3.53/1.61	WORST_CASE(Omega(n^1), O(n^1))
3.53/1.62	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.53/1.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.53/1.62	
3.53/1.62	
3.53/1.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.53/1.62	
3.53/1.62	(0) CpxTRS
3.53/1.62	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.53/1.62	(2) CpxTRS
3.53/1.62	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 48 ms]
3.53/1.62	    (4) BOUNDS(1, n^1)
3.53/1.62	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.53/1.62	(6) TRS for Loop Detection
3.53/1.62	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.53/1.62	    (8) BEST
3.53/1.62	        (9) proven lower bound
3.53/1.62	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.53/1.62	            (11) BOUNDS(n^1, INF)
3.53/1.62	        (12) TRS for Loop Detection
3.53/1.62	
3.53/1.62	
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(0)
3.53/1.62	Obligation:
3.53/1.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.53/1.62	
3.53/1.62	
3.53/1.62	The TRS R consists of the following rules:
3.53/1.62	
3.53/1.62	   not(true) -> false
3.53/1.62	   not(false) -> true
3.53/1.62	   odd(0) -> false
3.53/1.62	   odd(s(x)) -> not(odd(x))
3.53/1.62	   +(x, 0) -> x
3.53/1.62	   +(x, s(y)) -> s(+(x, y))
3.53/1.62	   +(s(x), y) -> s(+(x, y))
3.53/1.62	
3.53/1.62	S is empty.
3.53/1.62	Rewrite Strategy: INNERMOST
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.53/1.62	transformed relative TRS to TRS
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(2)
3.53/1.62	Obligation:
3.53/1.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.53/1.62	
3.53/1.62	
3.53/1.62	The TRS R consists of the following rules:
3.53/1.62	
3.53/1.62	   not(true) -> false
3.53/1.62	   not(false) -> true
3.53/1.62	   odd(0) -> false
3.53/1.62	   odd(s(x)) -> not(odd(x))
3.53/1.62	   +(x, 0) -> x
3.53/1.62	   +(x, s(y)) -> s(+(x, y))
3.53/1.62	   +(s(x), y) -> s(+(x, y))
3.53/1.62	
3.53/1.62	S is empty.
3.53/1.62	Rewrite Strategy: INNERMOST
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.53/1.62	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 2. 
3.53/1.62	
3.53/1.62	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.53/1.62	final states : [1, 2, 3]
3.53/1.62	transitions: 
3.53/1.62	true0() -> 0
3.53/1.62	false0() -> 0
3.53/1.62	00() -> 0
3.53/1.62	s0(0) -> 0
3.53/1.62	not0(0) -> 1
3.53/1.62	odd0(0) -> 2
3.53/1.62	+0(0, 0) -> 3
3.53/1.62	false1() -> 1
3.53/1.62	true1() -> 1
3.53/1.62	false1() -> 2
3.53/1.62	odd1(0) -> 4
3.53/1.62	not1(4) -> 2
3.53/1.62	+1(0, 0) -> 5
3.53/1.62	s1(5) -> 3
3.53/1.62	false1() -> 4
3.53/1.62	not1(4) -> 4
3.53/1.62	s1(5) -> 5
3.53/1.62	true2() -> 2
3.53/1.62	true2() -> 4
3.53/1.62	false2() -> 2
3.53/1.62	false2() -> 4
3.53/1.62	0 -> 3
3.53/1.62	0 -> 5
3.53/1.62	
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(4)
3.53/1.62	BOUNDS(1, n^1)
3.53/1.62	
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.53/1.62	Transformed a relative TRS into a decreasing-loop problem.
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(6)
3.53/1.62	Obligation:
3.53/1.62	Analyzing the following TRS for decreasing loops:
3.53/1.62	
3.53/1.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.53/1.62	
3.53/1.62	
3.53/1.62	The TRS R consists of the following rules:
3.53/1.62	
3.53/1.62	   not(true) -> false
3.53/1.62	   not(false) -> true
3.53/1.62	   odd(0) -> false
3.53/1.62	   odd(s(x)) -> not(odd(x))
3.53/1.62	   +(x, 0) -> x
3.53/1.62	   +(x, s(y)) -> s(+(x, y))
3.53/1.62	   +(s(x), y) -> s(+(x, y))
3.53/1.62	
3.53/1.62	S is empty.
3.53/1.62	Rewrite Strategy: INNERMOST
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.53/1.62	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.53/1.62	
3.53/1.62	The rewrite sequence
3.53/1.62	
3.53/1.62	+(x, s(y)) ->^+ s(+(x, y))
3.53/1.62	
3.53/1.62	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.53/1.62	
3.53/1.62	The pumping substitution is [y / s(y)].
3.53/1.62	
3.53/1.62	The result substitution is [ ].
3.53/1.62	
3.53/1.62	
3.53/1.62	
3.53/1.62	
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(8)
3.53/1.62	Complex Obligation (BEST)
3.53/1.62	
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(9)
3.53/1.62	Obligation:
3.53/1.62	Proved the lower bound n^1 for the following obligation:
3.53/1.62	
3.53/1.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.53/1.62	
3.53/1.62	
3.53/1.62	The TRS R consists of the following rules:
3.53/1.62	
3.53/1.62	   not(true) -> false
3.53/1.62	   not(false) -> true
3.53/1.62	   odd(0) -> false
3.53/1.62	   odd(s(x)) -> not(odd(x))
3.53/1.62	   +(x, 0) -> x
3.53/1.62	   +(x, s(y)) -> s(+(x, y))
3.53/1.62	   +(s(x), y) -> s(+(x, y))
3.53/1.62	
3.53/1.62	S is empty.
3.53/1.62	Rewrite Strategy: INNERMOST
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(10) LowerBoundPropagationProof (FINISHED)
3.53/1.62	Propagated lower bound.
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(11)
3.53/1.62	BOUNDS(n^1, INF)
3.53/1.62	
3.53/1.62	----------------------------------------
3.53/1.62	
3.53/1.62	(12)
3.53/1.62	Obligation:
3.53/1.62	Analyzing the following TRS for decreasing loops:
3.53/1.62	
3.53/1.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.53/1.62	
3.53/1.62	
3.53/1.62	The TRS R consists of the following rules:
3.53/1.62	
3.53/1.62	   not(true) -> false
3.53/1.62	   not(false) -> true
3.53/1.62	   odd(0) -> false
3.53/1.62	   odd(s(x)) -> not(odd(x))
3.53/1.62	   +(x, 0) -> x
3.53/1.62	   +(x, s(y)) -> s(+(x, y))
3.53/1.62	   +(s(x), y) -> s(+(x, y))
3.53/1.62	
3.53/1.62	S is empty.
3.53/1.62	Rewrite Strategy: INNERMOST
3.53/1.65	EOF