8.25/3.70	YES
10.15/4.27	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.15/4.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.15/4.27	
10.15/4.27	
10.15/4.27	H-Termination with start terms of the given HASKELL could be proven:
10.15/4.27	
10.15/4.27	(0) HASKELL
10.15/4.27	(1) BR [EQUIVALENT, 0 ms]
10.15/4.27	(2) HASKELL
10.15/4.27	(3) COR [EQUIVALENT, 0 ms]
10.15/4.27	(4) HASKELL
10.15/4.27	(5) Narrow [SOUND, 0 ms]
10.15/4.27	(6) QDP
10.15/4.27	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.15/4.27	(8) YES
10.15/4.27	
10.15/4.27	
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(0)
10.15/4.27	Obligation:
10.15/4.27	mainModule Main 
10.15/4.27	module Main where {
10.15/4.27	import qualified Prelude;
10.15/4.27	data List a = Cons a (List a)  | Nil ;
10.15/4.27	
10.15/4.27	data MyBool = MyTrue  | MyFalse ;
10.15/4.27	
10.15/4.27	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.15/4.27	
10.15/4.27	data Main.Nat = Succ Main.Nat  | Zero ;
10.15/4.27	
10.15/4.27	data Ordering = LT  | EQ  | GT ;
10.15/4.27	
10.15/4.27	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.15/4.27	compareMyInt = primCmpInt;
10.15/4.27	
10.15/4.27	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.15/4.27	esEsOrdering LT LT = MyTrue;
10.15/4.27	esEsOrdering LT EQ = MyFalse;
10.15/4.27	esEsOrdering LT GT = MyFalse;
10.15/4.27	esEsOrdering EQ LT = MyFalse;
10.15/4.27	esEsOrdering EQ EQ = MyTrue;
10.15/4.27	esEsOrdering EQ GT = MyFalse;
10.15/4.27	esEsOrdering GT LT = MyFalse;
10.15/4.27	esEsOrdering GT EQ = MyFalse;
10.15/4.27	esEsOrdering GT GT = MyTrue;
10.15/4.27	
10.15/4.27	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.15/4.27	fsEsOrdering x y = not (esEsOrdering x y);
10.15/4.27	
10.15/4.27	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.15/4.27	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.15/4.27	
10.15/4.27	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.15/4.27	msMyInt = primMinusInt;
10.15/4.27	
10.15/4.27	not :: MyBool  ->  MyBool;
10.15/4.27	not MyTrue = MyFalse;
10.15/4.27	not MyFalse = MyTrue;
10.15/4.27	
10.15/4.27	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.15/4.27	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.15/4.27	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.15/4.27	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.15/4.27	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.15/4.27	
10.15/4.27	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.15/4.27	primCmpNat Main.Zero Main.Zero = EQ;
10.15/4.27	primCmpNat Main.Zero (Main.Succ y) = LT;
10.15/4.27	primCmpNat (Main.Succ x) Main.Zero = GT;
10.15/4.27	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.15/4.27	
10.15/4.27	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.15/4.27	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.15/4.27	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.15/4.27	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.15/4.27	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.15/4.27	
10.15/4.27	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.15/4.27	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.15/4.27	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.15/4.27	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.15/4.27	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.15/4.27	
10.15/4.27	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.15/4.27	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.15/4.27	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.15/4.27	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.15/4.27	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.15/4.27	
10.15/4.27	repeat :: a  ->  List a;
10.15/4.27	repeat x = repeatXs x;
10.15/4.27	
10.15/4.27	repeatXs wx = Cons wx (repeatXs wx);
10.15/4.27	
10.15/4.27	replicate :: MyInt  ->  a  ->  List a;
10.15/4.27	replicate n x = take n (repeat x);
10.15/4.27	
10.15/4.27	take :: MyInt  ->  List a  ->  List a;
10.15/4.27	take n vv = take3 n vv;
10.15/4.27	take vw Nil = take1 vw Nil;
10.15/4.27	take n (Cons x xs) = take0 n (Cons x xs);
10.15/4.27	
10.15/4.27	take0 n (Cons x xs) = Cons x (take (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs);
10.15/4.27	
10.15/4.27	take1 vw Nil = Nil;
10.15/4.27	take1 vz wu = take0 vz wu;
10.15/4.27	
10.15/4.27	take2 n vv MyTrue = Nil;
10.15/4.27	take2 n vv MyFalse = take1 n vv;
10.15/4.27	
10.15/4.27	take3 n vv = take2 n vv (ltEsMyInt n (Main.Pos Main.Zero));
10.15/4.27	take3 wv ww = take1 wv ww;
10.15/4.27	
10.15/4.27	}
10.15/4.27	
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(1) BR (EQUIVALENT)
10.15/4.27	Replaced joker patterns by fresh variables and removed binding patterns.
