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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_sum_2.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

((sum :: [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

((sum :: [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

(sum :: [Int] -> Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sum\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sum vx3\n",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl' (+) (fromInt (Pos Zero)) vx3\n",fontsize=16,color="burlywood",shape="box"];118[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 5[label="",style="solid", color="burlywood", weight=3];
119[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 6[label="",style="solid", color="burlywood", weight=3];
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6[label="foldl' (+) (fromInt (Pos Zero)) []\n",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="(foldl' (+) $! (+) fromInt (Pos Zero) vx30)\n",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="fromInt (Pos Zero)\n",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 Zero) vx30 `seq` foldl' (+) ((+) fromInt (Pos Zero) vx30))\n",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 Zero\n",fontsize=16,color="green",shape="box"];12 -> 8[label="",style="dashed", color="red", weight=0];
12[label="fromInt (Pos Zero)\n",fontsize=16,color="magenta"];13 -> 8[label="",style="dashed", color="red", weight=0];
13[label="fromInt (Pos Zero)\n",fontsize=16,color="magenta"];11[label="((+) vx4 vx30 `seq` foldl' (+) ((+) vx5 vx30))\n",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\n",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="enforceWHNF (WHNF (primPlusInt vx4 vx30)) (foldl' primPlusInt (primPlusInt vx5 vx30)) vx31\n",fontsize=16,color="burlywood",shape="triangle"];123[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 16[label="",style="solid", color="burlywood", weight=3];
124[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="enforceWHNF (WHNF (primPlusInt (Pos vx40) vx30)) (foldl' primPlusInt (primPlusInt vx5 vx30)) vx31\n",fontsize=16,color="burlywood",shape="box"];125[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 18[label="",style="solid", color="burlywood", weight=3];
126[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 19[label="",style="solid", color="burlywood", weight=3];
17[label="enforceWHNF (WHNF (primPlusInt (Neg vx40) vx30)) (foldl' primPlusInt (primPlusInt vx5 vx30)) vx31\n",fontsize=16,color="burlywood",shape="box"];127[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];17 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 20[label="",style="solid", color="burlywood", weight=3];
128[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];17 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 21[label="",style="solid", color="burlywood", weight=3];
18[label="enforceWHNF (WHNF (primPlusInt (Pos vx40) (Pos vx300))) (foldl' primPlusInt (primPlusInt vx5 (Pos vx300))) vx31\n",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="enforceWHNF (WHNF (primPlusInt (Pos vx40) (Neg vx300))) (foldl' primPlusInt (primPlusInt vx5 (Neg vx300))) vx31\n",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="enforceWHNF (WHNF (primPlusInt (Neg vx40) (Pos vx300))) (foldl' primPlusInt (primPlusInt vx5 (Pos vx300))) vx31\n",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="enforceWHNF (WHNF (primPlusInt (Neg vx40) (Neg vx300))) (foldl' primPlusInt (primPlusInt vx5 (Neg vx300))) vx31\n",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="enforceWHNF (WHNF (Pos (primPlusNat vx40 vx300))) (foldl' primPlusInt (Pos (primPlusNat vx40 vx300))) vx31\n",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="enforceWHNF (WHNF (primMinusNat vx40 vx300)) (foldl' primPlusInt (primMinusNat vx40 vx300)) vx31\n",fontsize=16,color="burlywood",shape="triangle"];129[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];23 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 27[label="",style="solid", color="burlywood", weight=3];
