4.27/1.86	WORST_CASE(Omega(n^1), O(n^1))
4.27/1.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.27/1.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.27/1.88	
4.27/1.88	
4.27/1.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.27/1.88	
4.27/1.88	(0) CpxTRS
4.27/1.88	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
4.27/1.88	(2) CpxTRS
4.27/1.88	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 228 ms]
4.27/1.88	    (4) BOUNDS(1, n^1)
4.27/1.88	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.27/1.88	(6) TRS for Loop Detection
4.27/1.88	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.27/1.88	    (8) BEST
4.27/1.88	        (9) proven lower bound
4.27/1.88	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
4.27/1.88	            (11) BOUNDS(n^1, INF)
4.27/1.88	        (12) TRS for Loop Detection
4.27/1.88	
4.27/1.88	
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(0)
4.27/1.88	Obligation:
4.27/1.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.27/1.88	
4.27/1.88	
4.27/1.88	The TRS R consists of the following rules:
4.27/1.88	
4.27/1.88	   comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0))
4.27/1.88	   comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0)
4.27/1.88	   map#2(Nil) -> Nil
4.27/1.88	   map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6))
4.27/1.88	   plus_x#1(0, x6) -> x6
4.27/1.88	   plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10))
4.27/1.88	   foldr_f#3(Nil, 0) -> 0
4.27/1.88	   foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24)
4.27/1.88	   foldr#3(Nil) -> id
4.27/1.88	   foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6))
4.27/1.88	   main(x3) -> foldr_f#3(map#2(x3), 0)
4.27/1.88	
4.27/1.88	S is empty.
4.27/1.88	Rewrite Strategy: INNERMOST
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(1) RelTrsToTrsProof (UPPER BOUND(ID))
4.27/1.88	transformed relative TRS to TRS
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(2)
4.27/1.88	Obligation:
4.27/1.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
4.27/1.88	
4.27/1.88	
4.27/1.88	The TRS R consists of the following rules:
4.27/1.88	
4.27/1.88	   comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0))
4.27/1.88	   comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0)
4.27/1.88	   map#2(Nil) -> Nil
4.27/1.88	   map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6))
4.27/1.88	   plus_x#1(0, x6) -> x6
4.27/1.88	   plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10))
4.27/1.88	   foldr_f#3(Nil, 0) -> 0
4.27/1.88	   foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24)
4.27/1.88	   foldr#3(Nil) -> id
4.27/1.88	   foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6))
4.27/1.88	   main(x3) -> foldr_f#3(map#2(x3), 0)
4.27/1.88	
4.27/1.88	S is empty.
4.27/1.88	Rewrite Strategy: INNERMOST
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(3) CpxTrsMatchBoundsTAProof (FINISHED)
4.27/1.88	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. 
4.27/1.88	
4.27/1.88	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
4.27/1.88	final states : [1, 2, 3, 4, 5, 6]
4.27/1.88	transitions: 
4.27/1.88	plus_x0(0) -> 0
4.27/1.88	comp_f_g0(0, 0) -> 0
4.27/1.88	00() -> 0
4.27/1.88	id0() -> 0
4.27/1.88	Nil0() -> 0
4.27/1.88	Cons0(0, 0) -> 0
4.27/1.88	S0(0) -> 0
4.27/1.88	comp_f_g#10(0, 0, 0) -> 1
4.27/1.88	map#20(0) -> 2
4.27/1.88	plus_x#10(0, 0) -> 3
4.27/1.88	foldr_f#30(0, 0) -> 4
4.27/1.88	foldr#30(0) -> 5
4.27/1.88	main0(0) -> 6
4.27/1.88	01() -> 8
4.27/1.88	comp_f_g#11(0, 0, 8) -> 7
4.27/1.88	plus_x#11(0, 7) -> 1
4.27/1.88	01() -> 9
4.27/1.88	plus_x#11(0, 9) -> 1
4.27/1.88	Nil1() -> 2
4.27/1.88	plus_x1(0) -> 10
4.27/1.88	map#21(0) -> 11
4.27/1.88	Cons1(10, 11) -> 2
4.27/1.88	plus_x#11(0, 0) -> 12
4.27/1.88	S1(12) -> 3
4.27/1.88	01() -> 4
4.27/1.88	foldr#31(0) -> 13
4.27/1.88	comp_f_g#11(0, 13, 0) -> 4
4.27/1.88	id1() -> 5
4.27/1.88	foldr#31(0) -> 14
4.27/1.88	comp_f_g1(0, 14) -> 5
4.27/1.88	map#21(0) -> 15
4.27/1.88	01() -> 16
4.27/1.88	foldr_f#31(15, 16) -> 6
4.27/1.88	plus_x#11(0, 7) -> 7
4.27/1.88	plus_x#11(0, 9) -> 7
4.27/1.88	Nil1() -> 11
4.27/1.88	Nil1() -> 15
4.27/1.88	Cons1(10, 11) -> 11
4.27/1.88	Cons1(10, 11) -> 15
4.27/1.88	plus_x#11(0, 7) -> 12
4.27/1.88	S1(12) -> 1
4.27/1.88	plus_x#11(0, 9) -> 12
4.27/1.88	S1(12) -> 12
4.27/1.88	id1() -> 13
4.27/1.88	id1() -> 14
4.27/1.88	comp_f_g1(0, 14) -> 13
4.27/1.88	comp_f_g1(0, 14) -> 14
4.27/1.88	comp_f_g#11(0, 14, 8) -> 7
4.27/1.88	plus_x#11(0, 7) -> 4
4.27/1.88	plus_x#11(0, 9) -> 4
4.27/1.88	S1(12) -> 7
4.27/1.88	02() -> 6
4.27/1.88	foldr#32(11) -> 17
4.27/1.88	comp_f_g#12(10, 17, 16) -> 6
4.27/1.88	id2() -> 17
4.27/1.88	foldr#32(11) -> 18
