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

YES 0.684 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

((notElem :: 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

((notElem :: Bool -> [Bool] -> Bool) :: Bool -> [Bool] -> 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

          ↳ Narrow

mainModule Main

(notElem :: Bool -> [Bool] -> Bool)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="notElem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="notElem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="notElem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="all . (/=)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="all ((/=) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="and . map ((/=) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="and (map ((/=) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (&&) True (map ((/=) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];34[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 10[label="",style="solid", color="burlywood", weight=3];
35[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr (&&) True (map ((/=) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (&&) True (map ((/=) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (&&) True (((/=) vx3 vx40) : map ((/=) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr (&&) True []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14 -> 16[label="",style="dashed", color="red", weight=0];
14[label="(&&) (/=) vx3 vx40 foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="True",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
17[label="foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="(&&) (/=) vx3 vx40 vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(&&) not (vx3 == vx40) vx5",fontsize=16,color="burlywood",shape="box"];38[label="vx3/False",fontsize=10,color="white",style="solid",shape="box"];19 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 20[label="",style="solid", color="burlywood", weight=3];
39[label="vx3/True",fontsize=10,color="white",style="solid",shape="box"];19 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 21[label="",style="solid", color="burlywood", weight=3];
20[label="(&&) not (False == vx40) vx5",fontsize=16,color="burlywood",shape="box"];40[label="vx40/False",fontsize=10,color="white",style="solid",shape="box"];20 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 22[label="",style="solid", color="burlywood", weight=3];
41[label="vx40/True",fontsize=10,color="white",style="solid",shape="box"];20 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 23[label="",style="solid", color="burlywood", weight=3];
21[label="(&&) not (True == vx40) vx5",fontsize=16,color="burlywood",shape="box"];42[label="vx40/False",fontsize=10,color="white",style="solid",shape="box"];21 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 24[label="",style="solid", color="burlywood", weight=3];
43[label="vx40/True",fontsize=10,color="white",style="solid",shape="box"];21 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 25[label="",style="solid", color="burlywood", weight=3];
22[label="(&&) not (False == False) vx5",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="(&&) not (False == True) vx5",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="(&&) not (True == False) vx5",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="(&&) not (True == True) vx5",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="(&&) not True vx5",fontsize=16,color="black",shape="triangle"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="(&&) not False vx5",fontsize=16,color="black",shape="triangle"];27 -> 31[label="",style="solid", color="black", weight=3];
28 -> 27[label="",style="dashed", color="red", weight=0];
28[label="(&&) not False vx5",fontsize=16,color="magenta"];29 -> 26[label="",style="dashed", color="red", weight=0];
29[label="(&&) not True vx5",fontsize=16,color="magenta"];30[label="(&&) False vx5",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
31[label="(&&) True vx5",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="False",fontsize=16,color="green",shape="box"];33[label="vx5",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

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

new_foldr(vx3, :(vx40, vx41)) → new_foldr(vx3, vx41)

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_foldr(vx3, :(vx40, vx41)) → new_foldr(vx3, vx41)
The graph contains the following edges 1 >= 1, 2 > 2