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(2)
10.15/4.27	Obligation:
10.15/4.27	mainModule Main 
10.15/4.27	module Main where {
10.15/4.27	import qualified Prelude;
10.15/4.27	data List a = Cons a (List a)  | Nil ;
10.15/4.27	
10.15/4.27	data MyBool = MyTrue  | MyFalse ;
10.15/4.27	
10.15/4.27	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.15/4.27	
10.15/4.27	data Main.Nat = Succ Main.Nat  | Zero ;
10.15/4.27	
10.15/4.27	data Ordering = LT  | EQ  | GT ;
10.15/4.27	
10.15/4.27	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.15/4.27	compareMyInt = primCmpInt;
10.15/4.27	
10.15/4.27	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.15/4.27	esEsOrdering LT LT = MyTrue;
10.15/4.27	esEsOrdering LT EQ = MyFalse;
10.15/4.27	esEsOrdering LT GT = MyFalse;
10.15/4.27	esEsOrdering EQ LT = MyFalse;
10.15/4.27	esEsOrdering EQ EQ = MyTrue;
10.15/4.27	esEsOrdering EQ GT = MyFalse;
10.15/4.27	esEsOrdering GT LT = MyFalse;
10.15/4.27	esEsOrdering GT EQ = MyFalse;
10.15/4.27	esEsOrdering GT GT = MyTrue;
10.15/4.27	
10.15/4.27	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.15/4.27	fsEsOrdering x y = not (esEsOrdering x y);
10.15/4.27	
10.15/4.27	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.15/4.27	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.15/4.27	
10.15/4.27	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.15/4.27	msMyInt = primMinusInt;
10.15/4.27	
10.15/4.27	not :: MyBool  ->  MyBool;
10.15/4.27	not MyTrue = MyFalse;
10.15/4.27	not MyFalse = MyTrue;
10.15/4.27	
10.15/4.27	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.15/4.27	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.15/4.27	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.15/4.27	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.15/4.27	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.15/4.27	
10.15/4.27	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.15/4.27	primCmpNat Main.Zero Main.Zero = EQ;
10.15/4.27	primCmpNat Main.Zero (Main.Succ y) = LT;
10.15/4.27	primCmpNat (Main.Succ x) Main.Zero = GT;
10.15/4.27	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.15/4.27	
10.15/4.27	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.15/4.27	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.15/4.27	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.15/4.27	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.15/4.27	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.15/4.27	
10.15/4.27	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.15/4.27	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.15/4.27	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.15/4.27	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.15/4.27	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.15/4.27	
10.15/4.27	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.15/4.27	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.15/4.27	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.15/4.27	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.15/4.27	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.15/4.27	
10.15/4.27	repeat :: a  ->  List a;
10.15/4.27	repeat x = repeatXs x;
10.15/4.27	
10.15/4.27	repeatXs wx = Cons wx (repeatXs wx);
10.15/4.27	
10.15/4.27	replicate :: MyInt  ->  a  ->  List a;
10.15/4.27	replicate n x = take n (repeat x);
10.15/4.27	
10.15/4.27	take :: MyInt  ->  List a  ->  List a;
10.15/4.27	take n vv = take3 n vv;
10.15/4.27	take vw Nil = take1 vw Nil;
10.15/4.27	take n (Cons x xs) = take0 n (Cons x xs);
10.15/4.27	
10.15/4.27	take0 n (Cons x xs) = Cons x (take (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs);
10.15/4.27	
10.15/4.27	take1 vw Nil = Nil;
10.15/4.27	take1 vz wu = take0 vz wu;
10.15/4.27	
10.15/4.27	take2 n vv MyTrue = Nil;
10.15/4.27	take2 n vv MyFalse = take1 n vv;
10.15/4.27	
10.15/4.27	take3 n vv = take2 n vv (ltEsMyInt n (Main.Pos Main.Zero));
