/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) DependencyGraphProof [EQUIVALENT, 0 ms]
        (21) AND
            (22) QDP
                (23) QDPSizeChangeProof [EQUIVALENT, 20 ms]
                (24) YES
            (25) QDP
                (26) MRRProof [EQUIVALENT, 0 ms]
                (27) QDP
                (28) PisEmptyProof [EQUIVALENT, 0 ms]
                (29) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Float = Float MyInt MyInt ;

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 b a = Tup2 b a ;

error :: a;
error = stop MyTrue;

floatProperFractionFloat (Float vv vw) = Tup2 (quotMyInt vv vw) (msFloat (Float vv vw) (fromIntFloat (quotMyInt vv vw)));

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));

primMinusFloat :: Float  ->  Float  ->  Float;
primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);

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;

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

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt vx vy = Main.error;

properFractionFloat :: Float  ->  Tup2 MyInt Float;
properFractionFloat = floatProperFractionFloat;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Float = Float MyInt MyInt ;

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 a b = Tup2 a b ;

error :: a;
error = stop MyTrue;

floatProperFractionFloat (Float vv vw) = Tup2 (quotMyInt vv vw) (msFloat (Float vv vw) (fromIntFloat (quotMyInt vv vw)));

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));

primMinusFloat :: Float  ->  Float  ->  Float;
primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);

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;

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

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt vx vy = Main.error;

properFractionFloat :: Float  ->  Tup2 MyInt Float;
properFractionFloat = floatProperFractionFloat;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}

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

(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 Float = Float MyInt MyInt ;

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 b a = Tup2 b a ;

error :: a;
error = stop MyTrue;

floatProperFractionFloat (Float vv vw) = Tup2 (quotMyInt vv vw) (msFloat (Float vv vw) (fromIntFloat (quotMyInt vv vw)));

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

msFloat :: Float  ->  Float  ->  Float;
msFloat = primMinusFloat;

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

primDivNatS :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primDivNatS Main.Zero Main.Zero = Main.error;
primDivNatS (Main.Succ x) Main.Zero = Main.error;
primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y);
primDivNatS Main.Zero (Main.Succ x) = Main.Zero;

primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y));
primDivNatS0 x y MyFalse = Main.Zero;

primGEqNatS :: Main.Nat  ->  Main.Nat  ->  MyBool;
primGEqNatS (Main.Succ x) Main.Zero = MyTrue;
primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y;
primGEqNatS Main.Zero (Main.Succ x) = MyFalse;
primGEqNatS Main.Zero Main.Zero = MyTrue;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (Main.Pos (Main.Succ Main.Zero));

primMinusFloat :: Float  ->  Float  ->  Float;
primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2);

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;

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

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primQuotInt :: MyInt  ->  MyInt  ->  MyInt;
primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y));
primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y));
primQuotInt vx vy = Main.error;

properFractionFloat :: Float  ->  Tup2 MyInt Float;
properFractionFloat = floatProperFractionFloat;

quotMyInt :: MyInt  ->  MyInt  ->  MyInt;
quotMyInt = primQuotInt;

