/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) TransformationProof [EQUIVALENT, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) TransformationProof [EQUIVALENT, 0 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) QDP
(15) UsableRulesProof [EQUIVALENT, 0 ms]
(16) QDP
(17) QReductionProof [EQUIVALENT, 0 ms]
(18) QDP
(19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(20) 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 ;

data Tup2 a b = Tup2 a b ;

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));

splitAt :: MyInt  ->  List a  ->  Tup2 (List a) (List a);
splitAt n xs = splitAt3 n xs;
splitAt vv Nil = splitAt1 vv Nil;
splitAt n (Cons x xs) = splitAt0 n (Cons x xs);

splitAt0 n (Cons x xs) = Tup2 (Cons x (splitAt0Xs' n xs)) (splitAt0Xs'' n xs);

splitAt0Vu42 wy wz = splitAt (msMyInt wy (Main.Pos (Main.Succ Main.Zero))) wz;

splitAt0Xs' wy wz = splitAt0Xs'0 wy wz (splitAt0Vu42 wy wz);

splitAt0Xs'' wy wz = splitAt0Xs''0 wy wz (splitAt0Vu42 wy wz);

splitAt0Xs''0 wy wz (Tup2 vw xs'') = xs'';

splitAt0Xs'0 wy wz (Tup2 xs' vx) = xs';

splitAt1 vv Nil = Tup2 Nil Nil;
splitAt1 wu wv = splitAt0 wu wv;

splitAt2 n xs MyTrue = Tup2 Nil xs;
splitAt2 n xs MyFalse = splitAt1 n xs;

splitAt3 n xs = splitAt2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
splitAt3 ww wx = splitAt1 ww wx;

}

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

(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 ;

data Tup2 b a = Tup2 b a ;

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));

splitAt :: MyInt  ->  List a  ->  Tup2 (List a) (List a);
splitAt n xs = splitAt3 n xs;
splitAt vv Nil = splitAt1 vv Nil;
splitAt n (Cons x xs) = splitAt0 n (Cons x xs);

splitAt0 n (Cons x xs) = Tup2 (Cons x (splitAt0Xs' n xs)) (splitAt0Xs'' n xs);

splitAt0Vu42 wy wz = splitAt (msMyInt wy (Main.Pos (Main.Succ Main.Zero))) wz;

splitAt0Xs' wy wz = splitAt0Xs'0 wy wz (splitAt0Vu42 wy wz);

splitAt0Xs'' wy wz = splitAt0Xs''0 wy wz (splitAt0Vu42 wy wz);

splitAt0Xs''0 wy wz (Tup2 vw xs'') = xs'';

splitAt0Xs'0 wy wz (Tup2 xs' vx) = xs';

splitAt1 vv Nil = Tup2 Nil Nil;
splitAt1 wu wv = splitAt0 wu wv;

splitAt2 n xs MyTrue = Tup2 Nil xs;
splitAt2 n xs MyFalse = splitAt1 n xs;

splitAt3 n xs = splitAt2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
splitAt3 ww wx = splitAt1 ww wx;

}

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

(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 ;

data Tup2 b a = Tup2 b a ;

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));

splitAt :: MyInt  ->  List a  ->  Tup2 (List a) (List a);
splitAt n xs = splitAt3 n xs;
splitAt vv Nil = splitAt1 vv Nil;
splitAt n (Cons x xs) = splitAt0 n (Cons x xs);

splitAt0 n (Cons x xs) = Tup2 (Cons x (splitAt0Xs' n xs)) (splitAt0Xs'' n xs);

splitAt0Vu42 wy wz = splitAt (msMyInt wy (Main.Pos (Main.Succ Main.Zero))) wz;

splitAt0Xs' wy wz = splitAt0Xs'0 wy wz (splitAt0Vu42 wy wz);

splitAt0Xs'' wy wz = splitAt0Xs''0 wy wz (splitAt0Vu42 wy wz);

splitAt0Xs''0 wy wz (Tup2 vw xs'') = xs'';

splitAt0Xs'0 wy wz (Tup2 xs' vx) = xs';

splitAt1 vv Nil = Tup2 Nil Nil;
splitAt1 wu wv = splitAt0 wu wv;

splitAt2 n xs MyTrue = Tup2 Nil xs;
splitAt2 n xs MyFalse = splitAt1 n xs;

