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

YES 0.788 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

((elem :: Int -> [Int] -> Bool) :: Int -> [Int] -> Bool)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((elem :: Int -> [Int] -> Bool) :: Int -> [Int] -> 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

(elem :: Int -> [Int] -> Bool)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="elem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="elem vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="elem vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="any . (==)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="any ((==) vz3) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="or . map ((==) vz3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="or (map ((==) vz3) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (||) False (map ((==) vz3) vz4)",fontsize=16,color="burlywood",shape="triangle"];76[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 10[label="",style="solid", color="burlywood", weight=3];
77[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr (||) False (map ((==) vz3) (vz40 : vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (||) False (map ((==) vz3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (||) False (((==) vz3 vz40) : map ((==) vz3) vz41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr (||) False []",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="(||) (==) vz3 vz40 foldr (||) False (map ((==) vz3) vz41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="False",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
17[label="foldr (||) False (map ((==) vz3) vz41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="(||) (==) vz3 vz40 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
18[label="vz41",fontsize=16,color="green",shape="box"];19[label="(||) primEqInt vz3 vz40 vz5",fontsize=16,color="burlywood",shape="box"];80[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];19 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 20[label="",style="solid", color="burlywood", weight=3];
81[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];19 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 21[label="",style="solid", color="burlywood", weight=3];
20[label="(||) primEqInt (Pos vz30) vz40 vz5",fontsize=16,color="burlywood",shape="box"];82[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];20 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 22[label="",style="solid", color="burlywood", weight=3];
83[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 23[label="",style="solid", color="burlywood", weight=3];
21[label="(||) primEqInt (Neg vz30) vz40 vz5",fontsize=16,color="burlywood",shape="box"];84[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];21 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 24[label="",style="solid", color="burlywood", weight=3];
85[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 25[label="",style="solid", color="burlywood", weight=3];
22[label="(||) primEqInt (Pos (Succ vz300)) vz40 vz5",fontsize=16,color="burlywood",shape="box"];86[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];22 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 26[label="",style="solid", color="burlywood", weight=3];
87[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];22 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 27[label="",style="solid", color="burlywood", weight=3];
23[label="(||) primEqInt (Pos Zero) vz40 vz5",fontsize=16,color="burlywood",shape="box"];88[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 28[label="",style="solid", color="burlywood", weight=3];
89[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 29[label="",style="solid", color="burlywood", weight=3];
24[label="(||) primEqInt (Neg (Succ vz300)) vz40 vz5",fontsize=16,color="burlywood",shape="box"];90[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 30[label="",style="solid", color="burlywood", weight=3];
91[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 31[label="",style="solid", color="burlywood", weight=3];
25[label="(||) primEqInt (Neg Zero) vz40 vz5",fontsize=16,color="burlywood",shape="box"];92[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 32[label="",style="solid", color="burlywood", weight=3];
93[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 33[label="",style="solid", color="burlywood", weight=3];
26[label="(||) primEqInt (Pos (Succ vz300)) (Pos vz400) vz5",fontsize=16,color="burlywood",shape="box"];94[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 34[label="",style="solid", color="burlywood", weight=3];
95[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 35[label="",style="solid", color="burlywood", weight=3];
27[label="(||) primEqInt (Pos (Succ vz300)) (Neg vz400) vz5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="(||) primEqInt (Pos Zero) (Pos vz400) vz5",fontsize=16,color="burlywood",shape="box"];96[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 37[label="",style="solid", color="burlywood", weight=3];
97[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 38[label="",style="solid", color="burlywood", weight=3];
29[label="(||) primEqInt (Pos Zero) (Neg vz400) vz5",fontsize=16,color="burlywood",shape="box"];98[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 39[label="",style="solid", color="burlywood", weight=3];
99[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 40[label="",style="solid", color="burlywood", weight=3];
30[label="(||) primEqInt (Neg (Succ vz300)) (Pos vz400) vz5",fontsize=16,color="black",shape="box"];30 -> 41[label="",style="solid", color="black", weight=3];
31[label="(||) primEqInt (Neg (Succ vz300)) (Neg vz400) vz5",fontsize=16,color="burlywood",shape="box"];100[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];31 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 42[label="",style="solid", color="burlywood", weight=3];
101[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 43[label="",style="solid", color="burlywood", weight=3];
32[label="(||) primEqInt (Neg Zero) (Pos vz400) vz5",fontsize=16,color="burlywood",shape="box"];102[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];32 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 44[label="",style="solid", color="burlywood", weight=3];
103[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 45[label="",style="solid", color="burlywood", weight=3];
