H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 0.6890000000000001 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.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

      ↳ COR

mainModule Main

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

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ 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="(-)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(-) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(-) vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primMinusFloat vx3 vx4",fontsize=16,color="burlywood",shape="box"];86[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];5 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 6[label="",style="solid", color="burlywood", weight=3];
6[label="primMinusFloat (Float vx30 vx31) vx4",fontsize=16,color="burlywood",shape="box"];87[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 7[label="",style="solid", color="burlywood", weight=3];
7[label="primMinusFloat (Float vx30 vx31) (Float vx40 vx41)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="Float (vx30 - vx40) (vx31 * vx41)",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="green", weight=3];
8 -> 10[label="",style="dashed", color="green", weight=3];
9[label="vx30 - vx40",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="vx31 * vx41",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="primMinusInt vx30 vx40",fontsize=16,color="burlywood",shape="box"];88[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 13[label="",style="solid", color="burlywood", weight=3];
89[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="primMulInt vx31 vx41",fontsize=16,color="burlywood",shape="box"];90[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];12 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 15[label="",style="solid", color="burlywood", weight=3];
91[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];12 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 16[label="",style="solid", color="burlywood", weight=3];
13[label="primMinusInt (Pos vx300) vx40",fontsize=16,color="burlywood",shape="box"];92[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 17[label="",style="solid", color="burlywood", weight=3];
93[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 18[label="",style="solid", color="burlywood", weight=3];
14[label="primMinusInt (Neg vx300) vx40",fontsize=16,color="burlywood",shape="box"];94[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 19[label="",style="solid", color="burlywood", weight=3];
95[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 20[label="",style="solid", color="burlywood", weight=3];
15[label="primMulInt (Pos vx310) vx41",fontsize=16,color="burlywood",shape="box"];96[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];15 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 21[label="",style="solid", color="burlywood", weight=3];
97[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];15 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 22[label="",style="solid", color="burlywood", weight=3];
16[label="primMulInt (Neg vx310) vx41",fontsize=16,color="burlywood",shape="box"];98[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];16 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 23[label="",style="solid", color="burlywood", weight=3];
99[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];16 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 24[label="",style="solid", color="burlywood", weight=3];
17[label="primMinusInt (Pos vx300) (Pos vx400)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primMinusInt (Pos vx300) (Neg vx400)",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
19[label="primMinusInt (Neg vx300) (Pos vx400)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="primMinusInt (Neg vx300) (Neg vx400)",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3];
21[label="primMulInt (Pos vx310) (Pos vx410)",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="primMulInt (Pos vx310) (Neg vx410)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
23[label="primMulInt (Neg vx310) (Pos vx410)",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
24[label="primMulInt (Neg vx310) (Neg vx410)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
25[label="primMinusNat vx300 vx400",fontsize=16,color="burlywood",shape="triangle"];100[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];25 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 33[label="",style="solid", color="burlywood", weight=3];
101[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 34[label="",style="solid", color="burlywood", weight=3];
26[label="Pos (primPlusNat vx300 vx400)",fontsize=16,color="green",shape="box"];26 -> 35[label="",style="dashed", color="green", weight=3];
27[label="Neg (primPlusNat vx300 vx400)",fontsize=16,color="green",shape="box"];27 -> 36[label="",style="dashed", color="green", weight=3];
28 -> 25[label="",style="dashed", color="red", weight=0];
28[label="primMinusNat vx400 vx300",fontsize=16,color="magenta"];28 -> 37[label="",style="dashed", color="magenta", weight=3];
28 -> 38[label="",style="dashed", color="magenta", weight=3];
29[label="Pos (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];29 -> 39[label="",style="dashed", color="green", weight=3];
30[label="Neg (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];30 -> 40[label="",style="dashed", color="green", weight=3];
31[label="Neg (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];31 -> 41[label="",style="dashed", color="green", weight=3];
32[label="Pos (primMulNat vx310 vx410)",fontsize=16,color="green",shape="box"];32 -> 42[label="",style="dashed", color="green", weight=3];
33[label="primMinusNat (Succ vx3000) vx400",fontsize=16,color="burlywood",shape="box"];103[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 43[label="",style="solid", color="burlywood", weight=3];
104[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 44[label="",style="solid", color="burlywood", weight=3];
34[label="primMinusNat Zero vx400",fontsize=16,color="burlywood",shape="box"];105[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];34 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 45[label="",style="solid", color="burlywood", weight=3];
106[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 46[label="",style="solid", color="burlywood", weight=3];
35[label="primPlusNat vx300 vx400",fontsize=16,color="burlywood",shape="triangle"];107[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 47[label="",style="solid", color="burlywood", weight=3];
108[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 48[label="",style="solid", color="burlywood", weight=3];
36 -> 35[label="",style="dashed", color="red", weight=0];
