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

YES 0.735 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

((product :: [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

((product :: [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

          ↳ NumRed

mainModule Main

((product :: [Float] -> Float) :: [Float] -> Float)

module Main where

import qualified Prelude

Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(product :: [Float] -> Float)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="product",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="product vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl' (*) (fromInt (Pos (Succ Zero))) vx3",fontsize=16,color="burlywood",shape="box"];94[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 5[label="",style="solid", color="burlywood", weight=3];
95[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl' (*) (fromInt (Pos (Succ Zero))) (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl' (*) (fromInt (Pos (Succ Zero))) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="(foldl' (*) $! (*) fromInt (Pos (Succ Zero)) vx30)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="((*) fromInt (Pos (Succ Zero)) vx30 `seq` foldl' (*) ((*) fromInt (Pos (Succ Zero)) vx30))",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
9 -> 13[label="",style="dashed", color="magenta", weight=3];
10[label="primIntToFloat (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
12 -> 8[label="",style="dashed", color="red", weight=0];
12[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];13 -> 8[label="",style="dashed", color="red", weight=0];
13[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];11[label="((*) vx4 vx30 `seq` foldl' (*) ((*) vx5 vx30))",fontsize=16,color="black",shape="triangle"];11 -> 15[label="",style="solid", color="black", weight=3];
14[label="Float (Pos (Succ Zero)) (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];15[label="enforceWHNF (WHNF ((*) vx4 vx30)) (foldl' (*) ((*) vx5 vx30)) vx31",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="enforceWHNF (WHNF (primMulFloat vx4 vx30)) (foldl' primMulFloat (primMulFloat vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="triangle"];99[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];16 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 17[label="",style="solid", color="burlywood", weight=3];
17[label="enforceWHNF (WHNF (primMulFloat (Float vx40 vx41) vx30)) (foldl' primMulFloat (primMulFloat vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="box"];100[label="vx30/Float vx300 vx301",fontsize=10,color="white",style="solid",shape="box"];17 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 18[label="",style="solid", color="burlywood", weight=3];
18[label="enforceWHNF (WHNF (primMulFloat (Float vx40 vx41) (Float vx300 vx301))) (foldl' primMulFloat (primMulFloat vx5 (Float vx300 vx301))) vx31",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="enforceWHNF (WHNF (Float (vx40 * vx300) (vx41 * vx301))) (foldl' primMulFloat (Float (vx40 * vx300) (vx41 * vx301))) vx31",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="foldl' primMulFloat (Float (vx40 * vx300) (vx41 * vx301)) vx31",fontsize=16,color="burlywood",shape="box"];101[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];20 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 21[label="",style="solid", color="burlywood", weight=3];
102[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];20 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 22[label="",style="solid", color="burlywood", weight=3];
21[label="foldl' primMulFloat (Float (vx40 * vx300) (vx41 * vx301)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="foldl' primMulFloat (Float (vx40 * vx300) (vx41 * vx301)) []",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="(foldl' primMulFloat $! primMulFloat (Float (vx40 * vx300) (vx41 * vx301)) vx310)",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="Float (vx40 * vx300) (vx41 * vx301)",fontsize=16,color="green",shape="box"];24 -> 26[label="",style="dashed", color="green", weight=3];
24 -> 27[label="",style="dashed", color="green", weight=3];
25[label="(primMulFloat (Float (vx40 * vx300) (vx41 * vx301)) vx310 `seq` foldl' primMulFloat (primMulFloat (Float (vx40 * vx300) (vx41 * vx301)) vx310))",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
26[label="vx40 * vx300",fontsize=16,color="black",shape="triangle"];26 -> 29[label="",style="solid", color="black", weight=3];
27 -> 26[label="",style="dashed", color="red", weight=0];
27[label="vx41 * vx301",fontsize=16,color="magenta"];27 -> 30[label="",style="dashed", color="magenta", weight=3];
27 -> 31[label="",style="dashed", color="magenta", weight=3];
28 -> 16[label="",style="dashed", color="red", weight=0];
28[label="enforceWHNF (WHNF (primMulFloat (Float (vx40 * vx300) (vx41 * vx301)) vx310)) (foldl' primMulFloat (primMulFloat (Float (vx40 * vx300) (vx41 * vx301)) vx310)) vx311",fontsize=16,color="magenta"];28 -> 32[label="",style="dashed", color="magenta", weight=3];
28 -> 33[label="",style="dashed", color="magenta", weight=3];
28 -> 34[label="",style="dashed", color="magenta", weight=3];
28 -> 35[label="",style="dashed", color="magenta", weight=3];