10.15/4.27	take3 wv ww = take1 wv ww;
10.15/4.27	
10.15/4.27	}
10.15/4.27	
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(3) COR (EQUIVALENT)
10.15/4.27	Cond Reductions:
10.15/4.27	The following Function with conditions
10.15/4.27	"undefined |Falseundefined;
10.15/4.27	"
10.15/4.27	is transformed to
10.15/4.27	"undefined  = undefined1;
10.15/4.27	"
10.15/4.27	"undefined0 True = undefined;
10.15/4.27	"
10.15/4.27	"undefined1  = undefined0 False;
10.15/4.27	"
10.15/4.27	
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(4)
10.15/4.27	Obligation:
10.15/4.27	mainModule Main 
10.15/4.27	module Main where {
10.15/4.27	import qualified Prelude;
10.15/4.27	data List a = Cons a (List a)  | Nil ;
10.15/4.27	
10.15/4.27	data MyBool = MyTrue  | MyFalse ;
10.15/4.27	
10.15/4.27	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.15/4.27	
10.15/4.27	data Main.Nat = Succ Main.Nat  | Zero ;
10.15/4.27	
10.15/4.27	data Ordering = LT  | EQ  | GT ;
10.15/4.27	
10.15/4.27	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.15/4.27	compareMyInt = primCmpInt;
10.15/4.27	
10.15/4.27	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.15/4.27	esEsOrdering LT LT = MyTrue;
10.15/4.27	esEsOrdering LT EQ = MyFalse;
10.15/4.27	esEsOrdering LT GT = MyFalse;
10.15/4.27	esEsOrdering EQ LT = MyFalse;
10.15/4.27	esEsOrdering EQ EQ = MyTrue;
10.15/4.27	esEsOrdering EQ GT = MyFalse;
10.15/4.27	esEsOrdering GT LT = MyFalse;
10.15/4.27	esEsOrdering GT EQ = MyFalse;
10.15/4.27	esEsOrdering GT GT = MyTrue;
10.15/4.27	
10.15/4.27	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.15/4.27	fsEsOrdering x y = not (esEsOrdering x y);
10.15/4.27	
10.15/4.27	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.15/4.27	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.15/4.27	
10.15/4.27	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.15/4.27	msMyInt = primMinusInt;
10.15/4.27	
10.15/4.27	not :: MyBool  ->  MyBool;
10.15/4.27	not MyTrue = MyFalse;
10.15/4.27	not MyFalse = MyTrue;
10.15/4.27	
10.15/4.27	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.15/4.27	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.15/4.27	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.15/4.27	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.15/4.27	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.15/4.27	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.15/4.27	
10.15/4.27	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.15/4.27	primCmpNat Main.Zero Main.Zero = EQ;
10.15/4.27	primCmpNat Main.Zero (Main.Succ y) = LT;
10.15/4.27	primCmpNat (Main.Succ x) Main.Zero = GT;
10.15/4.27	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.15/4.27	
10.15/4.27	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.15/4.27	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.15/4.27	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.15/4.27	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.15/4.27	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.15/4.27	
10.15/4.27	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.15/4.27	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.15/4.27	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.15/4.27	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.15/4.27	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.15/4.27	
10.15/4.27	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.15/4.27	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.15/4.27	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.15/4.27	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.15/4.27	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.15/4.27	
10.15/4.27	repeat :: a  ->  List a;
10.15/4.27	repeat x = repeatXs x;
10.15/4.27	
10.15/4.27	repeatXs wx = Cons wx (repeatXs wx);
10.15/4.27	
10.15/4.27	replicate :: MyInt  ->  a  ->  List a;
10.15/4.27	replicate n x = take n (repeat x);
10.15/4.27	
