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

YES 0.427 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

((rangeSize :: ((),()) -> Int) :: ((),()) -> Int)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.
Binding Reductions:
The bind variable of the following binding Pattern

r@(vv,vw)

is replaced by the following term

(vv,vw)

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((rangeSize :: ((),()) -> Int) :: ((),()) -> Int)

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

rangeSize (vv,vw)

| null (range (vv,vw))

= 0

| otherwise

= index (vv,vw) vw + 1

is transformed to

rangeSize (vv,vw) = rangeSize2 (vv,vw)

rangeSize1 vv vw True = 0

rangeSize1 vv vw False = rangeSize0 vv vw otherwise

rangeSize0 vv vw True = index (vv,vw) vw + 1

rangeSize2 (vv,vw) = rangeSize1 vv vw (null (range (vv,vw)))

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

((rangeSize :: ((),()) -> Int) :: ((),()) -> 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

(rangeSize :: ((),()) -> Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="rangeSize",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="rangeSize ww3",fontsize=16,color="burlywood",shape="triangle"];19[label="ww3/(ww30,ww31)",fontsize=10,color="white",style="solid",shape="box"];3 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="rangeSize (ww30,ww31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="rangeSize2 (ww30,ww31)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="rangeSize1 ww30 ww31 (null (range (ww30,ww31)))",fontsize=16,color="burlywood",shape="box"];20[label="ww30/()",fontsize=10,color="white",style="solid",shape="box"];6 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 7[label="",style="solid", color="burlywood", weight=3];
7[label="rangeSize1 () ww31 (null (range ((),ww31)))",fontsize=16,color="burlywood",shape="box"];21[label="ww31/()",fontsize=10,color="white",style="solid",shape="box"];7 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 8[label="",style="solid", color="burlywood", weight=3];
8[label="rangeSize1 () () (null (range ((),())))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="rangeSize1 () () (null (() : []))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="rangeSize1 () () False",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="rangeSize0 () () otherwise",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="rangeSize0 () () True",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="index ((),()) () + Pos (Succ Zero)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="primPlusInt (index ((),()) ()) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="primPlusInt (Pos Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="Pos (primPlusNat Zero (Succ Zero))",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="green", weight=3];
17[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="Succ Zero",fontsize=16,color="green",shape="box"];}

rangeSize1	vv vw True	= 0
rangeSize1	vv vw False	= rangeSize0 vv vw otherwise