splitAt3 n xs = splitAt2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
splitAt3 ww wx = splitAt1 ww wx;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="splitAt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="splitAt xu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="splitAt xu3 xu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="splitAt3 xu3 xu4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="splitAt2 xu3 xu4 (ltEsMyInt xu3 (Pos Zero))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="splitAt2 xu3 xu4 (fsEsOrdering (compareMyInt xu3 (Pos Zero)) GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="splitAt2 xu3 xu4 (not (esEsOrdering (compareMyInt xu3 (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="splitAt2 xu3 xu4 (not (esEsOrdering (primCmpInt xu3 (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];64[label="xu3/Pos xu30",fontsize=10,color="white",style="solid",shape="box"];9 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 10[label="",style="solid", color="burlywood", weight=3];
65[label="xu3/Neg xu30",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="splitAt2 (Pos xu30) xu4 (not (esEsOrdering (primCmpInt (Pos xu30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];66[label="xu30/Succ xu300",fontsize=10,color="white",style="solid",shape="box"];10 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 12[label="",style="solid", color="burlywood", weight=3];
67[label="xu30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="splitAt2 (Neg xu30) xu4 (not (esEsOrdering (primCmpInt (Neg xu30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];68[label="xu30/Succ xu300",fontsize=10,color="white",style="solid",shape="box"];11 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 14[label="",style="solid", color="burlywood", weight=3];
69[label="xu30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="splitAt2 (Pos (Succ xu300)) xu4 (not (esEsOrdering (primCmpInt (Pos (Succ xu300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="splitAt2 (Pos Zero) xu4 (not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="splitAt2 (Neg (Succ xu300)) xu4 (not (esEsOrdering (primCmpInt (Neg (Succ xu300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="splitAt2 (Neg Zero) xu4 (not (esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="splitAt2 (Pos (Succ xu300)) xu4 (not (esEsOrdering (primCmpNat (Succ xu300) Zero) GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="splitAt2 (Pos Zero) xu4 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="splitAt2 (Neg (Succ xu300)) xu4 (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="splitAt2 (Neg Zero) xu4 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="splitAt2 (Pos (Succ xu300)) xu4 (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="splitAt2 (Pos Zero) xu4 (not MyFalse)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="splitAt2 (Neg (Succ xu300)) xu4 (not MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="splitAt2 (Neg Zero) xu4 (not MyFalse)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="splitAt2 (Pos (Succ xu300)) xu4 (not MyTrue)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="splitAt2 (Pos Zero) xu4 MyTrue",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="splitAt2 (Neg (Succ xu300)) xu4 MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="splitAt2 (Neg Zero) xu4 MyTrue",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="splitAt2 (Pos (Succ xu300)) xu4 MyFalse",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="Tup2 Nil xu4",fontsize=16,color="green",shape="box"];30[label="Tup2 Nil xu4",fontsize=16,color="green",shape="box"];31[label="Tup2 Nil xu4",fontsize=16,color="green",shape="box"];32[label="splitAt1 (Pos (Succ xu300)) xu4",fontsize=16,color="burlywood",shape="box"];70[label="xu4/Cons xu40 xu41",fontsize=10,color="white",style="solid",shape="box"];32 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 33[label="",style="solid", color="burlywood", weight=3];
71[label="xu4/Nil",fontsize=10,color="white",style="solid",shape="box"];32 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 34[label="",style="solid", color="burlywood", weight=3];
33[label="splitAt1 (Pos (Succ xu300)) (Cons xu40 xu41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="splitAt1 (Pos (Succ xu300)) Nil",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="splitAt0 (Pos (Succ xu300)) (Cons xu40 xu41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="Tup2 Nil Nil",fontsize=16,color="green",shape="box"];37[label="Tup2 (Cons xu40 (splitAt0Xs' (Pos (Succ xu300)) xu41)) (splitAt0Xs'' (Pos (Succ xu300)) xu41)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
37 -> 39[label="",style="dashed", color="green", weight=3];
38[label="splitAt0Xs' (Pos (Succ xu300)) xu41",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="splitAt0Xs'' (Pos (Succ xu300)) xu41",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40 -> 44[label="",style="dashed", color="red", weight=0];
40[label="splitAt0Xs'0 (Pos (Succ xu300)) xu41 (splitAt0Vu42 (Pos (Succ xu300)) xu41)",fontsize=16,color="magenta"];40 -> 45[label="",style="dashed", color="magenta", weight=3];
41 -> 49[label="",style="dashed", color="red", weight=0];
41[label="splitAt0Xs''0 (Pos (Succ xu300)) xu41 (splitAt0Vu42 (Pos (Succ xu300)) xu41)",fontsize=16,color="magenta"];41 -> 50[label="",style="dashed", color="magenta", weight=3];
45[label="splitAt0Vu42 (Pos (Succ xu300)) xu41",fontsize=16,color="black",shape="triangle"];45 -> 47[label="",style="solid", color="black", weight=3];
44[label="splitAt0Xs'0 (Pos (Succ xu300)) xu41 xu5",fontsize=16,color="burlywood",shape="triangle"];72[label="xu5/Tup2 xu50 xu51",fontsize=10,color="white",style="solid",shape="box"];44 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 48[label="",style="solid", color="burlywood", weight=3];
50 -> 45[label="",style="dashed", color="red", weight=0];
50[label="splitAt0Vu42 (Pos (Succ xu300)) xu41",fontsize=16,color="magenta"];49[label="splitAt0Xs''0 (Pos (Succ xu300)) xu41 xu6",fontsize=16,color="burlywood",shape="triangle"];73[label="xu6/Tup2 xu60 xu61",fontsize=10,color="white",style="solid",shape="box"];49 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 52[label="",style="solid", color="burlywood", weight=3];
47 -> 4[label="",style="dashed", color="red", weight=0];
47[label="splitAt (msMyInt (Pos (Succ xu300)) (Pos (Succ Zero))) xu41",fontsize=16,color="magenta"];47 -> 53[label="",style="dashed", color="magenta", weight=3];
47 -> 54[label="",style="dashed", color="magenta", weight=3];
48[label="splitAt0Xs'0 (Pos (Succ xu300)) xu41 (Tup2 xu50 xu51)",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
52[label="splitAt0Xs''0 (Pos (Succ xu300)) xu41 (Tup2 xu60 xu61)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="msMyInt (Pos (Succ xu300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="xu41",fontsize=16,color="green",shape="box"];55[label="xu50",fontsize=16,color="green",shape="box"];56[label="xu61",fontsize=16,color="green",shape="box"];57[label="primMinusInt (Pos (Succ xu300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];57 -> 58[label="",style="solid", color="black", weight=3];
58[label="primMinusNat (Succ xu300) (Succ Zero)",fontsize=16,color="black",shape="box"];58 -> 59[label="",style="solid", color="black", weight=3];
59[label="primMinusNat xu300 Zero",fontsize=16,color="burlywood",shape="box"];74[label="xu300/Succ xu3000",fontsize=10,color="white",style="solid",shape="box"];59 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 60[label="",style="solid", color="burlywood", weight=3];
75[label="xu300/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 61[label="",style="solid", color="burlywood", weight=3];
60[label="primMinusNat (Succ xu3000) Zero",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
61[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="Pos (Succ xu3000)",fontsize=16,color="green",shape="box"];63[label="Pos Zero",fontsize=16,color="green",shape="box"];}

