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

YES 0.533 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

((abs :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((abs :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int)

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

The following Function with conditions

absReal x

| x >= 0

= x

| otherwise

= `negate` x

is transformed to

absReal x = absReal2 x

absReal1 x True = x

absReal1 x False = absReal0 x otherwise

absReal0 x True = `negate` x

absReal2 x = absReal1 x (x >= 0)

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

((abs :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int)

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

(abs :: Ratio Int -> Ratio Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="abs",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="abs vx3",fontsize=16,color="burlywood",shape="triangle"];44[label="vx3/vx30 :% vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="abs (vx30 :% vx31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="abs vx30 :% vx31",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
6[label="abs vx30",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="absReal vx30",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="absReal2 vx30",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="absReal1 vx30 (vx30 >= fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="absReal1 vx30 (compare vx30 (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="absReal1 vx30 (not (compare vx30 (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="absReal1 vx30 (not (primCmpInt vx30 (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];45[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];12 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 13[label="",style="solid", color="burlywood", weight=3];
46[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];12 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="absReal1 (Pos vx300) (not (primCmpInt (Pos vx300) (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];47[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];13 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 15[label="",style="solid", color="burlywood", weight=3];
48[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="absReal1 (Neg vx300) (not (primCmpInt (Neg vx300) (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];49[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];14 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 17[label="",style="solid", color="burlywood", weight=3];
50[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="absReal1 (Pos (Succ vx3000)) (not (primCmpInt (Pos (Succ vx3000)) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="absReal1 (Pos Zero) (not (primCmpInt (Pos Zero) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="absReal1 (Neg (Succ vx3000)) (not (primCmpInt (Neg (Succ vx3000)) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="absReal1 (Neg Zero) (not (primCmpInt (Neg Zero) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="absReal1 (Pos (Succ vx3000)) (not (primCmpInt (Pos (Succ vx3000)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="absReal1 (Pos Zero) (not (primCmpInt (Pos Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="absReal1 (Neg (Succ vx3000)) (not (primCmpInt (Neg (Succ vx3000)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="absReal1 (Neg Zero) (not (primCmpInt (Neg Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="absReal1 (Pos (Succ vx3000)) (not (primCmpNat (Succ vx3000) Zero == LT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="absReal1 (Pos Zero) (not (EQ == LT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="absReal1 (Neg (Succ vx3000)) (not (LT == LT))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="absReal1 (Neg Zero) (not (EQ == LT))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="absReal1 (Pos (Succ vx3000)) (not (GT == LT))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="absReal1 (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="absReal1 (Neg (Succ vx3000)) (not True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="absReal1 (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="absReal1 (Pos (Succ vx3000)) (not False)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="absReal1 (Pos Zero) True",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="absReal1 (Neg (Succ vx3000)) False",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="absReal1 (Neg Zero) True",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="absReal1 (Pos (Succ vx3000)) True",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="Pos Zero",fontsize=16,color="green",shape="box"];37[label="absReal0 (Neg (Succ vx3000)) otherwise",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
38[label="Neg Zero",fontsize=16,color="green",shape="box"];39[label="Pos (Succ vx3000)",fontsize=16,color="green",shape="box"];40[label="absReal0 (Neg (Succ vx3000)) True",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
41[label="`negate` Neg (Succ vx3000)",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42[label="primNegInt (Neg (Succ vx3000))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
43[label="Pos (Succ vx3000)",fontsize=16,color="green",shape="box"];}