/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 [EQUIVALENT, 14 ms]
(6) 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 Tup0 = Tup0 ;

data Tup2 a b = Tup2 a b ;

indexTup0 :: Tup2 Tup0 Tup0  ->  Tup0  ->  MyInt;
indexTup0 (Tup2 Tup0 Tup0) Tup0 = Main.Pos Main.Zero;

null :: List a  ->  MyBool;
null Nil = MyTrue;
null (Cons vx vy) = MyFalse;

otherwise :: MyBool;
otherwise = MyTrue;

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;

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

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

rangeSize0 vv vw MyTrue = psMyInt (indexTup0 (Tup2 vv vw) vw) (Main.Pos (Main.Succ Main.Zero));

rangeSize1 vv vw MyTrue = Main.Pos Main.Zero;
rangeSize1 vv vw MyFalse = rangeSize0 vv vw otherwise;

rangeSize2 (Tup2 vv vw) = rangeSize1 vv vw (null (rangeTup0 (Tup2 vv vw)));

rangeSizeTup0 :: Tup2 Tup0 Tup0  ->  MyInt;
rangeSizeTup0 (Tup2 vv vw) = rangeSize2 (Tup2 vv vw);

rangeTup0 :: Tup2 Tup0 Tup0  ->  List Tup0;
rangeTup0 (Tup2 Tup0 Tup0) = Cons Tup0 Nil;

}

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

(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 Tup0 = Tup0 ;

data Tup2 b a = Tup2 b a ;

indexTup0 :: Tup2 Tup0 Tup0  ->  Tup0  ->  MyInt;
indexTup0 (Tup2 Tup0 Tup0) Tup0 = Main.Pos Main.Zero;

null :: List a  ->  MyBool;
null Nil = MyTrue;
null (Cons vx vy) = MyFalse;

otherwise :: MyBool;
otherwise = MyTrue;

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;

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

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

rangeSize0 vv vw MyTrue = psMyInt (indexTup0 (Tup2 vv vw) vw) (Main.Pos (Main.Succ Main.Zero));

rangeSize1 vv vw MyTrue = Main.Pos Main.Zero;
rangeSize1 vv vw MyFalse = rangeSize0 vv vw otherwise;

rangeSize2 (Tup2 vv vw) = rangeSize1 vv vw (null (rangeTup0 (Tup2 vv vw)));

rangeSizeTup0 :: Tup2 Tup0 Tup0  ->  MyInt;
rangeSizeTup0 (Tup2 vv vw) = rangeSize2 (Tup2 vv vw);

rangeTup0 :: Tup2 Tup0 Tup0  ->  List Tup0;
rangeTup0 (Tup2 Tup0 Tup0) = Cons Tup0 Nil;

}

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

(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 Tup0 = Tup0 ;

data Tup2 a b = Tup2 a b ;

indexTup0 :: Tup2 Tup0 Tup0  ->  Tup0  ->  MyInt;
indexTup0 (Tup2 Tup0 Tup0) Tup0 = Main.Pos Main.Zero;

null :: List a  ->  MyBool;
null Nil = MyTrue;
null (Cons vx vy) = MyFalse;

otherwise :: MyBool;
otherwise = MyTrue;

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;

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

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

rangeSize0 vv vw MyTrue = psMyInt (indexTup0 (Tup2 vv vw) vw) (Main.Pos (Main.Succ Main.Zero));

rangeSize1 vv vw MyTrue = Main.Pos Main.Zero;
rangeSize1 vv vw MyFalse = rangeSize0 vv vw otherwise;

rangeSize2 (Tup2 vv vw) = rangeSize1 vv vw (null (rangeTup0 (Tup2 vv vw)));

rangeSizeTup0 :: Tup2 Tup0 Tup0  ->  MyInt;
rangeSizeTup0 (Tup2 vv vw) = rangeSize2 (Tup2 vv vw);

rangeTup0 :: Tup2 Tup0 Tup0  ->  List Tup0;
rangeTup0 (Tup2 Tup0 Tup0) = Cons Tup0 Nil;

}

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

(5) Narrow (EQUIVALENT)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="rangeSizeTup0",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="rangeSizeTup0 wv3",fontsize=16,color="burlywood",shape="triangle"];19[label="wv3/Tup2 wv30 wv31",fontsize=10,color="white",style="solid",shape="box"];3 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="rangeSizeTup0 (Tup2 wv30 wv31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="rangeSize2 (Tup2 wv30 wv31)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="rangeSize1 wv30 wv31 (null (rangeTup0 (Tup2 wv30 wv31)))",fontsize=16,color="burlywood",shape="box"];20[label="wv30/Tup0",fontsize=10,color="white",style="solid",shape="box"];6 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 7[label="",style="solid", color="burlywood", weight=3];
7[label="rangeSize1 Tup0 wv31 (null (rangeTup0 (Tup2 Tup0 wv31)))",fontsize=16,color="burlywood",shape="box"];21[label="wv31/Tup0",fontsize=10,color="white",style="solid",shape="box"];7 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 8[label="",style="solid", color="burlywood", weight=3];
8[label="rangeSize1 Tup0 Tup0 (null (rangeTup0 (Tup2 Tup0 Tup0)))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="rangeSize1 Tup0 Tup0 (null (Cons Tup0 Nil))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="rangeSize1 Tup0 Tup0 MyFalse",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="rangeSize0 Tup0 Tup0 otherwise",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="rangeSize0 Tup0 Tup0 MyTrue",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="psMyInt (indexTup0 (Tup2 Tup0 Tup0) Tup0) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="primPlusInt (indexTup0 (Tup2 Tup0 Tup0) Tup0) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="primPlusInt (Pos Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="Pos (primPlusNat Zero (Succ Zero))",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="green", weight=3];
17[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="Succ Zero",fontsize=16,color="green",shape="box"];}

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

(6)
YES