29[label="primMulInt vx40 vx300",fontsize=16,color="burlywood",shape="box"];105[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];29 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 36[label="",style="solid", color="burlywood", weight=3];
106[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];29 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 37[label="",style="solid", color="burlywood", weight=3];
30[label="vx301",fontsize=16,color="green",shape="box"];31[label="vx41",fontsize=16,color="green",shape="box"];32[label="Float (vx40 * vx300) (vx41 * vx301)",fontsize=16,color="green",shape="box"];32 -> 38[label="",style="dashed", color="green", weight=3];
32 -> 39[label="",style="dashed", color="green", weight=3];
33[label="vx311",fontsize=16,color="green",shape="box"];34[label="Float (vx40 * vx300) (vx41 * vx301)",fontsize=16,color="green",shape="box"];34 -> 40[label="",style="dashed", color="green", weight=3];
34 -> 41[label="",style="dashed", color="green", weight=3];
35[label="vx310",fontsize=16,color="green",shape="box"];36[label="primMulInt (Pos vx400) vx300",fontsize=16,color="burlywood",shape="box"];107[label="vx300/Pos vx3000",fontsize=10,color="white",style="solid",shape="box"];36 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 42[label="",style="solid", color="burlywood", weight=3];
108[label="vx300/Neg vx3000",fontsize=10,color="white",style="solid",shape="box"];36 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 43[label="",style="solid", color="burlywood", weight=3];
37[label="primMulInt (Neg vx400) vx300",fontsize=16,color="burlywood",shape="box"];109[label="vx300/Pos vx3000",fontsize=10,color="white",style="solid",shape="box"];37 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 44[label="",style="solid", color="burlywood", weight=3];
110[label="vx300/Neg vx3000",fontsize=10,color="white",style="solid",shape="box"];37 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 45[label="",style="solid", color="burlywood", weight=3];
38 -> 26[label="",style="dashed", color="red", weight=0];
38[label="vx40 * vx300",fontsize=16,color="magenta"];39 -> 26[label="",style="dashed", color="red", weight=0];
39[label="vx41 * vx301",fontsize=16,color="magenta"];39 -> 46[label="",style="dashed", color="magenta", weight=3];
39 -> 47[label="",style="dashed", color="magenta", weight=3];
40 -> 26[label="",style="dashed", color="red", weight=0];
40[label="vx40 * vx300",fontsize=16,color="magenta"];41 -> 26[label="",style="dashed", color="red", weight=0];
41[label="vx41 * vx301",fontsize=16,color="magenta"];41 -> 48[label="",style="dashed", color="magenta", weight=3];
41 -> 49[label="",style="dashed", color="magenta", weight=3];
42[label="primMulInt (Pos vx400) (Pos vx3000)",fontsize=16,color="black",shape="box"];42 -> 50[label="",style="solid", color="black", weight=3];
43[label="primMulInt (Pos vx400) (Neg vx3000)",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3];
44[label="primMulInt (Neg vx400) (Pos vx3000)",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3];
45[label="primMulInt (Neg vx400) (Neg vx3000)",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3];
46[label="vx301",fontsize=16,color="green",shape="box"];47[label="vx41",fontsize=16,color="green",shape="box"];48[label="vx301",fontsize=16,color="green",shape="box"];49[label="vx41",fontsize=16,color="green",shape="box"];50[label="Pos (primMulNat vx400 vx3000)",fontsize=16,color="green",shape="box"];50 -> 54[label="",style="dashed", color="green", weight=3];
51[label="Neg (primMulNat vx400 vx3000)",fontsize=16,color="green",shape="box"];51 -> 55[label="",style="dashed", color="green", weight=3];
52[label="Neg (primMulNat vx400 vx3000)",fontsize=16,color="green",shape="box"];52 -> 56[label="",style="dashed", color="green", weight=3];
53[label="Pos (primMulNat vx400 vx3000)",fontsize=16,color="green",shape="box"];53 -> 57[label="",style="dashed", color="green", weight=3];
54[label="primMulNat vx400 vx3000",fontsize=16,color="burlywood",shape="triangle"];115[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];54 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 58[label="",style="solid", color="burlywood", weight=3];
116[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 59[label="",style="solid", color="burlywood", weight=3];
55 -> 54[label="",style="dashed", color="red", weight=0];
55[label="primMulNat vx400 vx3000",fontsize=16,color="magenta"];55 -> 60[label="",style="dashed", color="magenta", weight=3];
56 -> 54[label="",style="dashed", color="red", weight=0];
56[label="primMulNat vx400 vx3000",fontsize=16,color="magenta"];56 -> 61[label="",style="dashed", color="magenta", weight=3];
57 -> 54[label="",style="dashed", color="red", weight=0];
57[label="primMulNat vx400 vx3000",fontsize=16,color="magenta"];57 -> 62[label="",style="dashed", color="magenta", weight=3];
57 -> 63[label="",style="dashed", color="magenta", weight=3];
58[label="primMulNat (Succ vx4000) vx3000",fontsize=16,color="burlywood",shape="box"];120[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];58 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 64[label="",style="solid", color="burlywood", weight=3];
121[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 65[label="",style="solid", color="burlywood", weight=3];
