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

YES 0.761 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 :: [Int] -> Int) :: [Int] -> Int)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

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

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 :: [Int] -> Int) :: [Int] -> Int)

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 :: [Int] -> Int)

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="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];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 -> 14[label="",style="solid", color="black", weight=3];
14[label="enforceWHNF (WHNF ((*) vx4 vx30)) (foldl' (*) ((*) vx5 vx30)) vx31",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="enforceWHNF (WHNF (primMulInt vx4 vx30)) (foldl' primMulInt (primMulInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="triangle"];99[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 16[label="",style="solid", color="burlywood", weight=3];
100[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="enforceWHNF (WHNF (primMulInt (Pos vx40) vx30)) (foldl' primMulInt (primMulInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="box"];101[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 18[label="",style="solid", color="burlywood", weight=3];
102[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 19[label="",style="solid", color="burlywood", weight=3];
17[label="enforceWHNF (WHNF (primMulInt (Neg vx40) vx30)) (foldl' primMulInt (primMulInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="box"];103[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];17 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 20[label="",style="solid", color="burlywood", weight=3];
104[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];17 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 21[label="",style="solid", color="burlywood", weight=3];
18[label="enforceWHNF (WHNF (primMulInt (Pos vx40) (Pos vx300))) (foldl' primMulInt (primMulInt vx5 (Pos vx300))) vx31",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="enforceWHNF (WHNF (primMulInt (Pos vx40) (Neg vx300))) (foldl' primMulInt (primMulInt vx5 (Neg vx300))) vx31",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="enforceWHNF (WHNF (primMulInt (Neg vx40) (Pos vx300))) (foldl' primMulInt (primMulInt vx5 (Pos vx300))) vx31",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="enforceWHNF (WHNF (primMulInt (Neg vx40) (Neg vx300))) (foldl' primMulInt (primMulInt vx5 (Neg vx300))) vx31",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="enforceWHNF (WHNF (Pos (primMulNat vx40 vx300))) (foldl' primMulInt (Pos (primMulNat vx40 vx300))) vx31",fontsize=16,color="black",shape="triangle"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="enforceWHNF (WHNF (Neg (primMulNat vx40 vx300))) (foldl' primMulInt (Neg (primMulNat vx40 vx300))) vx31",fontsize=16,color="black",shape="triangle"];23 -> 27[label="",style="solid", color="black", weight=3];
24 -> 23[label="",style="dashed", color="red", weight=0];
24[label="enforceWHNF (WHNF (Neg (primMulNat vx40 vx300))) (foldl' primMulInt (Neg (primMulNat vx40 vx300))) vx31",fontsize=16,color="magenta"];24 -> 28[label="",style="dashed", color="magenta", weight=3];
24 -> 29[label="",style="dashed", color="magenta", weight=3];
25 -> 22[label="",style="dashed", color="red", weight=0];
25[label="enforceWHNF (WHNF (Pos (primMulNat vx40 vx300))) (foldl' primMulInt (Pos (primMulNat vx40 vx300))) vx31",fontsize=16,color="magenta"];25 -> 30[label="",style="dashed", color="magenta", weight=3];
25 -> 31[label="",style="dashed", color="magenta", weight=3];
26[label="foldl' primMulInt (Pos (primMulNat vx40 vx300)) vx31",fontsize=16,color="burlywood",shape="box"];107[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];26 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 32[label="",style="solid", color="burlywood", weight=3];
108[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 33[label="",style="solid", color="burlywood", weight=3];
27[label="foldl' primMulInt (Neg (primMulNat vx40 vx300)) vx31",fontsize=16,color="burlywood",shape="box"];109[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];27 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 34[label="",style="solid", color="burlywood", weight=3];
110[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];27 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 35[label="",style="solid", color="burlywood", weight=3];