4.27/1.88	comp_f_g2(10, 18) -> 17
4.27/1.88	id2() -> 18
4.27/1.88	comp_f_g2(10, 18) -> 18
4.27/1.88	02() -> 20
4.27/1.88	comp_f_g#12(10, 18, 20) -> 19
4.27/1.88	plus_x#12(0, 19) -> 6
4.27/1.88	02() -> 21
4.27/1.88	plus_x#12(0, 21) -> 6
4.27/1.88	plus_x#12(0, 19) -> 19
4.27/1.88	plus_x#12(0, 21) -> 19
4.27/1.88	plus_x#11(0, 19) -> 12
4.27/1.88	S1(12) -> 6
4.27/1.88	plus_x#11(0, 21) -> 12
4.27/1.88	S1(12) -> 19
4.27/1.88	0 -> 3
4.27/1.88	0 -> 12
4.27/1.88	7 -> 1
4.27/1.88	7 -> 12
4.27/1.88	7 -> 4
4.27/1.88	9 -> 1
4.27/1.88	9 -> 7
4.27/1.88	19 -> 6
4.27/1.88	19 -> 12
4.27/1.88	21 -> 6
4.27/1.88	21 -> 12
4.27/1.88	21 -> 19
4.27/1.88	
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(4)
4.27/1.88	BOUNDS(1, n^1)
4.27/1.88	
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.27/1.88	Transformed a relative TRS into a decreasing-loop problem.
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(6)
4.27/1.88	Obligation:
4.27/1.88	Analyzing the following TRS for decreasing loops:
4.27/1.88	
4.27/1.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.27/1.88	
4.27/1.88	
4.27/1.88	The TRS R consists of the following rules:
4.27/1.88	
4.27/1.88	   comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0))
4.27/1.88	   comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0)
4.27/1.88	   map#2(Nil) -> Nil
4.27/1.88	   map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6))
4.27/1.88	   plus_x#1(0, x6) -> x6
4.27/1.88	   plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10))
4.27/1.88	   foldr_f#3(Nil, 0) -> 0
4.27/1.88	   foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24)
4.27/1.88	   foldr#3(Nil) -> id
4.27/1.88	   foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6))
4.27/1.88	   main(x3) -> foldr_f#3(map#2(x3), 0)
4.27/1.88	
4.27/1.88	S is empty.
4.27/1.88	Rewrite Strategy: INNERMOST
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(7) DecreasingLoopProof (LOWER BOUND(ID))
4.27/1.88	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.27/1.88	
4.27/1.88	The rewrite sequence
4.27/1.88	
4.27/1.88	map#2(Cons(x16, x6)) ->^+ Cons(plus_x(x16), map#2(x6))
4.27/1.88	
4.27/1.88	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
4.27/1.88	
4.27/1.88	The pumping substitution is [x6 / Cons(x16, x6)].
4.27/1.88	
4.27/1.88	The result substitution is [ ].
4.27/1.88	
4.27/1.88	
4.27/1.88	
4.27/1.88	
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(8)
4.27/1.88	Complex Obligation (BEST)
4.27/1.88	
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(9)
4.27/1.88	Obligation:
4.27/1.88	Proved the lower bound n^1 for the following obligation:
4.27/1.88	
4.27/1.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.27/1.88	
4.27/1.88	
4.27/1.88	The TRS R consists of the following rules:
4.27/1.88	
4.27/1.88	   comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0))
4.27/1.88	   comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0)
4.27/1.88	   map#2(Nil) -> Nil
4.27/1.88	   map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6))
4.27/1.88	   plus_x#1(0, x6) -> x6
4.27/1.88	   plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10))
4.27/1.88	   foldr_f#3(Nil, 0) -> 0
4.27/1.88	   foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24)
4.27/1.88	   foldr#3(Nil) -> id
4.27/1.88	   foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6))
4.27/1.88	   main(x3) -> foldr_f#3(map#2(x3), 0)
4.27/1.88	
4.27/1.88	S is empty.
4.27/1.88	Rewrite Strategy: INNERMOST
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(10) LowerBoundPropagationProof (FINISHED)
4.27/1.88	Propagated lower bound.
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(11)
4.27/1.88	BOUNDS(n^1, INF)
4.27/1.88	
4.27/1.88	----------------------------------------
4.27/1.88	
4.27/1.88	(12)
4.27/1.88	Obligation:
4.27/1.88	Analyzing the following TRS for decreasing loops:
4.27/1.88	
4.27/1.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.27/1.88	
4.27/1.88	
4.27/1.88	The TRS R consists of the following rules:
4.27/1.88	
4.27/1.88	   comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0))
4.27/1.88	   comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0)
4.27/1.88	   map#2(Nil) -> Nil
4.27/1.88	   map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6))
4.27/1.88	   plus_x#1(0, x6) -> x6
4.27/1.88	   plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10))
4.27/1.88	   foldr_f#3(Nil, 0) -> 0
4.27/1.88	   foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24)
4.27/1.88	   foldr#3(Nil) -> id
4.27/1.88	   foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6))
4.27/1.88	   main(x3) -> foldr_f#3(map#2(x3), 0)
4.27/1.88	
4.27/1.88	S is empty.
4.27/1.88	Rewrite Strategy: INNERMOST
4.47/2.05	EOF