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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_LT_5.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

(((<) :: 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

      ↳ NumRed

mainModule Main

(((<) :: Int -> Int -> Bool) :: Int -> Int -> 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

((<) :: Int -> Int -> 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="(<) vx3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(<) vx3 vx4\n",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="compare vx3 vx4 == LT\n",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="primCmpInt vx3 vx4 == LT\n",fontsize=16,color="burlywood",shape="box"];70[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 7[label="",style="solid", color="burlywood", weight=3];
71[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="primCmpInt (Pos vx30) vx4 == LT\n",fontsize=16,color="burlywood",shape="box"];72[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];7 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 9[label="",style="solid", color="burlywood", weight=3];
73[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="primCmpInt (Neg vx30) vx4 == LT\n",fontsize=16,color="burlywood",shape="box"];74[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];8 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 11[label="",style="solid", color="burlywood", weight=3];
75[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 12[label="",style="solid", color="burlywood", weight=3];
9[label="primCmpInt (Pos (Succ vx300)) vx4 == LT\n",fontsize=16,color="burlywood",shape="box"];76[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];9 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 13[label="",style="solid", color="burlywood", weight=3];
77[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];9 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 14[label="",style="solid", color="burlywood", weight=3];
10[label="primCmpInt (Pos Zero) vx4 == LT\n",fontsize=16,color="burlywood",shape="box"];78[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 15[label="",style="solid", color="burlywood", weight=3];
79[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 16[label="",style="solid", color="burlywood", weight=3];
11[label="primCmpInt (Neg (Succ vx300)) vx4 == LT\n",fontsize=16,color="burlywood",shape="box"];80[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 17[label="",style="solid", color="burlywood", weight=3];
81[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 18[label="",style="solid", color="burlywood", weight=3];
12[label="primCmpInt (Neg Zero) vx4 == LT\n",fontsize=16,color="burlywood",shape="box"];82[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 19[label="",style="solid", color="burlywood", weight=3];
83[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 20[label="",style="solid", color="burlywood", weight=3];
13[label="primCmpInt (Pos (Succ vx300)) (Pos vx40) == LT\n",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="primCmpInt (Pos (Succ vx300)) (Neg vx40) == LT\n",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="primCmpInt (Pos Zero) (Pos vx40) == LT\n",fontsize=16,color="burlywood",shape="box"];84[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 23[label="",style="solid", color="burlywood", weight=3];
85[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 24[label="",style="solid", color="burlywood", weight=3];
16[label="primCmpInt (Pos Zero) (Neg vx40) == LT\n",fontsize=16,color="burlywood",shape="box"];86[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];16 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 25[label="",style="solid", color="burlywood", weight=3];
87[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 26[label="",style="solid", color="burlywood", weight=3];
17[label="primCmpInt (Neg (Succ vx300)) (Pos vx40) == LT\n",fontsize=16,color="black",shape="box"];17 -> 27[label="",style="solid", color="black", weight=3];
18[label="primCmpInt (Neg (Succ vx300)) (Neg vx40) == LT\n",fontsize=16,color="black",shape="box"];18 -> 28[label="",style="solid", color="black", weight=3];
19[label="primCmpInt (Neg Zero) (Pos vx40) == LT\n",fontsize=16,color="burlywood",shape="box"];88[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];19 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 29[label="",style="solid", color="burlywood", weight=3];
89[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 30[label="",style="solid", color="burlywood", weight=3];
20[label="primCmpInt (Neg Zero) (Neg vx40) == LT\n",fontsize=16,color="burlywood",shape="box"];90[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];20 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 31[label="",style="solid", color="burlywood", weight=3];
91[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 32[label="",style="solid", color="burlywood", weight=3];
21[label="primCmpNat (Succ vx300) vx40 == LT\n",fontsize=16,color="burlywood",shape="triangle"];92[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];21 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 33[label="",style="solid", color="burlywood", weight=3];
93[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 34[label="",style="solid", color="burlywood", weight=3];
22[label="GT == LT\n",fontsize=16,color="black",shape="triangle"];22 -> 35[label="",style="solid", color="black", weight=3];
23[label="primCmpInt (Pos Zero) (Pos (Succ vx400)) == LT\n",fontsize=16,color="black",shape="box"];23 -> 36[label="",style="solid", color="black", weight=3];
24[label="primCmpInt (Pos Zero) (Pos Zero) == LT\n",fontsize=16,color="black",shape="box"];24 -> 37[label="",style="solid", color="black", weight=3];
25[label="primCmpInt (Pos Zero) (Neg (Succ vx400)) == LT\n",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3];
26[label="primCmpInt (Pos Zero) (Neg Zero) == LT\n",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
27[label="LT == LT\n",fontsize=16,color="black",shape="triangle"];27 -> 40[label="",style="solid", color="black", weight=3];
