3.05/2.57	WORST_CASE(Omega(n^1), O(n^1))
3.05/2.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.05/2.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.05/2.58	
3.05/2.58	
3.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/2.58	
3.05/2.58	(0) CpxTRS
3.05/2.58	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.05/2.58	(2) CpxTRS
3.05/2.58	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 40 ms]
3.05/2.58	    (4) BOUNDS(1, n^1)
3.05/2.58	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.05/2.58	(6) TRS for Loop Detection
3.05/2.58	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.05/2.58	    (8) BEST
3.05/2.58	        (9) proven lower bound
3.05/2.58	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.05/2.58	            (11) BOUNDS(n^1, INF)
3.05/2.58	        (12) TRS for Loop Detection
3.05/2.58	
3.05/2.58	
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(0)
3.05/2.58	Obligation:
3.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/2.58	
3.05/2.58	
3.05/2.58	The TRS R consists of the following rules:
3.05/2.58	
3.05/2.58	   plus#2(0, x12) -> x12
3.05/2.58	   plus#2(S(x4), x2) -> S(plus#2(x4, x2))
3.05/2.58	   fold#3(Nil) -> 0
3.05/2.58	   fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2))
3.05/2.58	   main(x1) -> fold#3(x1)
3.05/2.58	
3.05/2.58	S is empty.
3.05/2.58	Rewrite Strategy: INNERMOST
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.05/2.58	transformed relative TRS to TRS
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(2)
3.05/2.58	Obligation:
3.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.05/2.58	
3.05/2.58	
3.05/2.58	The TRS R consists of the following rules:
3.05/2.58	
3.05/2.58	   plus#2(0, x12) -> x12
3.05/2.58	   plus#2(S(x4), x2) -> S(plus#2(x4, x2))
3.05/2.58	   fold#3(Nil) -> 0
3.05/2.58	   fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2))
3.05/2.58	   main(x1) -> fold#3(x1)
3.05/2.58	
3.05/2.58	S is empty.
3.05/2.58	Rewrite Strategy: INNERMOST
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.05/2.58	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. 
3.05/2.58	
3.05/2.58	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.05/2.58	final states : [1, 2, 3]
3.05/2.58	transitions: 
3.05/2.58	00() -> 0
3.05/2.58	S0(0) -> 0
3.05/2.58	Nil0() -> 0
3.05/2.58	Cons0(0, 0) -> 0
3.05/2.58	plus#20(0, 0) -> 1
3.05/2.58	fold#30(0) -> 2
3.05/2.58	main0(0) -> 3
3.05/2.58	plus#21(0, 0) -> 4
3.05/2.58	S1(4) -> 1
3.05/2.58	01() -> 2
3.05/2.58	fold#31(0) -> 5
3.05/2.58	plus#21(0, 5) -> 2
3.05/2.58	fold#31(0) -> 3
3.05/2.58	plus#21(0, 5) -> 4
3.05/2.58	S1(4) -> 2
3.05/2.58	S1(4) -> 4
3.05/2.58	01() -> 3
3.05/2.58	01() -> 5
3.05/2.58	plus#21(0, 5) -> 3
3.05/2.58	plus#21(0, 5) -> 5
3.05/2.58	S1(4) -> 3
3.05/2.58	S1(4) -> 5
3.05/2.58	0 -> 1
3.05/2.58	0 -> 4
3.05/2.58	5 -> 2
3.05/2.58	5 -> 3
3.05/2.58	5 -> 4
3.05/2.58	
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(4)
3.05/2.58	BOUNDS(1, n^1)
3.05/2.58	
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.05/2.58	Transformed a relative TRS into a decreasing-loop problem.
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(6)
3.05/2.58	Obligation:
3.05/2.58	Analyzing the following TRS for decreasing loops:
3.05/2.58	
3.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/2.58	
3.05/2.58	
3.05/2.58	The TRS R consists of the following rules:
3.05/2.58	
3.05/2.58	   plus#2(0, x12) -> x12
3.05/2.58	   plus#2(S(x4), x2) -> S(plus#2(x4, x2))
3.05/2.58	   fold#3(Nil) -> 0
3.05/2.58	   fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2))
3.05/2.58	   main(x1) -> fold#3(x1)
3.05/2.58	
3.05/2.58	S is empty.
3.05/2.58	Rewrite Strategy: INNERMOST
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.05/2.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.05/2.58	
3.05/2.58	The rewrite sequence
3.05/2.58	
3.05/2.58	fold#3(Cons(x4, x2)) ->^+ plus#2(x4, fold#3(x2))
3.05/2.58	
3.05/2.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.05/2.58	
3.05/2.58	The pumping substitution is [x2 / Cons(x4, x2)].
3.05/2.58	
3.05/2.58	The result substitution is [ ].
3.05/2.58	
3.05/2.58	
3.05/2.58	
3.05/2.58	
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(8)
3.05/2.58	Complex Obligation (BEST)
3.05/2.58	
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(9)
3.05/2.58	Obligation:
3.05/2.58	Proved the lower bound n^1 for the following obligation:
3.05/2.58	
3.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/2.58	
3.05/2.58	
3.05/2.58	The TRS R consists of the following rules:
3.05/2.58	
3.05/2.58	   plus#2(0, x12) -> x12
3.05/2.58	   plus#2(S(x4), x2) -> S(plus#2(x4, x2))
3.05/2.58	   fold#3(Nil) -> 0
3.05/2.58	   fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2))
3.05/2.58	   main(x1) -> fold#3(x1)
3.05/2.58	
3.05/2.58	S is empty.
3.05/2.58	Rewrite Strategy: INNERMOST
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(10) LowerBoundPropagationProof (FINISHED)
3.05/2.58	Propagated lower bound.
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(11)
3.05/2.58	BOUNDS(n^1, INF)
3.05/2.58	
3.05/2.58	----------------------------------------
3.05/2.58	
3.05/2.58	(12)
3.05/2.58	Obligation:
3.05/2.58	Analyzing the following TRS for decreasing loops:
3.05/2.58	
3.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/2.58	
3.05/2.58	
3.05/2.58	The TRS R consists of the following rules:
3.05/2.58	
3.05/2.58	   plus#2(0, x12) -> x12
3.05/2.58	   plus#2(S(x4), x2) -> S(plus#2(x4, x2))
3.05/2.58	   fold#3(Nil) -> 0
3.05/2.58	   fold#3(Cons(x4, x2)) -> plus#2(x4, fold#3(x2))
3.05/2.58	   main(x1) -> fold#3(x1)
3.05/2.58	
3.05/2.58	S is empty.
3.05/2.58	Rewrite Strategy: INNERMOST
3.43/2.61	EOF