130[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 28[label="",style="solid", color="burlywood", weight=3];
24 -> 23[label="",style="dashed", color="red", weight=0];
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24 -> 30[label="",style="dashed", color="magenta", weight=3];
25[label="enforceWHNF (WHNF (Neg (primPlusNat vx40 vx300))) (foldl' primPlusInt (Neg (primPlusNat vx40 vx300))) vx31\n",fontsize=16,color="black",shape="box"];25 -> 31[label="",style="solid", color="black", weight=3];
26[label="foldl' primPlusInt (Pos (primPlusNat vx40 vx300)) vx31\n",fontsize=16,color="burlywood",shape="box"];132[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];26 -> 132[label="",style="solid", color="burlywood", weight=9];
132 -> 32[label="",style="solid", color="burlywood", weight=3];
133[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 133[label="",style="solid", color="burlywood", weight=9];
133 -> 33[label="",style="solid", color="burlywood", weight=3];
27[label="enforceWHNF (WHNF (primMinusNat (Succ vx400) vx300)) (foldl' primPlusInt (primMinusNat (Succ vx400) vx300)) vx31\n",fontsize=16,color="burlywood",shape="box"];134[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];27 -> 134[label="",style="solid", color="burlywood", weight=9];
134 -> 34[label="",style="solid", color="burlywood", weight=3];
135[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 135[label="",style="solid", color="burlywood", weight=9];
135 -> 35[label="",style="solid", color="burlywood", weight=3];
28[label="enforceWHNF (WHNF (primMinusNat Zero vx300)) (foldl' primPlusInt (primMinusNat Zero vx300)) vx31\n",fontsize=16,color="burlywood",shape="box"];136[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];28 -> 136[label="",style="solid", color="burlywood", weight=9];
136 -> 36[label="",style="solid", color="burlywood", weight=3];
137[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 137[label="",style="solid", color="burlywood", weight=9];
137 -> 37[label="",style="solid", color="burlywood", weight=3];
29[label="vx300\n",fontsize=16,color="green",shape="box"];30[label="vx40\n",fontsize=16,color="green",shape="box"];31[label="foldl' primPlusInt (Neg (primPlusNat vx40 vx300)) vx31\n",fontsize=16,color="burlywood",shape="box"];138[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];31 -> 138[label="",style="solid", color="burlywood", weight=9];
138 -> 38[label="",style="solid", color="burlywood", weight=3];
139[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];31 -> 139[label="",style="solid", color="burlywood", weight=9];
139 -> 39[label="",style="solid", color="burlywood", weight=3];
32[label="foldl' primPlusInt (Pos (primPlusNat vx40 vx300)) (vx310 : vx311)\n",fontsize=16,color="black",shape="box"];32 -> 40[label="",style="solid", color="black", weight=3];
33[label="foldl' primPlusInt (Pos (primPlusNat vx40 vx300)) []\n",fontsize=16,color="black",shape="box"];33 -> 41[label="",style="solid", color="black", weight=3];
34[label="enforceWHNF (WHNF (primMinusNat (Succ vx400) (Succ vx3000))) (foldl' primPlusInt (primMinusNat (Succ vx400) (Succ vx3000))) vx31\n",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
