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

YES 0.799 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 :: Ratio Int -> [Ratio Int] -> Bool) :: Ratio Int -> [Ratio Int] -> Bool)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

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

(notElem :: Ratio Int -> [Ratio Int] -> 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 vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="notElem vz3 vz4",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 ((/=) vz3) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="and . map ((/=) vz3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="and (map ((/=) vz3) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (&&) True (map ((/=) vz3) vz4)",fontsize=16,color="burlywood",shape="triangle"];139[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 139[label="",style="solid", color="burlywood", weight=9];
139 -> 10[label="",style="solid", color="burlywood", weight=3];
140[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 140[label="",style="solid", color="burlywood", weight=9];
140 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr (&&) True (map ((/=) vz3) (vz40 : vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (&&) True (map ((/=) vz3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (&&) True (((/=) vz3 vz40) : map ((/=) vz3) vz41)",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="(&&) (/=) vz3 vz40 foldr (&&) True (map ((/=) vz3) vz41)",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 ((/=) 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="(&&) not (vz3 == vz40) vz5",fontsize=16,color="burlywood",shape="box"];143[label="vz3/vz30 :% vz31",fontsize=10,color="white",style="solid",shape="box"];19 -> 143[label="",style="solid", color="burlywood", weight=9];
143 -> 20[label="",style="solid", color="burlywood", weight=3];
20[label="(&&) not (vz30 :% vz31 == vz40) vz5",fontsize=16,color="burlywood",shape="box"];144[label="vz40/vz400 :% vz401",fontsize=10,color="white",style="solid",shape="box"];20 -> 144[label="",style="solid", color="burlywood", weight=9];
144 -> 21[label="",style="solid", color="burlywood", weight=3];
21[label="(&&) not (vz30 :% vz31 == vz400 :% vz401) vz5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="(&&) not (vz30 == vz400 && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="(&&) not (primEqInt vz30 vz400 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];145[label="vz30/Pos vz300",fontsize=10,color="white",style="solid",shape="box"];23 -> 145[label="",style="solid", color="burlywood", weight=9];
145 -> 24[label="",style="solid", color="burlywood", weight=3];
146[label="vz30/Neg vz300",fontsize=10,color="white",style="solid",shape="box"];23 -> 146[label="",style="solid", color="burlywood", weight=9];
146 -> 25[label="",style="solid", color="burlywood", weight=3];
24[label="(&&) not (primEqInt (Pos vz300) vz400 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];147[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];24 -> 147[label="",style="solid", color="burlywood", weight=9];
147 -> 26[label="",style="solid", color="burlywood", weight=3];
148[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 148[label="",style="solid", color="burlywood", weight=9];
148 -> 27[label="",style="solid", color="burlywood", weight=3];
25[label="(&&) not (primEqInt (Neg vz300) vz400 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];149[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];25 -> 149[label="",style="solid", color="burlywood", weight=9];
149 -> 28[label="",style="solid", color="burlywood", weight=3];
150[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 150[label="",style="solid", color="burlywood", weight=9];
150 -> 29[label="",style="solid", color="burlywood", weight=3];
26[label="(&&) not (primEqInt (Pos (Succ vz3000)) vz400 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];151[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 151[label="",style="solid", color="burlywood", weight=9];
151 -> 30[label="",style="solid", color="burlywood", weight=3];
152[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 152[label="",style="solid", color="burlywood", weight=9];
152 -> 31[label="",style="solid", color="burlywood", weight=3];