10.15/4.27	take :: MyInt  ->  List a  ->  List a;
10.15/4.27	take n vv = take3 n vv;
10.15/4.27	take vw Nil = take1 vw Nil;
10.15/4.27	take n (Cons x xs) = take0 n (Cons x xs);
10.15/4.27	
10.15/4.27	take0 n (Cons x xs) = Cons x (take (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs);
10.15/4.27	
10.15/4.27	take1 vw Nil = Nil;
10.15/4.27	take1 vz wu = take0 vz wu;
10.15/4.27	
10.15/4.27	take2 n vv MyTrue = Nil;
10.15/4.27	take2 n vv MyFalse = take1 n vv;
10.15/4.27	
10.15/4.27	take3 n vv = take2 n vv (ltEsMyInt n (Main.Pos Main.Zero));
10.15/4.27	take3 wv ww = take1 wv ww;
10.15/4.27	
10.15/4.27	}
10.15/4.27	
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(5) Narrow (SOUND)
10.15/4.27	Haskell To QDPs
10.15/4.27	
10.15/4.27	digraph dp_graph {
10.15/4.27	node [outthreshold=100, inthreshold=100];1[label="replicate",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.15/4.27	3[label="replicate wy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.15/4.27	4[label="replicate wy3 wy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.15/4.27	5[label="take wy3 (repeat wy4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.15/4.27	6[label="take3 wy3 (repeat wy4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.15/4.27	7[label="take2 wy3 (repeat wy4) (ltEsMyInt wy3 (Pos Zero))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.15/4.27	8[label="take2 wy3 (repeat wy4) (fsEsOrdering (compareMyInt wy3 (Pos Zero)) GT)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.15/4.27	9[label="take2 wy3 (repeat wy4) (not (esEsOrdering (compareMyInt wy3 (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10.15/4.27	10[label="take2 wy3 (repeat wy4) (not (esEsOrdering (primCmpInt wy3 (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];61[label="wy3/Pos wy30",fontsize=10,color="white",style="solid",shape="box"];10 -> 61[label="",style="solid", color="burlywood", weight=9];
10.15/4.27	61 -> 11[label="",style="solid", color="burlywood", weight=3];
10.15/4.27	62[label="wy3/Neg wy30",fontsize=10,color="white",style="solid",shape="box"];10 -> 62[label="",style="solid", color="burlywood", weight=9];
10.15/4.27	62 -> 12[label="",style="solid", color="burlywood", weight=3];
10.15/4.27	11[label="take2 (Pos wy30) (repeat wy4) (not (esEsOrdering (primCmpInt (Pos wy30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];63[label="wy30/Succ wy300",fontsize=10,color="white",style="solid",shape="box"];11 -> 63[label="",style="solid", color="burlywood", weight=9];
10.15/4.27	63 -> 13[label="",style="solid", color="burlywood", weight=3];
10.15/4.27	64[label="wy30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 64[label="",style="solid", color="burlywood", weight=9];
10.15/4.27	64 -> 14[label="",style="solid", color="burlywood", weight=3];
10.15/4.27	12[label="take2 (Neg wy30) (repeat wy4) (not (esEsOrdering (primCmpInt (Neg wy30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];65[label="wy30/Succ wy300",fontsize=10,color="white",style="solid",shape="box"];12 -> 65[label="",style="solid", color="burlywood", weight=9];
10.15/4.27	65 -> 15[label="",style="solid", color="burlywood", weight=3];
10.15/4.27	66[label="wy30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 66[label="",style="solid", color="burlywood", weight=9];
10.15/4.27	66 -> 16[label="",style="solid", color="burlywood", weight=3];
10.15/4.27	13[label="take2 (Pos (Succ wy300)) (repeat wy4) (not (esEsOrdering (primCmpInt (Pos (Succ wy300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
10.15/4.27	14[label="take2 (Pos Zero) (repeat wy4) (not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
10.15/4.27	15[label="take2 (Neg (Succ wy300)) (repeat wy4) (not (esEsOrdering (primCmpInt (Neg (Succ wy300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
10.15/4.27	16[label="take2 (Neg Zero) (repeat wy4) (not (esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