35[label="enforceWHNF (WHNF (primMinusNat (Succ vx400) Zero)) (foldl' primPlusInt (primMinusNat (Succ vx400) Zero)) vx31\n",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
36[label="enforceWHNF (WHNF (primMinusNat Zero (Succ vx3000))) (foldl' primPlusInt (primMinusNat Zero (Succ vx3000))) vx31\n",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
37[label="enforceWHNF (WHNF (primMinusNat Zero Zero)) (foldl' primPlusInt (primMinusNat Zero Zero)) vx31\n",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3];
38[label="foldl' primPlusInt (Neg (primPlusNat vx40 vx300)) (vx310 : vx311)\n",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3];
39[label="foldl' primPlusInt (Neg (primPlusNat vx40 vx300)) []\n",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3];
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41[label="Pos (primPlusNat vx40 vx300)\n",fontsize=16,color="green",shape="box"];41 -> 49[label="",style="dashed", color="green", weight=3];
42 -> 23[label="",style="dashed", color="red", weight=0];
42[label="enforceWHNF (WHNF (primMinusNat vx400 vx3000)) (foldl' primPlusInt (primMinusNat vx400 vx3000)) vx31\n",fontsize=16,color="magenta"];42 -> 50[label="",style="dashed", color="magenta", weight=3];
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43[label="enforceWHNF (WHNF (Pos (Succ vx400))) (foldl' primPlusInt (Pos (Succ vx400))) vx31\n",fontsize=16,color="black",shape="box"];43 -> 52[label="",style="solid", color="black", weight=3];
44[label="enforceWHNF (WHNF (Neg (Succ vx3000))) (foldl' primPlusInt (Neg (Succ vx3000))) vx31\n",fontsize=16,color="black",shape="box"];44 -> 53[label="",style="solid", color="black", weight=3];
45[label="enforceWHNF (WHNF (Pos Zero)) (foldl' primPlusInt (Pos Zero)) vx31\n",fontsize=16,color="black",shape="box"];45 -> 54[label="",style="solid", color="black", weight=3];
46[label="(foldl' primPlusInt $! primPlusInt (Neg (primPlusNat vx40 vx300)) vx310)\n",fontsize=16,color="black",shape="box"];46 -> 55[label="",style="solid", color="black", weight=3];
47[label="Neg (primPlusNat vx40 vx300)\n",fontsize=16,color="green",shape="box"];47 -> 56[label="",style="dashed", color="green", weight=3];
48[label="(primPlusInt (Pos (primPlusNat vx40 vx300)) vx310 `seq` foldl' primPlusInt (primPlusInt (Pos (primPlusNat vx40 vx300)) vx310))\n",fontsize=16,color="black",shape="box"];48 -> 57[label="",style="solid", color="black", weight=3];
49[label="primPlusNat vx40 vx300\n",fontsize=16,color="burlywood",shape="triangle"];141[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];49 -> 141[label="",style="solid", color="burlywood", weight=9];
141 -> 58[label="",style="solid", color="burlywood", weight=3];
142[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 142[label="",style="solid", color="burlywood", weight=9];
142 -> 59[label="",style="solid", color="burlywood", weight=3];
50[label="vx400\n",fontsize=16,color="green",shape="box"];51[label="vx3000\n",fontsize=16,color="green",shape="box"];52[label="foldl' primPlusInt (Pos (Succ vx400)) vx31\n",fontsize=16,color="burlywood",shape="box"];143[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];52 -> 143[label="",style="solid", color="burlywood", weight=9];
143 -> 60[label="",style="solid", color="burlywood", weight=3];
144[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];52 -> 144[label="",style="solid", color="burlywood", weight=9];
144 -> 61[label="",style="solid", color="burlywood", weight=3];
53[label="foldl' primPlusInt (Neg (Succ vx3000)) vx31\n",fontsize=16,color="burlywood",shape="box"];145[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];53 -> 145[label="",style="solid", color="burlywood", weight=9];