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

stop :: MyBool  ->  a;
stop MyFalse = stop MyFalse;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="properFractionFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="properFractionFloat wv3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="floatProperFractionFloat wv3",fontsize=16,color="burlywood",shape="box"];344[label="wv3/Float wv30 wv31",fontsize=10,color="white",style="solid",shape="box"];4 -> 344[label="",style="solid", color="burlywood", weight=9];
344 -> 5[label="",style="solid", color="burlywood", weight=3];
5[label="floatProperFractionFloat (Float wv30 wv31)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="Tup2 (quotMyInt wv30 wv31) (msFloat (Float wv30 wv31) (fromIntFloat (quotMyInt wv30 wv31)))",fontsize=16,color="green",shape="box"];6 -> 7[label="",style="dashed", color="green", weight=3];
6 -> 8[label="",style="dashed", color="green", weight=3];
7[label="quotMyInt wv30 wv31",fontsize=16,color="black",shape="triangle"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 10[label="",style="dashed", color="red", weight=0];
8[label="msFloat (Float wv30 wv31) (fromIntFloat (quotMyInt wv30 wv31))",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="primQuotInt wv30 wv31",fontsize=16,color="burlywood",shape="box"];345[label="wv30/Pos wv300",fontsize=10,color="white",style="solid",shape="box"];9 -> 345[label="",style="solid", color="burlywood", weight=9];
345 -> 12[label="",style="solid", color="burlywood", weight=3];
346[label="wv30/Neg wv300",fontsize=10,color="white",style="solid",shape="box"];9 -> 346[label="",style="solid", color="burlywood", weight=9];
346 -> 13[label="",style="solid", color="burlywood", weight=3];
11 -> 7[label="",style="dashed", color="red", weight=0];
11[label="quotMyInt wv30 wv31",fontsize=16,color="magenta"];10[label="msFloat (Float wv30 wv31) (fromIntFloat wv4)",fontsize=16,color="black",shape="triangle"];10 -> 14[label="",style="solid", color="black", weight=3];
12[label="primQuotInt (Pos wv300) wv31",fontsize=16,color="burlywood",shape="box"];347[label="wv31/Pos wv310",fontsize=10,color="white",style="solid",shape="box"];12 -> 347[label="",style="solid", color="burlywood", weight=9];
347 -> 15[label="",style="solid", color="burlywood", weight=3];
348[label="wv31/Neg wv310",fontsize=10,color="white",style="solid",shape="box"];12 -> 348[label="",style="solid", color="burlywood", weight=9];
348 -> 16[label="",style="solid", color="burlywood", weight=3];
13[label="primQuotInt (Neg wv300) wv31",fontsize=16,color="burlywood",shape="box"];349[label="wv31/Pos wv310",fontsize=10,color="white",style="solid",shape="box"];13 -> 349[label="",style="solid", color="burlywood", weight=9];
349 -> 17[label="",style="solid", color="burlywood", weight=3];
350[label="wv31/Neg wv310",fontsize=10,color="white",style="solid",shape="box"];13 -> 350[label="",style="solid", color="burlywood", weight=9];
350 -> 18[label="",style="solid", color="burlywood", weight=3];
14[label="primMinusFloat (Float wv30 wv31) (fromIntFloat wv4)",fontsize=16,color="black",shape="box"];14 -> 19[label="",style="solid", color="black", weight=3];
15[label="primQuotInt (Pos wv300) (Pos wv310)",fontsize=16,color="burlywood",shape="box"];351[label="wv310/Succ wv3100",fontsize=10,color="white",style="solid",shape="box"];15 -> 351[label="",style="solid", color="burlywood", weight=9];
351 -> 20[label="",style="solid", color="burlywood", weight=3];
352[label="wv310/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 352[label="",style="solid", color="burlywood", weight=9];
352 -> 21[label="",style="solid", color="burlywood", weight=3];
16[label="primQuotInt (Pos wv300) (Neg wv310)",fontsize=16,color="burlywood",shape="box"];353[label="wv310/Succ wv3100",fontsize=10,color="white",style="solid",shape="box"];16 -> 353[label="",style="solid", color="burlywood", weight=9];
353 -> 22[label="",style="solid", color="burlywood", weight=3];
354[label="wv310/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 354[label="",style="solid", color="burlywood", weight=9];
354 -> 23[label="",style="solid", color="burlywood", weight=3];
17[label="primQuotInt (Neg wv300) (Pos wv310)",fontsize=16,color="burlywood",shape="box"];355[label="wv310/Succ wv3100",fontsize=10,color="white",style="solid",shape="box"];17 -> 355[label="",style="solid", color="burlywood", weight=9];
355 -> 24[label="",style="solid", color="burlywood", weight=3];
356[label="wv310/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 356[label="",style="solid", color="burlywood", weight=9];
356 -> 25[label="",style="solid", color="burlywood", weight=3];
