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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_STARSTAR_1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

(((**) :: Float -> Float -> Float) :: Float -> Float -> Float)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

mainModule Main

(((**) :: Float -> Float -> Float) :: Float -> Float -> Float)

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

((**) :: Float -> Float -> Float)

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="exp (log vx3 * vx4)\n",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="primExpFloat (log vx3 * vx4)\n",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="terminator (log vx3 * vx4)\n",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="ter0m (log vx3 * vx4)\n",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="green", weight=3];
9[label="log vx3 * vx4\n",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10 -> 14[label="",style="dashed", color="red", weight=0];
10[label="primMulFloat (log vx3) vx4\n",fontsize=16,color="magenta"];10 -> 15[label="",style="dashed", color="magenta", weight=3];
15[label="log vx3\n",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
14[label="primMulFloat vx5 vx4\n",fontsize=16,color="burlywood",shape="triangle"];83[label="vx5/Float vx50 vx51",fontsize=10,color="white",style="solid",shape="box"];14 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 20[label="",style="solid", color="burlywood", weight=3];
19[label="primLogFloat vx3\n",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
20[label="primMulFloat (Float vx50 vx51) vx4\n",fontsize=16,color="burlywood",shape="box"];84[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];20 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 23[label="",style="solid", color="burlywood", weight=3];
22[label="terminator vx3\n",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="primMulFloat (Float vx50 vx51) (Float vx40 vx41)\n",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="ter2m vx3\n",fontsize=16,color="green",shape="box"];24 -> 26[label="",style="dashed", color="green", weight=3];
25[label="Float (vx50 * vx40) (vx51 * vx41)\n",fontsize=16,color="green",shape="box"];25 -> 27[label="",style="dashed", color="green", weight=3];
25 -> 28[label="",style="dashed", color="green", weight=3];
26[label="vx3\n",fontsize=16,color="green",shape="box"];27[label="vx50 * vx40\n",fontsize=16,color="black",shape="triangle"];27 -> 29[label="",style="solid", color="black", weight=3];
28 -> 27[label="",style="dashed", color="red", weight=0];
28[label="vx51 * vx41\n",fontsize=16,color="magenta"];28 -> 30[label="",style="dashed", color="magenta", weight=3];
28 -> 31[label="",style="dashed", color="magenta", weight=3];
29[label="primMulInt vx50 vx40\n",fontsize=16,color="burlywood",shape="box"];86[label="vx50/Pos vx500",fontsize=10,color="white",style="solid",shape="box"];29 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 32[label="",style="solid", color="burlywood", weight=3];
87[label="vx50/Neg vx500",fontsize=10,color="white",style="solid",shape="box"];29 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 33[label="",style="solid", color="burlywood", weight=3];
30[label="vx41\n",fontsize=16,color="green",shape="box"];31[label="vx51\n",fontsize=16,color="green",shape="box"];32[label="primMulInt (Pos vx500) vx40\n",fontsize=16,color="burlywood",shape="box"];88[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];32 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 34[label="",style="solid", color="burlywood", weight=3];
89[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];32 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 35[label="",style="solid", color="burlywood", weight=3];
33[label="primMulInt (Neg vx500) vx40\n",fontsize=16,color="burlywood",shape="box"];90[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];33 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 36[label="",style="solid", color="burlywood", weight=3];
91[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];33 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 37[label="",style="solid", color="burlywood", weight=3];
34[label="primMulInt (Pos vx500) (Pos vx400)\n",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="primMulInt (Pos vx500) (Neg vx400)\n",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="primMulInt (Neg vx500) (Pos vx400)\n",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="primMulInt (Neg vx500) (Neg vx400)\n",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="Pos (primMulNat vx500 vx400)\n",fontsize=16,color="green",shape="box"];38 -> 42[label="",style="dashed", color="green", weight=3];
39[label="Neg (primMulNat vx500 vx400)\n",fontsize=16,color="green",shape="box"];39 -> 43[label="",style="dashed", color="green", weight=3];
40[label="Neg (primMulNat vx500 vx400)\n",fontsize=16,color="green",shape="box"];40 -> 44[label="",style="dashed", color="green", weight=3];
41[label="Pos (primMulNat vx500 vx400)\n",fontsize=16,color="green",shape="box"];41 -> 45[label="",style="dashed", color="green", weight=3];
42[label="primMulNat vx500 vx400\n",fontsize=16,color="burlywood",shape="triangle"];92[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];42 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 46[label="",style="solid", color="burlywood", weight=3];
93[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 47[label="",style="solid", color="burlywood", weight=3];
43 -> 42[label="",style="dashed", color="red", weight=0];
43[label="primMulNat vx500 vx400\n",fontsize=16,color="magenta"];43 -> 48[label="",style="dashed", color="magenta", weight=3];
44 -> 42[label="",style="dashed", color="red", weight=0];
44[label="primMulNat vx500 vx400\n",fontsize=16,color="magenta"];44 -> 49[label="",style="dashed", color="magenta", weight=3];
45 -> 42[label="",style="dashed", color="red", weight=0];
45[label="primMulNat vx500 vx400\n",fontsize=16,color="magenta"];45 -> 50[label="",style="dashed", color="magenta", weight=3];
45 -> 51[label="",style="dashed", color="magenta", weight=3];
