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

YES 0.787 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

((isAscii :: Char -> Bool) :: Char -> Bool)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((isAscii :: Char -> Bool) :: Char -> Bool)

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

((isAscii :: Char -> Bool) :: Char -> Bool)

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

(isAscii :: Char -> Bool)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="isAscii",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="isAscii vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4 -> 5[label="",style="dashed", color="red", weight=0];
4[label="fromEnum vx3 < Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="magenta"];4 -> 6[label="",style="dashed", color="magenta", weight=3];
4 -> 7[label="",style="dashed", color="magenta", weight=3];
6[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];7[label="vx3",fontsize=16,color="green",shape="box"];5[label="fromEnum vx5 < Pos (Succ vx6)\n  Enum a",fontsize=16,color="black",shape="triangle"];5 -> 8[label="",style="solid", color="black", weight=3];
8[label="compare (fromEnum vx5) (Pos (Succ vx6)) == LT\n  Enum a",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9 -> 10[label="",style="dashed", color="red", weight=0];
9[label="primCmpInt (fromEnum vx5) (Pos (Succ vx6)) == LT\n  Enum a",fontsize=16,color="magenta"];9 -> 11[label="",style="dashed", color="magenta", weight=3];
9 -> 12[label="",style="dashed", color="magenta", weight=3];
11[label="fromEnum vx5\n  Enum a",fontsize=16,color="blue",shape="box"];62[label="fromEnum :: Double -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 62[label="",style="solid", color="blue", weight=9];
62 -> 13[label="",style="solid", color="blue", weight=3];
63[label="fromEnum :: Bool -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 63[label="",style="solid", color="blue", weight=9];
63 -> 14[label="",style="solid", color="blue", weight=3];
64[label="fromEnum :: Float -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 64[label="",style="solid", color="blue", weight=9];
64 -> 15[label="",style="solid", color="blue", weight=3];
65[label="fromEnum :: Integer -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 65[label="",style="solid", color="blue", weight=9];
65 -> 16[label="",style="solid", color="blue", weight=3];
66[label="fromEnum :: (Ratio a) -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 66[label="",style="solid", color="blue", weight=9];
66 -> 17[label="",style="solid", color="blue", weight=3];
67[label="fromEnum :: Ordering -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 67[label="",style="solid", color="blue", weight=9];
67 -> 18[label="",style="solid", color="blue", weight=3];
68[label="fromEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 68[label="",style="solid", color="blue", weight=9];
68 -> 19[label="",style="solid", color="blue", weight=3];
69[label="fromEnum :: () -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="blue", weight=9];
69 -> 20[label="",style="solid", color="blue", weight=3];
70[label="fromEnum :: Char -> Int",fontsize=10,color="white",style="solid",shape="box"];11 -> 70[label="",style="solid", color="blue", weight=9];
70 -> 21[label="",style="solid", color="blue", weight=3];
12[label="vx6",fontsize=16,color="green",shape="box"];10[label="primCmpInt vx10 (Pos (Succ vx11)) == LT",fontsize=16,color="burlywood",shape="triangle"];71[label="vx10/Pos vx100",fontsize=10,color="white",style="solid",shape="box"];10 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 22[label="",style="solid", color="burlywood", weight=3];
72[label="vx10/Neg vx100",fontsize=10,color="white",style="solid",shape="box"];10 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 23[label="",style="solid", color="burlywood", weight=3];
13[label="fromEnum vx5",fontsize=16,color="black",shape="box"];13 -> 24[label="",style="solid", color="black", weight=3];
14[label="fromEnum vx5",fontsize=16,color="black",shape="box"];14 -> 25[label="",style="solid", color="black", weight=3];
15[label="fromEnum vx5",fontsize=16,color="black",shape="box"];15 -> 26[label="",style="solid", color="black", weight=3];
16[label="fromEnum vx5",fontsize=16,color="black",shape="box"];16 -> 27[label="",style="solid", color="black", weight=3];
17[label="fromEnum vx5\n  Integral a",fontsize=16,color="black",shape="box"];17 -> 28[label="",style="solid", color="black", weight=3];
18[label="fromEnum vx5",fontsize=16,color="black",shape="box"];18 -> 29[label="",style="solid", color="black", weight=3];
19[label="fromEnum vx5",fontsize=16,color="black",shape="box"];19 -> 30[label="",style="solid", color="black", weight=3];
20[label="fromEnum vx5",fontsize=16,color="black",shape="box"];20 -> 31[label="",style="solid", color="black", weight=3];
21[label="fromEnum vx5",fontsize=16,color="black",shape="box"];21 -> 32[label="",style="solid", color="black", weight=3];
22[label="primCmpInt (Pos vx100) (Pos (Succ vx11)) == LT",fontsize=16,color="burlywood",shape="box"];73[label="vx100/Succ vx1000",fontsize=10,color="white",style="solid",shape="box"];22 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 33[label="",style="solid", color="burlywood", weight=3];