27[label="(&&) not (primEqInt (Pos Zero) vz400 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];153[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 153[label="",style="solid", color="burlywood", weight=9];
153 -> 32[label="",style="solid", color="burlywood", weight=3];
154[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 154[label="",style="solid", color="burlywood", weight=9];
154 -> 33[label="",style="solid", color="burlywood", weight=3];
28[label="(&&) not (primEqInt (Neg (Succ vz3000)) vz400 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];155[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 155[label="",style="solid", color="burlywood", weight=9];
155 -> 34[label="",style="solid", color="burlywood", weight=3];
156[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 156[label="",style="solid", color="burlywood", weight=9];
156 -> 35[label="",style="solid", color="burlywood", weight=3];
29[label="(&&) not (primEqInt (Neg Zero) vz400 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];157[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 157[label="",style="solid", color="burlywood", weight=9];
157 -> 36[label="",style="solid", color="burlywood", weight=3];
158[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 158[label="",style="solid", color="burlywood", weight=9];
158 -> 37[label="",style="solid", color="burlywood", weight=3];
30[label="(&&) not (primEqInt (Pos (Succ vz3000)) (Pos vz4000) && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];159[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];30 -> 159[label="",style="solid", color="burlywood", weight=9];
159 -> 38[label="",style="solid", color="burlywood", weight=3];
160[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 160[label="",style="solid", color="burlywood", weight=9];
160 -> 39[label="",style="solid", color="burlywood", weight=3];
31[label="(&&) not (primEqInt (Pos (Succ vz3000)) (Neg vz4000) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="(&&) not (primEqInt (Pos Zero) (Pos vz4000) && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];161[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];32 -> 161[label="",style="solid", color="burlywood", weight=9];
161 -> 41[label="",style="solid", color="burlywood", weight=3];
162[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 162[label="",style="solid", color="burlywood", weight=9];
162 -> 42[label="",style="solid", color="burlywood", weight=3];
33[label="(&&) not (primEqInt (Pos Zero) (Neg vz4000) && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];163[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];33 -> 163[label="",style="solid", color="burlywood", weight=9];
163 -> 43[label="",style="solid", color="burlywood", weight=3];
164[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 164[label="",style="solid", color="burlywood", weight=9];
164 -> 44[label="",style="solid", color="burlywood", weight=3];
34[label="(&&) not (primEqInt (Neg (Succ vz3000)) (Pos vz4000) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];34 -> 45[label="",style="solid", color="black", weight=3];
35[label="(&&) not (primEqInt (Neg (Succ vz3000)) (Neg vz4000) && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];165[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];35 -> 165[label="",style="solid", color="burlywood", weight=9];
165 -> 46[label="",style="solid", color="burlywood", weight=3];
166[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 166[label="",style="solid", color="burlywood", weight=9];
166 -> 47[label="",style="solid", color="burlywood", weight=3];
36[label="(&&) not (primEqInt (Neg Zero) (Pos vz4000) && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];167[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];36 -> 167[label="",style="solid", color="burlywood", weight=9];
167 -> 48[label="",style="solid", color="burlywood", weight=3];
168[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 168[label="",style="solid", color="burlywood", weight=9];
168 -> 49[label="",style="solid", color="burlywood", weight=3];
37[label="(&&) not (primEqInt (Neg Zero) (Neg vz4000) && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];169[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];37 -> 169[label="",style="solid", color="burlywood", weight=9];
169 -> 50[label="",style="solid", color="burlywood", weight=3];
