8.34/3.60	YES
10.65/4.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.65/4.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.65/4.16	
10.65/4.16	
10.65/4.16	H-Termination with start terms of the given HASKELL could be proven:
10.65/4.16	
10.65/4.16	(0) HASKELL
10.65/4.16	(1) BR [EQUIVALENT, 0 ms]
10.65/4.16	(2) HASKELL
10.65/4.16	(3) COR [EQUIVALENT, 0 ms]
10.65/4.16	(4) HASKELL
10.65/4.16	(5) LetRed [EQUIVALENT, 0 ms]
10.65/4.16	(6) HASKELL
10.65/4.16	(7) NumRed [SOUND, 6 ms]
10.65/4.16	(8) HASKELL
10.65/4.16	(9) Narrow [SOUND, 0 ms]
10.65/4.16	(10) QDP
10.65/4.16	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.65/4.16	(12) YES
10.65/4.16	
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(0)
10.65/4.16	Obligation:
10.65/4.16	mainModule Main 
10.65/4.16	module Main where {
10.65/4.16	import qualified Prelude;
10.65/4.16	}
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(1) BR (EQUIVALENT)
10.65/4.16	Replaced joker patterns by fresh variables and removed binding patterns.
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(2)
10.65/4.16	Obligation:
10.65/4.16	mainModule Main 
10.65/4.16	module Main where {
10.65/4.16	import qualified Prelude;
10.65/4.16	}
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(3) COR (EQUIVALENT)
10.65/4.16	Cond Reductions:
10.65/4.16	The following Function with conditions
10.65/4.16	"undefined |Falseundefined;
10.65/4.16	"
10.65/4.16	is transformed to
10.65/4.16	"undefined  = undefined1;
10.65/4.16	"
10.65/4.16	"undefined0 True = undefined;
10.65/4.16	"
10.65/4.16	"undefined1  = undefined0 False;
10.65/4.16	"
10.65/4.16	The following Function with conditions
10.65/4.16	"take n vx|n <= 0[];
10.65/4.16	take vy [] = [];
10.65/4.16	take n (x : xs) = x : take (n - 1) xs;
10.65/4.16	"
10.65/4.16	is transformed to
10.65/4.16	"take n vx = take3 n vx;
10.65/4.16	take vy [] = take1 vy [];
10.65/4.16	take n (x : xs) = take0 n (x : xs);
10.65/4.16	"
10.65/4.16	"take0 n (x : xs) = x : take (n - 1) xs;
10.65/4.16	"
10.65/4.16	"take1 vy [] = [];
10.65/4.16	take1 wv ww = take0 wv ww;
10.65/4.16	"
10.65/4.16	"take2 n vx True = [];
10.65/4.16	take2 n vx False = take1 n vx;
10.65/4.16	"
10.65/4.16	"take3 n vx = take2 n vx (n <= 0);
10.65/4.16	take3 wx wy = take1 wx wy;
10.65/4.16	"
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(4)
10.65/4.16	Obligation:
10.65/4.16	mainModule Main 
10.65/4.16	module Main where {
10.65/4.16	import qualified Prelude;
10.65/4.16	}
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(5) LetRed (EQUIVALENT)
10.65/4.16	Let/Where Reductions:
10.65/4.16	The bindings of the following Let/Where expression
10.65/4.16	"xs where {
10.65/4.16	xs  = x : xs;
10.65/4.16	} 
10.65/4.16	"
10.65/4.16	are unpacked to the following functions on top level
10.65/4.16	"repeatXs wz = wz : repeatXs wz;
10.65/4.16	"
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(6)
10.65/4.16	Obligation:
10.65/4.16	mainModule Main 
10.65/4.16	module Main where {
10.65/4.16	import qualified Prelude;
10.65/4.16	}
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(7) NumRed (SOUND)
10.65/4.16	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(8)
10.65/4.16	Obligation:
10.65/4.16	mainModule Main 
10.65/4.16	module Main where {
10.65/4.16	import qualified Prelude;
10.65/4.16	}
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(9) Narrow (SOUND)
10.65/4.16	Haskell To QDPs
10.65/4.16	
10.65/4.16	digraph dp_graph {
10.65/4.16	node [outthreshold=100, inthreshold=100];1[label="replicate",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.65/4.16	3[label="replicate xu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.65/4.16	4[label="replicate xu3 xu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.65/4.16	5[label="take xu3 (repeat xu4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.65/4.16	6[label="take3 xu3 (repeat xu4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.65/4.16	7[label="take2 xu3 (repeat xu4) (xu3 <= Pos Zero)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.65/4.16	8[label="take2 xu3 (repeat xu4) (compare xu3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.65/4.16	9[label="take2 xu3 (repeat xu4) (not (compare xu3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10.65/4.16	10[label="take2 xu3 (repeat xu4) (not (primCmpInt xu3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];61[label="xu3/Pos xu30",fontsize=10,color="white",style="solid",shape="box"];10 -> 61[label="",style="solid", color="burlywood", weight=9];
10.65/4.16	61 -> 11[label="",style="solid", color="burlywood", weight=3];
10.65/4.16	62[label="xu3/Neg xu30",fontsize=10,color="white",style="solid",shape="box"];10 -> 62[label="",style="solid", color="burlywood", weight=9];
