H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 0.431 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

((fromIntegral :: Num a => Int -> a) :: Num a => Int -> a)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((fromIntegral :: Num a => Int -> a) :: Num a => Int -> a)

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

((fromIntegral :: Num a => Int -> a) :: Num a => Int -> a)

module Main where

import qualified Prelude

Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(fromIntegral :: Num a => Int -> a)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="fromIntegral\n  Num a",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="fromIntegral vx3\n  Num a",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="fromInteger . toInteger\n  Num a",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="fromInteger (toInteger vx3)\n  Num a",fontsize=16,color="blue",shape="box"];51[label="fromInteger :: Integer -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];5 -> 51[label="",style="solid", color="blue", weight=9];
51 -> 6[label="",style="solid", color="blue", weight=3];
52[label="fromInteger :: Integer -> Double",fontsize=10,color="white",style="solid",shape="box"];5 -> 52[label="",style="solid", color="blue", weight=9];
52 -> 7[label="",style="solid", color="blue", weight=3];
53[label="fromInteger :: Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];5 -> 53[label="",style="solid", color="blue", weight=9];
53 -> 8[label="",style="solid", color="blue", weight=3];
54[label="fromInteger :: Integer -> Int",fontsize=10,color="white",style="solid",shape="box"];5 -> 54[label="",style="solid", color="blue", weight=9];
54 -> 9[label="",style="solid", color="blue", weight=3];
55[label="fromInteger :: Integer -> Float",fontsize=10,color="white",style="solid",shape="box"];5 -> 55[label="",style="solid", color="blue", weight=9];
55 -> 10[label="",style="solid", color="blue", weight=3];
6[label="fromInteger (toInteger vx3)\n  Integral a",fontsize=16,color="black",shape="box"];6 -> 11[label="",style="solid", color="black", weight=3];
7[label="fromInteger (toInteger vx3)",fontsize=16,color="black",shape="box"];7 -> 12[label="",style="solid", color="black", weight=3];
8[label="fromInteger (toInteger vx3)",fontsize=16,color="black",shape="box"];8 -> 36[label="",style="solid", color="black", weight=3];
9[label="fromInteger (toInteger vx3)",fontsize=16,color="black",shape="box"];9 -> 37[label="",style="solid", color="black", weight=3];
10[label="fromInteger (toInteger vx3)",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3];
11[label="fromInteger (toInteger vx3) :% fromInt (Pos (Succ Zero))\n  Integral a",fontsize=16,color="green",shape="box"];11 -> 16[label="",style="dashed", color="green", weight=3];
11 -> 17[label="",style="dashed", color="green", weight=3];
12[label="primIntegerToDouble (toInteger vx3)",fontsize=16,color="black",shape="box"];12 -> 18[label="",style="solid", color="black", weight=3];
36 -> 24[label="",style="dashed", color="red", weight=0];
36[label="toInteger vx3",fontsize=16,color="magenta"];37 -> 30[label="",style="dashed", color="red", weight=0];
37[label="fromInteger (Integer vx3)",fontsize=16,color="magenta"];37 -> 46[label="",style="dashed", color="magenta", weight=3];
15 -> 21[label="",style="dashed", color="red", weight=0];
15[label="primIntegerToFloat (toInteger vx3)",fontsize=16,color="magenta"];15 -> 22[label="",style="dashed", color="magenta", weight=3];
16[label="fromInteger (toInteger vx3)\n  Integral a",fontsize=16,color="blue",shape="box"];59[label="fromInteger :: Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];16 -> 59[label="",style="solid", color="blue", weight=9];
59 -> 38[label="",style="solid", color="blue", weight=3];
60[label="fromInteger :: Integer -> Int",fontsize=10,color="white",style="solid",shape="box"];16 -> 60[label="",style="solid", color="blue", weight=9];
60 -> 39[label="",style="solid", color="blue", weight=3];
17[label="fromInt (Pos (Succ Zero))\n  Integral a",fontsize=16,color="blue",shape="box"];61[label="fromInt :: -> Int Integer",fontsize=10,color="white",style="solid",shape="box"];17 -> 61[label="",style="solid", color="blue", weight=9];
61 -> 31[label="",style="solid", color="blue", weight=3];
62[label="fromInt :: -> Int Int",fontsize=10,color="white",style="solid",shape="box"];17 -> 62[label="",style="solid", color="blue", weight=9];
62 -> 32[label="",style="solid", color="blue", weight=3];
18[label="primIntegerToDouble (Integer vx3)",fontsize=16,color="black",shape="box"];18 -> 33[label="",style="solid", color="black", weight=3];
24[label="toInteger vx3",fontsize=16,color="black",shape="triangle"];24 -> 28[label="",style="solid", color="black", weight=3];
46[label="Integer vx3",fontsize=16,color="green",shape="box"];30[label="fromInteger vx5",fontsize=16,color="burlywood",shape="triangle"];63[label="vx5/Integer vx50",fontsize=10,color="white",style="solid",shape="box"];30 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 41[label="",style="solid", color="burlywood", weight=3];
22 -> 24[label="",style="dashed", color="red", weight=0];
22[label="toInteger vx3",fontsize=16,color="magenta"];21[label="primIntegerToFloat vx4",fontsize=16,color="burlywood",shape="triangle"];65[label="vx4/Integer vx40",fontsize=10,color="white",style="solid",shape="box"];21 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 34[label="",style="solid", color="burlywood", weight=3];
38 -> 29[label="",style="dashed", color="red", weight=0];
38[label="fromInteger (toInteger vx3)",fontsize=16,color="magenta"];38 -> 47[label="",style="dashed", color="magenta", weight=3];
39 -> 30[label="",style="dashed", color="red", weight=0];
39[label="fromInteger (toInteger vx3)",fontsize=16,color="magenta"];39 -> 48[label="",style="dashed", color="magenta", weight=3];
31[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];31 -> 42[label="",style="solid", color="black", weight=3];
32[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];32 -> 43[label="",style="solid", color="black", weight=3];
33[label="primIntToDouble vx3",fontsize=16,color="black",shape="box"];33 -> 44[label="",style="solid", color="black", weight=3];
28[label="Integer vx3",fontsize=16,color="green",shape="box"];41[label="fromInteger (Integer vx50)",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3];
34[label="primIntegerToFloat (Integer vx40)",fontsize=16,color="black",shape="box"];34 -> 45[label="",style="solid", color="black", weight=3];
47 -> 24[label="",style="dashed", color="red", weight=0];
47[label="toInteger vx3",fontsize=16,color="magenta"];29[label="fromInteger vx5",fontsize=16,color="black",shape="triangle"];29 -> 40[label="",style="solid", color="black", weight=3];
48 -> 24[label="",style="dashed", color="red", weight=0];
48[label="toInteger vx3",fontsize=16,color="magenta"];42[label="Integer (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];43[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];44[label="Double vx3 (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];49[label="vx50",fontsize=16,color="green",shape="box"];45[label="primIntToFloat vx40",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
40[label="vx5",fontsize=16,color="green",shape="box"];50[label="Float vx40 (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];}