H-Termination proof of ../tpdb/FP/full_haskell/Prelude_notElem

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_notElem_3.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

mainModule Main

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

module Main where

import qualified Prelude

Num Reduction:All numbers are transformed to thier corresponding representation with Succ, Pred and Zero.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ HASKELL

          ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="notElem\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="notElem vz3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="notElem vz3 vz4\n",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="all . (/=)\n",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="all ((/=) vz3) vz4\n",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="and . map ((/=) vz3)\n",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="and (map ((/=) vz3) vz4)\n",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (&&) True (map ((/=) vz3) vz4)\n",fontsize=16,color="burlywood",shape="triangle"];82[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 10[label="",style="solid", color="burlywood", weight=3];
83[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr (&&) True (map ((/=) vz3) (vz40 : vz41))\n",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (&&) True (map ((/=) vz3) [])\n",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (&&) True (((/=) vz3 vz40) : map ((/=) vz3) vz41)\n",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr (&&) True []\n",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)\n",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="True\n",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
17[label="foldr (&&) True (map ((/=) vz3) vz41)\n",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="(&&) (/=) vz3 vz40 vz5\n",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
18[label="vz41\n",fontsize=16,color="green",shape="box"];19[label="(&&) not (vz3 == vz40) vz5\n",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="(&&) not (primEqChar vz3 vz40) vz5\n",fontsize=16,color="burlywood",shape="box"];86[label="vz3/Char vz30",fontsize=10,color="white",style="solid",shape="box"];20 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 21[label="",style="solid", color="burlywood", weight=3];
21[label="(&&) not (primEqChar (Char vz30) vz40) vz5\n",fontsize=16,color="burlywood",shape="box"];87[label="vz40/Char vz400",fontsize=10,color="white",style="solid",shape="box"];21 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 22[label="",style="solid", color="burlywood", weight=3];
22[label="(&&) not (primEqChar (Char vz30) (Char vz400)) vz5\n",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="(&&) not (primEqInt vz30 vz400) vz5\n",fontsize=16,color="burlywood",shape="box"];88[label="vz30/Pos vz300",fontsize=10,color="white",style="solid",shape="box"];23 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 24[label="",style="solid", color="burlywood", weight=3];
89[label="vz30/Neg vz300",fontsize=10,color="white",style="solid",shape="box"];23 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 25[label="",style="solid", color="burlywood", weight=3];
24[label="(&&) not (primEqInt (Pos vz300) vz400) vz5\n",fontsize=16,color="burlywood",shape="box"];90[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];24 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 26[label="",style="solid", color="burlywood", weight=3];
91[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 27[label="",style="solid", color="burlywood", weight=3];
25[label="(&&) not (primEqInt (Neg vz300) vz400) vz5\n",fontsize=16,color="burlywood",shape="box"];92[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];25 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 28[label="",style="solid", color="burlywood", weight=3];
93[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 29[label="",style="solid", color="burlywood", weight=3];
26[label="(&&) not (primEqInt (Pos (Succ vz3000)) vz400) vz5\n",fontsize=16,color="burlywood",shape="box"];94[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 30[label="",style="solid", color="burlywood", weight=3];
95[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 31[label="",style="solid", color="burlywood", weight=3];
27[label="(&&) not (primEqInt (Pos Zero) vz400) vz5\n",fontsize=16,color="burlywood",shape="box"];96[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 32[label="",style="solid", color="burlywood", weight=3];
97[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 33[label="",style="solid", color="burlywood", weight=3];
28[label="(&&) not (primEqInt (Neg (Succ vz3000)) vz400) vz5\n",fontsize=16,color="burlywood",shape="box"];98[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 34[label="",style="solid", color="burlywood", weight=3];