170[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 170[label="",style="solid", color="burlywood", weight=9];
170 -> 51[label="",style="solid", color="burlywood", weight=3];
38[label="(&&) not (primEqInt (Pos (Succ vz3000)) (Pos (Succ vz40000)) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
39[label="(&&) not (primEqInt (Pos (Succ vz3000)) (Pos Zero) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
40[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="black",shape="triangle"];40 -> 54[label="",style="solid", color="black", weight=3];
41[label="(&&) not (primEqInt (Pos Zero) (Pos (Succ vz40000)) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3];
42[label="(&&) not (primEqInt (Pos Zero) (Pos Zero) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3];
43[label="(&&) not (primEqInt (Pos Zero) (Neg (Succ vz40000)) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
44[label="(&&) not (primEqInt (Pos Zero) (Neg Zero) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3];
45 -> 40[label="",style="dashed", color="red", weight=0];
45[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];46[label="(&&) not (primEqInt (Neg (Succ vz3000)) (Neg (Succ vz40000)) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3];
47[label="(&&) not (primEqInt (Neg (Succ vz3000)) (Neg Zero) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3];
48[label="(&&) not (primEqInt (Neg Zero) (Pos (Succ vz40000)) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];48 -> 61[label="",style="solid", color="black", weight=3];
49[label="(&&) not (primEqInt (Neg Zero) (Pos Zero) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];49 -> 62[label="",style="solid", color="black", weight=3];
50[label="(&&) not (primEqInt (Neg Zero) (Neg (Succ vz40000)) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];50 -> 63[label="",style="solid", color="black", weight=3];
51[label="(&&) not (primEqInt (Neg Zero) (Neg Zero) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];51 -> 64[label="",style="solid", color="black", weight=3];
52[label="(&&) not (primEqNat vz3000 vz40000 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="triangle"];172[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];52 -> 172[label="",style="solid", color="burlywood", weight=9];
172 -> 65[label="",style="solid", color="burlywood", weight=3];
173[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 173[label="",style="solid", color="burlywood", weight=9];
173 -> 66[label="",style="solid", color="burlywood", weight=3];
53 -> 40[label="",style="dashed", color="red", weight=0];
53[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];54[label="(&&) not False vz5",fontsize=16,color="black",shape="triangle"];54 -> 67[label="",style="solid", color="black", weight=3];
55 -> 40[label="",style="dashed", color="red", weight=0];
55[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];56[label="(&&) not (True && vz31 == vz401) vz5",fontsize=16,color="black",shape="triangle"];56 -> 68[label="",style="solid", color="black", weight=3];
57 -> 40[label="",style="dashed", color="red", weight=0];
57[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];58 -> 56[label="",style="dashed", color="red", weight=0];
58[label="(&&) not (True && vz31 == vz401) vz5",fontsize=16,color="magenta"];59 -> 52[label="",style="dashed", color="red", weight=0];
59[label="(&&) not (primEqNat vz3000 vz40000 && vz31 == vz401) vz5",fontsize=16,color="magenta"];59 -> 69[label="",style="dashed", color="magenta", weight=3];
59 -> 70[label="",style="dashed", color="magenta", weight=3];
60 -> 40[label="",style="dashed", color="red", weight=0];
60[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];61 -> 40[label="",style="dashed", color="red", weight=0];
61[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];62 -> 56[label="",style="dashed", color="red", weight=0];
62[label="(&&) not (True && vz31 == vz401) vz5",fontsize=16,color="magenta"];63 -> 40[label="",style="dashed", color="red", weight=0];
63[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];64 -> 56[label="",style="dashed", color="red", weight=0];
64[label="(&&) not (True && vz31 == vz401) vz5",fontsize=16,color="magenta"];65[label="(&&) not (primEqNat (Succ vz30000) vz40000 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];184[label="vz40000/Succ vz400000",fontsize=10,color="white",style="solid",shape="box"];65 -> 184[label="",style="solid", color="burlywood", weight=9];
