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


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


MAYBE
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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 not be shown:

(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) MRRProof [EQUIVALENT, 121 ms]
        (9) QDP
        (10) NonTerminationLoopProof [COMPLETE, 0 ms]
        (11) NO
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
(15) Narrow [COMPLETE, 0 ms]
(16) TRUE


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

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

data List a = Cons a (List a)  | Nil ;

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

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

data Main.WHNF a = WHNF a ;

dsEm :: (b  ->  a)  ->  b  ->  a;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF a  ->  b  ->  b;
enforceWHNF (Main.WHNF x) y = y;

enumFromFloat :: Float  ->  List Float;
enumFromFloat = numericEnumFrom;

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

numericEnumFrom n = Cons n (dsEm numericEnumFrom (psFloat n (fromIntFloat (Main.Pos (Main.Succ Main.Zero)))));

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

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;

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);

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

seq :: a  ->  b  ->  b;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

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

}

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

(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 List a = Cons a (List a)  | Nil ;

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

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

data Main.WHNF a = WHNF a ;

dsEm :: (a  ->  b)  ->  a  ->  b;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF a  ->  b  ->  b;
enforceWHNF (Main.WHNF x) y = y;

enumFromFloat :: Float  ->  List Float;
enumFromFloat = numericEnumFrom;

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

numericEnumFrom n = Cons n (dsEm numericEnumFrom (psFloat n (fromIntFloat (Main.Pos (Main.Succ Main.Zero)))));

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

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;

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);

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

seq :: a  ->  b  ->  b;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

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

}

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

(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 List a = Cons a (List a)  | Nil ;

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

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

data Main.WHNF a = WHNF a ;

dsEm :: (a  ->  b)  ->  a  ->  b;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF a  ->  b  ->  b;
enforceWHNF (Main.WHNF x) y = y;

enumFromFloat :: Float  ->  List Float;
enumFromFloat = numericEnumFrom;

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

numericEnumFrom n = Cons n (dsEm numericEnumFrom (psFloat n (fromIntFloat (Main.Pos (Main.Succ Main.Zero)))));

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

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;

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);

