3.94/1.78	WORST_CASE(Omega(n^1), O(n^1))
4.23/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.23/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.23/1.79	
4.23/1.79	
4.23/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.23/1.79	
4.23/1.79	(0) CpxTRS
4.23/1.79	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
4.23/1.79	(2) CpxTRS
4.23/1.79	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 94 ms]
4.23/1.79	    (4) BOUNDS(1, n^1)
4.23/1.79	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.23/1.79	(6) TRS for Loop Detection
4.23/1.79	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.23/1.79	    (8) BEST
4.23/1.79	        (9) proven lower bound
4.23/1.79	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
4.23/1.79	            (11) BOUNDS(n^1, INF)
4.23/1.79	        (12) TRS for Loop Detection
4.23/1.79	
4.23/1.79	
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(0)
4.23/1.79	Obligation:
4.23/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.23/1.79	
4.23/1.79	
4.23/1.79	The TRS R consists of the following rules:
4.23/1.79	
4.23/1.79	   a__zeros -> cons(0, zeros)
4.23/1.79	   a__tail(cons(X, XS)) -> mark(XS)
4.23/1.79	   mark(zeros) -> a__zeros
4.23/1.79	   mark(tail(X)) -> a__tail(mark(X))
4.23/1.79	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.23/1.79	   mark(0) -> 0
4.23/1.79	   a__zeros -> zeros
4.23/1.79	   a__tail(X) -> tail(X)
4.23/1.79	
4.23/1.79	S is empty.
4.23/1.79	Rewrite Strategy: FULL
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(1) RelTrsToTrsProof (UPPER BOUND(ID))
4.23/1.79	transformed relative TRS to TRS
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(2)
4.23/1.79	Obligation:
4.23/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
4.23/1.79	
4.23/1.79	
4.23/1.79	The TRS R consists of the following rules:
4.23/1.79	
4.23/1.79	   a__zeros -> cons(0, zeros)
4.23/1.79	   a__tail(cons(X, XS)) -> mark(XS)
4.23/1.79	   mark(zeros) -> a__zeros
4.23/1.79	   mark(tail(X)) -> a__tail(mark(X))
4.23/1.79	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.23/1.79	   mark(0) -> 0
4.23/1.79	   a__zeros -> zeros
4.23/1.79	   a__tail(X) -> tail(X)
4.23/1.79	
4.23/1.79	S is empty.
4.23/1.79	Rewrite Strategy: FULL
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(3) CpxTrsMatchBoundsTAProof (FINISHED)
4.23/1.79	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. 
4.23/1.79	
4.23/1.79	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
4.23/1.79	final states : [1, 2, 3]
4.23/1.79	transitions: 
4.23/1.79	cons0(0, 0) -> 0
4.23/1.79	00() -> 0
4.23/1.79	zeros0() -> 0
4.23/1.79	tail0(0) -> 0
4.23/1.79	a__zeros0() -> 1
4.23/1.79	a__tail0(0) -> 2
4.23/1.79	mark0(0) -> 3
4.23/1.79	01() -> 4
4.23/1.79	zeros1() -> 5
4.23/1.79	cons1(4, 5) -> 1
4.23/1.79	mark1(0) -> 2
4.23/1.79	a__zeros1() -> 3
4.23/1.79	mark1(0) -> 6
4.23/1.79	a__tail1(6) -> 3
4.23/1.79	mark1(0) -> 7
4.23/1.79	cons1(7, 0) -> 3
4.23/1.79	01() -> 3
4.23/1.79	zeros1() -> 1
4.23/1.79	tail1(0) -> 2
4.23/1.79	02() -> 8
4.23/1.79	zeros2() -> 9
4.23/1.79	cons2(8, 9) -> 3
4.23/1.79	a__zeros1() -> 2
4.23/1.79	a__zeros1() -> 6
4.23/1.79	a__zeros1() -> 7
4.23/1.79	a__tail1(6) -> 2
4.23/1.79	a__tail1(6) -> 6
4.23/1.79	a__tail1(6) -> 7
4.23/1.79	cons1(7, 0) -> 2
4.23/1.79	cons1(7, 0) -> 6
4.23/1.79	cons1(7, 0) -> 7
4.23/1.79	01() -> 2
4.23/1.79	01() -> 6
4.23/1.79	01() -> 7
4.23/1.79	zeros2() -> 3
4.23/1.79	tail2(6) -> 3
4.23/1.79	cons2(8, 9) -> 2
4.23/1.79	cons2(8, 9) -> 6
4.23/1.79	cons2(8, 9) -> 7
4.23/1.79	mark2(0) -> 2
4.23/1.79	mark2(0) -> 3
4.23/1.79	mark2(0) -> 6
4.23/1.79	mark2(0) -> 7
4.23/1.79	zeros2() -> 2
4.23/1.79	zeros2() -> 6
4.23/1.79	zeros2() -> 7
4.23/1.79	tail2(6) -> 2
4.23/1.79	tail2(6) -> 6
4.23/1.79	tail2(6) -> 7
4.23/1.79	mark2(9) -> 2
4.23/1.79	mark2(9) -> 3
4.23/1.79	mark2(9) -> 6
4.23/1.79	mark2(9) -> 7
4.23/1.79	a__zeros3() -> 2
4.23/1.79	a__zeros3() -> 3
4.23/1.79	a__zeros3() -> 6
4.23/1.79	a__zeros3() -> 7
4.23/1.79	04() -> 10
4.23/1.79	zeros4() -> 11
4.23/1.79	cons4(10, 11) -> 2
4.23/1.79	cons4(10, 11) -> 3
4.23/1.79	cons4(10, 11) -> 6
4.23/1.79	cons4(10, 11) -> 7
4.23/1.79	zeros4() -> 2
4.23/1.79	zeros4() -> 3
4.23/1.79	zeros4() -> 6
4.23/1.79	zeros4() -> 7
4.23/1.79	mark2(11) -> 2
4.23/1.79	mark2(11) -> 3
4.23/1.79	mark2(11) -> 6
4.23/1.79	mark2(11) -> 7
4.23/1.79	
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(4)
4.23/1.79	BOUNDS(1, n^1)
4.23/1.79	
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.23/1.79	Transformed a relative TRS into a decreasing-loop problem.