184 -> 71[label="",style="solid", color="burlywood", weight=3];
185[label="vz40000/Zero",fontsize=10,color="white",style="solid",shape="box"];65 -> 185[label="",style="solid", color="burlywood", weight=9];
185 -> 72[label="",style="solid", color="burlywood", weight=3];
66[label="(&&) not (primEqNat Zero vz40000 && vz31 == vz401) vz5",fontsize=16,color="burlywood",shape="box"];186[label="vz40000/Succ vz400000",fontsize=10,color="white",style="solid",shape="box"];66 -> 186[label="",style="solid", color="burlywood", weight=9];
186 -> 73[label="",style="solid", color="burlywood", weight=3];
187[label="vz40000/Zero",fontsize=10,color="white",style="solid",shape="box"];66 -> 187[label="",style="solid", color="burlywood", weight=9];
187 -> 74[label="",style="solid", color="burlywood", weight=3];
67[label="(&&) True vz5",fontsize=16,color="black",shape="box"];67 -> 75[label="",style="solid", color="black", weight=3];
68[label="(&&) not (vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];68 -> 76[label="",style="solid", color="black", weight=3];
69[label="vz3000",fontsize=16,color="green",shape="box"];70[label="vz40000",fontsize=16,color="green",shape="box"];71[label="(&&) not (primEqNat (Succ vz30000) (Succ vz400000) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];71 -> 77[label="",style="solid", color="black", weight=3];
72[label="(&&) not (primEqNat (Succ vz30000) Zero && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];72 -> 78[label="",style="solid", color="black", weight=3];
73[label="(&&) not (primEqNat Zero (Succ vz400000) && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];73 -> 79[label="",style="solid", color="black", weight=3];
74[label="(&&) not (primEqNat Zero Zero && vz31 == vz401) vz5",fontsize=16,color="black",shape="box"];74 -> 80[label="",style="solid", color="black", weight=3];
75[label="vz5",fontsize=16,color="green",shape="box"];76[label="(&&) not (primEqInt vz31 vz401) vz5",fontsize=16,color="burlywood",shape="box"];188[label="vz31/Pos vz310",fontsize=10,color="white",style="solid",shape="box"];76 -> 188[label="",style="solid", color="burlywood", weight=9];
188 -> 81[label="",style="solid", color="burlywood", weight=3];
189[label="vz31/Neg vz310",fontsize=10,color="white",style="solid",shape="box"];76 -> 189[label="",style="solid", color="burlywood", weight=9];
189 -> 82[label="",style="solid", color="burlywood", weight=3];
77 -> 52[label="",style="dashed", color="red", weight=0];
77[label="(&&) not (primEqNat vz30000 vz400000 && vz31 == vz401) vz5",fontsize=16,color="magenta"];77 -> 83[label="",style="dashed", color="magenta", weight=3];
77 -> 84[label="",style="dashed", color="magenta", weight=3];
78 -> 40[label="",style="dashed", color="red", weight=0];
78[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];79 -> 40[label="",style="dashed", color="red", weight=0];
79[label="(&&) not (False && vz31 == vz401) vz5",fontsize=16,color="magenta"];80 -> 56[label="",style="dashed", color="red", weight=0];
80[label="(&&) not (True && vz31 == vz401) vz5",fontsize=16,color="magenta"];81[label="(&&) not (primEqInt (Pos vz310) vz401) vz5",fontsize=16,color="burlywood",shape="box"];194[label="vz310/Succ vz3100",fontsize=10,color="white",style="solid",shape="box"];81 -> 194[label="",style="solid", color="burlywood", weight=9];
194 -> 85[label="",style="solid", color="burlywood", weight=3];
195[label="vz310/Zero",fontsize=10,color="white",style="solid",shape="box"];81 -> 195[label="",style="solid", color="burlywood", weight=9];
195 -> 86[label="",style="solid", color="burlywood", weight=3];
82[label="(&&) not (primEqInt (Neg vz310) vz401) vz5",fontsize=16,color="burlywood",shape="box"];196[label="vz310/Succ vz3100",fontsize=10,color="white",style="solid",shape="box"];82 -> 196[label="",style="solid", color="burlywood", weight=9];
196 -> 87[label="",style="solid", color="burlywood", weight=3];
197[label="vz310/Zero",fontsize=10,color="white",style="solid",shape="box"];82 -> 197[label="",style="solid", color="burlywood", weight=9];
197 -> 88[label="",style="solid", color="burlywood", weight=3];
83[label="vz30000",fontsize=16,color="green",shape="box"];84[label="vz400000",fontsize=16,color="green",shape="box"];85[label="(&&) not (primEqInt (Pos (Succ vz3100)) vz401) vz5",fontsize=16,color="burlywood",shape="box"];198[label="vz401/Pos vz4010",fontsize=10,color="white",style="solid",shape="box"];85 -> 198[label="",style="solid", color="burlywood", weight=9];
