9.25/3.90	YES
11.54/4.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.54/4.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.54/4.57	
11.54/4.57	
11.54/4.57	H-Termination with start terms of the given HASKELL could be proven:
11.54/4.57	
11.54/4.57	(0) HASKELL
11.54/4.57	(1) LR [EQUIVALENT, 0 ms]
11.54/4.57	(2) HASKELL
11.54/4.57	(3) BR [EQUIVALENT, 0 ms]
11.54/4.57	(4) HASKELL
11.54/4.57	(5) COR [EQUIVALENT, 0 ms]
11.54/4.57	(6) HASKELL
11.54/4.57	(7) LetRed [EQUIVALENT, 5 ms]
11.54/4.57	(8) HASKELL
11.54/4.57	(9) NumRed [SOUND, 0 ms]
11.54/4.57	(10) HASKELL
11.54/4.57	(11) Narrow [SOUND, 0 ms]
11.54/4.57	(12) QDP
11.54/4.57	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.54/4.57	(14) YES
11.54/4.57	
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(0)
11.54/4.57	Obligation:
11.54/4.57	mainModule Main 
11.54/4.57	module Maybe where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Main where {
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Monad where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Prelude;
11.54/4.57	replicateM_ :: Monad a => Int  ->  a b  ->  a ();
11.54/4.57	replicateM_ n x = sequence_ (replicate n x);
11.54/4.57	
11.54/4.57	}
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(1) LR (EQUIVALENT)
11.54/4.57	Lambda Reductions:
11.54/4.57	The following Lambda expression
11.54/4.57	"\_->q"
11.54/4.57	is transformed to
11.54/4.57	"gtGt0 q _ = q;
11.54/4.57	"
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(2)
11.54/4.57	Obligation:
11.54/4.57	mainModule Main 
11.54/4.57	module Maybe where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Main where {
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Monad where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Prelude;
11.54/4.57	replicateM_ :: Monad a => Int  ->  a b  ->  a ();
11.54/4.57	replicateM_ n x = sequence_ (replicate n x);
11.54/4.57	
11.54/4.57	}
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(3) BR (EQUIVALENT)
11.54/4.57	Replaced joker patterns by fresh variables and removed binding patterns.
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(4)
11.54/4.57	Obligation:
11.54/4.57	mainModule Main 
11.54/4.57	module Maybe where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Main where {
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Monad where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Prelude;
11.54/4.57	replicateM_ :: Monad a => Int  ->  a b  ->  a ();
11.54/4.57	replicateM_ n x = sequence_ (replicate n x);
11.54/4.57	
11.54/4.57	}
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(5) COR (EQUIVALENT)
11.54/4.57	Cond Reductions:
11.54/4.57	The following Function with conditions
11.54/4.57	"undefined |Falseundefined;
11.54/4.57	"
11.54/4.57	is transformed to
11.54/4.57	"undefined  = undefined1;
11.54/4.57	"
11.54/4.57	"undefined0 True = undefined;
11.54/4.57	"
11.54/4.57	"undefined1  = undefined0 False;
11.54/4.57	"
11.54/4.57	The following Function with conditions
11.54/4.57	"take n vy|n <= 0[];
11.54/4.57	take vz [] = [];
11.54/4.57	take n (x : xs) = x : take (n - 1) xs;
11.54/4.57	"
11.54/4.57	is transformed to
11.54/4.57	"take n vy = take3 n vy;
11.54/4.57	take vz [] = take1 vz [];
11.54/4.57	take n (x : xs) = take0 n (x : xs);
11.54/4.57	"
11.54/4.57	"take0 n (x : xs) = x : take (n - 1) xs;
11.54/4.57	"
11.54/4.57	"take1 vz [] = [];
11.54/4.57	take1 wx wy = take0 wx wy;
11.54/4.57	"
11.54/4.57	"take2 n vy True = [];
11.54/4.57	take2 n vy False = take1 n vy;
11.54/4.57	"
11.54/4.57	"take3 n vy = take2 n vy (n <= 0);
11.54/4.57	take3 wz xu = take1 wz xu;
11.54/4.57	"
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(6)
11.54/4.57	Obligation:
11.54/4.57	mainModule Main 
11.54/4.57	module Maybe where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Main where {
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Monad where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Prelude;
11.54/4.57	replicateM_ :: Monad b => Int  ->  b a  ->  b ();
11.54/4.57	replicateM_ n x = sequence_ (replicate n x);
11.54/4.57	
11.54/4.57	}
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(7) LetRed (EQUIVALENT)
11.54/4.57	Let/Where Reductions:
11.54/4.57	The bindings of the following Let/Where expression
11.54/4.57	"xs where {
11.54/4.57	xs  = x : xs;
11.54/4.57	} 
11.54/4.57	"
11.54/4.57	are unpacked to the following functions on top level
11.54/4.57	"repeatXs xv = xv : repeatXs xv;
11.54/4.57	"
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(8)
11.54/4.57	Obligation:
11.54/4.57	mainModule Main 
11.54/4.57	module Maybe where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Main where {
