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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_EQEQ_3.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

(((==) :: 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

(((==) :: 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

((==) :: Char -> Char -> Bool)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(==)\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(==) vz3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(==) vz3 vz4\n",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primEqChar vz3 vz4\n",fontsize=16,color="burlywood",shape="box"];63[label="vz3/Char vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 6[label="",style="solid", color="burlywood", weight=3];
6[label="primEqChar (Char vz30) vz4\n",fontsize=16,color="burlywood",shape="box"];64[label="vz4/Char vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 7[label="",style="solid", color="burlywood", weight=3];
7[label="primEqChar (Char vz30) (Char vz40)\n",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="primEqInt vz30 vz40\n",fontsize=16,color="burlywood",shape="box"];65[label="vz30/Pos vz300",fontsize=10,color="white",style="solid",shape="box"];8 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 9[label="",style="solid", color="burlywood", weight=3];
66[label="vz30/Neg vz300",fontsize=10,color="white",style="solid",shape="box"];8 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 10[label="",style="solid", color="burlywood", weight=3];
9[label="primEqInt (Pos vz300) vz40\n",fontsize=16,color="burlywood",shape="box"];67[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];9 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 11[label="",style="solid", color="burlywood", weight=3];
68[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 12[label="",style="solid", color="burlywood", weight=3];
10[label="primEqInt (Neg vz300) vz40\n",fontsize=16,color="burlywood",shape="box"];69[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];10 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 13[label="",style="solid", color="burlywood", weight=3];
70[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 14[label="",style="solid", color="burlywood", weight=3];
11[label="primEqInt (Pos (Succ vz3000)) vz40\n",fontsize=16,color="burlywood",shape="box"];71[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 15[label="",style="solid", color="burlywood", weight=3];
72[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 16[label="",style="solid", color="burlywood", weight=3];
12[label="primEqInt (Pos Zero) vz40\n",fontsize=16,color="burlywood",shape="box"];73[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];12 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 17[label="",style="solid", color="burlywood", weight=3];
74[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];12 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 18[label="",style="solid", color="burlywood", weight=3];
13[label="primEqInt (Neg (Succ vz3000)) vz40\n",fontsize=16,color="burlywood",shape="box"];75[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];13 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 19[label="",style="solid", color="burlywood", weight=3];
76[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];13 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 20[label="",style="solid", color="burlywood", weight=3];
14[label="primEqInt (Neg Zero) vz40\n",fontsize=16,color="burlywood",shape="box"];77[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];14 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 21[label="",style="solid", color="burlywood", weight=3];
78[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];14 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 22[label="",style="solid", color="burlywood", weight=3];
15[label="primEqInt (Pos (Succ vz3000)) (Pos vz400)\n",fontsize=16,color="burlywood",shape="box"];79[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];15 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 23[label="",style="solid", color="burlywood", weight=3];
80[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 24[label="",style="solid", color="burlywood", weight=3];
16[label="primEqInt (Pos (Succ vz3000)) (Neg vz400)\n",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
17[label="primEqInt (Pos Zero) (Pos vz400)\n",fontsize=16,color="burlywood",shape="box"];81[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];17 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 26[label="",style="solid", color="burlywood", weight=3];
82[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 27[label="",style="solid", color="burlywood", weight=3];
18[label="primEqInt (Pos Zero) (Neg vz400)\n",fontsize=16,color="burlywood",shape="box"];83[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];18 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 28[label="",style="solid", color="burlywood", weight=3];
84[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 29[label="",style="solid", color="burlywood", weight=3];
19[label="primEqInt (Neg (Succ vz3000)) (Pos vz400)\n",fontsize=16,color="black",shape="box"];19 -> 30[label="",style="solid", color="black", weight=3];
20[label="primEqInt (Neg (Succ vz3000)) (Neg vz400)\n",fontsize=16,color="burlywood",shape="box"];85[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];20 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 31[label="",style="solid", color="burlywood", weight=3];
86[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 32[label="",style="solid", color="burlywood", weight=3];
21[label="primEqInt (Neg Zero) (Pos vz400)\n",fontsize=16,color="burlywood",shape="box"];87[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];21 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 33[label="",style="solid", color="burlywood", weight=3];