198 -> 89[label="",style="solid", color="burlywood", weight=3];
199[label="vz401/Neg vz4010",fontsize=10,color="white",style="solid",shape="box"];85 -> 199[label="",style="solid", color="burlywood", weight=9];
199 -> 90[label="",style="solid", color="burlywood", weight=3];
86[label="(&&) not (primEqInt (Pos Zero) vz401) vz5",fontsize=16,color="burlywood",shape="box"];200[label="vz401/Pos vz4010",fontsize=10,color="white",style="solid",shape="box"];86 -> 200[label="",style="solid", color="burlywood", weight=9];
200 -> 91[label="",style="solid", color="burlywood", weight=3];
201[label="vz401/Neg vz4010",fontsize=10,color="white",style="solid",shape="box"];86 -> 201[label="",style="solid", color="burlywood", weight=9];
201 -> 92[label="",style="solid", color="burlywood", weight=3];
87[label="(&&) not (primEqInt (Neg (Succ vz3100)) vz401) vz5",fontsize=16,color="burlywood",shape="box"];202[label="vz401/Pos vz4010",fontsize=10,color="white",style="solid",shape="box"];87 -> 202[label="",style="solid", color="burlywood", weight=9];
202 -> 93[label="",style="solid", color="burlywood", weight=3];
203[label="vz401/Neg vz4010",fontsize=10,color="white",style="solid",shape="box"];87 -> 203[label="",style="solid", color="burlywood", weight=9];
203 -> 94[label="",style="solid", color="burlywood", weight=3];
88[label="(&&) not (primEqInt (Neg Zero) vz401) vz5",fontsize=16,color="burlywood",shape="box"];204[label="vz401/Pos vz4010",fontsize=10,color="white",style="solid",shape="box"];88 -> 204[label="",style="solid", color="burlywood", weight=9];
204 -> 95[label="",style="solid", color="burlywood", weight=3];
205[label="vz401/Neg vz4010",fontsize=10,color="white",style="solid",shape="box"];88 -> 205[label="",style="solid", color="burlywood", weight=9];
205 -> 96[label="",style="solid", color="burlywood", weight=3];
89[label="(&&) not (primEqInt (Pos (Succ vz3100)) (Pos vz4010)) vz5",fontsize=16,color="burlywood",shape="box"];206[label="vz4010/Succ vz40100",fontsize=10,color="white",style="solid",shape="box"];89 -> 206[label="",style="solid", color="burlywood", weight=9];
206 -> 97[label="",style="solid", color="burlywood", weight=3];
207[label="vz4010/Zero",fontsize=10,color="white",style="solid",shape="box"];89 -> 207[label="",style="solid", color="burlywood", weight=9];
207 -> 98[label="",style="solid", color="burlywood", weight=3];
90[label="(&&) not (primEqInt (Pos (Succ vz3100)) (Neg vz4010)) vz5",fontsize=16,color="black",shape="box"];90 -> 99[label="",style="solid", color="black", weight=3];
91[label="(&&) not (primEqInt (Pos Zero) (Pos vz4010)) vz5",fontsize=16,color="burlywood",shape="box"];208[label="vz4010/Succ vz40100",fontsize=10,color="white",style="solid",shape="box"];91 -> 208[label="",style="solid", color="burlywood", weight=9];
208 -> 100[label="",style="solid", color="burlywood", weight=3];
209[label="vz4010/Zero",fontsize=10,color="white",style="solid",shape="box"];91 -> 209[label="",style="solid", color="burlywood", weight=9];
209 -> 101[label="",style="solid", color="burlywood", weight=3];
92[label="(&&) not (primEqInt (Pos Zero) (Neg vz4010)) vz5",fontsize=16,color="burlywood",shape="box"];210[label="vz4010/Succ vz40100",fontsize=10,color="white",style="solid",shape="box"];92 -> 210[label="",style="solid", color="burlywood", weight=9];
210 -> 102[label="",style="solid", color="burlywood", weight=3];
211[label="vz4010/Zero",fontsize=10,color="white",style="solid",shape="box"];92 -> 211[label="",style="solid", color="burlywood", weight=9];
211 -> 103[label="",style="solid", color="burlywood", weight=3];
93[label="(&&) not (primEqInt (Neg (Succ vz3100)) (Pos vz4010)) vz5",fontsize=16,color="black",shape="box"];93 -> 104[label="",style="solid", color="black", weight=3];
94[label="(&&) not (primEqInt (Neg (Succ vz3100)) (Neg vz4010)) vz5",fontsize=16,color="burlywood",shape="box"];212[label="vz4010/Succ vz40100",fontsize=10,color="white",style="solid",shape="box"];94 -> 212[label="",style="solid", color="burlywood", weight=9];
212 -> 105[label="",style="solid", color="burlywood", weight=3];
213[label="vz4010/Zero",fontsize=10,color="white",style="solid",shape="box"];94 -> 213[label="",style="solid", color="burlywood", weight=9];