18[label="primQuotInt (Neg wv300) (Neg wv310)",fontsize=16,color="burlywood",shape="box"];357[label="wv310/Succ wv3100",fontsize=10,color="white",style="solid",shape="box"];18 -> 357[label="",style="solid", color="burlywood", weight=9];
357 -> 26[label="",style="solid", color="burlywood", weight=3];
358[label="wv310/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 358[label="",style="solid", color="burlywood", weight=9];
358 -> 27[label="",style="solid", color="burlywood", weight=3];
19[label="primMinusFloat (Float wv30 wv31) (primIntToFloat wv4)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
20[label="primQuotInt (Pos wv300) (Pos (Succ wv3100))",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="primQuotInt (Pos wv300) (Pos Zero)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="primQuotInt (Pos wv300) (Neg (Succ wv3100))",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="primQuotInt (Pos wv300) (Neg Zero)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="primQuotInt (Neg wv300) (Pos (Succ wv3100))",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="primQuotInt (Neg wv300) (Pos Zero)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="primQuotInt (Neg wv300) (Neg (Succ wv3100))",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="primQuotInt (Neg wv300) (Neg Zero)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="primMinusFloat (Float wv30 wv31) (Float wv4 (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
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366 -> 61[label="",style="solid", color="burlywood", weight=3];
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367 -> 62[label="",style="solid", color="burlywood", weight=3];
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71[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3];
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81[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];81 -> 95[label="",style="solid", color="black", weight=3];
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380 -> 97[label="",style="solid", color="burlywood", weight=3];
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89[label="primMulNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];89 -> 105[label="",style="solid", color="black", weight=3];
90[label="wv310",fontsize=16,color="green",shape="box"];282[label="wv31000",fontsize=16,color="green",shape="box"];283[label="wv30000",fontsize=16,color="green",shape="box"];284[label="wv30000",fontsize=16,color="green",shape="box"];285[label="wv31000",fontsize=16,color="green",shape="box"];281[label="primDivNatS0 (Succ wv21) (Succ wv22) (primGEqNatS wv23 wv24)",fontsize=16,color="burlywood",shape="triangle"];385[label="wv23/Succ wv230",fontsize=10,color="white",style="solid",shape="box"];281 -> 385[label="",style="solid", color="burlywood", weight=9];
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97[label="primMinusNat (Succ wv3000) Zero",fontsize=16,color="black",shape="box"];97 -> 113[label="",style="solid", color="black", weight=3];
98[label="primMinusNat Zero (Succ wv400)",fontsize=16,color="black",shape="box"];98 -> 114[label="",style="solid", color="black", weight=3];
99[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];99 -> 115[label="",style="solid", color="black", weight=3];
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117[label="primPlusNat (Succ wv3000) Zero",fontsize=16,color="black",shape="box"];117 -> 133[label="",style="solid", color="black", weight=3];
118[label="primPlusNat Zero (Succ wv400)",fontsize=16,color="black",shape="box"];118 -> 134[label="",style="solid", color="black", weight=3];
119[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];119 -> 135[label="",style="solid", color="black", weight=3];
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127[label="Zero",fontsize=16,color="green",shape="box"];128[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];128 -> 143[label="",style="solid", color="black", weight=3];
129[label="Zero",fontsize=16,color="green",shape="box"];130[label="wv400",fontsize=16,color="green",shape="box"];131[label="wv3000",fontsize=16,color="green",shape="box"];132[label="Succ (Succ (primPlusNat wv3000 wv400))",fontsize=16,color="green",shape="box"];132 -> 144[label="",style="dashed", color="green", weight=3];
133[label="Succ wv3000",fontsize=16,color="green",shape="box"];134[label="Succ wv400",fontsize=16,color="green",shape="box"];135[label="Zero",fontsize=16,color="green",shape="box"];136[label="wv3100",fontsize=16,color="green",shape="box"];320 -> 281[label="",style="dashed", color="red", weight=0];