46[label="primMulNat (Succ vx5000) vx400\n",fontsize=16,color="burlywood",shape="box"];97[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];46 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 52[label="",style="solid", color="burlywood", weight=3];
98[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 53[label="",style="solid", color="burlywood", weight=3];
47[label="primMulNat Zero vx400\n",fontsize=16,color="burlywood",shape="box"];99[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];47 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 54[label="",style="solid", color="burlywood", weight=3];
100[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 55[label="",style="solid", color="burlywood", weight=3];
48[label="vx400\n",fontsize=16,color="green",shape="box"];49[label="vx500\n",fontsize=16,color="green",shape="box"];50[label="vx400\n",fontsize=16,color="green",shape="box"];51[label="vx500\n",fontsize=16,color="green",shape="box"];52[label="primMulNat (Succ vx5000) (Succ vx4000)\n",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="primMulNat (Succ vx5000) Zero\n",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="primMulNat Zero (Succ vx4000)\n",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55[label="primMulNat Zero Zero\n",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
56 -> 60[label="",style="dashed", color="red", weight=0];
56[label="primPlusNat (primMulNat vx5000 (Succ vx4000)) (Succ vx4000)\n",fontsize=16,color="magenta"];56 -> 61[label="",style="dashed", color="magenta", weight=3];
57[label="Zero\n",fontsize=16,color="green",shape="box"];58[label="Zero\n",fontsize=16,color="green",shape="box"];59[label="Zero\n",fontsize=16,color="green",shape="box"];61 -> 42[label="",style="dashed", color="red", weight=0];
61[label="primMulNat vx5000 (Succ vx4000)\n",fontsize=16,color="magenta"];61 -> 62[label="",style="dashed", color="magenta", weight=3];
61 -> 63[label="",style="dashed", color="magenta", weight=3];
60[label="primPlusNat vx6 (Succ vx4000)\n",fontsize=16,color="burlywood",shape="triangle"];103[label="vx6/Succ vx60",fontsize=10,color="white",style="solid",shape="box"];60 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 64[label="",style="solid", color="burlywood", weight=3];
104[label="vx6/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 65[label="",style="solid", color="burlywood", weight=3];
62[label="Succ vx4000\n",fontsize=16,color="green",shape="box"];63[label="vx5000\n",fontsize=16,color="green",shape="box"];64[label="primPlusNat (Succ vx60) (Succ vx4000)\n",fontsize=16,color="black",shape="box"];64 -> 66[label="",style="solid", color="black", weight=3];
65[label="primPlusNat Zero (Succ vx4000)\n",fontsize=16,color="black",shape="box"];65 -> 67[label="",style="solid", color="black", weight=3];
66[label="Succ (Succ (primPlusNat vx60 vx4000))\n",fontsize=16,color="green",shape="box"];66 -> 68[label="",style="dashed", color="green", weight=3];
67[label="Succ vx4000\n",fontsize=16,color="green",shape="box"];68[label="primPlusNat vx60 vx4000\n",fontsize=16,color="burlywood",shape="triangle"];105[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];68 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 69[label="",style="solid", color="burlywood", weight=3];
106[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];68 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 70[label="",style="solid", color="burlywood", weight=3];
69[label="primPlusNat (Succ vx600) vx4000\n",fontsize=16,color="burlywood",shape="box"];107[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];69 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 71[label="",style="solid", color="burlywood", weight=3];
108[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];69 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 72[label="",style="solid", color="burlywood", weight=3];
70[label="primPlusNat Zero vx4000\n",fontsize=16,color="burlywood",shape="box"];109[label="vx4000/Succ vx40000",fontsize=10,color="white",style="solid",shape="box"];70 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 73[label="",style="solid", color="burlywood", weight=3];
110[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];70 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 74[label="",style="solid", color="burlywood", weight=3];
71[label="primPlusNat (Succ vx600) (Succ vx40000)\n",fontsize=16,color="black",shape="box"];71 -> 75[label="",style="solid", color="black", weight=3];
72[label="primPlusNat (Succ vx600) Zero\n",fontsize=16,color="black",shape="box"];72 -> 76[label="",style="solid", color="black", weight=3];
73[label="primPlusNat Zero (Succ vx40000)\n",fontsize=16,color="black",shape="box"];73 -> 77[label="",style="solid", color="black", weight=3];
74[label="primPlusNat Zero Zero\n",fontsize=16,color="black",shape="box"];74 -> 78[label="",style="solid", color="black", weight=3];
75[label="Succ (Succ (primPlusNat vx600 vx40000))\n",fontsize=16,color="green",shape="box"];75 -> 79[label="",style="dashed", color="green", weight=3];
76[label="Succ vx600\n",fontsize=16,color="green",shape="box"];77[label="Succ vx40000\n",fontsize=16,color="green",shape="box"];78[label="Zero\n",fontsize=16,color="green",shape="box"];79 -> 68[label="",style="dashed", color="red", weight=0];
79[label="primPlusNat vx600 vx40000\n",fontsize=16,color="magenta"];79 -> 80[label="",style="dashed", color="magenta", weight=3];
79 -> 81[label="",style="dashed", color="magenta", weight=3];
80[label="vx40000\n",fontsize=16,color="green",shape="box"];81[label="vx600\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_primMulNat(Succ(vx5000), Succ(vx4000)) → new_primMulNat(vx5000, Succ(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_primMulNat(Succ(vx5000), Succ(vx4000)) → new_primMulNat(vx5000, Succ(vx4000))
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_primPlusNat(Succ(vx600), Succ(vx40000)) → new_primPlusNat(vx600, vx40000)

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

new_primPlusNat(Succ(vx600), Succ(vx40000)) → new_primPlusNat(vx600, vx40000)
The graph contains the following edges 1 > 1, 2 > 2