3.68/1.76	WORST_CASE(Omega(n^1), O(n^1))
4.00/1.76	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.00/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.00/1.76	
4.00/1.76	
4.00/1.76	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.00/1.76	
4.00/1.76	(0) CpxRelTRS
4.00/1.76	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 144 ms]
4.00/1.76	(2) CpxRelTRS
4.00/1.76	(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
4.00/1.76	(4) CpxTRS
4.00/1.76	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 33 ms]
4.00/1.76	    (6) BOUNDS(1, n^1)
4.00/1.76	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.00/1.76	(8) TRS for Loop Detection
4.00/1.76	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.00/1.76	    (10) BEST
4.00/1.76	        (11) proven lower bound
4.00/1.76	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
4.00/1.76	            (13) BOUNDS(n^1, INF)
4.00/1.76	        (14) TRS for Loop Detection
4.00/1.76	
4.00/1.76	
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(0)
4.00/1.76	Obligation:
4.00/1.76	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.00/1.76	
4.00/1.76	
4.00/1.76	The TRS R consists of the following rules:
4.00/1.76	
4.00/1.76	   add0(S(x), x2) -> +(S(0), add0(x2, x))
4.00/1.76	   add0(0, x2) -> x2
4.00/1.76	
4.00/1.76	The (relative) TRS S consists of the following rules:
4.00/1.76	
4.00/1.76	   +(x, S(0)) -> S(x)
4.00/1.76	   +(S(0), y) -> S(y)
4.00/1.76	
4.00/1.76	Rewrite Strategy: INNERMOST
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
4.00/1.76	proved termination of relative rules
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(2)
4.00/1.76	Obligation:
4.00/1.76	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.00/1.76	
4.00/1.76	
4.00/1.76	The TRS R consists of the following rules:
4.00/1.76	
4.00/1.76	   add0(S(x), x2) -> +(S(0), add0(x2, x))
4.00/1.76	   add0(0, x2) -> x2
4.00/1.76	
4.00/1.76	The (relative) TRS S consists of the following rules:
4.00/1.76	
4.00/1.76	   +(x, S(0)) -> S(x)
4.00/1.76	   +(S(0), y) -> S(y)
4.00/1.76	
4.00/1.76	Rewrite Strategy: INNERMOST
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(3) RelTrsToTrsProof (UPPER BOUND(ID))
4.00/1.76	transformed relative TRS to TRS
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(4)
4.00/1.76	Obligation:
4.00/1.76	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
4.00/1.76	
4.00/1.76	
4.00/1.76	The TRS R consists of the following rules:
4.00/1.76	
4.00/1.76	   add0(S(x), x2) -> +(S(0), add0(x2, x))
4.00/1.76	   add0(0, x2) -> x2
4.00/1.76	   +(x, S(0)) -> S(x)
4.00/1.76	   +(S(0), y) -> S(y)
4.00/1.76	
4.00/1.76	S is empty.
4.00/1.76	Rewrite Strategy: INNERMOST
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(5) CpxTrsMatchBoundsTAProof (FINISHED)
4.00/1.76	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. 
4.00/1.76	
4.00/1.76	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
4.00/1.76	final states : [1, 2]
4.00/1.76	transitions: 
4.00/1.76	S0(0) -> 0
4.00/1.76	00() -> 0
4.00/1.76	add00(0, 0) -> 1
4.00/1.76	+0(0, 0) -> 2
4.00/1.76	01() -> 4
4.00/1.76	S1(4) -> 3
4.00/1.76	add01(0, 0) -> 5
4.00/1.76	+1(3, 5) -> 1
4.00/1.76	S1(0) -> 2
4.00/1.76	+1(3, 5) -> 5
4.00/1.76	S2(5) -> 1
4.00/1.76	S1(3) -> 1
4.00/1.76	S1(3) -> 5
4.00/1.76	S2(5) -> 5
4.00/1.76	0 -> 1
4.00/1.76	0 -> 5
4.00/1.76	
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(6)
4.00/1.76	BOUNDS(1, n^1)
4.00/1.76	
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.00/1.76	Transformed a relative TRS into a decreasing-loop problem.
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(8)
4.00/1.76	Obligation:
4.00/1.76	Analyzing the following TRS for decreasing loops:
4.00/1.76	
4.00/1.76	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.00/1.76	
4.00/1.76	
4.00/1.76	The TRS R consists of the following rules:
4.00/1.76	
4.00/1.76	   add0(S(x), x2) -> +(S(0), add0(x2, x))
4.00/1.76	   add0(0, x2) -> x2
4.00/1.76	
4.00/1.76	The (relative) TRS S consists of the following rules:
4.00/1.76	
4.00/1.76	   +(x, S(0)) -> S(x)
4.00/1.76	   +(S(0), y) -> S(y)
4.00/1.76	
4.00/1.76	Rewrite Strategy: INNERMOST
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(9) DecreasingLoopProof (LOWER BOUND(ID))
4.00/1.76	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.00/1.76	
4.00/1.76	The rewrite sequence
4.00/1.76	
4.00/1.76	add0(S(x), S(x1_0)) ->^+ +(S(0), +(S(0), add0(x, x1_0)))
4.00/1.76	
4.00/1.76	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
4.00/1.76	
4.00/1.76	The pumping substitution is [x / S(x), x1_0 / S(x1_0)].
4.00/1.76	
4.00/1.76	The result substitution is [ ].
4.00/1.76	
4.00/1.76	
4.00/1.76	
4.00/1.76	
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(10)
4.00/1.76	Complex Obligation (BEST)
4.00/1.76	
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(11)
4.00/1.76	Obligation:
4.00/1.76	Proved the lower bound n^1 for the following obligation:
4.00/1.76	
4.00/1.76	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.00/1.76	
4.00/1.76	
4.00/1.76	The TRS R consists of the following rules:
4.00/1.76	
4.00/1.76	   add0(S(x), x2) -> +(S(0), add0(x2, x))
4.00/1.76	   add0(0, x2) -> x2
4.00/1.76	
4.00/1.76	The (relative) TRS S consists of the following rules:
4.00/1.76	
4.00/1.76	   +(x, S(0)) -> S(x)
4.00/1.76	   +(S(0), y) -> S(y)
4.00/1.76	
4.00/1.76	Rewrite Strategy: INNERMOST
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(12) LowerBoundPropagationProof (FINISHED)
4.00/1.76	Propagated lower bound.
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(13)
4.00/1.76	BOUNDS(n^1, INF)
4.00/1.76	
4.00/1.76	----------------------------------------
4.00/1.76	
4.00/1.76	(14)
4.00/1.76	Obligation:
4.00/1.76	Analyzing the following TRS for decreasing loops:
4.00/1.76	
4.00/1.76	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.00/1.76	
4.00/1.76	
4.00/1.76	The TRS R consists of the following rules:
4.00/1.76	
4.00/1.76	   add0(S(x), x2) -> +(S(0), add0(x2, x))
4.00/1.76	   add0(0, x2) -> x2
4.00/1.76	
4.00/1.76	The (relative) TRS S consists of the following rules:
4.00/1.76	
4.00/1.76	   +(x, S(0)) -> S(x)
4.00/1.76	   +(S(0), y) -> S(y)
4.00/1.76	
4.00/1.76	Rewrite Strategy: INNERMOST
4.00/1.79	EOF