28[label="vx40",fontsize=16,color="green",shape="box"];29[label="vx300",fontsize=16,color="green",shape="box"];30[label="vx40",fontsize=16,color="green",shape="box"];31[label="vx300",fontsize=16,color="green",shape="box"];32[label="foldl' primMulInt (Pos (primMulNat vx40 vx300)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="foldl' primMulInt (Pos (primMulNat vx40 vx300)) []",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="foldl' primMulInt (Neg (primMulNat vx40 vx300)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="foldl' primMulInt (Neg (primMulNat vx40 vx300)) []",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="(foldl' primMulInt $! primMulInt (Pos (primMulNat vx40 vx300)) vx310)",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="Pos (primMulNat vx40 vx300)",fontsize=16,color="green",shape="box"];37 -> 41[label="",style="dashed", color="green", weight=3];
38[label="(foldl' primMulInt $! primMulInt (Neg (primMulNat vx40 vx300)) vx310)",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="Neg (primMulNat vx40 vx300)",fontsize=16,color="green",shape="box"];39 -> 43[label="",style="dashed", color="green", weight=3];
40[label="(primMulInt (Pos (primMulNat vx40 vx300)) vx310 `seq` foldl' primMulInt (primMulInt (Pos (primMulNat vx40 vx300)) vx310))",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="primMulNat vx40 vx300",fontsize=16,color="burlywood",shape="triangle"];111[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];41 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 45[label="",style="solid", color="burlywood", weight=3];
112[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 46[label="",style="solid", color="burlywood", weight=3];
42 -> 47[label="",style="dashed", color="red", weight=0];
42[label="(primMulInt (Neg (primMulNat vx40 vx300)) vx310 `seq` foldl' primMulInt (primMulInt (Neg (primMulNat vx40 vx300)) vx310))",fontsize=16,color="magenta"];42 -> 48[label="",style="dashed", color="magenta", weight=3];
42 -> 49[label="",style="dashed", color="magenta", weight=3];
43 -> 41[label="",style="dashed", color="red", weight=0];
43[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];43 -> 50[label="",style="dashed", color="magenta", weight=3];
44 -> 15[label="",style="dashed", color="red", weight=0];
44[label="enforceWHNF (WHNF (primMulInt (Pos (primMulNat vx40 vx300)) vx310)) (foldl' primMulInt (primMulInt (Pos (primMulNat vx40 vx300)) vx310)) vx311",fontsize=16,color="magenta"];44 -> 51[label="",style="dashed", color="magenta", weight=3];
44 -> 52[label="",style="dashed", color="magenta", weight=3];
44 -> 53[label="",style="dashed", color="magenta", weight=3];
44 -> 54[label="",style="dashed", color="magenta", weight=3];
45[label="primMulNat (Succ vx400) vx300",fontsize=16,color="burlywood",shape="box"];116[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];45 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 55[label="",style="solid", color="burlywood", weight=3];
117[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 56[label="",style="solid", color="burlywood", weight=3];
46[label="primMulNat Zero vx300",fontsize=16,color="burlywood",shape="box"];118[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];46 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 57[label="",style="solid", color="burlywood", weight=3];
119[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 58[label="",style="solid", color="burlywood", weight=3];
48 -> 41[label="",style="dashed", color="red", weight=0];
48[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];48 -> 59[label="",style="dashed", color="magenta", weight=3];
49 -> 41[label="",style="dashed", color="red", weight=0];
49[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];49 -> 60[label="",style="dashed", color="magenta", weight=3];
47[label="(primMulInt (Neg vx6) vx310 `seq` foldl' primMulInt (primMulInt (Neg vx7) vx310))",fontsize=16,color="black",shape="triangle"];47 -> 61[label="",style="solid", color="black", weight=3];
50[label="vx300",fontsize=16,color="green",shape="box"];51[label="vx310",fontsize=16,color="green",shape="box"];52[label="vx311",fontsize=16,color="green",shape="box"];53[label="Pos (primMulNat vx40 vx300)",fontsize=16,color="green",shape="box"];53 -> 62[label="",style="dashed", color="green", weight=3];
54[label="Pos (primMulNat vx40 vx300)",fontsize=16,color="green",shape="box"];54 -> 63[label="",style="dashed", color="green", weight=3];
55[label="primMulNat (Succ vx400) (Succ vx3000)",fontsize=16,color="black",shape="box"];55 -> 64[label="",style="solid", color="black", weight=3];
56[label="primMulNat (Succ vx400) Zero",fontsize=16,color="black",shape="box"];56 -> 65[label="",style="solid", color="black", weight=3];
57[label="primMulNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];57 -> 66[label="",style="solid", color="black", weight=3];
58[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];58 -> 67[label="",style="solid", color="black", weight=3];
59[label="vx300",fontsize=16,color="green",shape="box"];60[label="vx300",fontsize=16,color="green",shape="box"];61 -> 15[label="",style="dashed", color="red", weight=0];