33[label="(||) primEqInt (Neg Zero) (Neg vz400) vz5",fontsize=16,color="burlywood",shape="box"];104[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 46[label="",style="solid", color="burlywood", weight=3];
105[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 47[label="",style="solid", color="burlywood", weight=3];
34[label="(||) primEqInt (Pos (Succ vz300)) (Pos (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
35[label="(||) primEqInt (Pos (Succ vz300)) (Pos Zero) vz5",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
36[label="(||) False vz5",fontsize=16,color="black",shape="triangle"];36 -> 50[label="",style="solid", color="black", weight=3];
37[label="(||) primEqInt (Pos Zero) (Pos (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
38[label="(||) primEqInt (Pos Zero) (Pos Zero) vz5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
39[label="(||) primEqInt (Pos Zero) (Neg (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
40[label="(||) primEqInt (Pos Zero) (Neg Zero) vz5",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
41 -> 36[label="",style="dashed", color="red", weight=0];
41[label="(||) False vz5",fontsize=16,color="magenta"];42[label="(||) primEqInt (Neg (Succ vz300)) (Neg (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];42 -> 55[label="",style="solid", color="black", weight=3];
43[label="(||) primEqInt (Neg (Succ vz300)) (Neg Zero) vz5",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3];
44[label="(||) primEqInt (Neg Zero) (Pos (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];44 -> 57[label="",style="solid", color="black", weight=3];
45[label="(||) primEqInt (Neg Zero) (Pos Zero) vz5",fontsize=16,color="black",shape="box"];45 -> 58[label="",style="solid", color="black", weight=3];
46[label="(||) primEqInt (Neg Zero) (Neg (Succ vz4000)) vz5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3];
47[label="(||) primEqInt (Neg Zero) (Neg Zero) vz5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3];
48[label="(||) primEqNat vz300 vz4000 vz5",fontsize=16,color="burlywood",shape="triangle"];107[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];48 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 61[label="",style="solid", color="burlywood", weight=3];
108[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 62[label="",style="solid", color="burlywood", weight=3];
49 -> 36[label="",style="dashed", color="red", weight=0];
49[label="(||) False vz5",fontsize=16,color="magenta"];50[label="vz5",fontsize=16,color="green",shape="box"];51 -> 36[label="",style="dashed", color="red", weight=0];
51[label="(||) False vz5",fontsize=16,color="magenta"];52[label="(||) True vz5",fontsize=16,color="black",shape="triangle"];52 -> 63[label="",style="solid", color="black", weight=3];
53 -> 36[label="",style="dashed", color="red", weight=0];
53[label="(||) False vz5",fontsize=16,color="magenta"];54 -> 52[label="",style="dashed", color="red", weight=0];
54[label="(||) True vz5",fontsize=16,color="magenta"];55 -> 48[label="",style="dashed", color="red", weight=0];
55[label="(||) primEqNat vz300 vz4000 vz5",fontsize=16,color="magenta"];55 -> 64[label="",style="dashed", color="magenta", weight=3];
55 -> 65[label="",style="dashed", color="magenta", weight=3];
56 -> 36[label="",style="dashed", color="red", weight=0];
56[label="(||) False vz5",fontsize=16,color="magenta"];57 -> 36[label="",style="dashed", color="red", weight=0];
57[label="(||) False vz5",fontsize=16,color="magenta"];58 -> 52[label="",style="dashed", color="red", weight=0];
58[label="(||) True vz5",fontsize=16,color="magenta"];59 -> 36[label="",style="dashed", color="red", weight=0];
59[label="(||) False vz5",fontsize=16,color="magenta"];60 -> 52[label="",style="dashed", color="red", weight=0];
60[label="(||) True vz5",fontsize=16,color="magenta"];61[label="(||) primEqNat (Succ vz3000) vz4000 vz5",fontsize=16,color="burlywood",shape="box"];119[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];61 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 66[label="",style="solid", color="burlywood", weight=3];
120[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];61 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 67[label="",style="solid", color="burlywood", weight=3];
62[label="(||) primEqNat Zero vz4000 vz5",fontsize=16,color="burlywood",shape="box"];121[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];62 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 68[label="",style="solid", color="burlywood", weight=3];
122[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 69[label="",style="solid", color="burlywood", weight=3];
63[label="True",fontsize=16,color="green",shape="box"];64[label="vz4000",fontsize=16,color="green",shape="box"];65[label="vz300",fontsize=16,color="green",shape="box"];66[label="(||) primEqNat (Succ vz3000) (Succ vz40000) vz5",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
67[label="(||) primEqNat (Succ vz3000) Zero vz5",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
68[label="(||) primEqNat Zero (Succ vz40000) vz5",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="(||) primEqNat Zero Zero vz5",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
70 -> 48[label="",style="dashed", color="red", weight=0];
70[label="(||) primEqNat vz3000 vz40000 vz5",fontsize=16,color="magenta"];70 -> 74[label="",style="dashed", color="magenta", weight=3];
70 -> 75[label="",style="dashed", color="magenta", weight=3];
71 -> 36[label="",style="dashed", color="red", weight=0];
71[label="(||) False vz5",fontsize=16,color="magenta"];72 -> 36[label="",style="dashed", color="red", weight=0];
72[label="(||) False vz5",fontsize=16,color="magenta"];73 -> 52[label="",style="dashed", color="red", weight=0];
73[label="(||) True vz5",fontsize=16,color="magenta"];74[label="vz40000",fontsize=16,color="green",shape="box"];75[label="vz3000",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_pePe(Succ(vz3000), Succ(vz40000), vz5) → new_pePe(vz3000, vz40000, vz5)

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_pePe(Succ(vz3000), Succ(vz40000), vz5) → new_pePe(vz3000, vz40000, vz5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_foldr(vz3, :(vz40, vz41)) → new_foldr(vz3, vz41)

From the DPs we obtained the following set of size-change graphs:

new_foldr(vz3, :(vz40, vz41)) → new_foldr(vz3, vz41)
The graph contains the following edges 1 >= 1, 2 > 2