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

YES 0.494 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

(((<) :: Bool -> Bool -> Bool) :: Bool -> Bool -> Bool)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

(((<) :: Bool -> Bool -> Bool) :: Bool -> Bool -> Bool)

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

compare x y

| x == y

= EQ

| x <= y

= LT

| otherwise

= GT

is transformed to

compare x y = compare3 x y

compare2 x y True = EQ

compare2 x y False = compare1 x y (x <= y)

compare0 x y True = GT

compare1 x y True = LT

compare1 x y False = compare0 x y otherwise

compare3 x y = compare2 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

((<) :: Bool -> Bool -> Bool)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(<)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(<) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(<) vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="compare vx3 vx4 == LT",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="compare3 vx3 vx4 == LT",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="compare2 vx3 vx4 (vx3 == vx4) == LT",fontsize=16,color="burlywood",shape="box"];31[label="vx3/False",fontsize=10,color="white",style="solid",shape="box"];7 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 8[label="",style="solid", color="burlywood", weight=3];
32[label="vx3/True",fontsize=10,color="white",style="solid",shape="box"];7 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="compare2 False vx4 (False == vx4) == LT",fontsize=16,color="burlywood",shape="box"];33[label="vx4/False",fontsize=10,color="white",style="solid",shape="box"];8 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 10[label="",style="solid", color="burlywood", weight=3];
34[label="vx4/True",fontsize=10,color="white",style="solid",shape="box"];8 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 11[label="",style="solid", color="burlywood", weight=3];
9[label="compare2 True vx4 (True == vx4) == LT",fontsize=16,color="burlywood",shape="box"];35[label="vx4/False",fontsize=10,color="white",style="solid",shape="box"];9 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 12[label="",style="solid", color="burlywood", weight=3];
36[label="vx4/True",fontsize=10,color="white",style="solid",shape="box"];9 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 13[label="",style="solid", color="burlywood", weight=3];
10[label="compare2 False False (False == False) == LT",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
11[label="compare2 False True (False == True) == LT",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3];
12[label="compare2 True False (True == False) == LT",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="compare2 True True (True == True) == LT",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="compare2 False False True == LT",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="compare2 False True False == LT",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="compare2 True False False == LT",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="compare2 True True True == LT",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="EQ == LT",fontsize=16,color="black",shape="triangle"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="compare1 False True (False <= True) == LT",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="compare1 True False (True <= False) == LT",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21 -> 18[label="",style="dashed", color="red", weight=0];
21[label="EQ == LT",fontsize=16,color="magenta"];22[label="False",fontsize=16,color="green",shape="box"];23[label="compare1 False True True == LT",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="compare1 True False False == LT",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="LT == LT",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="compare0 True False otherwise == LT",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="True",fontsize=16,color="green",shape="box"];28[label="compare0 True False True == LT",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3];
29[label="GT == LT",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
30[label="False",fontsize=16,color="green",shape="box"];}

compare2	x y True	= EQ
compare2	x y False	= compare1 x y (x <= y)

compare1	x y True	= LT
compare1	x y False	= compare0 x y otherwise