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

YES 0.447 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

((fromRational :: Fractional a => Ratio Integer -> a) :: Fractional a => Ratio Integer -> a)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((fromRational :: Fractional a => Ratio Integer -> a) :: Fractional a => Ratio Integer -> 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

((fromRational :: Fractional a => Ratio Integer -> a) :: Fractional a => Ratio Integer -> 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

(fromRational :: Fractional a => Ratio Integer -> a)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="fromRational\n  Fractional a",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="fromRational vx3\n  Fractional a",fontsize=16,color="blue",shape="box"];32[label="fromRational :: (Ratio Integer) -> Double",fontsize=10,color="white",style="solid",shape="box"];3 -> 32[label="",style="solid", color="blue", weight=9];
32 -> 4[label="",style="solid", color="blue", weight=3];
33[label="fromRational :: (Ratio Integer) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];3 -> 33[label="",style="solid", color="blue", weight=9];
33 -> 5[label="",style="solid", color="blue", weight=3];
34[label="fromRational :: (Ratio Integer) -> Float",fontsize=10,color="white",style="solid",shape="box"];3 -> 34[label="",style="solid", color="blue", weight=9];
34 -> 6[label="",style="solid", color="blue", weight=3];
4[label="fromRational vx3",fontsize=16,color="black",shape="box"];4 -> 7[label="",style="solid", color="black", weight=3];
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35 -> 8[label="",style="solid", color="burlywood", weight=3];
6[label="fromRational vx3",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="primRationalToDouble vx3",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="fromRational (vx30 :% vx31)\n  Integral a",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="primRationalToFloat vx3",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
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36 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="fromInteger vx30 :% fromInteger vx31\n  Integral a",fontsize=16,color="green",shape="box"];11 -> 14[label="",style="dashed", color="green", weight=3];
11 -> 15[label="",style="dashed", color="green", weight=3];
12[label="rationalToFloat vx3",fontsize=16,color="burlywood",shape="box"];37[label="vx3/vx30 :% vx31",fontsize=10,color="white",style="solid",shape="box"];12 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 16[label="",style="solid", color="burlywood", weight=3];
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38 -> 17[label="",style="solid", color="burlywood", weight=3];
14[label="fromInteger vx30\n  Integral a",fontsize=16,color="blue",shape="box"];39[label="fromInteger :: Integer -> Int",fontsize=10,color="white",style="solid",shape="box"];14 -> 39[label="",style="solid", color="blue", weight=9];
39 -> 18[label="",style="solid", color="blue", weight=3];
40[label="fromInteger :: Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];14 -> 40[label="",style="solid", color="blue", weight=9];
40 -> 19[label="",style="solid", color="blue", weight=3];
15[label="fromInteger vx31\n  Integral a",fontsize=16,color="blue",shape="box"];41[label="fromInteger :: Integer -> Int",fontsize=10,color="white",style="solid",shape="box"];15 -> 41[label="",style="solid", color="blue", weight=9];
41 -> 20[label="",style="solid", color="blue", weight=3];
42[label="fromInteger :: Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];15 -> 42[label="",style="solid", color="blue", weight=9];
42 -> 21[label="",style="solid", color="blue", weight=3];
16[label="rationalToFloat (vx30 :% vx31)",fontsize=16,color="burlywood",shape="box"];43[label="vx30/Integer vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 22[label="",style="solid", color="burlywood", weight=3];
17[label="rationalToDouble (Integer vx300 :% vx31)",fontsize=16,color="burlywood",shape="box"];44[label="vx31/Integer vx310",fontsize=10,color="white",style="solid",shape="box"];17 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 23[label="",style="solid", color="burlywood", weight=3];
18[label="fromInteger vx30",fontsize=16,color="burlywood",shape="triangle"];45[label="vx30/Integer vx300",fontsize=10,color="white",style="solid",shape="box"];18 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 24[label="",style="solid", color="burlywood", weight=3];
19[label="fromInteger vx30",fontsize=16,color="black",shape="triangle"];19 -> 25[label="",style="solid", color="black", weight=3];
20 -> 18[label="",style="dashed", color="red", weight=0];
20[label="fromInteger vx31",fontsize=16,color="magenta"];20 -> 26[label="",style="dashed", color="magenta", weight=3];
21 -> 19[label="",style="dashed", color="red", weight=0];
21[label="fromInteger vx31",fontsize=16,color="magenta"];21 -> 27[label="",style="dashed", color="magenta", weight=3];
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48 -> 28[label="",style="solid", color="burlywood", weight=3];
23[label="rationalToDouble (Integer vx300 :% Integer vx310)",fontsize=16,color="black",shape="box"];23 -> 29[label="",style="solid", color="black", weight=3];
24[label="fromInteger (Integer vx300)",fontsize=16,color="black",shape="box"];24 -> 30[label="",style="solid", color="black", weight=3];
25[label="vx30",fontsize=16,color="green",shape="box"];26[label="vx31",fontsize=16,color="green",shape="box"];27[label="vx31",fontsize=16,color="green",shape="box"];28[label="rationalToFloat (Integer vx300 :% Integer vx310)",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3];
29[label="Double vx300 vx310",fontsize=16,color="green",shape="box"];30[label="vx300",fontsize=16,color="green",shape="box"];31[label="Float vx300 vx310",fontsize=16,color="green",shape="box"];}