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

YES 0.48 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

((min :: () -> () -> ()) :: () -> () -> ())

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((min :: () -> () -> ()) :: () -> () -> ())

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

min x y

| x <= y

= x

| otherwise

= y

is transformed to

min x y = min2 x y

min0 x y True = y

min1 x y True = x

min1 x y False = min0 x y otherwise

min2 x y = min1 x y (x <= y)

The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(min :: () -> () -> ())

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="min",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="min vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="min vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="min2 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="min1 vx3 vx4 (vx3 <= vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="min1 vx3 vx4 (compare vx3 vx4 /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="min1 vx3 vx4 (not (compare vx3 vx4 == GT))",fontsize=16,color="burlywood",shape="box"];15[label="vx3/()",fontsize=10,color="white",style="solid",shape="box"];8 -> 15[label="",style="solid", color="burlywood", weight=9];
15 -> 9[label="",style="solid", color="burlywood", weight=3];
9[label="min1 () vx4 (not (compare () vx4 == GT))",fontsize=16,color="burlywood",shape="box"];16[label="vx4/()",fontsize=10,color="white",style="solid",shape="box"];9 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 10[label="",style="solid", color="burlywood", weight=3];
10[label="min1 () () (not (compare () () == GT))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="min1 () () (not (EQ == GT))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="min1 () () (not False)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="min1 () () True",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="()",fontsize=16,color="green",shape="box"];}