59[label="primMulNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];122[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];59 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 66[label="",style="solid", color="burlywood", weight=3];
123[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 67[label="",style="solid", color="burlywood", weight=3];
60[label="vx3000",fontsize=16,color="green",shape="box"];61[label="vx400",fontsize=16,color="green",shape="box"];62[label="vx400",fontsize=16,color="green",shape="box"];63[label="vx3000",fontsize=16,color="green",shape="box"];64[label="primMulNat (Succ vx4000) (Succ vx30000)",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
65[label="primMulNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
66[label="primMulNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
67[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
68 -> 72[label="",style="dashed", color="red", weight=0];
68[label="primPlusNat (primMulNat vx4000 (Succ vx30000)) (Succ vx30000)",fontsize=16,color="magenta"];68 -> 73[label="",style="dashed", color="magenta", weight=3];
69[label="Zero",fontsize=16,color="green",shape="box"];70[label="Zero",fontsize=16,color="green",shape="box"];71[label="Zero",fontsize=16,color="green",shape="box"];73 -> 54[label="",style="dashed", color="red", weight=0];
73[label="primMulNat vx4000 (Succ vx30000)",fontsize=16,color="magenta"];73 -> 74[label="",style="dashed", color="magenta", weight=3];
73 -> 75[label="",style="dashed", color="magenta", weight=3];
72[label="primPlusNat vx6 (Succ vx30000)",fontsize=16,color="burlywood",shape="triangle"];126[label="vx6/Succ vx60",fontsize=10,color="white",style="solid",shape="box"];72 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 76[label="",style="solid", color="burlywood", weight=3];
127[label="vx6/Zero",fontsize=10,color="white",style="solid",shape="box"];72 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 77[label="",style="solid", color="burlywood", weight=3];
74[label="vx4000",fontsize=16,color="green",shape="box"];75[label="Succ vx30000",fontsize=16,color="green",shape="box"];76[label="primPlusNat (Succ vx60) (Succ vx30000)",fontsize=16,color="black",shape="box"];76 -> 78[label="",style="solid", color="black", weight=3];
77[label="primPlusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];77 -> 79[label="",style="solid", color="black", weight=3];
78[label="Succ (Succ (primPlusNat vx60 vx30000))",fontsize=16,color="green",shape="box"];78 -> 80[label="",style="dashed", color="green", weight=3];
79[label="Succ vx30000",fontsize=16,color="green",shape="box"];80[label="primPlusNat vx60 vx30000",fontsize=16,color="burlywood",shape="triangle"];128[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];80 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 81[label="",style="solid", color="burlywood", weight=3];
129[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];80 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 82[label="",style="solid", color="burlywood", weight=3];
81[label="primPlusNat (Succ vx600) vx30000",fontsize=16,color="burlywood",shape="box"];130[label="vx30000/Succ vx300000",fontsize=10,color="white",style="solid",shape="box"];81 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 83[label="",style="solid", color="burlywood", weight=3];
131[label="vx30000/Zero",fontsize=10,color="white",style="solid",shape="box"];81 -> 131[label="",style="solid", color="burlywood", weight=9];
131 -> 84[label="",style="solid", color="burlywood", weight=3];
82[label="primPlusNat Zero vx30000",fontsize=16,color="burlywood",shape="box"];132[label="vx30000/Succ vx300000",fontsize=10,color="white",style="solid",shape="box"];82 -> 132[label="",style="solid", color="burlywood", weight=9];
132 -> 85[label="",style="solid", color="burlywood", weight=3];
133[label="vx30000/Zero",fontsize=10,color="white",style="solid",shape="box"];82 -> 133[label="",style="solid", color="burlywood", weight=9];
133 -> 86[label="",style="solid", color="burlywood", weight=3];
83[label="primPlusNat (Succ vx600) (Succ vx300000)",fontsize=16,color="black",shape="box"];83 -> 87[label="",style="solid", color="black", weight=3];
84[label="primPlusNat (Succ vx600) Zero",fontsize=16,color="black",shape="box"];84 -> 88[label="",style="solid", color="black", weight=3];
85[label="primPlusNat Zero (Succ vx300000)",fontsize=16,color="black",shape="box"];85 -> 89[label="",style="solid", color="black", weight=3];
86[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];86 -> 90[label="",style="solid", color="black", weight=3];
87[label="Succ (Succ (primPlusNat vx600 vx300000))",fontsize=16,color="green",shape="box"];87 -> 91[label="",style="dashed", color="green", weight=3];
88[label="Succ vx600",fontsize=16,color="green",shape="box"];89[label="Succ vx300000",fontsize=16,color="green",shape="box"];90[label="Zero",fontsize=16,color="green",shape="box"];91 -> 80[label="",style="dashed", color="red", weight=0];
91[label="primPlusNat vx600 vx300000",fontsize=16,color="magenta"];91 -> 92[label="",style="dashed", color="magenta", weight=3];
91 -> 93[label="",style="dashed", color="magenta", weight=3];
92[label="vx300000",fontsize=16,color="green",shape="box"];93[label="vx600",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_primPlusNat(Succ(vx600), Succ(vx300000)) → new_primPlusNat(vx600, vx300000)

