4.38/2.30	WORST_CASE(Omega(n^1), O(n^1))
4.38/2.32	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.38/2.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.38/2.32	
4.38/2.32	
4.38/2.32	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.38/2.32	
4.38/2.32	(0) CpxTRS
4.38/2.32	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
4.38/2.32	(2) CpxTRS
4.38/2.32	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 117 ms]
4.38/2.32	    (4) BOUNDS(1, n^1)
4.38/2.32	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.38/2.32	(6) TRS for Loop Detection
4.38/2.32	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.38/2.32	    (8) BEST
4.38/2.32	        (9) proven lower bound
4.38/2.32	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
4.38/2.32	            (11) BOUNDS(n^1, INF)
4.38/2.32	        (12) TRS for Loop Detection
4.38/2.32	
4.38/2.32	
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(0)
4.38/2.32	Obligation:
4.38/2.32	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.38/2.32	
4.38/2.32	
4.38/2.32	The TRS R consists of the following rules:
4.38/2.32	
4.38/2.32	   a__f(0) -> cons(0, f(s(0)))
4.38/2.32	   a__f(s(0)) -> a__f(a__p(s(0)))
4.38/2.32	   a__p(s(0)) -> 0
4.38/2.32	   mark(f(X)) -> a__f(mark(X))
4.38/2.32	   mark(p(X)) -> a__p(mark(X))
4.38/2.32	   mark(0) -> 0
4.38/2.32	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.38/2.32	   mark(s(X)) -> s(mark(X))
4.38/2.32	   a__f(X) -> f(X)
4.38/2.32	   a__p(X) -> p(X)
4.38/2.32	
4.38/2.32	S is empty.
4.38/2.32	Rewrite Strategy: FULL
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(1) RelTrsToTrsProof (UPPER BOUND(ID))
4.38/2.32	transformed relative TRS to TRS
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(2)
4.38/2.32	Obligation:
4.38/2.32	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
4.38/2.32	
4.38/2.32	
4.38/2.32	The TRS R consists of the following rules:
4.38/2.32	
4.38/2.32	   a__f(0) -> cons(0, f(s(0)))
4.38/2.32	   a__f(s(0)) -> a__f(a__p(s(0)))
4.38/2.32	   a__p(s(0)) -> 0
4.38/2.32	   mark(f(X)) -> a__f(mark(X))
4.38/2.32	   mark(p(X)) -> a__p(mark(X))
4.38/2.32	   mark(0) -> 0
4.38/2.32	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.38/2.32	   mark(s(X)) -> s(mark(X))
4.38/2.32	   a__f(X) -> f(X)
4.38/2.32	   a__p(X) -> p(X)
4.38/2.32	
4.38/2.32	S is empty.
4.38/2.32	Rewrite Strategy: FULL
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(3) CpxTrsMatchBoundsTAProof (FINISHED)
4.38/2.32	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 3. 