145 -> 62[label="",style="solid", color="burlywood", weight=3];
146[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];53 -> 146[label="",style="solid", color="burlywood", weight=9];
146 -> 63[label="",style="solid", color="burlywood", weight=3];
54[label="foldl' primPlusInt (Pos Zero) vx31\n",fontsize=16,color="burlywood",shape="box"];147[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];54 -> 147[label="",style="solid", color="burlywood", weight=9];
147 -> 64[label="",style="solid", color="burlywood", weight=3];
148[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];54 -> 148[label="",style="solid", color="burlywood", weight=9];
148 -> 65[label="",style="solid", color="burlywood", weight=3];
55 -> 66[label="",style="dashed", color="red", weight=0];
55[label="(primPlusInt (Neg (primPlusNat vx40 vx300)) vx310 `seq` foldl' primPlusInt (primPlusInt (Neg (primPlusNat vx40 vx300)) vx310))\n",fontsize=16,color="magenta"];55 -> 67[label="",style="dashed", color="magenta", weight=3];
55 -> 68[label="",style="dashed", color="magenta", weight=3];
56 -> 49[label="",style="dashed", color="red", weight=0];
56[label="primPlusNat vx40 vx300\n",fontsize=16,color="magenta"];56 -> 69[label="",style="dashed", color="magenta", weight=3];
56 -> 70[label="",style="dashed", color="magenta", weight=3];
57 -> 15[label="",style="dashed", color="red", weight=0];
57[label="enforceWHNF (WHNF (primPlusInt (Pos (primPlusNat vx40 vx300)) vx310)) (foldl' primPlusInt (primPlusInt (Pos (primPlusNat vx40 vx300)) vx310)) vx311\n",fontsize=16,color="magenta"];57 -> 71[label="",style="dashed", color="magenta", weight=3];
57 -> 72[label="",style="dashed", color="magenta", weight=3];
57 -> 73[label="",style="dashed", color="magenta", weight=3];
57 -> 74[label="",style="dashed", color="magenta", weight=3];
58[label="primPlusNat (Succ vx400) vx300\n",fontsize=16,color="burlywood",shape="box"];152[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];58 -> 152[label="",style="solid", color="burlywood", weight=9];
152 -> 75[label="",style="solid", color="burlywood", weight=3];
153[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 153[label="",style="solid", color="burlywood", weight=9];
153 -> 76[label="",style="solid", color="burlywood", weight=3];
59[label="primPlusNat Zero vx300\n",fontsize=16,color="burlywood",shape="box"];154[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];59 -> 154[label="",style="solid", color="burlywood", weight=9];
154 -> 77[label="",style="solid", color="burlywood", weight=3];
155[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 155[label="",style="solid", color="burlywood", weight=9];
155 -> 78[label="",style="solid", color="burlywood", weight=3];
60[label="foldl' primPlusInt (Pos (Succ vx400)) (vx310 : vx311)\n",fontsize=16,color="black",shape="box"];60 -> 79[label="",style="solid", color="black", weight=3];
61[label="foldl' primPlusInt (Pos (Succ vx400)) []\n",fontsize=16,color="black",shape="box"];61 -> 80[label="",style="solid", color="black", weight=3];
62[label="foldl' primPlusInt (Neg (Succ vx3000)) (vx310 : vx311)\n",fontsize=16,color="black",shape="box"];62 -> 81[label="",style="solid", color="black", weight=3];
63[label="foldl' primPlusInt (Neg (Succ vx3000)) []\n",fontsize=16,color="black",shape="box"];63 -> 82[label="",style="solid", color="black", weight=3];
64[label="foldl' primPlusInt (Pos Zero) (vx310 : vx311)\n",fontsize=16,color="black",shape="box"];64 -> 83[label="",style="solid", color="black", weight=3];
