/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) UsableRulesProof [EQUIVALENT, 0 ms]
(12) QDP
(13) QReductionProof [EQUIVALENT, 0 ms]
(14) QDP
(15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(16) 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));

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));

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));

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="take",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="take wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="take wx3 wx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="take3 wx3 wx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="take2 wx3 wx4 (ltEsMyInt wx3 (Pos Zero))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="take2 wx3 wx4 (fsEsOrdering (compareMyInt wx3 (Pos Zero)) GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="take2 wx3 wx4 (not (esEsOrdering (compareMyInt wx3 (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="take2 wx3 wx4 (not (esEsOrdering (primCmpInt wx3 (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];48[label="wx3/Pos wx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 10[label="",style="solid", color="burlywood", weight=3];
49[label="wx3/Neg wx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="take2 (Pos wx30) wx4 (not (esEsOrdering (primCmpInt (Pos wx30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];50[label="wx30/Succ wx300",fontsize=10,color="white",style="solid",shape="box"];10 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 12[label="",style="solid", color="burlywood", weight=3];
51[label="wx30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="take2 (Neg wx30) wx4 (not (esEsOrdering (primCmpInt (Neg wx30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];52[label="wx30/Succ wx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 14[label="",style="solid", color="burlywood", weight=3];
53[label="wx30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="take2 (Pos (Succ wx300)) wx4 (not (esEsOrdering (primCmpInt (Pos (Succ wx300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="take2 (Pos Zero) wx4 (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="take2 (Neg (Succ wx300)) wx4 (not (esEsOrdering (primCmpInt (Neg (Succ wx300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="take2 (Neg Zero) wx4 (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="take2 (Pos (Succ wx300)) wx4 (not (esEsOrdering (primCmpNat (Succ wx300) Zero) GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="take2 (Pos Zero) wx4 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="take2 (Neg (Succ wx300)) wx4 (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="take2 (Neg Zero) wx4 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="take2 (Pos (Succ wx300)) wx4 (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="take2 (Pos Zero) wx4 (not MyFalse)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="take2 (Neg (Succ wx300)) wx4 (not MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="take2 (Neg Zero) wx4 (not MyFalse)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="take2 (Pos (Succ wx300)) wx4 (not MyTrue)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="take2 (Pos Zero) wx4 MyTrue",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="take2 (Neg (Succ wx300)) wx4 MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="take2 (Neg Zero) wx4 MyTrue",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="take2 (Pos (Succ wx300)) wx4 MyFalse",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="Nil",fontsize=16,color="green",shape="box"];30[label="Nil",fontsize=16,color="green",shape="box"];31[label="Nil",fontsize=16,color="green",shape="box"];32[label="take1 (Pos (Succ wx300)) wx4",fontsize=16,color="burlywood",shape="box"];54[label="wx4/Cons wx40 wx41",fontsize=10,color="white",style="solid",shape="box"];32 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 33[label="",style="solid", color="burlywood", weight=3];
55[label="wx4/Nil",fontsize=10,color="white",style="solid",shape="box"];32 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 34[label="",style="solid", color="burlywood", weight=3];
33[label="take1 (Pos (Succ wx300)) (Cons wx40 wx41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="take1 (Pos (Succ wx300)) Nil",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="take0 (Pos (Succ wx300)) (Cons wx40 wx41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="Nil",fontsize=16,color="green",shape="box"];37[label="Cons wx40 (take (msMyInt (Pos (Succ wx300)) (Pos (Succ Zero))) wx41)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
38 -> 4[label="",style="dashed", color="red", weight=0];
38[label="take (msMyInt (Pos (Succ wx300)) (Pos (Succ Zero))) wx41",fontsize=16,color="magenta"];38 -> 39[label="",style="dashed", color="magenta", weight=3];
38 -> 40[label="",style="dashed", color="magenta", weight=3];
39[label="wx41",fontsize=16,color="green",shape="box"];40[label="msMyInt (Pos (Succ wx300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
41[label="primMinusInt (Pos (Succ wx300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42[label="primMinusNat (Succ wx300) (Succ Zero)",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
43[label="primMinusNat wx300 Zero",fontsize=16,color="burlywood",shape="box"];56[label="wx300/Succ wx3000",fontsize=10,color="white",style="solid",shape="box"];43 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 44[label="",style="solid", color="burlywood", weight=3];
57[label="wx300/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 45[label="",style="solid", color="burlywood", weight=3];
44[label="primMinusNat (Succ wx3000) Zero",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
46[label="Pos (Succ wx3000)",fontsize=16,color="green",shape="box"];47[label="Pos Zero",fontsize=16,color="green",shape="box"];}

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

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

   new_take(Main.Pos(Main.Succ(wx300)), Cons(wx40, wx41), h) -> new_take(new_primMinusNat(wx300), wx41, h)

The TRS R consists of the following rules:

   new_primMinusNat(Main.Succ(wx3000)) -> Main.Pos(Main.Succ(wx3000))
   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_take(Main.Pos(Main.Succ(wx300)), Cons(wx40, wx41), h) -> new_take(new_primMinusNat(wx300), wx41, h) at position [0] we obtained the following new rules [LPAR04]:

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


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

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

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

The TRS R consists of the following rules:

   new_primMinusNat(Main.Succ(wx3000)) -> Main.Pos(Main.Succ(wx3000))
   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.
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(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
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(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   new_primMinusNat(Main.Succ(wx3000)) -> Main.Pos(Main.Succ(wx3000))
   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.
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(11) 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.
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(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_take(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_take(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.
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(13) 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))


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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(15) 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_take(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_take(Main.Pos(Main.Succ(x0)), y2, y3)
The graph contains the following edges 2 > 2, 3 >= 3


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(16)
YES