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

seq :: a  ->  b  ->  b;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="enumFromFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="enumFromFloat vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="numericEnumFrom vx3",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="Cons vx3 (dsEm numericEnumFrom (psFloat vx3 (fromIntFloat (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
6[label="dsEm numericEnumFrom (psFloat vx3 (fromIntFloat (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7 -> 8[label="",style="dashed", color="red", weight=0];
7[label="seq (psFloat vx3 (fromIntFloat (Pos (Succ Zero)))) (numericEnumFrom (psFloat vx3 (fromIntFloat (Pos (Succ Zero)))))",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
9 -> 4[label="",style="dashed", color="red", weight=0];
9[label="numericEnumFrom (psFloat vx3 (fromIntFloat (Pos (Succ Zero))))",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="seq (psFloat vx3 (fromIntFloat (Pos (Succ Zero)))) vx4",fontsize=16,color="black",shape="triangle"];8 -> 11[label="",style="solid", color="black", weight=3];
10[label="psFloat vx3 (fromIntFloat (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 13[label="",style="dashed", color="red", weight=0];
11[label="enforceWHNF (WHNF (psFloat vx3 (fromIntFloat (Pos (Succ Zero))))) vx4",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="primPlusFloat vx3 (fromIntFloat (Pos (Succ Zero)))",fontsize=16,color="burlywood",shape="box"];59[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];12 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 15[label="",style="solid", color="burlywood", weight=3];
14 -> 10[label="",style="dashed", color="red", weight=0];
14[label="psFloat vx3 (fromIntFloat (Pos (Succ Zero)))",fontsize=16,color="magenta"];13[label="enforceWHNF (WHNF vx5) vx4",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
15[label="primPlusFloat (Float vx30 vx31) (fromIntFloat (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="vx4",fontsize=16,color="green",shape="box"];17[label="primPlusFloat (Float vx30 vx31) (primIntToFloat (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="primPlusFloat (Float vx30 vx31) (Float (Pos (Succ Zero)) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="Float (psMyInt vx30 (Pos (Succ Zero))) (srMyInt vx31 (Pos (Succ Zero)))",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="green", weight=3];
19 -> 21[label="",style="dashed", color="green", weight=3];
20[label="psMyInt vx30 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="srMyInt vx31 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="primPlusInt vx30 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];60[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 24[label="",style="solid", color="burlywood", weight=3];
61[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 25[label="",style="solid", color="burlywood", weight=3];
23[label="primMulInt vx31 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];62[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];23 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 26[label="",style="solid", color="burlywood", weight=3];
63[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];23 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 27[label="",style="solid", color="burlywood", weight=3];
24[label="primPlusInt (Pos vx300) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="primPlusInt (Neg vx300) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="primMulInt (Pos vx310) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="primMulInt (Neg vx310) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="Pos (primPlusNat vx300 (Succ Zero))",fontsize=16,color="green",shape="box"];28 -> 32[label="",style="dashed", color="green", weight=3];
29[label="primMinusNat (Succ Zero) vx300",fontsize=16,color="burlywood",shape="box"];64[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];29 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 33[label="",style="solid", color="burlywood", weight=3];
65[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 34[label="",style="solid", color="burlywood", weight=3];
30[label="Pos (primMulNat vx310 (Succ Zero))",fontsize=16,color="green",shape="box"];30 -> 35[label="",style="dashed", color="green", weight=3];
31[label="Neg (primMulNat vx310 (Succ Zero))",fontsize=16,color="green",shape="box"];31 -> 36[label="",style="dashed", color="green", weight=3];
32[label="primPlusNat vx300 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];66[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];32 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 37[label="",style="solid", color="burlywood", weight=3];
67[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 38[label="",style="solid", color="burlywood", weight=3];
33[label="primMinusNat (Succ Zero) (Succ vx3000)",fontsize=16,color="black",shape="box"];33 -> 39[label="",style="solid", color="black", weight=3];
34[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3];
35[label="primMulNat vx310 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];68[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];35 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 41[label="",style="solid", color="burlywood", weight=3];
69[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 42[label="",style="solid", color="burlywood", weight=3];
36 -> 35[label="",style="dashed", color="red", weight=0];
36[label="primMulNat vx310 (Succ Zero)",fontsize=16,color="magenta"];36 -> 43[label="",style="dashed", color="magenta", weight=3];
37[label="primPlusNat (Succ vx3000) (Succ Zero)",fontsize=16,color="black",shape="box"];37 -> 44[label="",style="solid", color="black", weight=3];
38[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39[label="primMinusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];70[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];39 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 46[label="",style="solid", color="burlywood", weight=3];
71[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 47[label="",style="solid", color="burlywood", weight=3];
40[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];41[label="primMulNat (Succ vx3100) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
42[label="primMulNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
43[label="vx310",fontsize=16,color="green",shape="box"];44[label="Succ (Succ (primPlusNat vx3000 Zero))",fontsize=16,color="green",shape="box"];44 -> 50[label="",style="dashed", color="green", weight=3];
45[label="Succ Zero",fontsize=16,color="green",shape="box"];46[label="primMinusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
47[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];47 -> 52[label="",style="solid", color="black", weight=3];
48 -> 32[label="",style="dashed", color="red", weight=0];
48[label="primPlusNat (primMulNat vx3100 (Succ Zero)) (Succ Zero)",fontsize=16,color="magenta"];48 -> 53[label="",style="dashed", color="magenta", weight=3];
49[label="Zero",fontsize=16,color="green",shape="box"];50[label="primPlusNat vx3000 Zero",fontsize=16,color="burlywood",shape="box"];72[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];50 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 54[label="",style="solid", color="burlywood", weight=3];
73[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 55[label="",style="solid", color="burlywood", weight=3];
51[label="Neg (Succ vx30000)",fontsize=16,color="green",shape="box"];52[label="Pos Zero",fontsize=16,color="green",shape="box"];53 -> 35[label="",style="dashed", color="red", weight=0];
53[label="primMulNat vx3100 (Succ Zero)",fontsize=16,color="magenta"];53 -> 56[label="",style="dashed", color="magenta", weight=3];
54[label="primPlusNat (Succ vx30000) Zero",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
56[label="vx3100",fontsize=16,color="green",shape="box"];57[label="Succ vx30000",fontsize=16,color="green",shape="box"];58[label="Zero",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_numericEnumFrom(vx3) -> new_numericEnumFrom(new_psFloat(vx3))