99[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 35[label="",style="solid", color="burlywood", weight=3];
29[label="(&&) not (primEqInt (Neg Zero) vz400) vz5\n",fontsize=16,color="burlywood",shape="box"];100[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 36[label="",style="solid", color="burlywood", weight=3];
101[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 37[label="",style="solid", color="burlywood", weight=3];
30[label="(&&) not (primEqInt (Pos (Succ vz3000)) (Pos vz4000)) vz5\n",fontsize=16,color="burlywood",shape="box"];102[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];30 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 38[label="",style="solid", color="burlywood", weight=3];
103[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 39[label="",style="solid", color="burlywood", weight=3];
31[label="(&&) not (primEqInt (Pos (Succ vz3000)) (Neg vz4000)) vz5\n",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="(&&) not (primEqInt (Pos Zero) (Pos vz4000)) vz5\n",fontsize=16,color="burlywood",shape="box"];104[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];32 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 41[label="",style="solid", color="burlywood", weight=3];
105[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 42[label="",style="solid", color="burlywood", weight=3];
33[label="(&&) not (primEqInt (Pos Zero) (Neg vz4000)) vz5\n",fontsize=16,color="burlywood",shape="box"];106[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];33 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 43[label="",style="solid", color="burlywood", weight=3];
107[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 44[label="",style="solid", color="burlywood", weight=3];
34[label="(&&) not (primEqInt (Neg (Succ vz3000)) (Pos vz4000)) vz5\n",fontsize=16,color="black",shape="box"];34 -> 45[label="",style="solid", color="black", weight=3];
35[label="(&&) not (primEqInt (Neg (Succ vz3000)) (Neg vz4000)) vz5\n",fontsize=16,color="burlywood",shape="box"];108[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];35 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 46[label="",style="solid", color="burlywood", weight=3];
109[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 47[label="",style="solid", color="burlywood", weight=3];
36[label="(&&) not (primEqInt (Neg Zero) (Pos vz4000)) vz5\n",fontsize=16,color="burlywood",shape="box"];110[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];36 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 48[label="",style="solid", color="burlywood", weight=3];
111[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 49[label="",style="solid", color="burlywood", weight=3];
37[label="(&&) not (primEqInt (Neg Zero) (Neg vz4000)) vz5\n",fontsize=16,color="burlywood",shape="box"];112[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];37 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 50[label="",style="solid", color="burlywood", weight=3];
113[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 51[label="",style="solid", color="burlywood", weight=3];
38[label="(&&) not (primEqInt (Pos (Succ vz3000)) (Pos (Succ vz40000))) vz5\n",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
39[label="(&&) not (primEqInt (Pos (Succ vz3000)) (Pos Zero)) vz5\n",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
40[label="(&&) not False vz5\n",fontsize=16,color="black",shape="triangle"];40 -> 54[label="",style="solid", color="black", weight=3];
41[label="(&&) not (primEqInt (Pos Zero) (Pos (Succ vz40000))) vz5\n",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3];
42[label="(&&) not (primEqInt (Pos Zero) (Pos Zero)) vz5\n",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3];
43[label="(&&) not (primEqInt (Pos Zero) (Neg (Succ vz40000))) vz5\n",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
44[label="(&&) not (primEqInt (Pos Zero) (Neg Zero)) vz5\n",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 vz5\n",fontsize=16,color="magenta"];46[label="(&&) not (primEqInt (Neg (Succ vz3000)) (Neg (Succ vz40000))) vz5\n",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3];
47[label="(&&) not (primEqInt (Neg (Succ vz3000)) (Neg Zero)) vz5\n",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3];
48[label="(&&) not (primEqInt (Neg Zero) (Pos (Succ vz40000))) vz5\n",fontsize=16,color="black",shape="box"];48 -> 61[label="",style="solid", color="black", weight=3];
49[label="(&&) not (primEqInt (Neg Zero) (Pos Zero)) vz5\n",fontsize=16,color="black",shape="box"];49 -> 62[label="",style="solid", color="black", weight=3];
50[label="(&&) not (primEqInt (Neg Zero) (Neg (Succ vz40000))) vz5\n",fontsize=16,color="black",shape="box"];50 -> 63[label="",style="solid", color="black", weight=3];
