3.36/1.60	WORST_CASE(Omega(n^1), O(n^1))
3.36/1.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.36/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.36/1.61	
3.36/1.61	
3.36/1.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.61	
3.36/1.61	(0) CpxTRS
3.36/1.61	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.36/1.61	(2) CpxTRS
3.36/1.61	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 37 ms]
3.36/1.61	    (4) BOUNDS(1, n^1)
3.36/1.61	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.36/1.61	(6) TRS for Loop Detection
3.36/1.61	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.36/1.61	    (8) BEST
3.36/1.61	        (9) proven lower bound
3.36/1.61	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.36/1.61	            (11) BOUNDS(n^1, INF)
3.36/1.61	        (12) TRS for Loop Detection
3.36/1.61	
3.36/1.61	
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(0)
3.36/1.61	Obligation:
3.36/1.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.61	
3.36/1.61	
3.36/1.61	The TRS R consists of the following rules:
3.36/1.61	
3.36/1.61	   walk#1(Nil) -> walk_xs
3.36/1.61	   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
3.36/1.61	   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
3.36/1.61	   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
3.36/1.61	   main(Nil) -> Nil
3.36/1.61	   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)
3.36/1.61	
3.36/1.61	S is empty.
3.36/1.61	Rewrite Strategy: INNERMOST
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.36/1.61	transformed relative TRS to TRS
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(2)
3.36/1.61	Obligation:
3.36/1.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.36/1.61	
3.36/1.61	
3.36/1.61	The TRS R consists of the following rules:
3.36/1.61	
3.36/1.61	   walk#1(Nil) -> walk_xs
3.36/1.61	   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
3.36/1.61	   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
3.36/1.61	   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
3.36/1.61	   main(Nil) -> Nil
3.36/1.61	   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)
3.36/1.61	
3.36/1.61	S is empty.
3.36/1.61	Rewrite Strategy: INNERMOST
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.36/1.61	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. 
3.36/1.61	
3.36/1.61	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.36/1.61	final states : [1, 2, 3]
3.36/1.61	transitions: 
3.36/1.61	Nil0() -> 0
3.36/1.61	walk_xs0() -> 0
3.36/1.61	Cons0(0, 0) -> 0
3.36/1.61	comp_f_g0(0, 0) -> 0
3.36/1.61	walk_xs_30(0) -> 0
3.36/1.61	walk#10(0) -> 1
3.36/1.61	comp_f_g#10(0, 0, 0) -> 2
3.36/1.61	main0(0) -> 3
3.36/1.61	walk_xs1() -> 1
3.36/1.61	walk#11(0) -> 4
3.36/1.61	walk_xs_31(0) -> 5
3.36/1.61	comp_f_g1(4, 5) -> 1
3.36/1.61	Cons1(0, 0) -> 6
3.36/1.61	comp_f_g#11(0, 0, 6) -> 2
3.36/1.61	Cons1(0, 0) -> 2
3.36/1.61	Nil1() -> 3
3.36/1.61	walk#11(0) -> 7
3.36/1.61	walk_xs_31(0) -> 8
3.36/1.61	Nil1() -> 9
3.36/1.61	comp_f_g#11(7, 8, 9) -> 3
3.36/1.61	walk_xs1() -> 4
3.36/1.61	walk_xs1() -> 7
3.36/1.61	comp_f_g1(4, 5) -> 4
3.36/1.61	comp_f_g1(4, 5) -> 7
3.36/1.61	Cons1(0, 6) -> 6
3.36/1.61	Cons1(0, 6) -> 2
3.36/1.61	Cons2(0, 9) -> 10
3.36/1.61	comp_f_g#12(4, 5, 10) -> 3
3.36/1.61	Cons2(0, 9) -> 3
3.36/1.61	Cons2(0, 10) -> 10
3.36/1.61	Cons2(0, 10) -> 3
3.36/1.61	
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(4)
3.36/1.61	BOUNDS(1, n^1)
3.36/1.61	
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.36/1.61	Transformed a relative TRS into a decreasing-loop problem.
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(6)
3.36/1.61	Obligation:
3.36/1.61	Analyzing the following TRS for decreasing loops:
3.36/1.61	
3.36/1.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.61	
3.36/1.61	
3.36/1.61	The TRS R consists of the following rules:
3.36/1.61	
3.36/1.61	   walk#1(Nil) -> walk_xs
3.36/1.61	   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
3.36/1.61	   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
3.36/1.61	   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
3.36/1.61	   main(Nil) -> Nil
3.36/1.61	   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)
3.36/1.61	
3.36/1.61	S is empty.
3.36/1.61	Rewrite Strategy: INNERMOST
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.36/1.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.36/1.61	
3.36/1.61	The rewrite sequence
3.36/1.61	
3.36/1.61	walk#1(Cons(x4, x3)) ->^+ comp_f_g(walk#1(x3), walk_xs_3(x4))
3.36/1.61	
3.36/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.36/1.61	
3.36/1.61	The pumping substitution is [x3 / Cons(x4, x3)].
3.36/1.61	
3.36/1.61	The result substitution is [ ].
3.36/1.61	
3.36/1.61	
3.36/1.61	
3.36/1.61	
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(8)
3.36/1.61	Complex Obligation (BEST)
3.36/1.61	
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(9)
3.36/1.61	Obligation:
3.36/1.61	Proved the lower bound n^1 for the following obligation:
3.36/1.61	
3.36/1.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.61	
3.36/1.61	
3.36/1.61	The TRS R consists of the following rules:
3.36/1.61	
3.36/1.61	   walk#1(Nil) -> walk_xs
3.36/1.61	   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
3.36/1.61	   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
3.36/1.61	   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
3.36/1.61	   main(Nil) -> Nil
3.36/1.61	   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)
3.36/1.61	
3.36/1.61	S is empty.
3.36/1.61	Rewrite Strategy: INNERMOST
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(10) LowerBoundPropagationProof (FINISHED)
3.36/1.61	Propagated lower bound.
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(11)
3.36/1.61	BOUNDS(n^1, INF)
3.36/1.61	
3.36/1.61	----------------------------------------
3.36/1.61	
3.36/1.61	(12)
3.36/1.61	Obligation:
3.36/1.61	Analyzing the following TRS for decreasing loops:
3.36/1.61	
3.36/1.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.61	
3.36/1.61	
3.36/1.61	The TRS R consists of the following rules:
3.36/1.61	
3.36/1.61	   walk#1(Nil) -> walk_xs
3.36/1.61	   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
3.36/1.61	   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
3.36/1.61	   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
3.36/1.61	   main(Nil) -> Nil
3.36/1.61	   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)
3.36/1.61	
3.36/1.61	S is empty.
3.36/1.61	Rewrite Strategy: INNERMOST
3.36/1.65	EOF