The TRS R consists of the following rules:

   new_primPlusInt(Main.Neg(Main.Succ(Main.Zero))) -> Main.Pos(Main.Zero)
   new_primPlusInt(Main.Neg(Main.Zero)) -> Main.Pos(Main.Succ(Main.Zero))
   new_primMulInt(Main.Pos(vx310)) -> Main.Pos(new_primMulNat0(vx310))
   new_primPlusNat(Main.Zero) -> Main.Succ(Main.Zero)
   new_primPlusNat(Main.Succ(vx3000)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx3000)))
   new_psFloat(Float(vx30, vx31)) -> Float(new_primPlusInt(vx30), new_primMulInt(vx31))
   new_primPlusInt(Main.Pos(vx300)) -> Main.Pos(new_primPlusNat(vx300))
   new_primPlusInt(Main.Neg(Main.Succ(Main.Succ(vx30000)))) -> Main.Neg(Main.Succ(vx30000))
   new_primMulInt(Main.Neg(vx310)) -> Main.Neg(new_primMulNat0(vx310))
   new_primMulNat0(Main.Zero) -> Main.Zero
   new_primPlusNat0(Main.Succ(vx30000)) -> Main.Succ(vx30000)
   new_primMulNat0(Main.Succ(vx3100)) -> new_primPlusNat(new_primMulNat0(vx3100))
   new_primPlusNat0(Main.Zero) -> Main.Zero

The set Q consists of the following terms:

   new_primPlusNat0(Main.Zero)
   new_primPlusNat0(Main.Succ(x0))
   new_primMulNat0(Main.Succ(x0))
   new_primPlusInt(Main.Pos(x0))
   new_primPlusNat(Main.Succ(x0))
   new_psFloat(Float(x0, x1))
   new_primPlusNat(Main.Zero)
   new_primPlusInt(Main.Neg(Main.Zero))
   new_primPlusInt(Main.Neg(Main.Succ(Main.Zero)))
   new_primMulNat0(Main.Zero)
   new_primPlusInt(Main.Neg(Main.Succ(Main.Succ(x0))))
   new_primMulInt(Main.Neg(x0))
   new_primMulInt(Main.Pos(x0))

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

(8) 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 rules of the TRS R:

   new_primPlusInt(Main.Neg(Main.Succ(Main.Zero))) -> Main.Pos(Main.Zero)
   new_primPlusInt(Main.Neg(Main.Zero)) -> Main.Pos(Main.Succ(Main.Zero))

Used ordering: Polynomial interpretation [POLO]:

   POL(Float(x_1, x_2)) = 1 + x_1 + x_2
   POL(Main.Neg(x_1)) = 2*x_1
   POL(Main.Pos(x_1)) = x_1
   POL(Main.Succ(x_1)) = x_1
   POL(Main.Zero) = 2
   POL(new_numericEnumFrom(x_1)) = x_1
   POL(new_primMulInt(x_1)) = x_1
   POL(new_primMulNat0(x_1)) = x_1
   POL(new_primPlusInt(x_1)) = x_1
   POL(new_primPlusNat(x_1)) = x_1
   POL(new_primPlusNat0(x_1)) = x_1
   POL(new_psFloat(x_1)) = x_1


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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_numericEnumFrom(vx3) -> new_numericEnumFrom(new_psFloat(vx3))

The TRS R consists of the following rules:

   new_primMulInt(Main.Pos(vx310)) -> Main.Pos(new_primMulNat0(vx310))
   new_primPlusNat(Main.Zero) -> Main.Succ(Main.Zero)
   new_primPlusNat(Main.Succ(vx3000)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx3000)))
   new_psFloat(Float(vx30, vx31)) -> Float(new_primPlusInt(vx30), new_primMulInt(vx31))
   new_primPlusInt(Main.Pos(vx300)) -> Main.Pos(new_primPlusNat(vx300))
   new_primPlusInt(Main.Neg(Main.Succ(Main.Succ(vx30000)))) -> Main.Neg(Main.Succ(vx30000))
   new_primMulInt(Main.Neg(vx310)) -> Main.Neg(new_primMulNat0(vx310))
   new_primMulNat0(Main.Zero) -> Main.Zero
   new_primPlusNat0(Main.Succ(vx30000)) -> Main.Succ(vx30000)
   new_primMulNat0(Main.Succ(vx3100)) -> new_primPlusNat(new_primMulNat0(vx3100))
   new_primPlusNat0(Main.Zero) -> Main.Zero