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Monad where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Prelude;
11.54/4.57	replicateM_ :: Monad b => Int  ->  b a  ->  b ();
11.54/4.57	replicateM_ n x = sequence_ (replicate n x);
11.54/4.57	
11.54/4.57	}
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(9) NumRed (SOUND)
11.54/4.57	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(10)
11.54/4.57	Obligation:
11.54/4.57	mainModule Main 
11.54/4.57	module Maybe where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Main where {
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Monad;
11.54/4.57	import qualified Prelude;
11.54/4.57	}
11.54/4.57	module Monad where {
11.54/4.57	import qualified Main;
11.54/4.57	import qualified Maybe;
11.54/4.57	import qualified Prelude;
11.54/4.57	replicateM_ :: Monad a => Int  ->  a b  ->  a ();
11.54/4.57	replicateM_ n x = sequence_ (replicate n x);
11.54/4.57	
11.54/4.57	}
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(11) Narrow (SOUND)
11.54/4.57	Haskell To QDPs
11.54/4.57	
11.54/4.57	digraph dp_graph {
11.54/4.57	node [outthreshold=100, inthreshold=100];1[label="Monad.replicateM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.54/4.57	3[label="Monad.replicateM_ xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.54/4.57	4[label="Monad.replicateM_ xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
11.54/4.57	5[label="sequence_ (replicate xw3 xw4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.54/4.57	6[label="foldr (>>) (return ()) (replicate xw3 xw4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
11.54/4.57	7[label="foldr (>>) (return ()) (take xw3 (repeat xw4))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
11.54/4.57	8[label="foldr (>>) (return ()) (take3 xw3 (repeat xw4))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
11.54/4.57	9[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (xw3 <= Pos Zero))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
11.54/4.57	10[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (compare xw3 (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11.54/4.57	11[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (compare xw3 (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
11.54/4.57	12[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (primCmpInt xw3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];75[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 75[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	75 -> 13[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	76[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 76[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	76 -> 14[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	13[label="foldr (>>) (return ()) (take2 (Pos xw30) (repeat xw4) (not (primCmpInt (Pos xw30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];77[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	77 -> 15[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	78[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 78[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	78 -> 16[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	14[label="foldr (>>) (return ()) (take2 (Neg xw30) (repeat xw4) (not (primCmpInt (Neg xw30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];79[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];14 -> 79[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	79 -> 17[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	80[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	80 -> 18[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	15[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (primCmpInt (Pos (Succ xw300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
11.54/4.57	16[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
11.54/4.57	17[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not (primCmpInt (Neg (Succ xw300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
11.54/4.57	18[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
11.54/4.57	19[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (primCmpNat (Succ xw300) Zero == GT)))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
11.54/4.57	20[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
11.54/4.57	21[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not (LT == GT)))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
11.54/4.57	22[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
11.54/4.57	23[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (GT == GT)))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