65[label="foldl' primPlusInt (Pos Zero) []\n",fontsize=16,color="black",shape="box"];65 -> 84[label="",style="solid", color="black", weight=3];
67 -> 49[label="",style="dashed", color="red", weight=0];
67[label="primPlusNat vx40 vx300\n",fontsize=16,color="magenta"];67 -> 85[label="",style="dashed", color="magenta", weight=3];
67 -> 86[label="",style="dashed", color="magenta", weight=3];
68 -> 49[label="",style="dashed", color="red", weight=0];
68[label="primPlusNat vx40 vx300\n",fontsize=16,color="magenta"];68 -> 87[label="",style="dashed", color="magenta", weight=3];
68 -> 88[label="",style="dashed", color="magenta", weight=3];
66[label="(primPlusInt (Neg vx6) vx310 `seq` foldl' primPlusInt (primPlusInt (Neg vx7) vx310))\n",fontsize=16,color="black",shape="triangle"];66 -> 89[label="",style="solid", color="black", weight=3];
69[label="vx40\n",fontsize=16,color="green",shape="box"];70[label="vx300\n",fontsize=16,color="green",shape="box"];71[label="Pos (primPlusNat vx40 vx300)\n",fontsize=16,color="green",shape="box"];71 -> 90[label="",style="dashed", color="green", weight=3];
72[label="Pos (primPlusNat vx40 vx300)\n",fontsize=16,color="green",shape="box"];72 -> 91[label="",style="dashed", color="green", weight=3];
73[label="vx310\n",fontsize=16,color="green",shape="box"];74[label="vx311\n",fontsize=16,color="green",shape="box"];75[label="primPlusNat (Succ vx400) (Succ vx3000)\n",fontsize=16,color="black",shape="box"];75 -> 92[label="",style="solid", color="black", weight=3];
76[label="primPlusNat (Succ vx400) Zero\n",fontsize=16,color="black",shape="box"];76 -> 93[label="",style="solid", color="black", weight=3];
77[label="primPlusNat Zero (Succ vx3000)\n",fontsize=16,color="black",shape="box"];77 -> 94[label="",style="solid", color="black", weight=3];
78[label="primPlusNat Zero Zero\n",fontsize=16,color="black",shape="box"];78 -> 95[label="",style="solid", color="black", weight=3];
79[label="(foldl' primPlusInt $! primPlusInt (Pos (Succ vx400)) vx310)\n",fontsize=16,color="black",shape="box"];79 -> 96[label="",style="solid", color="black", weight=3];
80[label="Pos (Succ vx400)\n",fontsize=16,color="green",shape="box"];81[label="(foldl' primPlusInt $! primPlusInt (Neg (Succ vx3000)) vx310)\n",fontsize=16,color="black",shape="box"];81 -> 97[label="",style="solid", color="black", weight=3];
82[label="Neg (Succ vx3000)\n",fontsize=16,color="green",shape="box"];83[label="(foldl' primPlusInt $! primPlusInt (Pos Zero) vx310)\n",fontsize=16,color="black",shape="box"];83 -> 98[label="",style="solid", color="black", weight=3];
84[label="Pos Zero\n",fontsize=16,color="green",shape="box"];85[label="vx40\n",fontsize=16,color="green",shape="box"];86[label="vx300\n",fontsize=16,color="green",shape="box"];87[label="vx40\n",fontsize=16,color="green",shape="box"];88[label="vx300\n",fontsize=16,color="green",shape="box"];89 -> 15[label="",style="dashed", color="red", weight=0];
89[label="enforceWHNF (WHNF (primPlusInt (Neg vx6) vx310)) (foldl' primPlusInt (primPlusInt (Neg vx7) vx310)) vx311\n",fontsize=16,color="magenta"];89 -> 99[label="",style="dashed", color="magenta", weight=3];
89 -> 100[label="",style="dashed", color="magenta", weight=3];
89 -> 101[label="",style="dashed", color="magenta", weight=3];
89 -> 102[label="",style="dashed", color="magenta", weight=3];
90 -> 49[label="",style="dashed", color="red", weight=0];
90[label="primPlusNat vx40 vx300\n",fontsize=16,color="magenta"];91 -> 49[label="",style="dashed", color="red", weight=0];
91[label="primPlusNat vx40 vx300\n",fontsize=16,color="magenta"];92[label="Succ (Succ (primPlusNat vx400 vx3000))\n",fontsize=16,color="green",shape="box"];92 -> 103[label="",style="dashed", color="green", weight=3];