The set Q consists of the following terms:

   new_primPlusNat0(Main.Zero)
   new_primPlusNat0(Main.Succ(x0))
   new_primMulNat0(Main.Succ(x0))
   new_primPlusInt(Main.Pos(x0))
   new_primPlusNat(Main.Succ(x0))
   new_psFloat(Float(x0, x1))
   new_primPlusNat(Main.Zero)
   new_primPlusInt(Main.Neg(Main.Zero))
   new_primPlusInt(Main.Neg(Main.Succ(Main.Zero)))
   new_primMulNat0(Main.Zero)
   new_primPlusInt(Main.Neg(Main.Succ(Main.Succ(x0))))
   new_primMulInt(Main.Neg(x0))
   new_primMulInt(Main.Pos(x0))

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

(10) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = new_numericEnumFrom(vx3) evaluates to  t =new_numericEnumFrom(new_psFloat(vx3))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [vx3 / new_psFloat(vx3)]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom(vx3) to new_numericEnumFrom(new_psFloat(vx3)).




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

(11)
NO

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

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

   new_primMulNat(Main.Succ(vx3100)) -> new_primMulNat(vx3100)

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

(13) 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(vx3100)) -> new_primMulNat(vx3100)
The graph contains the following edges 1 > 1


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

(14)
YES

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

(15) Narrow (COMPLETE)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="enumFromFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="enumFromFloat vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="numericEnumFrom vx3",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="Cons vx3 (dsEm numericEnumFrom (psFloat vx3 (fromIntFloat (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
6[label="dsEm numericEnumFrom (psFloat vx3 (fromIntFloat (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7 -> 8[label="",style="dashed", color="red", weight=0];
7[label="seq (psFloat vx3 (fromIntFloat (Pos (Succ Zero)))) (numericEnumFrom (psFloat vx3 (fromIntFloat (Pos (Succ Zero)))))",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
9 -> 4[label="",style="dashed", color="red", weight=0];
9[label="numericEnumFrom (psFloat vx3 (fromIntFloat (Pos (Succ Zero))))",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="seq (psFloat vx3 (fromIntFloat (Pos (Succ Zero)))) vx4",fontsize=16,color="black",shape="triangle"];8 -> 11[label="",style="solid", color="black", weight=3];
10[label="psFloat vx3 (fromIntFloat (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 13[label="",style="dashed", color="red", weight=0];
11[label="enforceWHNF (WHNF (psFloat vx3 (fromIntFloat (Pos (Succ Zero))))) vx4",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="primPlusFloat vx3 (fromIntFloat (Pos (Succ Zero)))",fontsize=16,color="burlywood",shape="box"];59[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];12 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 15[label="",style="solid", color="burlywood", weight=3];
14 -> 10[label="",style="dashed", color="red", weight=0];
14[label="psFloat vx3 (fromIntFloat (Pos (Succ Zero)))",fontsize=16,color="magenta"];13[label="enforceWHNF (WHNF vx5) vx4",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
15[label="primPlusFloat (Float vx30 vx31) (fromIntFloat (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="vx4",fontsize=16,color="green",shape="box"];17[label="primPlusFloat (Float vx30 vx31) (primIntToFloat (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="primPlusFloat (Float vx30 vx31) (Float (Pos (Succ Zero)) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="Float (psMyInt vx30 (Pos (Succ Zero))) (srMyInt vx31 (Pos (Succ Zero)))",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="green", weight=3];
19 -> 21[label="",style="dashed", color="green", weight=3];
20[label="psMyInt vx30 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="srMyInt vx31 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="primPlusInt vx30 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];60[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 24[label="",style="solid", color="burlywood", weight=3];
61[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 25[label="",style="solid", color="burlywood", weight=3];
23[label="primMulInt vx31 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];62[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];23 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 26[label="",style="solid", color="burlywood", weight=3];
63[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];23 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 27[label="",style="solid", color="burlywood", weight=3];
