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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_even_1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

((even :: Int -> Bool) :: Int -> Bool)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

mainModule Main

((even :: Int -> Bool) :: 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

(even :: Int -> Bool)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="even\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="even vx3\n",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="primEvenInt vx3\n",fontsize=16,color="burlywood",shape="box"];18[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 5[label="",style="solid", color="burlywood", weight=3];
19[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="primEvenInt (Pos vx30)\n",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="primEvenInt (Neg vx30)\n",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="primEvenNat vx30\n",fontsize=16,color="burlywood",shape="triangle"];20[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];7 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 9[label="",style="solid", color="burlywood", weight=3];
21[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 10[label="",style="solid", color="burlywood", weight=3];
8 -> 7[label="",style="dashed", color="red", weight=0];
8[label="primEvenNat vx30\n",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="primEvenNat (Succ vx300)\n",fontsize=16,color="burlywood",shape="box"];23[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 12[label="",style="solid", color="burlywood", weight=3];
24[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 13[label="",style="solid", color="burlywood", weight=3];
10[label="primEvenNat Zero\n",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
11[label="vx30\n",fontsize=16,color="green",shape="box"];12[label="primEvenNat (Succ (Succ vx3000))\n",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13[label="primEvenNat (Succ Zero)\n",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="True\n",fontsize=16,color="green",shape="box"];15 -> 7[label="",style="dashed", color="red", weight=0];
15[label="primEvenNat vx3000\n",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
16[label="False\n",fontsize=16,color="green",shape="box"];17[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_primEvenNat(Succ(Succ(vx3000))) → new_primEvenNat(vx3000)

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_primEvenNat(Succ(Succ(vx3000))) → new_primEvenNat(vx3000)
The graph contains the following edges 1 > 1