11.54/4.57	24[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
11.54/4.57	25[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
11.54/4.57	26[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
11.54/4.57	27[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not True))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
11.54/4.57	28[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) True)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
11.54/4.57	29[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
11.54/4.57	30[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) True)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
11.54/4.57	31[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) False)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
11.54/4.57	32[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];32 -> 36[label="",style="solid", color="black", weight=3];
11.54/4.57	33 -> 32[label="",style="dashed", color="red", weight=0];
11.54/4.57	33[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];34 -> 32[label="",style="dashed", color="red", weight=0];
11.54/4.57	34[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];35[label="foldr (>>) (return ()) (take1 (Pos (Succ xw300)) (repeat xw4))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
11.54/4.57	36[label="return ()",fontsize=16,color="black",shape="triangle"];36 -> 38[label="",style="solid", color="black", weight=3];
11.54/4.57	37 -> 39[label="",style="dashed", color="red", weight=0];
11.54/4.57	37[label="foldr (>>) (return ()) (take1 (Pos (Succ xw300)) (repeatXs xw4))",fontsize=16,color="magenta"];37 -> 40[label="",style="dashed", color="magenta", weight=3];
11.54/4.57	38[label="Just ()",fontsize=16,color="green",shape="box"];40 -> 36[label="",style="dashed", color="red", weight=0];
11.54/4.57	40[label="return ()",fontsize=16,color="magenta"];39[label="foldr (>>) xw5 (take1 (Pos (Succ xw300)) (repeatXs xw4))",fontsize=16,color="black",shape="triangle"];39 -> 41[label="",style="solid", color="black", weight=3];
11.54/4.57	41[label="foldr (>>) xw5 (take1 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
11.54/4.57	42[label="foldr (>>) xw5 (take0 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
11.54/4.57	43[label="foldr (>>) xw5 (xw4 : take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
11.54/4.57	44[label="(>>) xw4 foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3];
11.54/4.57	45[label="xw4 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4)))",fontsize=16,color="burlywood",shape="box"];81[label="xw4/Nothing",fontsize=10,color="white",style="solid",shape="box"];45 -> 81[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	81 -> 46[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	82[label="xw4/Just xw40",fontsize=10,color="white",style="solid",shape="box"];45 -> 82[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	82 -> 47[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	46[label="Nothing >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs Nothing)))",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
11.54/4.57	47[label="Just xw40 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40))))",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
11.54/4.57	48[label="Nothing",fontsize=16,color="green",shape="box"];49[label="gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)))) xw40",fontsize=16,color="black",shape="box"];49 -> 50[label="",style="solid", color="black", weight=3];
11.54/4.57	50[label="foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)))",fontsize=16,color="black",shape="box"];50 -> 51[label="",style="solid", color="black", weight=3];
11.54/4.57	51[label="foldr (>>) xw5 (take3 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)))",fontsize=16,color="black",shape="box"];51 -> 52[label="",style="solid", color="black", weight=3];
11.54/4.57	52[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)) (Pos (Succ xw300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];52 -> 53[label="",style="solid", color="black", weight=3];
11.54/4.57	53[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)) (compare (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];53 -> 54[label="",style="solid", color="black", weight=3];
11.54/4.57	54[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)) (not (compare (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];54 -> 55[label="",style="solid", color="black", weight=3];
11.54/4.57	55[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)) (not (primCmpInt (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];55 -> 56[label="",style="solid", color="black", weight=3];