10.65/4.16	62 -> 12[label="",style="solid", color="burlywood", weight=3];
10.65/4.16	11[label="take2 (Pos xu30) (repeat xu4) (not (primCmpInt (Pos xu30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];63[label="xu30/Succ xu300",fontsize=10,color="white",style="solid",shape="box"];11 -> 63[label="",style="solid", color="burlywood", weight=9];
10.65/4.16	63 -> 13[label="",style="solid", color="burlywood", weight=3];
10.65/4.16	64[label="xu30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 64[label="",style="solid", color="burlywood", weight=9];
10.65/4.16	64 -> 14[label="",style="solid", color="burlywood", weight=3];
10.65/4.16	12[label="take2 (Neg xu30) (repeat xu4) (not (primCmpInt (Neg xu30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];65[label="xu30/Succ xu300",fontsize=10,color="white",style="solid",shape="box"];12 -> 65[label="",style="solid", color="burlywood", weight=9];
10.65/4.16	65 -> 15[label="",style="solid", color="burlywood", weight=3];
10.65/4.16	66[label="xu30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 66[label="",style="solid", color="burlywood", weight=9];
10.65/4.16	66 -> 16[label="",style="solid", color="burlywood", weight=3];
10.65/4.16	13[label="take2 (Pos (Succ xu300)) (repeat xu4) (not (primCmpInt (Pos (Succ xu300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
10.65/4.16	14[label="take2 (Pos Zero) (repeat xu4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
10.65/4.16	15[label="take2 (Neg (Succ xu300)) (repeat xu4) (not (primCmpInt (Neg (Succ xu300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
10.65/4.16	16[label="take2 (Neg Zero) (repeat xu4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
10.65/4.16	17[label="take2 (Pos (Succ xu300)) (repeat xu4) (not (primCmpNat (Succ xu300) Zero == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
10.65/4.16	18[label="take2 (Pos Zero) (repeat xu4) (not (EQ == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
10.65/4.16	19[label="take2 (Neg (Succ xu300)) (repeat xu4) (not (LT == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
10.65/4.16	20[label="take2 (Neg Zero) (repeat xu4) (not (EQ == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
10.65/4.16	21[label="take2 (Pos (Succ xu300)) (repeat xu4) (not (GT == GT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
10.65/4.16	22[label="take2 (Pos Zero) (repeat xu4) (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
10.65/4.16	23[label="take2 (Neg (Succ xu300)) (repeat xu4) (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
10.65/4.16	24[label="take2 (Neg Zero) (repeat xu4) (not False)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
10.65/4.16	25[label="take2 (Pos (Succ xu300)) (repeat xu4) (not True)",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
10.65/4.16	26[label="take2 (Pos Zero) (repeat xu4) True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
10.65/4.16	27[label="take2 (Neg (Succ xu300)) (repeat xu4) True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
10.65/4.16	28[label="take2 (Neg Zero) (repeat xu4) True",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.65/4.16	29[label="take2 (Pos (Succ xu300)) (repeat xu4) False",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
10.65/4.16	30[label="[]",fontsize=16,color="green",shape="box"];31[label="[]",fontsize=16,color="green",shape="box"];32[label="[]",fontsize=16,color="green",shape="box"];33[label="take1 (Pos (Succ xu300)) (repeat xu4)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
10.65/4.16	34[label="take1 (Pos (Succ xu300)) (repeatXs xu4)",fontsize=16,color="black",shape="triangle"];34 -> 35[label="",style="solid", color="black", weight=3];
10.65/4.16	35[label="take1 (Pos (Succ xu300)) (xu4 : repeatXs xu4)",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
10.65/4.16	36[label="take0 (Pos (Succ xu300)) (xu4 : repeatXs xu4)",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
10.65/4.16	37[label="xu4 : take (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
10.65/4.16	38[label="take (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4)",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
10.65/4.16	39[label="take3 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4)",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3];
10.65/4.16	40[label="take2 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4) (Pos (Succ xu300) - Pos (Succ Zero) <= Pos Zero)",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
10.65/4.16	41[label="take2 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4) (compare (Pos (Succ xu300) - Pos (Succ Zero)) (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