10.15/4.27	17[label="take2 (Pos (Succ wy300)) (repeat wy4) (not (esEsOrdering (primCmpNat (Succ wy300) Zero) GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
10.15/4.27	18[label="take2 (Pos Zero) (repeat wy4) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
10.15/4.27	19[label="take2 (Neg (Succ wy300)) (repeat wy4) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
10.15/4.27	20[label="take2 (Neg Zero) (repeat wy4) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
10.15/4.27	21[label="take2 (Pos (Succ wy300)) (repeat wy4) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
10.15/4.27	22[label="take2 (Pos Zero) (repeat wy4) (not MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
10.15/4.27	23[label="take2 (Neg (Succ wy300)) (repeat wy4) (not MyFalse)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
10.15/4.27	24[label="take2 (Neg Zero) (repeat wy4) (not MyFalse)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
10.15/4.27	25[label="take2 (Pos (Succ wy300)) (repeat wy4) (not MyTrue)",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
10.15/4.27	26[label="take2 (Pos Zero) (repeat wy4) MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
10.15/4.27	27[label="take2 (Neg (Succ wy300)) (repeat wy4) MyTrue",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
10.15/4.27	28[label="take2 (Neg Zero) (repeat wy4) MyTrue",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.15/4.27	29[label="take2 (Pos (Succ wy300)) (repeat wy4) MyFalse",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
10.15/4.27	30[label="Nil",fontsize=16,color="green",shape="box"];31[label="Nil",fontsize=16,color="green",shape="box"];32[label="Nil",fontsize=16,color="green",shape="box"];33[label="take1 (Pos (Succ wy300)) (repeat wy4)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
10.15/4.27	34[label="take1 (Pos (Succ wy300)) (repeatXs wy4)",fontsize=16,color="black",shape="triangle"];34 -> 35[label="",style="solid", color="black", weight=3];
10.15/4.27	35[label="take1 (Pos (Succ wy300)) (Cons wy4 (repeatXs wy4))",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
10.15/4.27	36[label="take0 (Pos (Succ wy300)) (Cons wy4 (repeatXs wy4))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
10.15/4.27	37[label="Cons wy4 (take (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (repeatXs wy4))",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
10.15/4.27	38[label="take (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (repeatXs wy4)",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
10.15/4.27	39[label="take3 (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (repeatXs wy4)",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3];
10.15/4.27	40[label="take2 (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (repeatXs wy4) (ltEsMyInt (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (Pos Zero))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
10.15/4.27	41[label="take2 (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (repeatXs wy4) (fsEsOrdering (compareMyInt (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (Pos Zero)) GT)",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
10.15/4.27	42[label="take2 (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (repeatXs wy4) (not (esEsOrdering (compareMyInt (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
10.15/4.27	43[label="take2 (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (repeatXs wy4) (not (esEsOrdering (primCmpInt (msMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
10.15/4.27	44[label="take2 (primMinusInt (Pos (Succ wy300)) (Pos (Succ Zero))) (repeatXs wy4) (not (esEsOrdering (primCmpInt (primMinusInt (Pos (Succ wy300)) (Pos (Succ Zero))) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3];
10.15/4.27	45[label="take2 (primMinusNat (Succ wy300) (Succ Zero)) (repeatXs wy4) (not (esEsOrdering (primCmpInt (primMinusNat (Succ wy300) (Succ Zero)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];45 -> 46[label="",style="solid", color="black", weight=3];