213 -> 106[label="",style="solid", color="burlywood", weight=3];
95[label="(&&) not (primEqInt (Neg Zero) (Pos vz4010)) vz5",fontsize=16,color="burlywood",shape="box"];214[label="vz4010/Succ vz40100",fontsize=10,color="white",style="solid",shape="box"];95 -> 214[label="",style="solid", color="burlywood", weight=9];
214 -> 107[label="",style="solid", color="burlywood", weight=3];
215[label="vz4010/Zero",fontsize=10,color="white",style="solid",shape="box"];95 -> 215[label="",style="solid", color="burlywood", weight=9];
215 -> 108[label="",style="solid", color="burlywood", weight=3];
96[label="(&&) not (primEqInt (Neg Zero) (Neg vz4010)) vz5",fontsize=16,color="burlywood",shape="box"];216[label="vz4010/Succ vz40100",fontsize=10,color="white",style="solid",shape="box"];96 -> 216[label="",style="solid", color="burlywood", weight=9];
216 -> 109[label="",style="solid", color="burlywood", weight=3];
217[label="vz4010/Zero",fontsize=10,color="white",style="solid",shape="box"];96 -> 217[label="",style="solid", color="burlywood", weight=9];
217 -> 110[label="",style="solid", color="burlywood", weight=3];
97[label="(&&) not (primEqInt (Pos (Succ vz3100)) (Pos (Succ vz40100))) vz5",fontsize=16,color="black",shape="box"];97 -> 111[label="",style="solid", color="black", weight=3];
98[label="(&&) not (primEqInt (Pos (Succ vz3100)) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];98 -> 112[label="",style="solid", color="black", weight=3];
99 -> 54[label="",style="dashed", color="red", weight=0];
99[label="(&&) not False vz5",fontsize=16,color="magenta"];100[label="(&&) not (primEqInt (Pos Zero) (Pos (Succ vz40100))) vz5",fontsize=16,color="black",shape="box"];100 -> 113[label="",style="solid", color="black", weight=3];
101[label="(&&) not (primEqInt (Pos Zero) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];101 -> 114[label="",style="solid", color="black", weight=3];
102[label="(&&) not (primEqInt (Pos Zero) (Neg (Succ vz40100))) vz5",fontsize=16,color="black",shape="box"];102 -> 115[label="",style="solid", color="black", weight=3];
103[label="(&&) not (primEqInt (Pos Zero) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];103 -> 116[label="",style="solid", color="black", weight=3];
104 -> 54[label="",style="dashed", color="red", weight=0];
104[label="(&&) not False vz5",fontsize=16,color="magenta"];105[label="(&&) not (primEqInt (Neg (Succ vz3100)) (Neg (Succ vz40100))) vz5",fontsize=16,color="black",shape="box"];105 -> 117[label="",style="solid", color="black", weight=3];
106[label="(&&) not (primEqInt (Neg (Succ vz3100)) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];106 -> 118[label="",style="solid", color="black", weight=3];
107[label="(&&) not (primEqInt (Neg Zero) (Pos (Succ vz40100))) vz5",fontsize=16,color="black",shape="box"];107 -> 119[label="",style="solid", color="black", weight=3];
108[label="(&&) not (primEqInt (Neg Zero) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];108 -> 120[label="",style="solid", color="black", weight=3];
109[label="(&&) not (primEqInt (Neg Zero) (Neg (Succ vz40100))) vz5",fontsize=16,color="black",shape="box"];109 -> 121[label="",style="solid", color="black", weight=3];
110[label="(&&) not (primEqInt (Neg Zero) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];110 -> 122[label="",style="solid", color="black", weight=3];
111[label="(&&) not (primEqNat vz3100 vz40100) vz5",fontsize=16,color="burlywood",shape="triangle"];220[label="vz3100/Succ vz31000",fontsize=10,color="white",style="solid",shape="box"];111 -> 220[label="",style="solid", color="burlywood", weight=9];
220 -> 123[label="",style="solid", color="burlywood", weight=3];
221[label="vz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];111 -> 221[label="",style="solid", color="burlywood", weight=9];
221 -> 124[label="",style="solid", color="burlywood", weight=3];
112 -> 54[label="",style="dashed", color="red", weight=0];
112[label="(&&) not False vz5",fontsize=16,color="magenta"];113 -> 54[label="",style="dashed", color="red", weight=0];
113[label="(&&) not False vz5",fontsize=16,color="magenta"];114[label="(&&) not True vz5",fontsize=16,color="black",shape="triangle"];114 -> 125[label="",style="solid", color="black", weight=3];
115 -> 54[label="",style="dashed", color="red", weight=0];