74[label="vx100/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 34[label="",style="solid", color="burlywood", weight=3];
23[label="primCmpInt (Neg vx100) (Pos (Succ vx11)) == LT",fontsize=16,color="burlywood",shape="box"];75[label="vx100/Succ vx1000",fontsize=10,color="white",style="solid",shape="box"];23 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 35[label="",style="solid", color="burlywood", weight=3];
76[label="vx100/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 36[label="",style="solid", color="burlywood", weight=3];
24[label="error []",fontsize=16,color="red",shape="box"];25[label="error []",fontsize=16,color="red",shape="box"];26[label="error []",fontsize=16,color="red",shape="box"];27[label="error []",fontsize=16,color="red",shape="box"];28[label="error []",fontsize=16,color="red",shape="box"];29[label="error []",fontsize=16,color="red",shape="box"];30[label="error []",fontsize=16,color="red",shape="box"];31[label="error []",fontsize=16,color="red",shape="box"];32[label="primCharToInt vx5",fontsize=16,color="burlywood",shape="box"];77[label="vx5/Char vx50",fontsize=10,color="white",style="solid",shape="box"];32 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 37[label="",style="solid", color="burlywood", weight=3];
33[label="primCmpInt (Pos (Succ vx1000)) (Pos (Succ vx11)) == LT",fontsize=16,color="black",shape="box"];33 -> 38[label="",style="solid", color="black", weight=3];
34[label="primCmpInt (Pos Zero) (Pos (Succ vx11)) == LT",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
35[label="primCmpInt (Neg (Succ vx1000)) (Pos (Succ vx11)) == LT",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
36[label="primCmpInt (Neg Zero) (Pos (Succ vx11)) == LT",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
37[label="primCharToInt (Char vx50)",fontsize=16,color="black",shape="box"];37 -> 42[label="",style="solid", color="black", weight=3];
38[label="primCmpNat (Succ vx1000) (Succ vx11) == LT",fontsize=16,color="black",shape="box"];38 -> 43[label="",style="solid", color="black", weight=3];
39[label="primCmpNat Zero (Succ vx11) == LT",fontsize=16,color="black",shape="box"];39 -> 44[label="",style="solid", color="black", weight=3];
40[label="LT == LT",fontsize=16,color="black",shape="triangle"];40 -> 45[label="",style="solid", color="black", weight=3];
41 -> 40[label="",style="dashed", color="red", weight=0];
41[label="LT == LT",fontsize=16,color="magenta"];42[label="Pos vx50",fontsize=16,color="green",shape="box"];43[label="primCmpNat vx1000 vx11 == LT",fontsize=16,color="burlywood",shape="triangle"];79[label="vx1000/Succ vx10000",fontsize=10,color="white",style="solid",shape="box"];43 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 46[label="",style="solid", color="burlywood", weight=3];
80[label="vx1000/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 47[label="",style="solid", color="burlywood", weight=3];
44 -> 40[label="",style="dashed", color="red", weight=0];
44[label="LT == LT",fontsize=16,color="magenta"];45[label="True",fontsize=16,color="green",shape="box"];46[label="primCmpNat (Succ vx10000) vx11 == LT",fontsize=16,color="burlywood",shape="box"];82[label="vx11/Succ vx110",fontsize=10,color="white",style="solid",shape="box"];46 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 48[label="",style="solid", color="burlywood", weight=3];
83[label="vx11/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 49[label="",style="solid", color="burlywood", weight=3];
47[label="primCmpNat Zero vx11 == LT",fontsize=16,color="burlywood",shape="box"];84[label="vx11/Succ vx110",fontsize=10,color="white",style="solid",shape="box"];47 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 50[label="",style="solid", color="burlywood", weight=3];
85[label="vx11/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 51[label="",style="solid", color="burlywood", weight=3];
48[label="primCmpNat (Succ vx10000) (Succ vx110) == LT",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3];
49[label="primCmpNat (Succ vx10000) Zero == LT",fontsize=16,color="black",shape="box"];49 -> 53[label="",style="solid", color="black", weight=3];
50[label="primCmpNat Zero (Succ vx110) == LT",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3];
51[label="primCmpNat Zero Zero == LT",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
52 -> 43[label="",style="dashed", color="red", weight=0];
52[label="primCmpNat vx10000 vx110 == LT",fontsize=16,color="magenta"];52 -> 56[label="",style="dashed", color="magenta", weight=3];
52 -> 57[label="",style="dashed", color="magenta", weight=3];
53[label="GT == LT",fontsize=16,color="black",shape="box"];53 -> 58[label="",style="solid", color="black", weight=3];
54 -> 40[label="",style="dashed", color="red", weight=0];
54[label="LT == LT",fontsize=16,color="magenta"];55[label="EQ == LT",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="vx10000",fontsize=16,color="green",shape="box"];57[label="vx110",fontsize=16,color="green",shape="box"];58[label="False",fontsize=16,color="green",shape="box"];59[label="False",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

new_esEs(Succ(vx10000), Succ(vx110)) → new_esEs(vx10000, vx110)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

new_esEs(Succ(vx10000), Succ(vx110)) → new_esEs(vx10000, vx110)
The graph contains the following edges 1 > 1, 2 > 2