61[label="enforceWHNF (WHNF (primMulInt (Neg vx6) vx310)) (foldl' primMulInt (primMulInt (Neg vx7) vx310)) vx311",fontsize=16,color="magenta"];61 -> 68[label="",style="dashed", color="magenta", weight=3];
61 -> 69[label="",style="dashed", color="magenta", weight=3];
61 -> 70[label="",style="dashed", color="magenta", weight=3];
61 -> 71[label="",style="dashed", color="magenta", weight=3];
62 -> 41[label="",style="dashed", color="red", weight=0];
62[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];63 -> 41[label="",style="dashed", color="red", weight=0];
63[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];64 -> 72[label="",style="dashed", color="red", weight=0];
64[label="primPlusNat (primMulNat vx400 (Succ vx3000)) (Succ vx3000)",fontsize=16,color="magenta"];64 -> 73[label="",style="dashed", color="magenta", weight=3];
65[label="Zero",fontsize=16,color="green",shape="box"];66[label="Zero",fontsize=16,color="green",shape="box"];67[label="Zero",fontsize=16,color="green",shape="box"];68[label="vx310",fontsize=16,color="green",shape="box"];69[label="vx311",fontsize=16,color="green",shape="box"];70[label="Neg vx7",fontsize=16,color="green",shape="box"];71[label="Neg vx6",fontsize=16,color="green",shape="box"];73 -> 41[label="",style="dashed", color="red", weight=0];
73[label="primMulNat vx400 (Succ vx3000)",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 vx8 (Succ vx3000)",fontsize=16,color="burlywood",shape="triangle"];127[label="vx8/Succ vx80",fontsize=10,color="white",style="solid",shape="box"];72 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 76[label="",style="solid", color="burlywood", weight=3];
128[label="vx8/Zero",fontsize=10,color="white",style="solid",shape="box"];72 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 77[label="",style="solid", color="burlywood", weight=3];
74[label="vx400",fontsize=16,color="green",shape="box"];75[label="Succ vx3000",fontsize=16,color="green",shape="box"];76[label="primPlusNat (Succ vx80) (Succ vx3000)",fontsize=16,color="black",shape="box"];76 -> 78[label="",style="solid", color="black", weight=3];
77[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];77 -> 79[label="",style="solid", color="black", weight=3];
78[label="Succ (Succ (primPlusNat vx80 vx3000))",fontsize=16,color="green",shape="box"];78 -> 80[label="",style="dashed", color="green", weight=3];
79[label="Succ vx3000",fontsize=16,color="green",shape="box"];80[label="primPlusNat vx80 vx3000",fontsize=16,color="burlywood",shape="triangle"];129[label="vx80/Succ vx800",fontsize=10,color="white",style="solid",shape="box"];80 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 81[label="",style="solid", color="burlywood", weight=3];
130[label="vx80/Zero",fontsize=10,color="white",style="solid",shape="box"];80 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 82[label="",style="solid", color="burlywood", weight=3];
81[label="primPlusNat (Succ vx800) vx3000",fontsize=16,color="burlywood",shape="box"];131[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];81 -> 131[label="",style="solid", color="burlywood", weight=9];
131 -> 83[label="",style="solid", color="burlywood", weight=3];
132[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];81 -> 132[label="",style="solid", color="burlywood", weight=9];
132 -> 84[label="",style="solid", color="burlywood", weight=3];
82[label="primPlusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];133[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];82 -> 133[label="",style="solid", color="burlywood", weight=9];
133 -> 85[label="",style="solid", color="burlywood", weight=3];
134[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];82 -> 134[label="",style="solid", color="burlywood", weight=9];
134 -> 86[label="",style="solid", color="burlywood", weight=3];
83[label="primPlusNat (Succ vx800) (Succ vx30000)",fontsize=16,color="black",shape="box"];83 -> 87[label="",style="solid", color="black", weight=3];
84[label="primPlusNat (Succ vx800) Zero",fontsize=16,color="black",shape="box"];84 -> 88[label="",style="solid", color="black", weight=3];
85[label="primPlusNat Zero (Succ vx30000)",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 vx800 vx30000))",fontsize=16,color="green",shape="box"];87 -> 91[label="",style="dashed", color="green", weight=3];
88[label="Succ vx800",fontsize=16,color="green",shape="box"];89[label="Succ vx30000",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 vx800 vx30000",fontsize=16,color="magenta"];91 -> 92[label="",style="dashed", color="magenta", weight=3];
91 -> 93[label="",style="dashed", color="magenta", weight=3];
92[label="vx800",fontsize=16,color="green",shape="box"];93[label="vx30000",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(vx800), Succ(vx30000)) → new_primPlusNat(vx800, vx30000)

