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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_MINUS_2.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

(((-) :: Int -> Int -> Int) :: Int -> Int -> Int)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

mainModule Main

(((-) :: Int -> Int -> Int) :: Int -> Int -> Int)

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 -> Int)

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="primMinusInt vx3 vx4\n",fontsize=16,color="burlywood",shape="box"];47[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];5 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 6[label="",style="solid", color="burlywood", weight=3];
48[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];5 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="primMinusInt (Pos vx30) vx4\n",fontsize=16,color="burlywood",shape="box"];49[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 8[label="",style="solid", color="burlywood", weight=3];
50[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="primMinusInt (Neg vx30) vx4\n",fontsize=16,color="burlywood",shape="box"];51[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];7 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 10[label="",style="solid", color="burlywood", weight=3];
52[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];7 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="primMinusInt (Pos vx30) (Pos vx40)\n",fontsize=16,color="black",shape="box"];8 -> 12[label="",style="solid", color="black", weight=3];
9[label="primMinusInt (Pos vx30) (Neg vx40)\n",fontsize=16,color="black",shape="box"];9 -> 13[label="",style="solid", color="black", weight=3];
10[label="primMinusInt (Neg vx30) (Pos vx40)\n",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
11[label="primMinusInt (Neg vx30) (Neg vx40)\n",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3];
12[label="primMinusNat vx30 vx40\n",fontsize=16,color="burlywood",shape="triangle"];53[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];12 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 16[label="",style="solid", color="burlywood", weight=3];
54[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 17[label="",style="solid", color="burlywood", weight=3];
13[label="Pos (primPlusNat vx30 vx40)\n",fontsize=16,color="green",shape="box"];13 -> 18[label="",style="dashed", color="green", weight=3];
14[label="Neg (primPlusNat vx30 vx40)\n",fontsize=16,color="green",shape="box"];14 -> 19[label="",style="dashed", color="green", weight=3];
15 -> 12[label="",style="dashed", color="red", weight=0];
15[label="primMinusNat vx40 vx30\n",fontsize=16,color="magenta"];15 -> 20[label="",style="dashed", color="magenta", weight=3];
15 -> 21[label="",style="dashed", color="magenta", weight=3];
16[label="primMinusNat (Succ vx300) vx40\n",fontsize=16,color="burlywood",shape="box"];56[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];16 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 22[label="",style="solid", color="burlywood", weight=3];
57[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 23[label="",style="solid", color="burlywood", weight=3];
17[label="primMinusNat Zero vx40\n",fontsize=16,color="burlywood",shape="box"];58[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];17 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 24[label="",style="solid", color="burlywood", weight=3];
59[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 25[label="",style="solid", color="burlywood", weight=3];
18[label="primPlusNat vx30 vx40\n",fontsize=16,color="burlywood",shape="triangle"];60[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];18 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 26[label="",style="solid", color="burlywood", weight=3];
61[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 27[label="",style="solid", color="burlywood", weight=3];
19 -> 18[label="",style="dashed", color="red", weight=0];
19[label="primPlusNat vx30 vx40\n",fontsize=16,color="magenta"];19 -> 28[label="",style="dashed", color="magenta", weight=3];
19 -> 29[label="",style="dashed", color="magenta", weight=3];
20[label="vx30\n",fontsize=16,color="green",shape="box"];21[label="vx40\n",fontsize=16,color="green",shape="box"];22[label="primMinusNat (Succ vx300) (Succ vx400)\n",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
23[label="primMinusNat (Succ vx300) Zero\n",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
24[label="primMinusNat Zero (Succ vx400)\n",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
25[label="primMinusNat Zero Zero\n",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
26[label="primPlusNat (Succ vx300) vx40\n",fontsize=16,color="burlywood",shape="box"];63[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];26 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 34[label="",style="solid", color="burlywood", weight=3];
64[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 35[label="",style="solid", color="burlywood", weight=3];
27[label="primPlusNat Zero vx40\n",fontsize=16,color="burlywood",shape="box"];65[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];27 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 36[label="",style="solid", color="burlywood", weight=3];
66[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 37[label="",style="solid", color="burlywood", weight=3];
28[label="vx40\n",fontsize=16,color="green",shape="box"];29[label="vx30\n",fontsize=16,color="green",shape="box"];30 -> 12[label="",style="dashed", color="red", weight=0];
30[label="primMinusNat vx300 vx400\n",fontsize=16,color="magenta"];30 -> 38[label="",style="dashed", color="magenta", weight=3];
30 -> 39[label="",style="dashed", color="magenta", weight=3];
31[label="Pos (Succ vx300)\n",fontsize=16,color="green",shape="box"];32[label="Neg (Succ vx400)\n",fontsize=16,color="green",shape="box"];33[label="Pos Zero\n",fontsize=16,color="green",shape="box"];34[label="primPlusNat (Succ vx300) (Succ vx400)\n",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3];
35[label="primPlusNat (Succ vx300) Zero\n",fontsize=16,color="black",shape="box"];35 -> 41[label="",style="solid", color="black", weight=3];
36[label="primPlusNat Zero (Succ vx400)\n",fontsize=16,color="black",shape="box"];36 -> 42[label="",style="solid", color="black", weight=3];
37[label="primPlusNat Zero Zero\n",fontsize=16,color="black",shape="box"];37 -> 43[label="",style="solid", color="black", weight=3];
38[label="vx400\n",fontsize=16,color="green",shape="box"];39[label="vx300\n",fontsize=16,color="green",shape="box"];40[label="Succ (Succ (primPlusNat vx300 vx400))\n",fontsize=16,color="green",shape="box"];40 -> 44[label="",style="dashed", color="green", weight=3];
41[label="Succ vx300\n",fontsize=16,color="green",shape="box"];42[label="Succ vx400\n",fontsize=16,color="green",shape="box"];43[label="Zero\n",fontsize=16,color="green",shape="box"];44 -> 18[label="",style="dashed", color="red", weight=0];
44[label="primPlusNat vx300 vx400\n",fontsize=16,color="magenta"];44 -> 45[label="",style="dashed", color="magenta", weight=3];
44 -> 46[label="",style="dashed", color="magenta", weight=3];
45[label="vx400\n",fontsize=16,color="green",shape="box"];46[label="vx300\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_primPlusNat(Succ(vx300), Succ(vx400)) → new_primPlusNat(vx300, vx400)

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_primPlusNat(Succ(vx300), Succ(vx400)) → new_primPlusNat(vx300, vx400)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_primMinusNat(Succ(vx300), Succ(vx400)) → new_primMinusNat(vx300, vx400)

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

new_primMinusNat(Succ(vx300), Succ(vx400)) → new_primMinusNat(vx300, vx400)
The graph contains the following edges 1 > 1, 2 > 2