -> 7
3.97/1.81	
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(4)
3.97/1.81	BOUNDS(1, n^1)
3.97/1.81	
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.97/1.81	Transformed a relative TRS into a decreasing-loop problem.
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(6)
3.97/1.81	Obligation:
3.97/1.81	Analyzing the following TRS for decreasing loops:
3.97/1.81	
3.97/1.81	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.97/1.81	
3.97/1.81	
3.97/1.81	The TRS R consists of the following rules:
3.97/1.81	
3.97/1.81	   a__zeros -> cons(0, zeros)
3.97/1.81	   a__tail(cons(X, XS)) -> mark(XS)
3.97/1.81	   mark(zeros) -> a__zeros
3.97/1.81	   mark(tail(X)) -> a__tail(mark(X))
3.97/1.81	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.97/1.81	   mark(0) -> 0
3.97/1.81	   a__zeros -> zeros
3.97/1.81	   a__tail(X) -> tail(X)
3.97/1.81	
3.97/1.81	S is empty.
3.97/1.81	Rewrite Strategy: INNERMOST
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.97/1.81	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.97/1.81	
3.97/1.81	The rewrite sequence
3.97/1.81	
3.97/1.81	mark(tail(X)) ->^+ a__tail(mark(X))
3.97/1.81	
3.97/1.81	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.97/1.81	
3.97/1.81	The pumping substitution is [X / tail(X)].
3.97/1.81	
3.97/1.81	The result substitution is [ ].
3.97/1.81	
3.97/1.81	
3.97/1.81	
3.97/1.81	
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(8)
3.97/1.81	Complex Obligation (BEST)
3.97/1.81	
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(9)
3.97/1.81	Obligation:
3.97/1.81	Proved the lower bound n^1 for the following obligation:
3.97/1.81	
3.97/1.81	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.97/1.81	
3.97/1.81	
3.97/1.81	The TRS R consists of the following rules:
3.97/1.81	
3.97/1.81	   a__zeros -> cons(0, zeros)
3.97/1.81	   a__tail(cons(X, XS)) -> mark(XS)
3.97/1.81	   mark(zeros) -> a__zeros
3.97/1.81	   mark(tail(X)) -> a__tail(mark(X))
3.97/1.81	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.97/1.81	   mark(0) -> 0
3.97/1.81	   a__zeros -> zeros
3.97/1.81	   a__tail(X) -> tail(X)
3.97/1.81	
3.97/1.81	S is empty.
3.97/1.81	Rewrite Strategy: INNERMOST
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(10) LowerBoundPropagationProof (FINISHED)
3.97/1.81	Propagated lower bound.
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(11)
3.97/1.81	BOUNDS(n^1, INF)
3.97/1.81	
3.97/1.81	----------------------------------------
3.97/1.81	
3.97/1.81	(12)
3.97/1.81	Obligation:
3.97/1.81	Analyzing the following TRS for decreasing loops:
3.97/1.81	
3.97/1.81	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.97/1.81	
3.97/1.81	
3.97/1.81	The TRS R consists of the following rules:
3.97/1.81	
3.97/1.81	   a__zeros -> cons(0, zeros)
3.97/1.81	   a__tail(cons(X, XS)) -> mark(XS)
3.97/1.81	   mark(zeros) -> a__zeros
3.97/1.81	   mark(tail(X)) -> a__tail(mark(X))
3.97/1.81	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.97/1.81	   mark(0) -> 0
3.97/1.81	   a__zeros -> zeros
3.97/1.81	   a__tail(X) -> tail(X)
3.97/1.81	
3.97/1.81	S is empty.
3.97/1.81	Rewrite Strategy: INNERMOST
3.97/1.85	EOF