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

YES 1.926 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

  ↳ IFR

mainModule Main

((div :: Int -> Int -> Int) :: Int -> Int -> Int)

module Main where

import qualified Prelude

If Reductions:
The following If expression

if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero

is transformed to

primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y))

primDivNatS0 x y False = Zero

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

mainModule Main

((div :: Int -> Int -> Int) :: Int -> Int -> Int)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((div :: Int -> Int -> 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

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(div :: Int -> Int -> Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="div",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="div vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="div vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primDivInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];288[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 288[label="",style="solid", color="burlywood", weight=9];
288 -> 6[label="",style="solid", color="burlywood", weight=3];
289[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="primDivInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];290[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 8[label="",style="solid", color="burlywood", weight=3];
291[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="primDivInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];292[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 292[label="",style="solid", color="burlywood", weight=9];
292 -> 10[label="",style="solid", color="burlywood", weight=3];
293[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 293[label="",style="solid", color="burlywood", weight=9];
293 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="primDivInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];294[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 294[label="",style="solid", color="burlywood", weight=9];
294 -> 12[label="",style="solid", color="burlywood", weight=3];
295[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 295[label="",style="solid", color="burlywood", weight=9];
295 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="primDivInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];296[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 296[label="",style="solid", color="burlywood", weight=9];
296 -> 14[label="",style="solid", color="burlywood", weight=3];
297[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 297[label="",style="solid", color="burlywood", weight=9];
297 -> 15[label="",style="solid", color="burlywood", weight=3];
10[label="primDivInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];298[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 298[label="",style="solid", color="burlywood", weight=9];
298 -> 16[label="",style="solid", color="burlywood", weight=3];
299[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 299[label="",style="solid", color="burlywood", weight=9];
299 -> 17[label="",style="solid", color="burlywood", weight=3];
11[label="primDivInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];300[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 300[label="",style="solid", color="burlywood", weight=9];
300 -> 18[label="",style="solid", color="burlywood", weight=3];
301[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 301[label="",style="solid", color="burlywood", weight=9];
301 -> 19[label="",style="solid", color="burlywood", weight=3];
12[label="primDivInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
13[label="primDivInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="primDivInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="primDivInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16[label="primDivInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="primDivInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primDivInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
19[label="primDivInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
21[label="error []",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="Neg (primDivNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3];
23 -> 21[label="",style="dashed", color="red", weight=0];
23[label="error []",fontsize=16,color="magenta"];24[label="Neg (primDivNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3];
25 -> 21[label="",style="dashed", color="red", weight=0];
25[label="error []",fontsize=16,color="magenta"];26[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3];
27 -> 21[label="",style="dashed", color="red", weight=0];
27[label="error []",fontsize=16,color="magenta"];28[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];305[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 305[label="",style="solid", color="burlywood", weight=9];
305 -> 33[label="",style="solid", color="burlywood", weight=3];
306[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 306[label="",style="solid", color="burlywood", weight=9];
306 -> 34[label="",style="solid", color="burlywood", weight=3];
29[label="error []",fontsize=16,color="red",shape="box"];30[label="primDivNatP vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];307[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 307[label="",style="solid", color="burlywood", weight=9];
307 -> 35[label="",style="solid", color="burlywood", weight=3];
308[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 308[label="",style="solid", color="burlywood", weight=9];
308 -> 36[label="",style="solid", color="burlywood", weight=3];
31 -> 30[label="",style="dashed", color="red", weight=0];
31[label="primDivNatP vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3];
31 -> 38[label="",style="dashed", color="magenta", weight=3];
32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 39[label="",style="dashed", color="magenta", weight=3];
32 -> 40[label="",style="dashed", color="magenta", weight=3];
33[label="primDivNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 41[label="",style="solid", color="black", weight=3];
34[label="primDivNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
35[label="primDivNatP (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
36[label="primDivNatP Zero (Succ vz400)",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
37[label="vz30",fontsize=16,color="green",shape="box"];38[label="vz400",fontsize=16,color="green",shape="box"];39[label="vz400",fontsize=16,color="green",shape="box"];40[label="vz30",fontsize=16,color="green",shape="box"];41[label="primDivNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];311[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];41 -> 311[label="",style="solid", color="burlywood", weight=9];
311 -> 45[label="",style="solid", color="burlywood", weight=3];
312[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 312[label="",style="solid", color="burlywood", weight=9];
312 -> 46[label="",style="solid", color="burlywood", weight=3];
42[label="Zero",fontsize=16,color="green",shape="box"];43[label="Succ (primDivNatP (primMinusNatS vz300 vz400) (Succ vz400))",fontsize=16,color="green",shape="box"];43 -> 47[label="",style="dashed", color="green", weight=3];
44[label="Zero",fontsize=16,color="green",shape="box"];45[label="primDivNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];313[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];45 -> 313[label="",style="solid", color="burlywood", weight=9];