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(vx600), Succ(vx300000)) → new_primPlusNat(vx600, vx300000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_primMulNat(Succ(vx4000), Succ(vx30000)) → new_primMulNat(vx4000, Succ(vx30000))

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

new_primMulNat(Succ(vx4000), Succ(vx30000)) → new_primMulNat(vx4000, Succ(vx30000))
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) → new_enforceWHNF(Float(new_sr(vx40, vx300), new_sr(vx41, vx301)), vx310, Float(new_sr(vx40, vx300), new_sr(vx41, vx301)), vx311)

The TRS R consists of the following rules:

new_primMulNat0(Zero, Zero) → Zero
new_sr(Neg(vx400), Neg(vx3000)) → Pos(new_primMulNat0(vx400, vx3000))
new_primPlusNat0(Succ(vx60), vx30000) → Succ(Succ(new_primPlusNat1(vx60, vx30000)))
new_primPlusNat1(Zero, Succ(vx300000)) → Succ(vx300000)
new_primPlusNat1(Succ(vx600), Zero) → Succ(vx600)
new_sr(Pos(vx400), Pos(vx3000)) → Pos(new_primMulNat0(vx400, vx3000))
new_primMulNat0(Succ(vx4000), Succ(vx30000)) → new_primPlusNat0(new_primMulNat0(vx4000, Succ(vx30000)), vx30000)
new_sr(Neg(vx400), Pos(vx3000)) → Neg(new_primMulNat0(vx400, vx3000))
new_sr(Pos(vx400), Neg(vx3000)) → Neg(new_primMulNat0(vx400, vx3000))
new_primPlusNat1(Succ(vx600), Succ(vx300000)) → Succ(Succ(new_primPlusNat1(vx600, vx300000)))
new_primMulNat0(Zero, Succ(vx30000)) → Zero
new_primMulNat0(Succ(vx4000), Zero) → Zero
new_primPlusNat0(Zero, vx30000) → Succ(vx30000)
new_primPlusNat1(Zero, Zero) → Zero

The set Q consists of the following terms:

new_sr(Pos(x0), Neg(x1))
new_sr(Neg(x0), Pos(x1))
new_sr(Neg(x0), Neg(x1))
new_primMulNat0(Succ(x0), Succ(x1))
new_primPlusNat1(Succ(x0), Succ(x1))
new_primPlusNat1(Zero, Zero)
new_primMulNat0(Zero, Zero)
new_sr(Pos(x0), Pos(x1))
new_primPlusNat1(Succ(x0), Zero)
new_primPlusNat0(Succ(x0), x1)
new_primPlusNat1(Zero, Succ(x0))
new_primPlusNat0(Zero, x0)
new_primMulNat0(Succ(x0), Zero)
new_primMulNat0(Zero, Succ(x0))

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_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) → new_enforceWHNF(Float(new_sr(vx40, vx300), new_sr(vx41, vx301)), vx310, Float(new_sr(vx40, vx300), new_sr(vx41, vx301)), vx311)
The graph contains the following edges 4 > 2, 4 > 4