88[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 34[label="",style="solid", color="burlywood", weight=3];
22[label="primEqInt (Neg Zero) (Neg vz400)\n",fontsize=16,color="burlywood",shape="box"];89[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];22 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 35[label="",style="solid", color="burlywood", weight=3];
90[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 36[label="",style="solid", color="burlywood", weight=3];
23[label="primEqInt (Pos (Succ vz3000)) (Pos (Succ vz4000))\n",fontsize=16,color="black",shape="box"];23 -> 37[label="",style="solid", color="black", weight=3];
24[label="primEqInt (Pos (Succ vz3000)) (Pos Zero)\n",fontsize=16,color="black",shape="box"];24 -> 38[label="",style="solid", color="black", weight=3];
25[label="False\n",fontsize=16,color="green",shape="box"];26[label="primEqInt (Pos Zero) (Pos (Succ vz4000))\n",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
27[label="primEqInt (Pos Zero) (Pos Zero)\n",fontsize=16,color="black",shape="box"];27 -> 40[label="",style="solid", color="black", weight=3];
28[label="primEqInt (Pos Zero) (Neg (Succ vz4000))\n",fontsize=16,color="black",shape="box"];28 -> 41[label="",style="solid", color="black", weight=3];
29[label="primEqInt (Pos Zero) (Neg Zero)\n",fontsize=16,color="black",shape="box"];29 -> 42[label="",style="solid", color="black", weight=3];
30[label="False\n",fontsize=16,color="green",shape="box"];31[label="primEqInt (Neg (Succ vz3000)) (Neg (Succ vz4000))\n",fontsize=16,color="black",shape="box"];31 -> 43[label="",style="solid", color="black", weight=3];
32[label="primEqInt (Neg (Succ vz3000)) (Neg Zero)\n",fontsize=16,color="black",shape="box"];32 -> 44[label="",style="solid", color="black", weight=3];
33[label="primEqInt (Neg Zero) (Pos (Succ vz4000))\n",fontsize=16,color="black",shape="box"];33 -> 45[label="",style="solid", color="black", weight=3];
34[label="primEqInt (Neg Zero) (Pos Zero)\n",fontsize=16,color="black",shape="box"];34 -> 46[label="",style="solid", color="black", weight=3];
35[label="primEqInt (Neg Zero) (Neg (Succ vz4000))\n",fontsize=16,color="black",shape="box"];35 -> 47[label="",style="solid", color="black", weight=3];
36[label="primEqInt (Neg Zero) (Neg Zero)\n",fontsize=16,color="black",shape="box"];36 -> 48[label="",style="solid", color="black", weight=3];
37[label="primEqNat vz3000 vz4000\n",fontsize=16,color="burlywood",shape="triangle"];91[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];37 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 49[label="",style="solid", color="burlywood", weight=3];
92[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 50[label="",style="solid", color="burlywood", weight=3];
38[label="False\n",fontsize=16,color="green",shape="box"];39[label="False\n",fontsize=16,color="green",shape="box"];40[label="True\n",fontsize=16,color="green",shape="box"];41[label="False\n",fontsize=16,color="green",shape="box"];42[label="True\n",fontsize=16,color="green",shape="box"];43 -> 37[label="",style="dashed", color="red", weight=0];
43[label="primEqNat vz3000 vz4000\n",fontsize=16,color="magenta"];43 -> 51[label="",style="dashed", color="magenta", weight=3];
43 -> 52[label="",style="dashed", color="magenta", weight=3];
44[label="False\n",fontsize=16,color="green",shape="box"];45[label="False\n",fontsize=16,color="green",shape="box"];46[label="True\n",fontsize=16,color="green",shape="box"];47[label="False\n",fontsize=16,color="green",shape="box"];48[label="True\n",fontsize=16,color="green",shape="box"];49[label="primEqNat (Succ vz30000) vz4000\n",fontsize=16,color="burlywood",shape="box"];94[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];49 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 53[label="",style="solid", color="burlywood", weight=3];
95[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 54[label="",style="solid", color="burlywood", weight=3];
50[label="primEqNat Zero vz4000\n",fontsize=16,color="burlywood",shape="box"];96[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];50 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 55[label="",style="solid", color="burlywood", weight=3];
97[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 56[label="",style="solid", color="burlywood", weight=3];
51[label="vz4000\n",fontsize=16,color="green",shape="box"];52[label="vz3000\n",fontsize=16,color="green",shape="box"];53[label="primEqNat (Succ vz30000) (Succ vz40000)\n",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="primEqNat (Succ vz30000) Zero\n",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55[label="primEqNat Zero (Succ vz40000)\n",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="primEqNat Zero Zero\n",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3];
57 -> 37[label="",style="dashed", color="red", weight=0];
57[label="primEqNat vz30000 vz40000\n",fontsize=16,color="magenta"];57 -> 61[label="",style="dashed", color="magenta", weight=3];
57 -> 62[label="",style="dashed", color="magenta", weight=3];
58[label="False\n",fontsize=16,color="green",shape="box"];59[label="False\n",fontsize=16,color="green",shape="box"];60[label="True\n",fontsize=16,color="green",shape="box"];61[label="vz40000\n",fontsize=16,color="green",shape="box"];62[label="vz30000\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

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

new_primEqNat(Succ(vz30000), Succ(vz40000)) → new_primEqNat(vz30000, vz40000)

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_primEqNat(Succ(vz30000), Succ(vz40000)) → new_primEqNat(vz30000, vz40000)
The graph contains the following edges 1 > 1, 2 > 2