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

YES 0.51 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

(((>=) :: Ordering -> Ordering -> Bool) :: Ordering -> Ordering -> Bool)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

(((>=) :: Ordering -> Ordering -> Bool) :: Ordering -> Ordering -> 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

compare0 x y True = GT

compare1 x y True = LT

compare1 x y False = compare0 x y otherwise

compare2 x y True = EQ

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

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

((>=) :: Ordering -> Ordering -> 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="not (compare vx3 vx4 == LT)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="not (compare3 vx3 vx4 == LT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="not (compare2 vx3 vx4 (vx3 == vx4) == LT)",fontsize=16,color="burlywood",shape="box"];62[label="vx3/LT",fontsize=10,color="white",style="solid",shape="box"];8 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 9[label="",style="solid", color="burlywood", weight=3];
63[label="vx3/EQ",fontsize=10,color="white",style="solid",shape="box"];8 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 10[label="",style="solid", color="burlywood", weight=3];
64[label="vx3/GT",fontsize=10,color="white",style="solid",shape="box"];8 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 11[label="",style="solid", color="burlywood", weight=3];
9[label="not (compare2 LT vx4 (LT == vx4) == LT)",fontsize=16,color="burlywood",shape="box"];65[label="vx4/LT",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 12[label="",style="solid", color="burlywood", weight=3];
66[label="vx4/EQ",fontsize=10,color="white",style="solid",shape="box"];9 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 13[label="",style="solid", color="burlywood", weight=3];
67[label="vx4/GT",fontsize=10,color="white",style="solid",shape="box"];9 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 14[label="",style="solid", color="burlywood", weight=3];
10[label="not (compare2 EQ vx4 (EQ == vx4) == LT)",fontsize=16,color="burlywood",shape="box"];68[label="vx4/LT",fontsize=10,color="white",style="solid",shape="box"];10 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 15[label="",style="solid", color="burlywood", weight=3];
69[label="vx4/EQ",fontsize=10,color="white",style="solid",shape="box"];10 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 16[label="",style="solid", color="burlywood", weight=3];
70[label="vx4/GT",fontsize=10,color="white",style="solid",shape="box"];10 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 17[label="",style="solid", color="burlywood", weight=3];
11[label="not (compare2 GT vx4 (GT == vx4) == LT)",fontsize=16,color="burlywood",shape="box"];71[label="vx4/LT",fontsize=10,color="white",style="solid",shape="box"];11 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 18[label="",style="solid", color="burlywood", weight=3];
72[label="vx4/EQ",fontsize=10,color="white",style="solid",shape="box"];11 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 19[label="",style="solid", color="burlywood", weight=3];
73[label="vx4/GT",fontsize=10,color="white",style="solid",shape="box"];11 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 20[label="",style="solid", color="burlywood", weight=3];
12[label="not (compare2 LT LT (LT == LT) == LT)",fontsize=16,color="black",shape="box"];12 -> 21[label="",style="solid", color="black", weight=3];
13[label="not (compare2 LT EQ (LT == EQ) == LT)",fontsize=16,color="black",shape="box"];13 -> 22[label="",style="solid", color="black", weight=3];
14[label="not (compare2 LT GT (LT == GT) == LT)",fontsize=16,color="black",shape="box"];14 -> 23[label="",style="solid", color="black", weight=3];
15[label="not (compare2 EQ LT (EQ == LT) == LT)",fontsize=16,color="black",shape="box"];15 -> 24[label="",style="solid", color="black", weight=3];
16[label="not (compare2 EQ EQ (EQ == EQ) == LT)",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
17[label="not (compare2 EQ GT (EQ == GT) == LT)",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
18[label="not (compare2 GT LT (GT == LT) == LT)",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
19[label="not (compare2 GT EQ (GT == EQ) == LT)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
20[label="not (compare2 GT GT (GT == GT) == LT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="not (compare2 LT LT True == LT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="not (compare2 LT EQ False == LT)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="not (compare2 LT GT False == LT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="not (compare2 EQ LT False == LT)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="not (compare2 EQ EQ True == LT)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="not (compare2 EQ GT False == LT)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="not (compare2 GT LT False == LT)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="not (compare2 GT EQ False == LT)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
29[label="not (compare2 GT GT True == LT)",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
30[label="not (EQ == LT)",fontsize=16,color="black",shape="triangle"];30 -> 39[label="",style="solid", color="black", weight=3];
31[label="not (compare1 LT EQ (LT <= EQ) == LT)",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="not (compare1 LT GT (LT <= GT) == LT)",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="not (compare1 EQ LT (EQ <= LT) == LT)",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
34 -> 30[label="",style="dashed", color="red", weight=0];
34[label="not (EQ == LT)",fontsize=16,color="magenta"];35[label="not (compare1 EQ GT (EQ <= GT) == LT)",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
36[label="not (compare1 GT LT (GT <= LT) == LT)",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
37[label="not (compare1 GT EQ (GT <= EQ) == LT)",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3];
38 -> 30[label="",style="dashed", color="red", weight=0];
38[label="not (EQ == LT)",fontsize=16,color="magenta"];39[label="not False",fontsize=16,color="black",shape="triangle"];39 -> 46[label="",style="solid", color="black", weight=3];
40[label="not (compare1 LT EQ True == LT)",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
41[label="not (compare1 LT GT True == LT)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
42[label="not (compare1 EQ LT False == LT)",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
43[label="not (compare1 EQ GT True == LT)",fontsize=16,color="black",shape="box"];43 -> 50[label="",style="solid", color="black", weight=3];
44[label="not (compare1 GT LT False == LT)",fontsize=16,color="black",shape="box"];44 -> 51[label="",style="solid", color="black", weight=3];
45[label="not (compare1 GT EQ False == LT)",fontsize=16,color="black",shape="box"];45 -> 52[label="",style="solid", color="black", weight=3];
46[label="True",fontsize=16,color="green",shape="box"];47[label="not (LT == LT)",fontsize=16,color="black",shape="triangle"];47 -> 53[label="",style="solid", color="black", weight=3];
48 -> 47[label="",style="dashed", color="red", weight=0];
48[label="not (LT == LT)",fontsize=16,color="magenta"];49[label="not (compare0 EQ LT otherwise == LT)",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
50 -> 47[label="",style="dashed", color="red", weight=0];
50[label="not (LT == LT)",fontsize=16,color="magenta"];51[label="not (compare0 GT LT otherwise == LT)",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
52[label="not (compare0 GT EQ otherwise == LT)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="not True",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="not (compare0 EQ LT True == LT)",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55[label="not (compare0 GT LT True == LT)",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="not (compare0 GT EQ True == LT)",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3];
57[label="False",fontsize=16,color="green",shape="box"];58[label="not (GT == LT)",fontsize=16,color="black",shape="triangle"];58 -> 61[label="",style="solid", color="black", weight=3];
59 -> 58[label="",style="dashed", color="red", weight=0];
59[label="not (GT == LT)",fontsize=16,color="magenta"];60 -> 58[label="",style="dashed", color="red", weight=0];
60[label="not (GT == LT)",fontsize=16,color="magenta"];61 -> 39[label="",style="dashed", color="red", weight=0];
61[label="not False",fontsize=16,color="magenta"];}

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

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