/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyBool = MyTrue  | MyFalse ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Ordering = LT  | EQ  | GT ;

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering x y);

ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

repeat :: a  ->  List a;
repeat x = repeatXs x;

repeatXs wx = Cons wx (repeatXs wx);

replicate :: MyInt  ->  a  ->  List a;
replicate n x = take n (repeat x);

take :: MyInt  ->  List a  ->  List a;
take n vv = take3 n vv;
take vw Nil = take1 vw Nil;
take n (Cons x xs) = take0 n (Cons x xs);

take0 n (Cons x xs) = Cons x (take (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs);

take1 vw Nil = Nil;
take1 vz wu = take0 vz wu;

take2 n vv MyTrue = Nil;
take2 n vv MyFalse = take1 n vv;

take3 n vv = take2 n vv (ltEsMyInt n (Main.Pos Main.Zero));
take3 wv ww = take1 wv ww;

}

----------------------------------------

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyBool = MyTrue  | MyFalse ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Ordering = LT  | EQ  | GT ;

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering x y);

ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

repeat :: a  ->  List a;
repeat x = repeatXs x;

repeatXs wx = Cons wx (repeatXs wx);

replicate :: MyInt  ->  a  ->  List a;
replicate n x = take n (repeat x);

take :: MyInt  ->  List a  ->  List a;
take n vv = take3 n vv;
take vw Nil = take1 vw Nil;
take n (Cons x xs) = take0 n (Cons x xs);

take0 n (Cons x xs) = Cons x (take (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs);

take1 vw Nil = Nil;
take1 vz wu = take0 vz wu;

take2 n vv MyTrue = Nil;
take2 n vv MyFalse = take1 n vv;

take3 n vv = take2 n vv (ltEsMyInt n (Main.Pos Main.Zero));
take3 wv ww = take1 wv ww;

}

----------------------------------------

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyBool = MyTrue  | MyFalse ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Ordering = LT  | EQ  | GT ;

compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
compareMyInt = primCmpInt;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = MyTrue;

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering x y);

ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;

msMyInt :: MyInt  ->  MyInt  ->  MyInt;
msMyInt = primMinusInt;

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
primCmpInt (Main.Pos x) (Main.Neg y) = GT;
primCmpInt (Main.Neg x) (Main.Pos y) = LT;
primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;

primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
primCmpNat Main.Zero Main.Zero = EQ;
primCmpNat Main.Zero (Main.Succ y) = LT;
primCmpNat (Main.Succ x) Main.Zero = GT;
primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;

primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

repeat :: a  ->  List a;
repeat x = repeatXs x;

repeatXs wx = Cons wx (repeatXs wx);

replicate :: MyInt  ->  a  ->  List a;
replicate n x = take n (repeat x);

take :: MyInt  ->  List a  ->  List a;
take n vv = take3 n vv;
take vw Nil = take1 vw Nil;
take n (Cons x xs) = take0 n (Cons x xs);

take0 n (Cons x xs) = Cons x (take (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs);

take1 vw Nil = Nil;
take1 vz wu = take0 vz wu;

take2 n vv MyTrue = Nil;
take2 n vv MyFalse = take1 n vv;

take3 n vv = take2 n vv (ltEsMyInt n (Main.Pos Main.Zero));
take3 wv ww = take1 wv ww;

}

----------------------------------------

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="replicate",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="replicate wy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="replicate wy3 wy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="take wy3 (repeat wy4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="take3 wy3 (repeat wy4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
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];
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];
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[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];
61 -> 11[label="",style="solid", color="burlywood", weight=3];
62[label="wy3/Neg wy30",fontsize=10,color="white",style="solid",shape="box"];10 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 12[label="",style="solid", color="burlywood", weight=3];
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];
63 -> 13[label="",style="solid", color="burlywood", weight=3];
64[label="wy30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 14[label="",style="solid", color="burlywood", weight=3];
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];
65 -> 15[label="",style="solid", color="burlywood", weight=3];
66[label="wy30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 16[label="",style="solid", color="burlywood", weight=3];
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];
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];
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];
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];
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];
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];
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];
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];
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];
22[label="take2 (Pos Zero) (repeat wy4) (not MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
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];
24[label="take2 (Neg Zero) (repeat wy4) (not MyFalse)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
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];
26[label="take2 (Pos Zero) (repeat wy4) MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="take2 (Neg (Succ wy300)) (repeat wy4) MyTrue",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="take2 (Neg Zero) (repeat wy4) MyTrue",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="take2 (Pos (Succ wy300)) (repeat wy4) MyFalse",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
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];
34[label="take1 (Pos (Succ wy300)) (repeatXs wy4)",fontsize=16,color="black",shape="triangle"];34 -> 35[label="",style="solid", color="black", weight=3];
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];
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];
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];
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];
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];
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];
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];
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];
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];
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];
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];
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];
67 -> 47[label="",style="solid", color="burlywood", weight=3];
68[label="wy300/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 48[label="",style="solid", color="burlywood", weight=3];
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];
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];
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];
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];
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];
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];
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];
54[label="take2 (Pos Zero) (repeatXs wy4) (not MyFalse)",fontsize=16,color="black",shape="box"];54 -> 56[label="",style="solid", color="black", weight=3];
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];
56[label="take2 (Pos Zero) (repeatXs wy4) MyTrue",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
57[label="take2 (Pos (Succ wy3000)) (repeatXs wy4) MyFalse",fontsize=16,color="black",shape="box"];57 -> 59[label="",style="solid", color="black", weight=3];
58[label="Nil",fontsize=16,color="green",shape="box"];59 -> 34[label="",style="dashed", color="red", weight=0];
59[label="take1 (Pos (Succ wy3000)) (repeatXs wy4)",fontsize=16,color="magenta"];59 -> 60[label="",style="dashed", color="magenta", weight=3];
60[label="wy3000",fontsize=16,color="green",shape="box"];}

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(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_take1(Main.Succ(wy3000), wy4, h) -> new_take1(wy3000, wy4, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(7) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*new_take1(Main.Succ(wy3000), wy4, h) -> new_take1(wy3000, wy4, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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(8)
YES