1120.94/291.49	WORST_CASE(Omega(n^1), ?)
1121.23/291.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1121.23/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1121.23/291.57	
1121.23/291.57	
1121.23/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.23/291.57	
1121.23/291.57	(0) CpxTRS
1121.23/291.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1121.23/291.57	(2) TRS for Loop Detection
1121.23/291.57	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1121.23/291.57	(4) BEST
1121.23/291.57	    (5) proven lower bound
1121.23/291.57	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1121.23/291.57	        (7) BOUNDS(n^1, INF)
1121.23/291.57	    (8) TRS for Loop Detection
1121.23/291.57	
1121.23/291.57	
1121.23/291.57	----------------------------------------
1121.23/291.57	
1121.23/291.57	(0)
1121.23/291.57	Obligation:
1121.23/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.23/291.57	
1121.23/291.57	
1121.23/291.57	The TRS R consists of the following rules:
1121.23/291.57	
1121.23/291.57	   divide_ys#1(x2, x1) -> Cons(take#2(x1, x2), Cons(drop#2(x1, x2), Nil))
1121.23/291.57	   cond_merge_ys_zs_2(True, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x4, merge#2(x3, Cons(x5, x6)))
1121.23/291.57	   cond_merge_ys_zs_2(False, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x2, merge#2(Cons(x7, x8), x1))
1121.23/291.58	   merge#2(Nil, x2) -> x2
1121.23/291.58	   merge#2(Cons(x4, x2), Nil) -> Cons(x4, x2)
1121.23/291.58	   merge#2(Cons(x8, x6), Cons(x4, x2)) -> cond_merge_ys_zs_2(leq#2(x8, x4), Cons(x8, x6), Cons(x4, x2), x8, x6, x4, x2)
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Nil) -> Nil
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x229, Nil)) -> Cons(x229, Nil)
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x51, Cons(x25, x33))) -> const_f#2(Cons(x51, Cons(x25, x33)), map#2(dc(map, divisible, mergesort_zs_3, divide, const_f), divide_ys#1(Cons(x51, Cons(x25, x33)), S(halve#1(length#1(x33))))))
1121.23/291.58	   drop#2(0, x2) -> x2
1121.23/291.58	   drop#2(S(0), Nil) -> bot[1]
1121.23/291.58	   drop#2(S(x10), Cons(x56, x64)) -> drop#2(x10, x64)
1121.23/291.58	   take#2(0, x2) -> Nil
1121.23/291.58	   take#2(S(0), Nil) -> Cons(bot[0], Nil)
1121.23/291.58	   take#2(S(x22), Cons(x56, x64)) -> Cons(x56, take#2(x22, x64))
1121.23/291.58	   halve#1(0) -> 0
1121.23/291.58	   halve#1(S(0)) -> S(0)
1121.23/291.58	   halve#1(S(S(x14))) -> S(halve#1(x14))
1121.23/291.58	   const_f#2(x3, Cons(x6, Cons(x4, x2))) -> merge#2(x6, x4)
1121.23/291.58	   leq#2(0, x16) -> True
1121.23/291.58	   leq#2(S(x20), 0) -> False
1121.23/291.58	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
1121.23/291.58	   length#1(Nil) -> 0
1121.23/291.58	   length#1(Cons(x6, x8)) -> S(length#1(x8))
1121.23/291.58	   map#2(dc(x2, x4, x6, x8, x10), Nil) -> Nil
1121.23/291.58	   map#2(dc(x6, x8, x10, x12, x14), Cons(x4, x2)) -> Cons(dc#1(x6, x8, x10, x12, x14, x4), map#2(dc(x6, x8, x10, x12, x14), x2))
1121.23/291.58	   main(x113) -> dc#1(map, divisible, mergesort_zs_3, divide, const_f, x113)
1121.23/291.58	
1121.23/291.58	S is empty.