11.54/4.57	56[label="foldr (>>) xw5 (take2 (primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))) (repeatXs (Just xw40)) (not (primCmpInt (primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];56 -> 57[label="",style="solid", color="black", weight=3];
11.54/4.57	57[label="foldr (>>) xw5 (take2 (primMinusNat (Succ xw300) (Succ Zero)) (repeatXs (Just xw40)) (not (primCmpInt (primMinusNat (Succ xw300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];57 -> 58[label="",style="solid", color="black", weight=3];
11.54/4.57	58[label="foldr (>>) xw5 (take2 (primMinusNat xw300 Zero) (repeatXs (Just xw40)) (not (primCmpInt (primMinusNat xw300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];83[label="xw300/Succ xw3000",fontsize=10,color="white",style="solid",shape="box"];58 -> 83[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	83 -> 59[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	84[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 84[label="",style="solid", color="burlywood", weight=9];
11.54/4.57	84 -> 60[label="",style="solid", color="burlywood", weight=3];
11.54/4.57	59[label="foldr (>>) xw5 (take2 (primMinusNat (Succ xw3000) Zero) (repeatXs (Just xw40)) (not (primCmpInt (primMinusNat (Succ xw3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3];
11.54/4.57	60[label="foldr (>>) xw5 (take2 (primMinusNat Zero Zero) (repeatXs (Just xw40)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
11.54/4.57	61[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) (not (primCmpInt (Pos (Succ xw3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
11.54/4.57	62[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (Just xw40)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];62 -> 64[label="",style="solid", color="black", weight=3];
11.54/4.57	63[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) (not (primCmpNat (Succ xw3000) Zero == GT)))",fontsize=16,color="black",shape="box"];63 -> 65[label="",style="solid", color="black", weight=3];
11.54/4.57	64[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (Just xw40)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];64 -> 66[label="",style="solid", color="black", weight=3];
11.54/4.57	65[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];65 -> 67[label="",style="solid", color="black", weight=3];
11.54/4.57	66[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (Just xw40)) (not False))",fontsize=16,color="black",shape="box"];66 -> 68[label="",style="solid", color="black", weight=3];
11.54/4.57	67[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) (not True))",fontsize=16,color="black",shape="box"];67 -> 69[label="",style="solid", color="black", weight=3];
11.54/4.57	68[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (Just xw40)) True)",fontsize=16,color="black",shape="box"];68 -> 70[label="",style="solid", color="black", weight=3];
11.54/4.57	69[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) False)",fontsize=16,color="black",shape="box"];69 -> 71[label="",style="solid", color="black", weight=3];
11.54/4.57	70[label="foldr (>>) xw5 []",fontsize=16,color="black",shape="box"];70 -> 72[label="",style="solid", color="black", weight=3];
11.54/4.57	71 -> 39[label="",style="dashed", color="red", weight=0];
11.54/4.57	71[label="foldr (>>) xw5 (take1 (Pos (Succ xw3000)) (repeatXs (Just xw40)))",fontsize=16,color="magenta"];71 -> 73[label="",style="dashed", color="magenta", weight=3];
11.54/4.57	71 -> 74[label="",style="dashed", color="magenta", weight=3];
11.54/4.57	72[label="xw5",fontsize=16,color="green",shape="box"];73[label="xw3000",fontsize=16,color="green",shape="box"];74[label="Just xw40",fontsize=16,color="green",shape="box"];}
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(12)
11.54/4.57	Obligation:
11.54/4.57	Q DP problem:
11.54/4.57	The TRS P consists of the following rules:
11.54/4.57	
11.54/4.57	   new_foldr(xw5, Succ(xw3000), Just(xw40), h) -> new_foldr(xw5, xw3000, Just(xw40), h)
11.54/4.57	
11.54/4.57	R is empty.
11.54/4.57	Q is empty.
11.54/4.57	We have to consider all minimal (P,Q,R)-chains.
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(13) QDPSizeChangeProof (EQUIVALENT)
11.54/4.57	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. 
11.54/4.57	
11.54/4.57	From the DPs we obtained the following set of size-change graphs:
11.54/4.57	*new_foldr(xw5, Succ(xw3000), Just(xw40), h) -> new_foldr(xw5, xw3000, Just(xw40), h)
11.54/4.57	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
11.54/4.57	
11.54/4.57	
11.54/4.57	----------------------------------------
11.54/4.57	
11.54/4.57	(14)
11.54/4.57	YES
11.83/4.61	EOF