24[label="primPlusInt (Pos vx300) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="primPlusInt (Neg vx300) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="primMulInt (Pos vx310) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="primMulInt (Neg vx310) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="Pos (primPlusNat vx300 (Succ Zero))",fontsize=16,color="green",shape="box"];28 -> 32[label="",style="dashed", color="green", weight=3];
29[label="primMinusNat (Succ Zero) vx300",fontsize=16,color="burlywood",shape="box"];64[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];29 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 33[label="",style="solid", color="burlywood", weight=3];
65[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 34[label="",style="solid", color="burlywood", weight=3];
30[label="Pos (primMulNat vx310 (Succ Zero))",fontsize=16,color="green",shape="box"];30 -> 35[label="",style="dashed", color="green", weight=3];
31[label="Neg (primMulNat vx310 (Succ Zero))",fontsize=16,color="green",shape="box"];31 -> 36[label="",style="dashed", color="green", weight=3];
32[label="primPlusNat vx300 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];66[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];32 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 37[label="",style="solid", color="burlywood", weight=3];
67[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 38[label="",style="solid", color="burlywood", weight=3];
33[label="primMinusNat (Succ Zero) (Succ vx3000)",fontsize=16,color="black",shape="box"];33 -> 39[label="",style="solid", color="black", weight=3];
34[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3];
35[label="primMulNat vx310 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];68[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];35 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 41[label="",style="solid", color="burlywood", weight=3];
69[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 42[label="",style="solid", color="burlywood", weight=3];
36 -> 35[label="",style="dashed", color="red", weight=0];
36[label="primMulNat vx310 (Succ Zero)",fontsize=16,color="magenta"];36 -> 43[label="",style="dashed", color="magenta", weight=3];
37[label="primPlusNat (Succ vx3000) (Succ Zero)",fontsize=16,color="black",shape="box"];37 -> 44[label="",style="solid", color="black", weight=3];
38[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39[label="primMinusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];70[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];39 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 46[label="",style="solid", color="burlywood", weight=3];
71[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 47[label="",style="solid", color="burlywood", weight=3];
40[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];41[label="primMulNat (Succ vx3100) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
42[label="primMulNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
43[label="vx310",fontsize=16,color="green",shape="box"];44[label="Succ (Succ (primPlusNat vx3000 Zero))",fontsize=16,color="green",shape="box"];44 -> 50[label="",style="dashed", color="green", weight=3];
45[label="Succ Zero",fontsize=16,color="green",shape="box"];46[label="primMinusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
47[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];47 -> 52[label="",style="solid", color="black", weight=3];
48 -> 32[label="",style="dashed", color="red", weight=0];
48[label="primPlusNat (primMulNat vx3100 (Succ Zero)) (Succ Zero)",fontsize=16,color="magenta"];48 -> 53[label="",style="dashed", color="magenta", weight=3];
49[label="Zero",fontsize=16,color="green",shape="box"];50[label="primPlusNat vx3000 Zero",fontsize=16,color="burlywood",shape="box"];72[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];50 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 54[label="",style="solid", color="burlywood", weight=3];
73[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 55[label="",style="solid", color="burlywood", weight=3];
51[label="Neg (Succ vx30000)",fontsize=16,color="green",shape="box"];52[label="Pos Zero",fontsize=16,color="green",shape="box"];53 -> 35[label="",style="dashed", color="red", weight=0];
53[label="primMulNat vx3100 (Succ Zero)",fontsize=16,color="magenta"];53 -> 56[label="",style="dashed", color="magenta", weight=3];
54[label="primPlusNat (Succ vx30000) Zero",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
56[label="vx3100",fontsize=16,color="green",shape="box"];57[label="Succ vx30000",fontsize=16,color="green",shape="box"];58[label="Zero",fontsize=16,color="green",shape="box"];}

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

(16)
TRUE