10.15/4.27	46[label="take2 (primMinusNat wy300 Zero) (repeatXs wy4) (not (esEsOrdering (primCmpInt (primMinusNat wy300 Zero) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];67[label="wy300/Succ wy3000",fontsize=10,color="white",style="solid",shape="box"];46 -> 67[label="",style="solid", color="burlywood", weight=9];
10.15/4.27	67 -> 47[label="",style="solid", color="burlywood", weight=3];
10.15/4.27	68[label="wy300/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 68[label="",style="solid", color="burlywood", weight=9];
10.15/4.27	68 -> 48[label="",style="solid", color="burlywood", weight=3];
10.15/4.27	47[label="take2 (primMinusNat (Succ wy3000) Zero) (repeatXs wy4) (not (esEsOrdering (primCmpInt (primMinusNat (Succ wy3000) Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
10.15/4.27	48[label="take2 (primMinusNat Zero Zero) (repeatXs wy4) (not (esEsOrdering (primCmpInt (primMinusNat Zero Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];48 -> 50[label="",style="solid", color="black", weight=3];
10.15/4.27	49[label="take2 (Pos (Succ wy3000)) (repeatXs wy4) (not (esEsOrdering (primCmpInt (Pos (Succ wy3000)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];49 -> 51[label="",style="solid", color="black", weight=3];
10.15/4.27	50[label="take2 (Pos Zero) (repeatXs wy4) (not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];50 -> 52[label="",style="solid", color="black", weight=3];
10.15/4.27	51[label="take2 (Pos (Succ wy3000)) (repeatXs wy4) (not (esEsOrdering (primCmpNat (Succ wy3000) Zero) GT))",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3];
10.15/4.27	52[label="take2 (Pos Zero) (repeatXs wy4) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];52 -> 54[label="",style="solid", color="black", weight=3];
10.15/4.27	53[label="take2 (Pos (Succ wy3000)) (repeatXs wy4) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];53 -> 55[label="",style="solid", color="black", weight=3];
10.15/4.27	54[label="take2 (Pos Zero) (repeatXs wy4) (not MyFalse)",fontsize=16,color="black",shape="box"];54 -> 56[label="",style="solid", color="black", weight=3];
10.15/4.27	55[label="take2 (Pos (Succ wy3000)) (repeatXs wy4) (not MyTrue)",fontsize=16,color="black",shape="box"];55 -> 57[label="",style="solid", color="black", weight=3];
10.15/4.27	56[label="take2 (Pos Zero) (repeatXs wy4) MyTrue",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
10.15/4.27	57[label="take2 (Pos (Succ wy3000)) (repeatXs wy4) MyFalse",fontsize=16,color="black",shape="box"];57 -> 59[label="",style="solid", color="black", weight=3];
10.15/4.27	58[label="Nil",fontsize=16,color="green",shape="box"];59 -> 34[label="",style="dashed", color="red", weight=0];
10.15/4.27	59[label="take1 (Pos (Succ wy3000)) (repeatXs wy4)",fontsize=16,color="magenta"];59 -> 60[label="",style="dashed", color="magenta", weight=3];
10.15/4.27	60[label="wy3000",fontsize=16,color="green",shape="box"];}
10.15/4.27	
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(6)
10.15/4.27	Obligation:
10.15/4.27	Q DP problem:
10.15/4.27	The TRS P consists of the following rules:
10.15/4.27	
10.15/4.27	   new_take1(Main.Succ(wy3000), wy4, h) -> new_take1(wy3000, wy4, h)
10.15/4.27	
10.15/4.27	R is empty.
10.15/4.27	Q is empty.
10.15/4.27	We have to consider all minimal (P,Q,R)-chains.
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(7) QDPSizeChangeProof (EQUIVALENT)
10.15/4.27	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. 
10.15/4.27	
10.15/4.27	From the DPs we obtained the following set of size-change graphs:
10.15/4.27	*new_take1(Main.Succ(wy3000), wy4, h) -> new_take1(wy3000, wy4, h)
10.15/4.27	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.15/4.27	
10.15/4.27	
10.15/4.27	----------------------------------------
10.15/4.27	
10.15/4.27	(8)
10.15/4.27	YES
10.32/4.57	EOF