4.38/2.32	
4.38/2.32	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
4.38/2.32	final states : [1, 2, 3]
4.38/2.32	transitions: 
4.38/2.32	00() -> 0
4.38/2.32	cons0(0, 0) -> 0
4.38/2.32	f0(0) -> 0
4.38/2.32	s0(0) -> 0
4.38/2.32	p0(0) -> 0
4.38/2.32	a__f0(0) -> 1
4.38/2.32	a__p0(0) -> 2
4.38/2.32	mark0(0) -> 3
4.38/2.32	01() -> 4
4.38/2.32	01() -> 7
4.38/2.32	s1(7) -> 6
4.38/2.32	f1(6) -> 5
4.38/2.32	cons1(4, 5) -> 1
4.38/2.32	s1(7) -> 9
4.38/2.32	a__p1(9) -> 8
4.38/2.32	a__f1(8) -> 1
4.38/2.32	01() -> 2
4.38/2.32	mark1(0) -> 10
4.38/2.32	a__f1(10) -> 3
4.38/2.32	mark1(0) -> 11
4.38/2.32	a__p1(11) -> 3
4.38/2.32	01() -> 3
4.38/2.32	mark1(0) -> 12
4.38/2.32	cons1(12, 0) -> 3
4.38/2.32	mark1(0) -> 13
4.38/2.32	s1(13) -> 3
4.38/2.32	f1(0) -> 1
4.38/2.32	p1(0) -> 2
4.38/2.32	02() -> 8
4.38/2.32	a__f1(10) -> 10
4.38/2.32	a__f1(10) -> 11
4.38/2.32	a__f1(10) -> 12
4.38/2.32	a__f1(10) -> 13
4.38/2.32	a__p1(11) -> 10
4.38/2.32	a__p1(11) -> 11
4.38/2.32	a__p1(11) -> 12
4.38/2.32	a__p1(11) -> 13
4.38/2.32	01() -> 10
4.38/2.32	01() -> 11
4.38/2.32	01() -> 12
4.38/2.32	01() -> 13
4.38/2.32	cons1(12, 0) -> 10
4.38/2.32	cons1(12, 0) -> 11
4.38/2.32	cons1(12, 0) -> 12
4.38/2.32	cons1(12, 0) -> 13
4.38/2.32	s1(13) -> 10
4.38/2.32	s1(13) -> 11
4.38/2.32	s1(13) -> 12
4.38/2.32	s1(13) -> 13
4.38/2.32	f2(8) -> 1
4.38/2.32	f2(10) -> 3
4.38/2.32	p2(11) -> 3
4.38/2.32	p2(9) -> 8
4.38/2.32	02() -> 14
4.38/2.32	02() -> 17
4.38/2.32	s2(17) -> 16
4.38/2.32	f2(16) -> 15
4.38/2.32	cons2(14, 15) -> 3
4.38/2.32	cons2(14, 15) -> 10
4.38/2.32	cons2(14, 15) -> 11
4.38/2.32	cons2(14, 15) -> 12
4.38/2.32	cons2(14, 15) -> 13
4.38/2.32	s2(17) -> 19
4.38/2.32	a__p2(19) -> 18
4.38/2.32	a__f2(18) -> 3
4.38/2.32	a__f2(18) -> 10
4.38/2.32	a__f2(18) -> 11
4.38/2.32	a__f2(18) -> 12
4.38/2.32	a__f2(18) -> 13
4.38/2.32	02() -> 3
4.38/2.32	02() -> 10
4.38/2.32	02() -> 11
4.38/2.32	02() -> 12
4.38/2.32	02() -> 13
4.38/2.32	f2(10) -> 10
4.38/2.32	f2(10) -> 11
4.38/2.32	f2(10) -> 12
4.38/2.32	f2(10) -> 13
4.38/2.32	p2(11) -> 10
4.38/2.32	p2(11) -> 11
4.38/2.32	p2(11) -> 12
4.38/2.32	p2(11) -> 13
4.38/2.32	cons2(14, 15) -> 1
4.38/2.32	03() -> 18
4.38/2.32	f3(18) -> 3
4.38/2.32	f3(18) -> 10
4.38/2.32	f3(18) -> 11
4.38/2.32	f3(18) -> 12
4.38/2.32	f3(18) -> 13
4.38/2.32	p3(19) -> 18
4.38/2.32	03() -> 20
4.38/2.32	03() -> 23
4.38/2.32	s3(23) -> 22
4.38/2.32	f3(22) -> 21
4.38/2.32	cons3(20, 21) -> 3
4.38/2.32	cons3(20, 21) -> 10
4.38/2.32	cons3(20, 21) -> 11
4.38/2.32	cons3(20, 21) -> 12
4.38/2.32	cons3(20, 21) -> 13
4.38/2.32	
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(4)
4.38/2.32	BOUNDS(1, n^1)
4.38/2.32	
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.38/2.32	Transformed a relative TRS into a decreasing-loop problem.