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320 -> 325[label="",style="dashed", color="magenta", weight=3];
321[label="primDivNatS0 (Succ wv21) (Succ wv22) MyTrue",fontsize=16,color="black",shape="triangle"];321 -> 326[label="",style="solid", color="black", weight=3];
322[label="primDivNatS0 (Succ wv21) (Succ wv22) MyFalse",fontsize=16,color="black",shape="box"];322 -> 327[label="",style="solid", color="black", weight=3];
323 -> 321[label="",style="dashed", color="red", weight=0];
323[label="primDivNatS0 (Succ wv21) (Succ wv22) MyTrue",fontsize=16,color="magenta"];142[label="Succ wv30000",fontsize=16,color="green",shape="box"];143[label="Zero",fontsize=16,color="green",shape="box"];144 -> 84[label="",style="dashed", color="red", weight=0];
144[label="primPlusNat wv3000 wv400",fontsize=16,color="magenta"];144 -> 151[label="",style="dashed", color="magenta", weight=3];
144 -> 152[label="",style="dashed", color="magenta", weight=3];
324[label="wv230",fontsize=16,color="green",shape="box"];325[label="wv240",fontsize=16,color="green",shape="box"];326[label="Succ (primDivNatS (primMinusNatS (Succ wv21) (Succ wv22)) (Succ (Succ wv22)))",fontsize=16,color="green",shape="box"];326 -> 328[label="",style="dashed", color="green", weight=3];
327[label="Zero",fontsize=16,color="green",shape="box"];151[label="wv400",fontsize=16,color="green",shape="box"];152[label="wv3000",fontsize=16,color="green",shape="box"];328 -> 38[label="",style="dashed", color="red", weight=0];
328[label="primDivNatS (primMinusNatS (Succ wv21) (Succ wv22)) (Succ (Succ wv22))",fontsize=16,color="magenta"];328 -> 329[label="",style="dashed", color="magenta", weight=3];
328 -> 330[label="",style="dashed", color="magenta", weight=3];
329[label="primMinusNatS (Succ wv21) (Succ wv22)",fontsize=16,color="black",shape="box"];329 -> 331[label="",style="solid", color="black", weight=3];
330[label="Succ wv22",fontsize=16,color="green",shape="box"];331[label="primMinusNatS wv21 wv22",fontsize=16,color="burlywood",shape="triangle"];395[label="wv21/Succ wv210",fontsize=10,color="white",style="solid",shape="box"];331 -> 395[label="",style="solid", color="burlywood", weight=9];
395 -> 332[label="",style="solid", color="burlywood", weight=3];
396[label="wv21/Zero",fontsize=10,color="white",style="solid",shape="box"];331 -> 396[label="",style="solid", color="burlywood", weight=9];
396 -> 333[label="",style="solid", color="burlywood", weight=3];
332[label="primMinusNatS (Succ wv210) wv22",fontsize=16,color="burlywood",shape="box"];397[label="wv22/Succ wv220",fontsize=10,color="white",style="solid",shape="box"];332 -> 397[label="",style="solid", color="burlywood", weight=9];
397 -> 334[label="",style="solid", color="burlywood", weight=3];
398[label="wv22/Zero",fontsize=10,color="white",style="solid",shape="box"];332 -> 398[label="",style="solid", color="burlywood", weight=9];
398 -> 335[label="",style="solid", color="burlywood", weight=3];
333[label="primMinusNatS Zero wv22",fontsize=16,color="burlywood",shape="box"];399[label="wv22/Succ wv220",fontsize=10,color="white",style="solid",shape="box"];333 -> 399[label="",style="solid", color="burlywood", weight=9];
399 -> 336[label="",style="solid", color="burlywood", weight=3];
400[label="wv22/Zero",fontsize=10,color="white",style="solid",shape="box"];333 -> 400[label="",style="solid", color="burlywood", weight=9];
400 -> 337[label="",style="solid", color="burlywood", weight=3];
334[label="primMinusNatS (Succ wv210) (Succ wv220)",fontsize=16,color="black",shape="box"];334 -> 338[label="",style="solid", color="black", weight=3];
335[label="primMinusNatS (Succ wv210) Zero",fontsize=16,color="black",shape="box"];335 -> 339[label="",style="solid", color="black", weight=3];
336[label="primMinusNatS Zero (Succ wv220)",fontsize=16,color="black",shape="box"];336 -> 340[label="",style="solid", color="black", weight=3];
337[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];337 -> 341[label="",style="solid", color="black", weight=3];
338 -> 331[label="",style="dashed", color="red", weight=0];
338[label="primMinusNatS wv210 wv220",fontsize=16,color="magenta"];338 -> 342[label="",style="dashed", color="magenta", weight=3];
338 -> 343[label="",style="dashed", color="magenta", weight=3];
339[label="Succ wv210",fontsize=16,color="green",shape="box"];340[label="Zero",fontsize=16,color="green",shape="box"];341[label="Zero",fontsize=16,color="green",shape="box"];342[label="wv220",fontsize=16,color="green",shape="box"];343[label="wv210",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMulNat(Main.Succ(wv3100)) -> new_primMulNat(wv3100)