1121.23/291.58	Rewrite Strategy: INNERMOST
1121.23/291.58	----------------------------------------
1121.23/291.58	
1121.23/291.58	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1121.23/291.58	Transformed a relative TRS into a decreasing-loop problem.
1121.23/291.58	----------------------------------------
1121.23/291.58	
1121.23/291.58	(2)
1121.23/291.58	Obligation:
1121.23/291.58	Analyzing the following TRS for decreasing loops:
1121.23/291.58	
1121.23/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.23/291.58	
1121.23/291.58	
1121.23/291.58	The TRS R consists of the following rules:
1121.23/291.58	
1121.23/291.58	   divide_ys#1(x2, x1) -> Cons(take#2(x1, x2), Cons(drop#2(x1, x2), Nil))
1121.23/291.58	   cond_merge_ys_zs_2(True, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x4, merge#2(x3, Cons(x5, x6)))
1121.23/291.58	   cond_merge_ys_zs_2(False, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x2, merge#2(Cons(x7, x8), x1))
1121.23/291.58	   merge#2(Nil, x2) -> x2
1121.23/291.58	   merge#2(Cons(x4, x2), Nil) -> Cons(x4, x2)
1121.23/291.58	   merge#2(Cons(x8, x6), Cons(x4, x2)) -> cond_merge_ys_zs_2(leq#2(x8, x4), Cons(x8, x6), Cons(x4, x2), x8, x6, x4, x2)
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Nil) -> Nil
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x229, Nil)) -> Cons(x229, Nil)
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x51, Cons(x25, x33))) -> const_f#2(Cons(x51, Cons(x25, x33)), map#2(dc(map, divisible, mergesort_zs_3, divide, const_f), divide_ys#1(Cons(x51, Cons(x25, x33)), S(halve#1(length#1(x33))))))
1121.23/291.58	   drop#2(0, x2) -> x2
1121.23/291.58	   drop#2(S(0), Nil) -> bot[1]
1121.23/291.58	   drop#2(S(x10), Cons(x56, x64)) -> drop#2(x10, x64)
1121.23/291.58	   take#2(0, x2) -> Nil
1121.23/291.58	   take#2(S(0), Nil) -> Cons(bot[0], Nil)
1121.23/291.58	   take#2(S(x22), Cons(x56, x64)) -> Cons(x56, take#2(x22, x64))
1121.23/291.58	   halve#1(0) -> 0
1121.23/291.58	   halve#1(S(0)) -> S(0)
1121.23/291.58	   halve#1(S(S(x14))) -> S(halve#1(x14))
1121.23/291.58	   const_f#2(x3, Cons(x6, Cons(x4, x2))) -> merge#2(x6, x4)
1121.23/291.58	   leq#2(0, x16) -> True
1121.23/291.58	   leq#2(S(x20), 0) -> False
1121.23/291.58	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
1121.23/291.58	   length#1(Nil) -> 0
1121.23/291.58	   length#1(Cons(x6, x8)) -> S(length#1(x8))
1121.23/291.58	   map#2(dc(x2, x4, x6, x8, x10), Nil) -> Nil
1121.23/291.58	   map#2(dc(x6, x8, x10, x12, x14), Cons(x4, x2)) -> Cons(dc#1(x6, x8, x10, x12, x14, x4), map#2(dc(x6, x8, x10, x12, x14), x2))
1121.23/291.58	   main(x113) -> dc#1(map, divisible, mergesort_zs_3, divide, const_f, x113)
1121.23/291.58	
1121.23/291.58	S is empty.
1121.23/291.58	Rewrite Strategy: INNERMOST
1121.23/291.58	----------------------------------------
1121.23/291.58	
1121.23/291.58	(3) DecreasingLoopProof (LOWER BOUND(ID))
1121.23/291.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1121.23/291.58	
1121.23/291.58	The rewrite sequence
1121.23/291.58	
1121.23/291.58	length#1(Cons(x6, x8)) ->^+ S(length#1(x8))
1121.23/291.58	
1121.23/291.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1121.23/291.58	
1121.23/291.58	The pumping substitution is [x8 / Cons(x6, x8)].