93[label="Succ vx400\n",fontsize=16,color="green",shape="box"];94[label="Succ vx3000\n",fontsize=16,color="green",shape="box"];95[label="Zero\n",fontsize=16,color="green",shape="box"];96[label="(primPlusInt (Pos (Succ vx400)) vx310 `seq` foldl' primPlusInt (primPlusInt (Pos (Succ vx400)) vx310))\n",fontsize=16,color="black",shape="box"];96 -> 104[label="",style="solid", color="black", weight=3];
97 -> 66[label="",style="dashed", color="red", weight=0];
97[label="(primPlusInt (Neg (Succ vx3000)) vx310 `seq` foldl' primPlusInt (primPlusInt (Neg (Succ vx3000)) vx310))\n",fontsize=16,color="magenta"];97 -> 105[label="",style="dashed", color="magenta", weight=3];
97 -> 106[label="",style="dashed", color="magenta", weight=3];
98[label="(primPlusInt (Pos Zero) vx310 `seq` foldl' primPlusInt (primPlusInt (Pos Zero) vx310))\n",fontsize=16,color="black",shape="box"];98 -> 107[label="",style="solid", color="black", weight=3];
99[label="Neg vx7\n",fontsize=16,color="green",shape="box"];100[label="Neg vx6\n",fontsize=16,color="green",shape="box"];101[label="vx310\n",fontsize=16,color="green",shape="box"];102[label="vx311\n",fontsize=16,color="green",shape="box"];103 -> 49[label="",style="dashed", color="red", weight=0];
103[label="primPlusNat vx400 vx3000\n",fontsize=16,color="magenta"];103 -> 108[label="",style="dashed", color="magenta", weight=3];
103 -> 109[label="",style="dashed", color="magenta", weight=3];
104 -> 15[label="",style="dashed", color="red", weight=0];
104[label="enforceWHNF (WHNF (primPlusInt (Pos (Succ vx400)) vx310)) (foldl' primPlusInt (primPlusInt (Pos (Succ vx400)) vx310)) vx311\n",fontsize=16,color="magenta"];104 -> 110[label="",style="dashed", color="magenta", weight=3];
104 -> 111[label="",style="dashed", color="magenta", weight=3];
104 -> 112[label="",style="dashed", color="magenta", weight=3];
104 -> 113[label="",style="dashed", color="magenta", weight=3];
105[label="Succ vx3000\n",fontsize=16,color="green",shape="box"];106[label="Succ vx3000\n",fontsize=16,color="green",shape="box"];107 -> 15[label="",style="dashed", color="red", weight=0];
107[label="enforceWHNF (WHNF (primPlusInt (Pos Zero) vx310)) (foldl' primPlusInt (primPlusInt (Pos Zero) vx310)) vx311\n",fontsize=16,color="magenta"];107 -> 114[label="",style="dashed", color="magenta", weight=3];
107 -> 115[label="",style="dashed", color="magenta", weight=3];
107 -> 116[label="",style="dashed", color="magenta", weight=3];
107 -> 117[label="",style="dashed", color="magenta", weight=3];
108[label="vx400\n",fontsize=16,color="green",shape="box"];109[label="vx3000\n",fontsize=16,color="green",shape="box"];110[label="Pos (Succ vx400)\n",fontsize=16,color="green",shape="box"];111[label="Pos (Succ vx400)\n",fontsize=16,color="green",shape="box"];112[label="vx310\n",fontsize=16,color="green",shape="box"];113[label="vx311\n",fontsize=16,color="green",shape="box"];114[label="Pos Zero\n",fontsize=16,color="green",shape="box"];115[label="Pos Zero\n",fontsize=16,color="green",shape="box"];116[label="vx310\n",fontsize=16,color="green",shape="box"];117[label="vx311\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(vx400), Succ(vx3000)) → new_primPlusNat(vx400, vx3000)

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(vx400), Succ(vx3000)) → new_primPlusNat(vx400, vx3000)
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_enforceWHNF(Pos(Zero), Neg(Zero), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
new_enforceWHNF(Pos(Succ(vx400)), Neg(Zero), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