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

(8) 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_primMulNat(Main.Succ(wv3100)) -> new_primMulNat(wv3100)
The graph contains the following edges 1 > 1


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

(9)
YES

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

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

   new_primPlusNat(Main.Succ(wv3000), Main.Succ(wv400)) -> new_primPlusNat(wv3000, wv400)

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

(11) 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_primPlusNat(Main.Succ(wv3000), Main.Succ(wv400)) -> new_primPlusNat(wv3000, wv400)
The graph contains the following edges 1 > 1, 2 > 2


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

(12)
YES

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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNat(Main.Succ(wv3000), Main.Succ(wv400)) -> new_primMinusNat(wv3000, wv400)

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

(14) 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_primMinusNat(Main.Succ(wv3000), Main.Succ(wv400)) -> new_primMinusNat(wv3000, wv400)
The graph contains the following edges 1 > 1, 2 > 2


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

(15)
YES

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

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

   new_primMinusNatS(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS(wv210, wv220)

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

(17) 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_primMinusNatS(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS(wv210, wv220)
The graph contains the following edges 1 > 1, 2 > 2


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

(18)
YES

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

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wv30000), Main.Zero)
   new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Succ(wv240)) -> new_primDivNatS0(wv21, wv22, wv230, wv240)
   new_primDivNatS00(wv21, wv22) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22))
   new_primDivNatS0(wv21, wv22, Main.Zero, Main.Zero) -> new_primDivNatS00(wv21, wv22)
   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Succ(wv31000)) -> new_primDivNatS0(wv30000, wv31000, wv30000, wv31000)
   new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22))
   new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero)

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
   new_primMinusNatS2 -> Main.Zero
   new_primMinusNatS1(wv30000) -> Main.Succ(wv30000)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
   new_primMinusNatS1(x0)
   new_primMinusNatS0(Main.Succ(x0), Main.Zero)

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

(20) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(21)
Complex Obligation (AND)

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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS0(wv21, wv22, Main.Zero, Main.Zero) -> new_primDivNatS00(wv21, wv22)
   new_primDivNatS00(wv21, wv22) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22))
   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Succ(wv31000)) -> new_primDivNatS0(wv30000, wv31000, wv30000, wv31000)
   new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Succ(wv240)) -> new_primDivNatS0(wv21, wv22, wv230, wv240)
   new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22))

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
   new_primMinusNatS2 -> Main.Zero
   new_primMinusNatS1(wv30000) -> Main.Succ(wv30000)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
   new_primMinusNatS1(x0)
   new_primMinusNatS0(Main.Succ(x0), Main.Zero)

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

(23) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(Main.Succ(x_1)) = 1 + x_1
   POL(Main.Zero) = 1
   POL(new_primMinusNatS0(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_primDivNatS00(wv21, wv22) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Succ(wv31000)) -> new_primDivNatS0(wv30000, wv31000, wv30000, wv31000) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4


*new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Succ(wv240)) -> new_primDivNatS0(wv21, wv22, wv230, wv240) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


*new_primDivNatS0(wv21, wv22, Main.Zero, Main.Zero) -> new_primDivNatS00(wv21, wv22) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1, 2 >= 2


*new_primDivNatS0(wv21, wv22, Main.Succ(wv230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wv21, wv22), Main.Succ(wv22)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)

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

(24)
YES

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

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wv30000), Main.Zero)

The TRS R consists of the following rules:

   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero
   new_primMinusNatS2 -> Main.Zero
   new_primMinusNatS1(wv30000) -> Main.Succ(wv30000)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
   new_primMinusNatS1(x0)
   new_primMinusNatS0(Main.Succ(x0), Main.Zero)

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

(26) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   new_primDivNatS(Main.Succ(Main.Succ(wv30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wv30000), Main.Zero)

Strictly oriented rules of the TRS R:

   new_primMinusNatS0(Main.Succ(wv210), Main.Succ(wv220)) -> new_primMinusNatS0(wv210, wv220)
   new_primMinusNatS0(Main.Succ(wv210), Main.Zero) -> Main.Succ(wv210)
   new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNatS0(Main.Zero, Main.Succ(wv220)) -> Main.Zero

Used ordering: Polynomial interpretation [POLO]:

   POL(Main.Succ(x_1)) = 1 + x_1
   POL(Main.Zero) = 2
   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
   POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(new_primMinusNatS1(x_1)) = 1 + x_1
   POL(new_primMinusNatS2) = 2


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

(27)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   new_primMinusNatS2 -> Main.Zero
   new_primMinusNatS1(wv30000) -> Main.Succ(wv30000)

The set Q consists of the following terms:

   new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNatS0(Main.Zero, Main.Zero)
   new_primMinusNatS2
   new_primMinusNatS0(Main.Zero, Main.Succ(x0))
   new_primMinusNatS1(x0)
   new_primMinusNatS0(Main.Succ(x0), Main.Zero)

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

(28) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(29)
YES