3.00/1.63	WORST_CASE(Omega(n^1), O(n^1))
3.41/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.41/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.64	
3.41/1.64	
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.41/1.64	
3.41/1.64	(0) CpxTRS
3.41/1.64	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.41/1.64	(2) CpxTRS
3.41/1.64	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 51 ms]
3.41/1.64	    (4) BOUNDS(1, n^1)
3.41/1.64	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.41/1.64	(6) TRS for Loop Detection
3.41/1.64	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.41/1.64	    (8) BEST
3.41/1.64	        (9) proven lower bound
3.41/1.64	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.41/1.64	            (11) BOUNDS(n^1, INF)
3.41/1.64	        (12) TRS for Loop Detection
3.41/1.64	
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(0)
3.41/1.64	Obligation:
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   f(0) -> s(0)
3.41/1.64	   f(s(0)) -> s(s(0))
3.41/1.64	   f(s(0)) -> *(s(s(0)), f(0))
3.41/1.64	   f(+(x, s(0))) -> +(s(s(0)), f(x))
3.41/1.64	   f(+(x, y)) -> *(f(x), f(y))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.41/1.64	transformed relative TRS to TRS
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(2)
3.41/1.64	Obligation:
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   f(0) -> s(0)
3.41/1.64	   f(s(0)) -> s(s(0))
3.41/1.64	   f(s(0)) -> *(s(s(0)), f(0))
3.41/1.64	   f(+(x, s(0))) -> +(s(s(0)), f(x))
3.41/1.64	   f(+(x, y)) -> *(f(x), f(y))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.41/1.64	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. 
3.41/1.64	
3.41/1.64	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.41/1.64	final states : [1]
3.41/1.64	transitions: 
3.41/1.64	00() -> 0
3.41/1.64	s0(0) -> 0
3.41/1.64	*0(0, 0) -> 0
3.41/1.64	+0(0, 0) -> 0
3.41/1.64	f0(0) -> 1
3.41/1.64	01() -> 2
3.41/1.64	s1(2) -> 1
3.41/1.64	s1(2) -> 3
3.41/1.64	s1(3) -> 1
3.41/1.64	s1(3) -> 4
3.41/1.64	01() -> 6
3.41/1.64	f1(6) -> 5
3.41/1.64	*1(4, 5) -> 1
3.41/1.64	s1(3) -> 7
3.41/1.64	f1(0) -> 8
3.41/1.64	+1(7, 8) -> 1
3.41/1.64	f1(0) -> 9
3.41/1.64	f1(0) -> 10
3.41/1.64	*1(9, 10) -> 1
3.41/1.64	s1(2) -> 8
3.41/1.64	s1(2) -> 9
3.41/1.64	s1(2) -> 10
3.41/1.64	02() -> 11
3.41/1.64	s2(11) -> 5
3.41/1.64	s1(3) -> 8
3.41/1.64	s1(3) -> 9
3.41/1.64	s1(3) -> 10
3.41/1.64	*1(4, 5) -> 8
3.41/1.64	*1(4, 5) -> 9
3.41/1.64	*1(4, 5) -> 10
3.41/1.64	+1(7, 8) -> 8
3.41/1.64	+1(7, 8) -> 9
3.41/1.64	+1(7, 8) -> 10
3.41/1.64	*1(9, 10) -> 8
3.41/1.64	*1(9, 10) -> 9
3.41/1.64	*1(9, 10) -> 10
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(4)
3.41/1.64	BOUNDS(1, n^1)
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.41/1.64	Transformed a relative TRS into a decreasing-loop problem.
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(6)
3.41/1.64	Obligation:
3.41/1.64	Analyzing the following TRS for decreasing loops:
3.41/1.64	
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   f(0) -> s(0)
3.41/1.64	   f(s(0)) -> s(s(0))
3.41/1.64	   f(s(0)) -> *(s(s(0)), f(0))
3.41/1.64	   f(+(x, s(0))) -> +(s(s(0)), f(x))
3.41/1.64	   f(+(x, y)) -> *(f(x), f(y))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.41/1.64	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.41/1.64	
3.41/1.64	The rewrite sequence
3.41/1.64	
3.41/1.64	f(+(x, y)) ->^+ *(f(x), f(y))
3.41/1.64	
3.41/1.64	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.41/1.64	
3.41/1.64	The pumping substitution is [x / +(x, y)].
3.41/1.64	
3.41/1.64	The result substitution is [ ].
3.41/1.64	
3.41/1.64	
3.41/1.64	
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(8)
3.41/1.64	Complex Obligation (BEST)
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(9)
3.41/1.64	Obligation:
3.41/1.64	Proved the lower bound n^1 for the following obligation:
3.41/1.64	
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   f(0) -> s(0)
3.41/1.64	   f(s(0)) -> s(s(0))
3.41/1.64	   f(s(0)) -> *(s(s(0)), f(0))
3.41/1.64	   f(+(x, s(0))) -> +(s(s(0)), f(x))
3.41/1.64	   f(+(x, y)) -> *(f(x), f(y))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(10) LowerBoundPropagationProof (FINISHED)
3.41/1.64	Propagated lower bound.
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(11)
3.41/1.64	BOUNDS(n^1, INF)
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(12)
3.41/1.64	Obligation:
3.41/1.64	Analyzing the following TRS for decreasing loops:
3.41/1.64	
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   f(0) -> s(0)
3.41/1.64	   f(s(0)) -> s(s(0))
3.41/1.64	   f(s(0)) -> *(s(s(0)), f(0))
3.41/1.64	   f(+(x, s(0))) -> +(s(s(0)), f(x))
3.41/1.64	   f(+(x, y)) -> *(f(x), f(y))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.67	EOF