115[label="(&&) not False vz5",fontsize=16,color="magenta"];116 -> 114[label="",style="dashed", color="red", weight=0];
116[label="(&&) not True vz5",fontsize=16,color="magenta"];117 -> 111[label="",style="dashed", color="red", weight=0];
117[label="(&&) not (primEqNat vz3100 vz40100) vz5",fontsize=16,color="magenta"];117 -> 126[label="",style="dashed", color="magenta", weight=3];
117 -> 127[label="",style="dashed", color="magenta", weight=3];
118 -> 54[label="",style="dashed", color="red", weight=0];
118[label="(&&) not False vz5",fontsize=16,color="magenta"];119 -> 54[label="",style="dashed", color="red", weight=0];
119[label="(&&) not False vz5",fontsize=16,color="magenta"];120 -> 114[label="",style="dashed", color="red", weight=0];
120[label="(&&) not True vz5",fontsize=16,color="magenta"];121 -> 54[label="",style="dashed", color="red", weight=0];
121[label="(&&) not False vz5",fontsize=16,color="magenta"];122 -> 114[label="",style="dashed", color="red", weight=0];
122[label="(&&) not True vz5",fontsize=16,color="magenta"];123[label="(&&) not (primEqNat (Succ vz31000) vz40100) vz5",fontsize=16,color="burlywood",shape="box"];232[label="vz40100/Succ vz401000",fontsize=10,color="white",style="solid",shape="box"];123 -> 232[label="",style="solid", color="burlywood", weight=9];
232 -> 128[label="",style="solid", color="burlywood", weight=3];
233[label="vz40100/Zero",fontsize=10,color="white",style="solid",shape="box"];123 -> 233[label="",style="solid", color="burlywood", weight=9];
233 -> 129[label="",style="solid", color="burlywood", weight=3];
124[label="(&&) not (primEqNat Zero vz40100) vz5",fontsize=16,color="burlywood",shape="box"];234[label="vz40100/Succ vz401000",fontsize=10,color="white",style="solid",shape="box"];124 -> 234[label="",style="solid", color="burlywood", weight=9];
234 -> 130[label="",style="solid", color="burlywood", weight=3];
235[label="vz40100/Zero",fontsize=10,color="white",style="solid",shape="box"];124 -> 235[label="",style="solid", color="burlywood", weight=9];
235 -> 131[label="",style="solid", color="burlywood", weight=3];
125[label="(&&) False vz5",fontsize=16,color="black",shape="box"];125 -> 132[label="",style="solid", color="black", weight=3];
126[label="vz3100",fontsize=16,color="green",shape="box"];127[label="vz40100",fontsize=16,color="green",shape="box"];128[label="(&&) not (primEqNat (Succ vz31000) (Succ vz401000)) vz5",fontsize=16,color="black",shape="box"];128 -> 133[label="",style="solid", color="black", weight=3];
129[label="(&&) not (primEqNat (Succ vz31000) Zero) vz5",fontsize=16,color="black",shape="box"];129 -> 134[label="",style="solid", color="black", weight=3];
130[label="(&&) not (primEqNat Zero (Succ vz401000)) vz5",fontsize=16,color="black",shape="box"];130 -> 135[label="",style="solid", color="black", weight=3];
131[label="(&&) not (primEqNat Zero Zero) vz5",fontsize=16,color="black",shape="box"];131 -> 136[label="",style="solid", color="black", weight=3];
132[label="False",fontsize=16,color="green",shape="box"];133 -> 111[label="",style="dashed", color="red", weight=0];
133[label="(&&) not (primEqNat vz31000 vz401000) vz5",fontsize=16,color="magenta"];133 -> 137[label="",style="dashed", color="magenta", weight=3];
133 -> 138[label="",style="dashed", color="magenta", weight=3];
134 -> 54[label="",style="dashed", color="red", weight=0];
134[label="(&&) not False vz5",fontsize=16,color="magenta"];135 -> 54[label="",style="dashed", color="red", weight=0];
135[label="(&&) not False vz5",fontsize=16,color="magenta"];136 -> 114[label="",style="dashed", color="red", weight=0];
136[label="(&&) not True vz5",fontsize=16,color="magenta"];137[label="vz31000",fontsize=16,color="green",shape="box"];138[label="vz401000",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

new_asAs(Succ(vz31000), Succ(vz401000), vz5) → new_asAs(vz31000, vz401000, 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_asAs(Succ(vz31000), Succ(vz401000), vz5) → new_asAs(vz31000, vz401000, vz5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_asAs0(Succ(vz30000), Succ(vz400000), vz31, vz401, vz5) → new_asAs0(vz30000, vz400000, vz31, vz401, vz5)

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

new_asAs0(Succ(vz30000), Succ(vz400000), vz31, vz401, vz5) → new_asAs0(vz30000, vz400000, vz31, vz401, vz5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ 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