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

YES 0.459 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

((recip :: Fractional a => a -> a) :: Fractional a => a -> a)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((recip :: Fractional a => a -> a) :: Fractional a => a -> 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

((recip :: Fractional a => a -> a) :: Fractional a => a -> 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

(recip :: Fractional a => a -> a)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="recip\n  Fractional a",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="recip vx3\n  Fractional a",fontsize=16,color="blue",shape="box"];42[label="recip :: Double -> Double",fontsize=10,color="white",style="solid",shape="box"];3 -> 42[label="",style="solid", color="blue", weight=9];
42 -> 4[label="",style="solid", color="blue", weight=3];
43[label="recip :: (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];3 -> 43[label="",style="solid", color="blue", weight=9];
43 -> 5[label="",style="solid", color="blue", weight=3];
44[label="recip :: Float -> Float",fontsize=10,color="white",style="solid",shape="box"];3 -> 44[label="",style="solid", color="blue", weight=9];
44 -> 6[label="",style="solid", color="blue", weight=3];
4[label="recip vx3",fontsize=16,color="black",shape="box"];4 -> 7[label="",style="solid", color="black", weight=3];
5[label="recip vx3\n  Integral a",fontsize=16,color="burlywood",shape="box"];45[label="vx3/vx30 :% vx31",fontsize=10,color="white",style="solid",shape="box"];5 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 8[label="",style="solid", color="burlywood", weight=3];
6[label="recip vx3",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="fromInt (Pos (Succ Zero)) / vx3",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="recip (vx30 :% vx31)\n  Integral a",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="fromInt (Pos (Succ Zero)) / vx3",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="primDivDouble (fromInt (Pos (Succ Zero))) vx3",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="vx31 :% vx30\n  Integral a",fontsize=16,color="green",shape="box"];12[label="primDivFloat (fromInt (Pos (Succ Zero))) vx3",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="primDivDouble (primIntToDouble (Pos (Succ Zero))) vx3",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="primDivFloat (primIntToFloat (Pos (Succ Zero))) vx3",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="primDivDouble (Double (Pos (Succ Zero)) (Pos (Succ Zero))) vx3",fontsize=16,color="burlywood",shape="box"];46[label="vx3/Double vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];15 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="primDivFloat (Float (Pos (Succ Zero)) (Pos (Succ Zero))) vx3",fontsize=16,color="burlywood",shape="box"];47[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];16 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 18[label="",style="solid", color="burlywood", weight=3];
17[label="primDivDouble (Double (Pos (Succ Zero)) (Pos (Succ Zero))) (Double vx30 vx31)",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="primDivFloat (Float (Pos (Succ Zero)) (Pos (Succ Zero))) (Float vx30 vx31)",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="Double (Pos (Succ Zero) * vx31) (Pos (Succ Zero) * vx30)",fontsize=16,color="green",shape="box"];19 -> 21[label="",style="dashed", color="green", weight=3];
19 -> 22[label="",style="dashed", color="green", weight=3];
20[label="Float (Pos (Succ Zero) * vx31) (Pos (Succ Zero) * vx30)",fontsize=16,color="green",shape="box"];20 -> 23[label="",style="dashed", color="green", weight=3];
20 -> 24[label="",style="dashed", color="green", weight=3];
21[label="Pos (Succ Zero) * vx31",fontsize=16,color="black",shape="triangle"];21 -> 25[label="",style="solid", color="black", weight=3];
22 -> 21[label="",style="dashed", color="red", weight=0];
22[label="Pos (Succ Zero) * vx30",fontsize=16,color="magenta"];22 -> 26[label="",style="dashed", color="magenta", weight=3];
23 -> 21[label="",style="dashed", color="red", weight=0];
23[label="Pos (Succ Zero) * vx31",fontsize=16,color="magenta"];23 -> 27[label="",style="dashed", color="magenta", weight=3];
24 -> 21[label="",style="dashed", color="red", weight=0];
24[label="Pos (Succ Zero) * vx30",fontsize=16,color="magenta"];24 -> 28[label="",style="dashed", color="magenta", weight=3];
25[label="primMulInt (Pos (Succ Zero)) vx31",fontsize=16,color="burlywood",shape="box"];51[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];25 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 29[label="",style="solid", color="burlywood", weight=3];
52[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];25 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 30[label="",style="solid", color="burlywood", weight=3];
26[label="vx30",fontsize=16,color="green",shape="box"];27[label="vx31",fontsize=16,color="green",shape="box"];28[label="vx30",fontsize=16,color="green",shape="box"];29[label="primMulInt (Pos (Succ Zero)) (Pos vx310)",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="primMulInt (Pos (Succ Zero)) (Neg vx310)",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
31[label="Pos (primMulNat (Succ Zero) vx310)",fontsize=16,color="green",shape="box"];31 -> 33[label="",style="dashed", color="green", weight=3];
32[label="Neg (primMulNat (Succ Zero) vx310)",fontsize=16,color="green",shape="box"];32 -> 34[label="",style="dashed", color="green", weight=3];
33[label="primMulNat (Succ Zero) vx310",fontsize=16,color="burlywood",shape="triangle"];53[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];33 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 35[label="",style="solid", color="burlywood", weight=3];
54[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 36[label="",style="solid", color="burlywood", weight=3];
34 -> 33[label="",style="dashed", color="red", weight=0];
34[label="primMulNat (Succ Zero) vx310",fontsize=16,color="magenta"];34 -> 37[label="",style="dashed", color="magenta", weight=3];
35[label="primMulNat (Succ Zero) (Succ vx3100)",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
36[label="primMulNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
37[label="vx310",fontsize=16,color="green",shape="box"];38[label="primPlusNat (primMulNat Zero (Succ vx3100)) (Succ vx3100)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="Zero",fontsize=16,color="green",shape="box"];40[label="primPlusNat Zero (Succ vx3100)",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
41[label="Succ vx3100",fontsize=16,color="green",shape="box"];}