28[label="primCmpNat vx40 (Succ vx300) == LT\n",fontsize=16,color="burlywood",shape="triangle"];94[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];28 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 41[label="",style="solid", color="burlywood", weight=3];
95[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 42[label="",style="solid", color="burlywood", weight=3];
29[label="primCmpInt (Neg Zero) (Pos (Succ vx400)) == LT\n",fontsize=16,color="black",shape="box"];29 -> 43[label="",style="solid", color="black", weight=3];
30[label="primCmpInt (Neg Zero) (Pos Zero) == LT\n",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
31[label="primCmpInt (Neg Zero) (Neg (Succ vx400)) == LT\n",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
32[label="primCmpInt (Neg Zero) (Neg Zero) == LT\n",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
33[label="primCmpNat (Succ vx300) (Succ vx400) == LT\n",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
34[label="primCmpNat (Succ vx300) Zero == LT\n",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
35[label="False\n",fontsize=16,color="green",shape="box"];36 -> 28[label="",style="dashed", color="red", weight=0];
36[label="primCmpNat Zero (Succ vx400) == LT\n",fontsize=16,color="magenta"];36 -> 49[label="",style="dashed", color="magenta", weight=3];
36 -> 50[label="",style="dashed", color="magenta", weight=3];
37[label="EQ == LT\n",fontsize=16,color="black",shape="triangle"];37 -> 51[label="",style="solid", color="black", weight=3];
38 -> 22[label="",style="dashed", color="red", weight=0];
38[label="GT == LT\n",fontsize=16,color="magenta"];39 -> 37[label="",style="dashed", color="red", weight=0];
39[label="EQ == LT\n",fontsize=16,color="magenta"];40[label="True\n",fontsize=16,color="green",shape="box"];41[label="primCmpNat (Succ vx400) (Succ vx300) == LT\n",fontsize=16,color="black",shape="box"];41 -> 52[label="",style="solid", color="black", weight=3];
42[label="primCmpNat Zero (Succ vx300) == LT\n",fontsize=16,color="black",shape="box"];42 -> 53[label="",style="solid", color="black", weight=3];
43 -> 27[label="",style="dashed", color="red", weight=0];
43[label="LT == LT\n",fontsize=16,color="magenta"];44 -> 37[label="",style="dashed", color="red", weight=0];
44[label="EQ == LT\n",fontsize=16,color="magenta"];45 -> 21[label="",style="dashed", color="red", weight=0];
45[label="primCmpNat (Succ vx400) Zero == LT\n",fontsize=16,color="magenta"];45 -> 54[label="",style="dashed", color="magenta", weight=3];
45 -> 55[label="",style="dashed", color="magenta", weight=3];
46 -> 37[label="",style="dashed", color="red", weight=0];
46[label="EQ == LT\n",fontsize=16,color="magenta"];47[label="primCmpNat vx300 vx400 == LT\n",fontsize=16,color="burlywood",shape="triangle"];103[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];47 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 56[label="",style="solid", color="burlywood", weight=3];
104[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 57[label="",style="solid", color="burlywood", weight=3];
48 -> 22[label="",style="dashed", color="red", weight=0];
48[label="GT == LT\n",fontsize=16,color="magenta"];49[label="vx400\n",fontsize=16,color="green",shape="box"];50[label="Zero\n",fontsize=16,color="green",shape="box"];51[label="False\n",fontsize=16,color="green",shape="box"];52 -> 47[label="",style="dashed", color="red", weight=0];
52[label="primCmpNat vx400 vx300 == LT\n",fontsize=16,color="magenta"];52 -> 58[label="",style="dashed", color="magenta", weight=3];
52 -> 59[label="",style="dashed", color="magenta", weight=3];
53 -> 27[label="",style="dashed", color="red", weight=0];
53[label="LT == LT\n",fontsize=16,color="magenta"];54[label="vx400\n",fontsize=16,color="green",shape="box"];55[label="Zero\n",fontsize=16,color="green",shape="box"];56[label="primCmpNat (Succ vx3000) vx400 == LT\n",fontsize=16,color="burlywood",shape="box"];108[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];56 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 60[label="",style="solid", color="burlywood", weight=3];
109[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 61[label="",style="solid", color="burlywood", weight=3];
57[label="primCmpNat Zero vx400 == LT\n",fontsize=16,color="burlywood",shape="box"];110[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];57 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 62[label="",style="solid", color="burlywood", weight=3];
111[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];57 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 63[label="",style="solid", color="burlywood", weight=3];
58[label="vx300\n",fontsize=16,color="green",shape="box"];59[label="vx400\n",fontsize=16,color="green",shape="box"];60[label="primCmpNat (Succ vx3000) (Succ vx4000) == LT\n",fontsize=16,color="black",shape="box"];60 -> 64[label="",style="solid", color="black", weight=3];
61[label="primCmpNat (Succ vx3000) Zero == LT\n",fontsize=16,color="black",shape="box"];61 -> 65[label="",style="solid", color="black", weight=3];
62[label="primCmpNat Zero (Succ vx4000) == LT\n",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3];
63[label="primCmpNat Zero Zero == LT\n",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
64 -> 47[label="",style="dashed", color="red", weight=0];
64[label="primCmpNat vx3000 vx4000 == LT\n",fontsize=16,color="magenta"];64 -> 68[label="",style="dashed", color="magenta", weight=3];
64 -> 69[label="",style="dashed", color="magenta", weight=3];
65 -> 22[label="",style="dashed", color="red", weight=0];
65[label="GT == LT\n",fontsize=16,color="magenta"];66 -> 27[label="",style="dashed", color="red", weight=0];
66[label="LT == LT\n",fontsize=16,color="magenta"];67 -> 37[label="",style="dashed", color="red", weight=0];
67[label="EQ == LT\n",fontsize=16,color="magenta"];68[label="vx4000\n",fontsize=16,color="green",shape="box"];69[label="vx3000\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_esEs(Succ(vx3000), Succ(vx4000)) → new_esEs(vx3000, vx4000)

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_esEs(Succ(vx3000), Succ(vx4000)) → new_esEs(vx3000, vx4000)
The graph contains the following edges 1 > 1, 2 > 2