36[label="primPlusNat vx300 vx400",fontsize=16,color="magenta"];36 -> 49[label="",style="dashed", color="magenta", weight=3];
36 -> 50[label="",style="dashed", color="magenta", weight=3];
37[label="vx400",fontsize=16,color="green",shape="box"];38[label="vx300",fontsize=16,color="green",shape="box"];39[label="primMulNat vx310 vx410",fontsize=16,color="burlywood",shape="triangle"];110[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];39 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 51[label="",style="solid", color="burlywood", weight=3];
111[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 52[label="",style="solid", color="burlywood", weight=3];
40 -> 39[label="",style="dashed", color="red", weight=0];
40[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];40 -> 53[label="",style="dashed", color="magenta", weight=3];
41 -> 39[label="",style="dashed", color="red", weight=0];
41[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];41 -> 54[label="",style="dashed", color="magenta", weight=3];
42 -> 39[label="",style="dashed", color="red", weight=0];
42[label="primMulNat vx310 vx410",fontsize=16,color="magenta"];42 -> 55[label="",style="dashed", color="magenta", weight=3];
42 -> 56[label="",style="dashed", color="magenta", weight=3];
43[label="primMinusNat (Succ vx3000) (Succ vx4000)",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
44[label="primMinusNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3];
45[label="primMinusNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3];
46[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3];
47[label="primPlusNat (Succ vx3000) vx400",fontsize=16,color="burlywood",shape="box"];115[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];47 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 61[label="",style="solid", color="burlywood", weight=3];
116[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 62[label="",style="solid", color="burlywood", weight=3];
48[label="primPlusNat Zero vx400",fontsize=16,color="burlywood",shape="box"];117[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];48 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 63[label="",style="solid", color="burlywood", weight=3];
118[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 64[label="",style="solid", color="burlywood", weight=3];
49[label="vx300",fontsize=16,color="green",shape="box"];50[label="vx400",fontsize=16,color="green",shape="box"];51[label="primMulNat (Succ vx3100) vx410",fontsize=16,color="burlywood",shape="box"];119[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];51 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 65[label="",style="solid", color="burlywood", weight=3];
120[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];51 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 66[label="",style="solid", color="burlywood", weight=3];
52[label="primMulNat Zero vx410",fontsize=16,color="burlywood",shape="box"];121[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];52 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 67[label="",style="solid", color="burlywood", weight=3];
122[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 68[label="",style="solid", color="burlywood", weight=3];
53[label="vx410",fontsize=16,color="green",shape="box"];54[label="vx310",fontsize=16,color="green",shape="box"];55[label="vx310",fontsize=16,color="green",shape="box"];56[label="vx410",fontsize=16,color="green",shape="box"];57 -> 25[label="",style="dashed", color="red", weight=0];
57[label="primMinusNat vx3000 vx4000",fontsize=16,color="magenta"];57 -> 69[label="",style="dashed", color="magenta", weight=3];
57 -> 70[label="",style="dashed", color="magenta", weight=3];
58[label="Pos (Succ vx3000)",fontsize=16,color="green",shape="box"];59[label="Neg (Succ vx4000)",fontsize=16,color="green",shape="box"];60[label="Pos Zero",fontsize=16,color="green",shape="box"];61[label="primPlusNat (Succ vx3000) (Succ vx4000)",fontsize=16,color="black",shape="box"];61 -> 71[label="",style="solid", color="black", weight=3];
62[label="primPlusNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];62 -> 72[label="",style="solid", color="black", weight=3];
63[label="primPlusNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];63 -> 73[label="",style="solid", color="black", weight=3];
64[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
65[label="primMulNat (Succ vx3100) (Succ vx4100)",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
66[label="primMulNat (Succ vx3100) Zero",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
67[label="primMulNat Zero (Succ vx4100)",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
68[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3];
69[label="vx3000",fontsize=16,color="green",shape="box"];70[label="vx4000",fontsize=16,color="green",shape="box"];71[label="Succ (Succ (primPlusNat vx3000 vx4000))",fontsize=16,color="green",shape="box"];71 -> 79[label="",style="dashed", color="green", weight=3];
72[label="Succ vx3000",fontsize=16,color="green",shape="box"];73[label="Succ vx4000",fontsize=16,color="green",shape="box"];74[label="Zero",fontsize=16,color="green",shape="box"];75 -> 35[label="",style="dashed", color="red", weight=0];
75[label="primPlusNat (primMulNat vx3100 (Succ vx4100)) (Succ vx4100)",fontsize=16,color="magenta"];75 -> 80[label="",style="dashed", color="magenta", weight=3];
75 -> 81[label="",style="dashed", color="magenta", weight=3];
76[label="Zero",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];78[label="Zero",fontsize=16,color="green",shape="box"];79 -> 35[label="",style="dashed", color="red", weight=0];
79[label="primPlusNat vx3000 vx4000",fontsize=16,color="magenta"];79 -> 82[label="",style="dashed", color="magenta", weight=3];
79 -> 83[label="",style="dashed", color="magenta", weight=3];
80 -> 39[label="",style="dashed", color="red", weight=0];
80[label="primMulNat vx3100 (Succ vx4100)",fontsize=16,color="magenta"];80 -> 84[label="",style="dashed", color="magenta", weight=3];
80 -> 85[label="",style="dashed", color="magenta", weight=3];
81[label="Succ vx4100",fontsize=16,color="green",shape="box"];82[label="vx3000",fontsize=16,color="green",shape="box"];83[label="vx4000",fontsize=16,color="green",shape="box"];84[label="vx3100",fontsize=16,color="green",shape="box"];85[label="Succ vx4100",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_primMulNat(Succ(vx3100), Succ(vx4100)) → new_primMulNat(vx3100, Succ(vx4100))

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

new_primMulNat(Succ(vx3100), Succ(vx4100)) → new_primMulNat(vx3100, Succ(vx4100))
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_primMinusNat(Succ(vx3000), Succ(vx4000)) → new_primMinusNat(vx3000, vx4000)

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

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