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

YES 0.515 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

((pred :: Ordering -> Ordering) :: Ordering -> Ordering)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((pred :: Ordering -> Ordering) :: Ordering -> Ordering)

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

toEnum 0 = LT

toEnum 1 = EQ

toEnum 2 = GT

is transformed to

toEnum wy = toEnum5 wy

toEnum wu = toEnum3 wu

toEnum vz = toEnum1 vz

toEnum0 True vz = GT

toEnum1 vz = toEnum0 (vz == 2) vz

toEnum2 True wu = EQ

toEnum2 wv ww = toEnum1 ww

toEnum3 wu = toEnum2 (wu == 1) wu

toEnum3 wx = toEnum1 wx

toEnum4 True wy = LT

toEnum4 wz xu = toEnum3 xu

toEnum5 wy = toEnum4 (wy == 0) wy

toEnum5 xv = toEnum3 xv

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

((pred :: Ordering -> Ordering) :: Ordering -> Ordering)

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

(pred :: Ordering -> Ordering)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="pred",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="pred xw3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="toEnum . (subtract (Pos (Succ Zero))) . fromEnum",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="toEnum ((subtract (Pos (Succ Zero))) . fromEnum)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="toEnum5 ((subtract (Pos (Succ Zero))) . fromEnum)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="toEnum4 ((subtract (Pos (Succ Zero))) . fromEnum == Pos Zero) ((subtract (Pos (Succ Zero))) . fromEnum)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="toEnum4 (primEqInt ((subtract (Pos (Succ Zero))) . fromEnum) (Pos Zero)) ((subtract (Pos (Succ Zero))) . fromEnum)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="toEnum4 (primEqInt (subtract (Pos (Succ Zero)) (fromEnum xw3)) (Pos Zero)) (subtract (Pos (Succ Zero)) (fromEnum xw3))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="toEnum4 (primEqInt (flip (-) (Pos (Succ Zero)) (fromEnum xw3)) (Pos Zero)) (flip (-) (Pos (Succ Zero)) (fromEnum xw3))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="toEnum4 (primEqInt ((-) fromEnum xw3 Pos (Succ Zero)) (Pos Zero)) ((-) fromEnum xw3 Pos (Succ Zero))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="toEnum4 (primEqInt (primMinusInt (fromEnum xw3) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (fromEnum xw3) (Pos (Succ Zero)))",fontsize=16,color="burlywood",shape="box"];46[label="xw3/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 13[label="",style="solid", color="burlywood", weight=3];
47[label="xw3/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 14[label="",style="solid", color="burlywood", weight=3];
48[label="xw3/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 15[label="",style="solid", color="burlywood", weight=3];
13[label="toEnum4 (primEqInt (primMinusInt (fromEnum LT) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (fromEnum LT) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="toEnum4 (primEqInt (primMinusInt (fromEnum EQ) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (fromEnum EQ) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="toEnum4 (primEqInt (primMinusInt (fromEnum GT) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (fromEnum GT) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
16[label="toEnum4 (primEqInt (primMinusInt (Pos Zero) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (Pos Zero) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
17[label="toEnum4 (primEqInt (primMinusInt (Pos (Succ Zero)) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (Pos (Succ Zero)) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
18[label="toEnum4 (primEqInt (primMinusInt (Pos (Succ (Succ Zero))) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (Pos (Succ (Succ Zero))) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
19[label="toEnum4 (primEqInt (primMinusNat Zero (Succ Zero)) (Pos Zero)) (primMinusNat Zero (Succ Zero))",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
20[label="toEnum4 (primEqInt (primMinusNat (Succ Zero) (Succ Zero)) (Pos Zero)) (primMinusNat (Succ Zero) (Succ Zero))",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3];
21[label="toEnum4 (primEqInt (primMinusNat (Succ (Succ Zero)) (Succ Zero)) (Pos Zero)) (primMinusNat (Succ (Succ Zero)) (Succ Zero))",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
22[label="toEnum4 (primEqInt (Neg (Succ Zero)) (Pos Zero)) (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
23[label="toEnum4 (primEqInt (primMinusNat Zero Zero) (Pos Zero)) (primMinusNat Zero Zero)",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="toEnum4 (primEqInt (primMinusNat (Succ Zero) Zero) (Pos Zero)) (primMinusNat (Succ Zero) Zero)",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
25[label="toEnum4 False (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
26[label="toEnum4 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero)",fontsize=16,color="black",shape="box"];26 -> 29[label="",style="solid", color="black", weight=3];
27[label="toEnum4 (primEqInt (Pos (Succ Zero)) (Pos Zero)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];27 -> 30[label="",style="solid", color="black", weight=3];
28[label="toEnum3 (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3];
29[label="toEnum4 True (Pos Zero)",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
30[label="toEnum4 False (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];30 -> 33[label="",style="solid", color="black", weight=3];
31[label="toEnum2 (Neg (Succ Zero) == Pos (Succ Zero)) (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3];
32[label="LT",fontsize=16,color="green",shape="box"];33[label="toEnum3 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="toEnum2 (primEqInt (Neg (Succ Zero)) (Pos (Succ Zero))) (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="toEnum2 (Pos (Succ Zero) == Pos (Succ Zero)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="toEnum2 False (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="toEnum2 (primEqInt (Pos (Succ Zero)) (Pos (Succ Zero))) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="toEnum1 (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="toEnum2 (primEqNat Zero Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="toEnum0 (Neg (Succ Zero) == Pos (Succ (Succ Zero))) (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3];
41[label="toEnum2 True (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="toEnum0 (primEqInt (Neg (Succ Zero)) (Pos (Succ (Succ Zero)))) (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="EQ",fontsize=16,color="green",shape="box"];44[label="toEnum0 False (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3];
45[label="error []",fontsize=16,color="red",shape="box"];}

toEnum	wy	= toEnum5 wy
toEnum	wu	= toEnum3 wu
toEnum	vz	= toEnum1 vz