10.65/4.16	42[label="take2 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4) (not (compare (Pos (Succ xu300) - Pos (Succ Zero)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
10.65/4.16	43[label="take2 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4) (not (primCmpInt (Pos (Succ xu300) - Pos (Succ Zero)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
10.65/4.16	44[label="take2 (primMinusInt (Pos (Succ xu300)) (Pos (Succ Zero))) (repeatXs xu4) (not (primCmpInt (primMinusInt (Pos (Succ xu300)) (Pos (Succ Zero))) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3];
10.65/4.16	45[label="take2 (primMinusNat (Succ xu300) (Succ Zero)) (repeatXs xu4) (not (primCmpInt (primMinusNat (Succ xu300) (Succ Zero)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];45 -> 46[label="",style="solid", color="black", weight=3];
10.65/4.16	46[label="take2 (primMinusNat xu300 Zero) (repeatXs xu4) (not (primCmpInt (primMinusNat xu300 Zero) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];67[label="xu300/Succ xu3000",fontsize=10,color="white",style="solid",shape="box"];46 -> 67[label="",style="solid", color="burlywood", weight=9];
10.65/4.16	67 -> 47[label="",style="solid", color="burlywood", weight=3];
10.65/4.16	68[label="xu300/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 68[label="",style="solid", color="burlywood", weight=9];
10.65/4.16	68 -> 48[label="",style="solid", color="burlywood", weight=3];
10.65/4.16	47[label="take2 (primMinusNat (Succ xu3000) Zero) (repeatXs xu4) (not (primCmpInt (primMinusNat (Succ xu3000) Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
10.65/4.16	48[label="take2 (primMinusNat Zero Zero) (repeatXs xu4) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];48 -> 50[label="",style="solid", color="black", weight=3];
10.65/4.16	49[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) (not (primCmpInt (Pos (Succ xu3000)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];49 -> 51[label="",style="solid", color="black", weight=3];
10.65/4.16	50[label="take2 (Pos Zero) (repeatXs xu4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];50 -> 52[label="",style="solid", color="black", weight=3];
10.65/4.16	51[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) (not (primCmpNat (Succ xu3000) Zero == GT))",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3];
10.65/4.16	52[label="take2 (Pos Zero) (repeatXs xu4) (not (EQ == GT))",fontsize=16,color="black",shape="box"];52 -> 54[label="",style="solid", color="black", weight=3];
10.65/4.16	53[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) (not (GT == GT))",fontsize=16,color="black",shape="box"];53 -> 55[label="",style="solid", color="black", weight=3];
10.65/4.16	54[label="take2 (Pos Zero) (repeatXs xu4) (not False)",fontsize=16,color="black",shape="box"];54 -> 56[label="",style="solid", color="black", weight=3];
10.65/4.16	55[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) (not True)",fontsize=16,color="black",shape="box"];55 -> 57[label="",style="solid", color="black", weight=3];
10.65/4.16	56[label="take2 (Pos Zero) (repeatXs xu4) True",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
10.65/4.16	57[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) False",fontsize=16,color="black",shape="box"];57 -> 59[label="",style="solid", color="black", weight=3];
10.65/4.16	58[label="[]",fontsize=16,color="green",shape="box"];59 -> 34[label="",style="dashed", color="red", weight=0];
10.65/4.16	59[label="take1 (Pos (Succ xu3000)) (repeatXs xu4)",fontsize=16,color="magenta"];59 -> 60[label="",style="dashed", color="magenta", weight=3];
10.65/4.16	60[label="xu3000",fontsize=16,color="green",shape="box"];}
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(10)
10.65/4.16	Obligation:
10.65/4.16	Q DP problem:
10.65/4.16	The TRS P consists of the following rules:
10.65/4.16	
10.65/4.16	   new_take1(Succ(xu3000), xu4, h) -> new_take1(xu3000, xu4, h)
10.65/4.16	
10.65/4.16	R is empty.
10.65/4.16	Q is empty.
10.65/4.16	We have to consider all minimal (P,Q,R)-chains.
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(11) QDPSizeChangeProof (EQUIVALENT)
10.65/4.16	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.65/4.16	
10.65/4.16	From the DPs we obtained the following set of size-change graphs:
10.65/4.16	*new_take1(Succ(xu3000), xu4, h) -> new_take1(xu3000, xu4, h)
10.65/4.16	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.65/4.16	
10.65/4.16	
10.65/4.16	----------------------------------------
10.65/4.16	
10.65/4.16	(12)
10.65/4.16	YES
10.65/4.24	EOF