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

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

   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
   new_splitAt0Vu42(xu300, xu41, h) -> new_splitAt(new_primMinusNat(xu300), xu41, h)
   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt(new_primMinusNat(xu300), xu41, h)

The TRS R consists of the following rules:

   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)

The set Q consists of the following terms:

   new_primMinusNat(Main.Zero)
   new_primMinusNat(Main.Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_splitAt0Vu42(xu300, xu41, h) -> new_splitAt(new_primMinusNat(xu300), xu41, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2),new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2))
   (new_splitAt0Vu42(Main.Zero, y1, y2) -> new_splitAt(Main.Pos(Main.Zero), y1, y2),new_splitAt0Vu42(Main.Zero, y1, y2) -> new_splitAt(Main.Pos(Main.Zero), y1, y2))


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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt(new_primMinusNat(xu300), xu41, h)
   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
   new_splitAt0Vu42(Main.Zero, y1, y2) -> new_splitAt(Main.Pos(Main.Zero), y1, y2)

The TRS R consists of the following rules:

   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)

The set Q consists of the following terms:

   new_primMinusNat(Main.Zero)
   new_primMinusNat(Main.Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt(new_primMinusNat(xu300), xu41, h)

The TRS R consists of the following rules:

   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)

The set Q consists of the following terms:

   new_primMinusNat(Main.Zero)
   new_primMinusNat(Main.Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt(new_primMinusNat(xu300), xu41, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3),new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3))
   (new_splitAt(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Zero), y2, y3),new_splitAt(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Zero), y2, y3))


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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
   new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)
   new_splitAt(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Zero), y2, y3)

The TRS R consists of the following rules:

   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)

The set Q consists of the following terms:

   new_primMinusNat(Main.Zero)
   new_primMinusNat(Main.Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
   new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)

The TRS R consists of the following rules:

   new_primMinusNat(Main.Succ(xu3000)) -> Main.Pos(Main.Succ(xu3000))
   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)

The set Q consists of the following terms:

   new_primMinusNat(Main.Zero)
   new_primMinusNat(Main.Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
   new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)

R is empty.
The set Q consists of the following terms:

   new_primMinusNat(Main.Zero)
   new_primMinusNat(Main.Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNat(Main.Zero)
   new_primMinusNat(Main.Succ(x0))


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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
   new_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
   new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)

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

(19) 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_splitAt0Vu42(Main.Succ(x0), y1, y2) -> new_splitAt(Main.Pos(Main.Succ(x0)), y1, y2)
The graph contains the following edges 2 >= 2, 3 >= 3


*new_splitAt(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_splitAt(Main.Pos(Main.Succ(x0)), y2, y3)
The graph contains the following edges 2 > 2, 3 >= 3


*new_splitAt(Main.Pos(Main.Succ(xu300)), Cons(xu40, xu41), h) -> new_splitAt0Vu42(xu300, xu41, h)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3


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

(20)
YES