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(6)
4.23/1.79	Obligation:
4.23/1.79	Analyzing the following TRS for decreasing loops:
4.23/1.79	
4.23/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.23/1.79	
4.23/1.79	
4.23/1.79	The TRS R consists of the following rules:
4.23/1.79	
4.23/1.79	   a__zeros -> cons(0, zeros)
4.23/1.79	   a__tail(cons(X, XS)) -> mark(XS)
4.23/1.79	   mark(zeros) -> a__zeros
4.23/1.79	   mark(tail(X)) -> a__tail(mark(X))
4.23/1.79	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.23/1.79	   mark(0) -> 0
4.23/1.79	   a__zeros -> zeros
4.23/1.79	   a__tail(X) -> tail(X)
4.23/1.79	
4.23/1.79	S is empty.
4.23/1.79	Rewrite Strategy: FULL
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(7) DecreasingLoopProof (LOWER BOUND(ID))
4.23/1.79	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.23/1.79	
4.23/1.79	The rewrite sequence
4.23/1.79	
4.23/1.79	mark(tail(X)) ->^+ a__tail(mark(X))
4.23/1.79	
4.23/1.79	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.23/1.79	
4.23/1.79	The pumping substitution is [X / tail(X)].
4.23/1.79	
4.23/1.79	The result substitution is [ ].
4.23/1.79	
4.23/1.79	
4.23/1.79	
4.23/1.79	
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(8)
4.23/1.79	Complex Obligation (BEST)
4.23/1.79	
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(9)
4.23/1.79	Obligation:
4.23/1.79	Proved the lower bound n^1 for the following obligation:
4.23/1.79	
4.23/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.23/1.79	
4.23/1.79	
4.23/1.79	The TRS R consists of the following rules:
4.23/1.79	
4.23/1.79	   a__zeros -> cons(0, zeros)
4.23/1.79	   a__tail(cons(X, XS)) -> mark(XS)
4.23/1.79	   mark(zeros) -> a__zeros
4.23/1.79	   mark(tail(X)) -> a__tail(mark(X))
4.23/1.79	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.23/1.79	   mark(0) -> 0
4.23/1.79	   a__zeros -> zeros
4.23/1.79	   a__tail(X) -> tail(X)
4.23/1.79	
4.23/1.79	S is empty.
4.23/1.79	Rewrite Strategy: FULL
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(10) LowerBoundPropagationProof (FINISHED)
4.23/1.79	Propagated lower bound.
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(11)
4.23/1.79	BOUNDS(n^1, INF)
4.23/1.79	
4.23/1.79	----------------------------------------
4.23/1.79	
4.23/1.79	(12)
4.23/1.79	Obligation:
4.23/1.79	Analyzing the following TRS for decreasing loops:
4.23/1.79	
4.23/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.23/1.79	
4.23/1.79	
4.23/1.79	The TRS R consists of the following rules:
4.23/1.79	
4.23/1.79	   a__zeros -> cons(0, zeros)
4.23/1.79	   a__tail(cons(X, XS)) -> mark(XS)
4.23/1.79	   mark(zeros) -> a__zeros
4.23/1.79	   mark(tail(X)) -> a__tail(mark(X))
4.23/1.79	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.23/1.79	   mark(0) -> 0
4.23/1.79	   a__zeros -> zeros
4.23/1.79	   a__tail(X) -> tail(X)
4.23/1.79	
4.23/1.79	S is empty.
4.23/1.79	Rewrite Strategy: FULL
4.26/2.36	EOF