3.36/1.55	WORST_CASE(Omega(n^1), O(n^1))
3.36/1.56	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.36/1.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.36/1.56	
3.36/1.56	
3.36/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.56	
3.36/1.56	(0) CpxTRS
3.36/1.56	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.36/1.56	(2) CpxTRS
3.36/1.56	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.36/1.56	    (4) BOUNDS(1, n^1)
3.36/1.56	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.36/1.56	(6) TRS for Loop Detection
3.36/1.56	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.36/1.56	    (8) BEST
3.36/1.56	        (9) proven lower bound
3.36/1.56	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.36/1.56	            (11) BOUNDS(n^1, INF)
3.36/1.56	        (12) TRS for Loop Detection
3.36/1.56	
3.36/1.56	
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(0)
3.36/1.56	Obligation:
3.36/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.56	
3.36/1.56	
3.36/1.56	The TRS R consists of the following rules:
3.36/1.56	
3.36/1.56	   odd(Cons(x, xs)) -> even(xs)
3.36/1.56	   odd(Nil) -> False
3.36/1.56	   even(Cons(x, xs)) -> odd(xs)
3.36/1.56	   notEmpty(Cons(x, xs)) -> True
3.36/1.56	   notEmpty(Nil) -> False
3.36/1.56	   even(Nil) -> True
3.36/1.56	   evenodd(x) -> even(x)
3.36/1.56	
3.36/1.56	S is empty.
3.36/1.56	Rewrite Strategy: INNERMOST
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.36/1.56	transformed relative TRS to TRS
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(2)
3.36/1.56	Obligation:
3.36/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.36/1.56	
3.36/1.56	
3.36/1.56	The TRS R consists of the following rules:
3.36/1.56	
3.36/1.56	   odd(Cons(x, xs)) -> even(xs)
3.36/1.56	   odd(Nil) -> False
3.36/1.56	   even(Cons(x, xs)) -> odd(xs)
3.36/1.56	   notEmpty(Cons(x, xs)) -> True
3.36/1.56	   notEmpty(Nil) -> False
3.36/1.56	   even(Nil) -> True
3.36/1.56	   evenodd(x) -> even(x)
3.36/1.56	
3.36/1.56	S is empty.
3.36/1.56	Rewrite Strategy: INNERMOST
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.36/1.56	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. 
3.36/1.56	
3.36/1.56	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.36/1.56	final states : [1, 2, 3, 4]
3.36/1.56	transitions: 
3.36/1.56	Cons0(0, 0) -> 0
3.36/1.56	Nil0() -> 0
3.36/1.56	False0() -> 0
3.36/1.56	True0() -> 0
3.36/1.56	odd0(0) -> 1
3.36/1.56	even0(0) -> 2
3.36/1.56	notEmpty0(0) -> 3
3.36/1.56	evenodd0(0) -> 4
3.36/1.56	even1(0) -> 1
3.36/1.56	False1() -> 1
3.36/1.56	odd1(0) -> 2
3.36/1.56	True1() -> 3
3.36/1.56	False1() -> 3
3.36/1.56	True1() -> 2
3.36/1.56	even1(0) -> 4
3.36/1.56	even1(0) -> 2
3.36/1.56	False1() -> 2
3.36/1.56	odd1(0) -> 1
3.36/1.56	odd1(0) -> 4
3.36/1.56	True1() -> 1
3.36/1.56	True1() -> 4
3.36/1.56	False1() -> 4
3.36/1.56	
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(4)
3.36/1.56	BOUNDS(1, n^1)
3.36/1.56	
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.36/1.56	Transformed a relative TRS into a decreasing-loop problem.
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(6)
3.36/1.56	Obligation:
3.36/1.56	Analyzing the following TRS for decreasing loops:
3.36/1.56	
3.36/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.56	
3.36/1.56	
3.36/1.56	The TRS R consists of the following rules:
3.36/1.56	
3.36/1.56	   odd(Cons(x, xs)) -> even(xs)
3.36/1.56	   odd(Nil) -> False
3.36/1.56	   even(Cons(x, xs)) -> odd(xs)
3.36/1.56	   notEmpty(Cons(x, xs)) -> True
3.36/1.56	   notEmpty(Nil) -> False
3.36/1.56	   even(Nil) -> True
3.36/1.56	   evenodd(x) -> even(x)
3.36/1.56	
3.36/1.56	S is empty.
3.36/1.56	Rewrite Strategy: INNERMOST
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.36/1.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.36/1.56	
3.36/1.56	The rewrite sequence
3.36/1.56	
3.36/1.56	even(Cons(x, Cons(x1_0, xs2_0))) ->^+ even(xs2_0)
3.36/1.56	
3.36/1.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.36/1.56	
3.36/1.56	The pumping substitution is [xs2_0 / Cons(x, Cons(x1_0, xs2_0))].
3.36/1.56	
3.36/1.56	The result substitution is [ ].
3.36/1.56	
3.36/1.56	
3.36/1.56	
3.36/1.56	
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(8)
3.36/1.56	Complex Obligation (BEST)
3.36/1.56	
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(9)
3.36/1.56	Obligation:
3.36/1.56	Proved the lower bound n^1 for the following obligation:
3.36/1.56	
3.36/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.56	
3.36/1.56	
3.36/1.56	The TRS R consists of the following rules:
3.36/1.56	
3.36/1.56	   odd(Cons(x, xs)) -> even(xs)
3.36/1.56	   odd(Nil) -> False
3.36/1.56	   even(Cons(x, xs)) -> odd(xs)
3.36/1.56	   notEmpty(Cons(x, xs)) -> True
3.36/1.56	   notEmpty(Nil) -> False
3.36/1.56	   even(Nil) -> True
3.36/1.56	   evenodd(x) -> even(x)
3.36/1.56	
3.36/1.56	S is empty.
3.36/1.56	Rewrite Strategy: INNERMOST
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(10) LowerBoundPropagationProof (FINISHED)
3.36/1.56	Propagated lower bound.
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(11)
3.36/1.56	BOUNDS(n^1, INF)
3.36/1.56	
3.36/1.56	----------------------------------------
3.36/1.56	
3.36/1.56	(12)
3.36/1.56	Obligation:
3.36/1.56	Analyzing the following TRS for decreasing loops:
3.36/1.56	
3.36/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.56	
3.36/1.56	
3.36/1.56	The TRS R consists of the following rules:
3.36/1.56	
3.36/1.56	   odd(Cons(x, xs)) -> even(xs)
3.36/1.56	   odd(Nil) -> False
3.36/1.56	   even(Cons(x, xs)) -> odd(xs)
3.36/1.56	   notEmpty(Cons(x, xs)) -> True
3.36/1.56	   notEmpty(Nil) -> False
3.36/1.56	   even(Nil) -> True
3.36/1.56	   evenodd(x) -> even(x)
3.36/1.56	
3.36/1.56	S is empty.
3.36/1.56	Rewrite Strategy: INNERMOST
3.55/1.59	EOF