1121.23/291.58	
1121.23/291.58	The result substitution is [ ].
1121.23/291.58	
1121.23/291.58	
1121.23/291.58	
1121.23/291.58	
1121.23/291.58	----------------------------------------
1121.23/291.58	
1121.23/291.58	(4)
1121.23/291.58	Complex Obligation (BEST)
1121.23/291.58	
1121.23/291.58	----------------------------------------
1121.23/291.58	
1121.23/291.58	(5)
1121.23/291.58	Obligation:
1121.23/291.58	Proved the lower bound n^1 for the following obligation:
1121.23/291.58	
1121.23/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.23/291.58	
1121.23/291.58	
1121.23/291.58	The TRS R consists of the following rules:
1121.23/291.58	
1121.23/291.58	   divide_ys#1(x2, x1) -> Cons(take#2(x1, x2), Cons(drop#2(x1, x2), Nil))
1121.23/291.58	   cond_merge_ys_zs_2(True, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x4, merge#2(x3, Cons(x5, x6)))
1121.23/291.58	   cond_merge_ys_zs_2(False, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x2, merge#2(Cons(x7, x8), x1))
1121.23/291.58	   merge#2(Nil, x2) -> x2
1121.23/291.58	   merge#2(Cons(x4, x2), Nil) -> Cons(x4, x2)
1121.23/291.58	   merge#2(Cons(x8, x6), Cons(x4, x2)) -> cond_merge_ys_zs_2(leq#2(x8, x4), Cons(x8, x6), Cons(x4, x2), x8, x6, x4, x2)
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Nil) -> Nil
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x229, Nil)) -> Cons(x229, Nil)
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x51, Cons(x25, x33))) -> const_f#2(Cons(x51, Cons(x25, x33)), map#2(dc(map, divisible, mergesort_zs_3, divide, const_f), divide_ys#1(Cons(x51, Cons(x25, x33)), S(halve#1(length#1(x33))))))
1121.23/291.58	   drop#2(0, x2) -> x2
1121.23/291.58	   drop#2(S(0), Nil) -> bot[1]
1121.23/291.58	   drop#2(S(x10), Cons(x56, x64)) -> drop#2(x10, x64)
1121.23/291.58	   take#2(0, x2) -> Nil
1121.23/291.58	   take#2(S(0), Nil) -> Cons(bot[0], Nil)
1121.23/291.58	   take#2(S(x22), Cons(x56, x64)) -> Cons(x56, take#2(x22, x64))
1121.23/291.58	   halve#1(0) -> 0
1121.23/291.58	   halve#1(S(0)) -> S(0)
1121.23/291.58	   halve#1(S(S(x14))) -> S(halve#1(x14))
1121.23/291.58	   const_f#2(x3, Cons(x6, Cons(x4, x2))) -> merge#2(x6, x4)
1121.23/291.58	   leq#2(0, x16) -> True
1121.23/291.58	   leq#2(S(x20), 0) -> False
1121.23/291.58	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
1121.23/291.58	   length#1(Nil) -> 0
1121.23/291.58	   length#1(Cons(x6, x8)) -> S(length#1(x8))
1121.23/291.58	   map#2(dc(x2, x4, x6, x8, x10), Nil) -> Nil
1121.23/291.58	   map#2(dc(x6, x8, x10, x12, x14), Cons(x4, x2)) -> Cons(dc#1(x6, x8, x10, x12, x14, x4), map#2(dc(x6, x8, x10, x12, x14), x2))
1121.23/291.58	   main(x113) -> dc#1(map, divisible, mergesort_zs_3, divide, const_f, x113)
1121.23/291.58	
1121.23/291.58	S is empty.