51[label="(&&) not (primEqInt (Neg Zero) (Neg Zero)) vz5\n",fontsize=16,color="black",shape="box"];51 -> 64[label="",style="solid", color="black", weight=3];
52[label="(&&) not (primEqNat vz3000 vz40000) vz5\n",fontsize=16,color="burlywood",shape="triangle"];115[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];52 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 65[label="",style="solid", color="burlywood", weight=3];
116[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 66[label="",style="solid", color="burlywood", weight=3];
53 -> 40[label="",style="dashed", color="red", weight=0];
53[label="(&&) not False vz5\n",fontsize=16,color="magenta"];54[label="(&&) True vz5\n",fontsize=16,color="black",shape="box"];54 -> 67[label="",style="solid", color="black", weight=3];
55 -> 40[label="",style="dashed", color="red", weight=0];
55[label="(&&) not False vz5\n",fontsize=16,color="magenta"];56[label="(&&) not True vz5\n",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 vz5\n",fontsize=16,color="magenta"];58 -> 56[label="",style="dashed", color="red", weight=0];
58[label="(&&) not True vz5\n",fontsize=16,color="magenta"];59 -> 52[label="",style="dashed", color="red", weight=0];
59[label="(&&) not (primEqNat vz3000 vz40000) vz5\n",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 vz5\n",fontsize=16,color="magenta"];61 -> 40[label="",style="dashed", color="red", weight=0];
61[label="(&&) not False vz5\n",fontsize=16,color="magenta"];62 -> 56[label="",style="dashed", color="red", weight=0];
62[label="(&&) not True vz5\n",fontsize=16,color="magenta"];63 -> 40[label="",style="dashed", color="red", weight=0];
63[label="(&&) not False vz5\n",fontsize=16,color="magenta"];64 -> 56[label="",style="dashed", color="red", weight=0];
64[label="(&&) not True vz5\n",fontsize=16,color="magenta"];65[label="(&&) not (primEqNat (Succ vz30000) vz40000) vz5\n",fontsize=16,color="burlywood",shape="box"];127[label="vz40000/Succ vz400000",fontsize=10,color="white",style="solid",shape="box"];65 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 71[label="",style="solid", color="burlywood", weight=3];
128[label="vz40000/Zero",fontsize=10,color="white",style="solid",shape="box"];65 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 72[label="",style="solid", color="burlywood", weight=3];
66[label="(&&) not (primEqNat Zero vz40000) vz5\n",fontsize=16,color="burlywood",shape="box"];129[label="vz40000/Succ vz400000",fontsize=10,color="white",style="solid",shape="box"];66 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 73[label="",style="solid", color="burlywood", weight=3];
130[label="vz40000/Zero",fontsize=10,color="white",style="solid",shape="box"];66 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 74[label="",style="solid", color="burlywood", weight=3];
67[label="vz5\n",fontsize=16,color="green",shape="box"];68[label="(&&) False vz5\n",fontsize=16,color="black",shape="box"];68 -> 75[label="",style="solid", color="black", weight=3];
69[label="vz3000\n",fontsize=16,color="green",shape="box"];70[label="vz40000\n",fontsize=16,color="green",shape="box"];71[label="(&&) not (primEqNat (Succ vz30000) (Succ vz400000)) vz5\n",fontsize=16,color="black",shape="box"];71 -> 76[label="",style="solid", color="black", weight=3];
72[label="(&&) not (primEqNat (Succ vz30000) Zero) vz5\n",fontsize=16,color="black",shape="box"];72 -> 77[label="",style="solid", color="black", weight=3];
73[label="(&&) not (primEqNat Zero (Succ vz400000)) vz5\n",fontsize=16,color="black",shape="box"];73 -> 78[label="",style="solid", color="black", weight=3];
74[label="(&&) not (primEqNat Zero Zero) vz5\n",fontsize=16,color="black",shape="box"];74 -> 79[label="",style="solid", color="black", weight=3];
75[label="False\n",fontsize=16,color="green",shape="box"];76 -> 52[label="",style="dashed", color="red", weight=0];
76[label="(&&) not (primEqNat vz30000 vz400000) vz5\n",fontsize=16,color="magenta"];76 -> 80[label="",style="dashed", color="magenta", weight=3];
76 -> 81[label="",style="dashed", color="magenta", weight=3];
77 -> 40[label="",style="dashed", color="red", weight=0];
77[label="(&&) not False vz5\n",fontsize=16,color="magenta"];78 -> 40[label="",style="dashed", color="red", weight=0];
78[label="(&&) not False vz5\n",fontsize=16,color="magenta"];79 -> 56[label="",style="dashed", color="red", weight=0];
79[label="(&&) not True vz5\n",fontsize=16,color="magenta"];80[label="vz30000\n",fontsize=16,color="green",shape="box"];81[label="vz400000\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_asAs(Succ(vz30000), Succ(vz400000), vz5) → new_asAs(vz30000, vz400000, 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(vz30000), Succ(vz400000), vz5) → new_asAs(vz30000, vz400000, vz5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ 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