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(vx800), Succ(vx30000)) → new_primPlusNat(vx800, vx30000)
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(vx400), Succ(vx3000)) → new_primMulNat(vx400, Succ(vx3000))

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

new_primMulNat(Succ(vx400), Succ(vx3000)) → new_primMulNat(vx400, Succ(vx3000))
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_enforceWHNF0(Neg(vx40), Pos(vx300), vx5, vx31) → new_enforceWHNF1(vx40, vx300, vx31)
new_enforceWHNF(vx40, vx300, :(vx310, vx311)) → new_enforceWHNF0(Pos(new_primMulNat0(vx40, vx300)), vx310, Pos(new_primMulNat0(vx40, vx300)), vx311)
new_enforceWHNF0(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) → new_enforceWHNF0(Pos(new_primMulNat0(vx40, vx300)), vx310, Pos(new_primMulNat0(vx40, vx300)), vx311)
new_enforceWHNF0(Pos(vx40), Neg(vx300), vx5, :(vx310, vx311)) → new_seq(new_primMulNat0(vx40, vx300), vx310, new_primMulNat0(vx40, vx300), vx311)
new_enforceWHNF1(vx40, vx300, :(vx310, vx311)) → new_seq(new_primMulNat0(vx40, vx300), vx310, new_primMulNat0(vx40, vx300), vx311)
new_enforceWHNF0(Neg(vx40), Neg(vx300), vx5, vx31) → new_enforceWHNF(vx40, vx300, vx31)
new_seq(vx6, vx310, vx7, vx311) → new_enforceWHNF0(Neg(vx6), vx310, Neg(vx7), vx311)

The TRS R consists of the following rules:

new_primMulNat0(Zero, Zero) → Zero
new_primMulNat0(Succ(vx400), Zero) → Zero
new_primMulNat0(Zero, Succ(vx3000)) → Zero
new_primPlusNat0(Zero, vx3000) → Succ(vx3000)
new_primPlusNat0(Succ(vx80), vx3000) → Succ(Succ(new_primPlusNat1(vx80, vx3000)))
new_primPlusNat1(Succ(vx800), Zero) → Succ(vx800)
new_primPlusNat1(Zero, Succ(vx30000)) → Succ(vx30000)
new_primMulNat0(Succ(vx400), Succ(vx3000)) → new_primPlusNat0(new_primMulNat0(vx400, Succ(vx3000)), vx3000)
new_primPlusNat1(Succ(vx800), Succ(vx30000)) → Succ(Succ(new_primPlusNat1(vx800, vx30000)))
new_primPlusNat1(Zero, Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat1(Succ(x0), Succ(x1))
new_primPlusNat1(Succ(x0), Zero)
new_primMulNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), x1)
new_primMulNat0(Succ(x0), Zero)
new_primPlusNat1(Zero, Succ(x0))
new_primPlusNat1(Zero, Zero)
new_primPlusNat0(Zero, x0)
new_primMulNat0(Succ(x0), Succ(x1))
new_primMulNat0(Zero, Zero)

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_enforceWHNF1(vx40, vx300, :(vx310, vx311)) → new_seq(new_primMulNat0(vx40, vx300), vx310, new_primMulNat0(vx40, vx300), vx311)
The graph contains the following edges 3 > 2, 3 > 4
new_seq(vx6, vx310, vx7, vx311) → new_enforceWHNF0(Neg(vx6), vx310, Neg(vx7), vx311)
The graph contains the following edges 2 >= 2, 4 >= 4
new_enforceWHNF0(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) → new_enforceWHNF0(Pos(new_primMulNat0(vx40, vx300)), vx310, Pos(new_primMulNat0(vx40, vx300)), vx311)
The graph contains the following edges 4 > 2, 4 > 4
new_enforceWHNF0(Pos(vx40), Neg(vx300), vx5, :(vx310, vx311)) → new_seq(new_primMulNat0(vx40, vx300), vx310, new_primMulNat0(vx40, vx300), vx311)
The graph contains the following edges 4 > 2, 4 > 4
new_enforceWHNF(vx40, vx300, :(vx310, vx311)) → new_enforceWHNF0(Pos(new_primMulNat0(vx40, vx300)), vx310, Pos(new_primMulNat0(vx40, vx300)), vx311)
The graph contains the following edges 3 > 2, 3 > 4
new_enforceWHNF0(Neg(vx40), Neg(vx300), vx5, vx31) → new_enforceWHNF(vx40, vx300, vx31)
The graph contains the following edges 1 > 1, 2 > 2, 4 >= 3
new_enforceWHNF0(Neg(vx40), Pos(vx300), vx5, vx31) → new_enforceWHNF1(vx40, vx300, vx31)
The graph contains the following edges 1 > 1, 2 > 2, 4 >= 3