1121.23/291.58	Rewrite Strategy: INNERMOST
1121.23/291.58	----------------------------------------
1121.23/291.58	
1121.23/291.58	(6) LowerBoundPropagationProof (FINISHED)
1121.23/291.58	Propagated lower bound.
1121.23/291.58	----------------------------------------
1121.23/291.58	
1121.23/291.58	(7)
1121.23/291.58	BOUNDS(n^1, INF)
1121.23/291.58	
1121.23/291.58	----------------------------------------
1121.23/291.58	
1121.23/291.58	(8)
1121.23/291.58	Obligation:
1121.23/291.58	Analyzing the following TRS for decreasing loops:
1121.23/291.58	
1121.23/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.23/291.58	
1121.23/291.58	
1121.23/291.58	The TRS R consists of the following rules:
1121.23/291.58	
1121.23/291.58	   divide_ys#1(x2, x1) -> Cons(take#2(x1, x2), Cons(drop#2(x1, x2), Nil))
1121.23/291.58	   cond_merge_ys_zs_2(True, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x4, merge#2(x3, Cons(x5, x6)))
1121.23/291.58	   cond_merge_ys_zs_2(False, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x2, merge#2(Cons(x7, x8), x1))
1121.23/291.58	   merge#2(Nil, x2) -> x2
1121.23/291.58	   merge#2(Cons(x4, x2), Nil) -> Cons(x4, x2)
1121.23/291.58	   merge#2(Cons(x8, x6), Cons(x4, x2)) -> cond_merge_ys_zs_2(leq#2(x8, x4), Cons(x8, x6), Cons(x4, x2), x8, x6, x4, x2)
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Nil) -> Nil
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x229, Nil)) -> Cons(x229, Nil)
1121.23/291.58	   dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x51, Cons(x25, x33))) -> const_f#2(Cons(x51, Cons(x25, x33)), map#2(dc(map, divisible, mergesort_zs_3, divide, const_f), divide_ys#1(Cons(x51, Cons(x25, x33)), S(halve#1(length#1(x33))))))
1121.23/291.58	   drop#2(0, x2) -> x2
1121.23/291.58	   drop#2(S(0), Nil) -> bot[1]
1121.23/291.58	   drop#2(S(x10), Cons(x56, x64)) -> drop#2(x10, x64)
1121.23/291.58	   take#2(0, x2) -> Nil
1121.23/291.58	   take#2(S(0), Nil) -> Cons(bot[0], Nil)
1121.23/291.58	   take#2(S(x22), Cons(x56, x64)) -> Cons(x56, take#2(x22, x64))
1121.23/291.58	   halve#1(0) -> 0
1121.23/291.58	   halve#1(S(0)) -> S(0)
1121.23/291.58	   halve#1(S(S(x14))) -> S(halve#1(x14))
1121.23/291.58	   const_f#2(x3, Cons(x6, Cons(x4, x2))) -> merge#2(x6, x4)
1121.23/291.58	   leq#2(0, x16) -> True
1121.23/291.58	   leq#2(S(x20), 0) -> False
1121.23/291.58	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
1121.23/291.58	   length#1(Nil) -> 0
1121.23/291.58	   length#1(Cons(x6, x8)) -> S(length#1(x8))
1121.23/291.58	   map#2(dc(x2, x4, x6, x8, x10), Nil) -> Nil
1121.23/291.58	   map#2(dc(x6, x8, x10, x12, x14), Cons(x4, x2)) -> Cons(dc#1(x6, x8, x10, x12, x14, x4), map#2(dc(x6, x8, x10, x12, x14), x2))
1121.23/291.58	   main(x113) -> dc#1(map, divisible, mergesort_zs_3, divide, const_f, x113)
1121.23/291.58	
1121.23/291.58	S is empty.
1121.23/291.58	Rewrite Strategy: INNERMOST
1121.50/291.65	EOF