new_enforceWHNF0(Zero, Zero, :(vx310, vx311)) → new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
new_enforceWHNF(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(new_primPlusNat0(vx40, vx300)), vx310, Pos(new_primPlusNat0(vx40, vx300)), vx311)
new_enforceWHNF0(Succ(vx400), Succ(vx3000), vx31) → new_enforceWHNF0(vx400, vx3000, vx31)
new_seq(vx6, vx310, vx7, vx311) → new_enforceWHNF(Neg(vx6), vx310, Neg(vx7), vx311)
new_enforceWHNF(Pos(Succ(vx400)), Neg(Succ(vx3000)), vx5, vx31) → new_enforceWHNF0(vx400, vx3000, vx31)
new_enforceWHNF(Neg(vx40), Neg(vx300), vx5, :(vx310, vx311)) → new_seq(new_primPlusNat0(vx40, vx300), vx310, new_primPlusNat0(vx40, vx300), vx311)
new_enforceWHNF(Pos(Zero), Neg(Succ(vx3000)), vx5, :(vx310, vx311)) → new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
new_enforceWHNF(Neg(vx40), Pos(vx300), vx5, vx31) → new_enforceWHNF0(vx300, vx40, vx31)
new_enforceWHNF0(Succ(vx400), Zero, :(vx310, vx311)) → new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
new_enforceWHNF0(Zero, Succ(vx3000), :(vx310, vx311)) → new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)

The TRS R consists of the following rules:

new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(vx400), Zero) → Succ(vx400)
new_primPlusNat0(Zero, Succ(vx3000)) → Succ(vx3000)
new_primPlusNat0(Succ(vx400), Succ(vx3000)) → Succ(Succ(new_primPlusNat0(vx400, vx3000)))

The set Q consists of the following terms:

new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Succ(x1))
new_primPlusNat0(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(Pos(Zero), Neg(Succ(vx3000)), vx5, :(vx310, vx311)) → new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
The graph contains the following edges 2 > 1, 4 > 2, 2 > 3, 4 > 4
new_enforceWHNF0(Zero, Zero, :(vx310, vx311)) → new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
The graph contains the following edges 3 > 2, 3 > 4
new_enforceWHNF(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(new_primPlusNat0(vx40, vx300)), vx310, Pos(new_primPlusNat0(vx40, vx300)), vx311)
The graph contains the following edges 4 > 2, 4 > 4
new_enforceWHNF(Pos(Zero), Neg(Zero), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
The graph contains the following edges 1 >= 1, 4 > 2, 1 >= 3, 4 > 4
new_enforceWHNF(Pos(Succ(vx400)), Neg(Zero), vx5, :(vx310, vx311)) → new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
The graph contains the following edges 1 >= 1, 4 > 2, 1 >= 3, 4 > 4
new_enforceWHNF(Pos(Succ(vx400)), Neg(Succ(vx3000)), vx5, vx31) → new_enforceWHNF0(vx400, vx3000, vx31)
The graph contains the following edges 1 > 1, 2 > 2, 4 >= 3
new_enforceWHNF0(Succ(vx400), Zero, :(vx310, vx311)) → new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
The graph contains the following edges 3 > 2, 3 > 4
new_enforceWHNF0(Succ(vx400), Succ(vx3000), vx31) → new_enforceWHNF0(vx400, vx3000, vx31)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
new_enforceWHNF(Neg(vx40), Pos(vx300), vx5, vx31) → new_enforceWHNF0(vx300, vx40, vx31)
The graph contains the following edges 2 > 1, 1 > 2, 4 >= 3
new_enforceWHNF0(Zero, Succ(vx3000), :(vx310, vx311)) → new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
The graph contains the following edges 2 >= 1, 3 > 2, 2 >= 3, 3 > 4
new_enforceWHNF(Neg(vx40), Neg(vx300), vx5, :(vx310, vx311)) → new_seq(new_primPlusNat0(vx40, vx300), vx310, new_primPlusNat0(vx40, vx300), vx311)
The graph contains the following edges 4 > 2, 4 > 4
new_seq(vx6, vx310, vx7, vx311) → new_enforceWHNF(Neg(vx6), vx310, Neg(vx7), vx311)
The graph contains the following edges 2 >= 2, 4 >= 4