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(6)
4.38/2.32	Obligation:
4.38/2.32	Analyzing the following TRS for decreasing loops:
4.38/2.32	
4.38/2.32	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.38/2.32	
4.38/2.32	
4.38/2.32	The TRS R consists of the following rules:
4.38/2.32	
4.38/2.32	   a__f(0) -> cons(0, f(s(0)))
4.38/2.32	   a__f(s(0)) -> a__f(a__p(s(0)))
4.38/2.32	   a__p(s(0)) -> 0
4.38/2.32	   mark(f(X)) -> a__f(mark(X))
4.38/2.32	   mark(p(X)) -> a__p(mark(X))
4.38/2.32	   mark(0) -> 0
4.38/2.32	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.38/2.32	   mark(s(X)) -> s(mark(X))
4.38/2.32	   a__f(X) -> f(X)
4.38/2.32	   a__p(X) -> p(X)
4.38/2.32	
4.38/2.32	S is empty.
4.38/2.32	Rewrite Strategy: FULL
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(7) DecreasingLoopProof (LOWER BOUND(ID))
4.38/2.32	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.38/2.32	
4.38/2.32	The rewrite sequence
4.38/2.32	
4.38/2.32	mark(f(X)) ->^+ a__f(mark(X))
4.38/2.32	
4.38/2.32	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.38/2.32	
4.38/2.32	The pumping substitution is [X / f(X)].
4.38/2.32	
4.38/2.32	The result substitution is [ ].
4.38/2.32	
4.38/2.32	
4.38/2.32	
4.38/2.32	
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(8)
4.38/2.32	Complex Obligation (BEST)
4.38/2.32	
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(9)
4.38/2.32	Obligation:
4.38/2.32	Proved the lower bound n^1 for the following obligation:
4.38/2.32	
4.38/2.32	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.38/2.32	
4.38/2.32	
4.38/2.32	The TRS R consists of the following rules:
4.38/2.32	
4.38/2.32	   a__f(0) -> cons(0, f(s(0)))
4.38/2.32	   a__f(s(0)) -> a__f(a__p(s(0)))
4.38/2.32	   a__p(s(0)) -> 0
4.38/2.32	   mark(f(X)) -> a__f(mark(X))
4.38/2.32	   mark(p(X)) -> a__p(mark(X))
4.38/2.32	   mark(0) -> 0
4.38/2.32	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.38/2.32	   mark(s(X)) -> s(mark(X))
4.38/2.32	   a__f(X) -> f(X)
4.38/2.32	   a__p(X) -> p(X)
4.38/2.32	
4.38/2.32	S is empty.
4.38/2.32	Rewrite Strategy: FULL
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(10) LowerBoundPropagationProof (FINISHED)
4.38/2.32	Propagated lower bound.
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(11)
4.38/2.32	BOUNDS(n^1, INF)
4.38/2.32	
4.38/2.32	----------------------------------------
4.38/2.32	
4.38/2.32	(12)
4.38/2.32	Obligation:
4.38/2.32	Analyzing the following TRS for decreasing loops:
4.38/2.32	
4.38/2.32	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.38/2.32	
4.38/2.32	
4.38/2.32	The TRS R consists of the following rules:
4.38/2.32	
4.38/2.32	   a__f(0) -> cons(0, f(s(0)))
4.38/2.32	   a__f(s(0)) -> a__f(a__p(s(0)))
4.38/2.32	   a__p(s(0)) -> 0
4.38/2.32	   mark(f(X)) -> a__f(mark(X))
4.38/2.32	   mark(p(X)) -> a__p(mark(X))
4.38/2.32	   mark(0) -> 0
4.38/2.32	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
4.38/2.32	   mark(s(X)) -> s(mark(X))
4.38/2.32	   a__f(X) -> f(X)
4.38/2.32	   a__p(X) -> p(X)
4.38/2.32	
4.38/2.32	S is empty.
4.38/2.32	Rewrite Strategy: FULL
4.51/2.35	EOF