313 -> 48[label="",style="solid", color="burlywood", weight=3];
314[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 314[label="",style="solid", color="burlywood", weight=9];
314 -> 49[label="",style="solid", color="burlywood", weight=3];
46[label="primDivNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];315[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];46 -> 315[label="",style="solid", color="burlywood", weight=9];
315 -> 50[label="",style="solid", color="burlywood", weight=3];
316[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 316[label="",style="solid", color="burlywood", weight=9];
316 -> 51[label="",style="solid", color="burlywood", weight=3];
47 -> 30[label="",style="dashed", color="red", weight=0];
47[label="primDivNatP (primMinusNatS vz300 vz400) (Succ vz400)",fontsize=16,color="magenta"];47 -> 52[label="",style="dashed", color="magenta", weight=3];
48[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];48 -> 53[label="",style="solid", color="black", weight=3];
49[label="primDivNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
50[label="primDivNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
51[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3];
52[label="primMinusNatS vz300 vz400",fontsize=16,color="burlywood",shape="triangle"];318[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];52 -> 318[label="",style="solid", color="burlywood", weight=9];
318 -> 57[label="",style="solid", color="burlywood", weight=3];
319[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 319[label="",style="solid", color="burlywood", weight=9];
319 -> 58[label="",style="solid", color="burlywood", weight=3];
53 -> 236[label="",style="dashed", color="red", weight=0];
53[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];53 -> 237[label="",style="dashed", color="magenta", weight=3];
53 -> 238[label="",style="dashed", color="magenta", weight=3];
53 -> 239[label="",style="dashed", color="magenta", weight=3];
53 -> 240[label="",style="dashed", color="magenta", weight=3];
54[label="primDivNatS0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];54 -> 61[label="",style="solid", color="black", weight=3];
55[label="primDivNatS0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];55 -> 62[label="",style="solid", color="black", weight=3];
56[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];56 -> 63[label="",style="solid", color="black", weight=3];
57[label="primMinusNatS (Succ vz3000) vz400",fontsize=16,color="burlywood",shape="box"];321[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];57 -> 321[label="",style="solid", color="burlywood", weight=9];
321 -> 64[label="",style="solid", color="burlywood", weight=3];
322[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];57 -> 322[label="",style="solid", color="burlywood", weight=9];
322 -> 65[label="",style="solid", color="burlywood", weight=3];
58[label="primMinusNatS Zero vz400",fontsize=16,color="burlywood",shape="box"];323[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];58 -> 323[label="",style="solid", color="burlywood", weight=9];
323 -> 66[label="",style="solid", color="burlywood", weight=3];
324[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 324[label="",style="solid", color="burlywood", weight=9];
324 -> 67[label="",style="solid", color="burlywood", weight=3];
237[label="vz3000",fontsize=16,color="green",shape="box"];238[label="vz4000",fontsize=16,color="green",shape="box"];239[label="vz4000",fontsize=16,color="green",shape="box"];240[label="vz3000",fontsize=16,color="green",shape="box"];236[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)",fontsize=16,color="burlywood",shape="triangle"];325[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];236 -> 325[label="",style="solid", color="burlywood", weight=9];
325 -> 269[label="",style="solid", color="burlywood", weight=3];
326[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];236 -> 326[label="",style="solid", color="burlywood", weight=9];
326 -> 270[label="",style="solid", color="burlywood", weight=3];
61[label="Succ (primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];61 -> 72[label="",style="dashed", color="green", weight=3];
62[label="Zero",fontsize=16,color="green",shape="box"];63[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];63 -> 73[label="",style="dashed", color="green", weight=3];
64[label="primMinusNatS (Succ vz3000) (Succ vz4000)",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
65[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
66[label="primMinusNatS Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
67[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
269[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)",fontsize=16,color="burlywood",shape="box"];327[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];269 -> 327[label="",style="solid", color="burlywood", weight=9];
327 -> 271[label="",style="solid", color="burlywood", weight=3];
328[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];269 -> 328[label="",style="solid", color="burlywood", weight=9];
328 -> 272[label="",style="solid", color="burlywood", weight=3];
270[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)",fontsize=16,color="burlywood",shape="box"];329[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];270 -> 329[label="",style="solid", color="burlywood", weight=9];
329 -> 273[label="",style="solid", color="burlywood", weight=3];
330[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 330[label="",style="solid", color="burlywood", weight=9];
330 -> 274[label="",style="solid", color="burlywood", weight=3];
72 -> 28[label="",style="dashed", color="red", weight=0];
72[label="primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];72 -> 82[label="",style="dashed", color="magenta", weight=3];
72 -> 83[label="",style="dashed", color="magenta", weight=3];
73 -> 28[label="",style="dashed", color="red", weight=0];
73[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];73 -> 84[label="",style="dashed", color="magenta", weight=3];
73 -> 85[label="",style="dashed", color="magenta", weight=3];
74 -> 52[label="",style="dashed", color="red", weight=0];
74[label="primMinusNatS vz3000 vz4000",fontsize=16,color="magenta"];74 -> 86[label="",style="dashed", color="magenta", weight=3];
74 -> 87[label="",style="dashed", color="magenta", weight=3];
75[label="Succ vz3000",fontsize=16,color="green",shape="box"];76[label="Zero",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];271[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))",fontsize=16,color="black",shape="box"];271 -> 275[label="",style="solid", color="black", weight=3];
272[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3];
273[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3];
274[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3];
82[label="Zero",fontsize=16,color="green",shape="box"];83 -> 52[label="",style="dashed", color="red", weight=0];
83[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="magenta"];83 -> 93[label="",style="dashed", color="magenta", weight=3];
83 -> 94[label="",style="dashed", color="magenta", weight=3];
84[label="Zero",fontsize=16,color="green",shape="box"];85 -> 52[label="",style="dashed", color="red", weight=0];
85[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];85 -> 95[label="",style="dashed", color="magenta", weight=3];
85 -> 96[label="",style="dashed", color="magenta", weight=3];
86[label="vz3000",fontsize=16,color="green",shape="box"];87[label="vz4000",fontsize=16,color="green",shape="box"];275 -> 236[label="",style="dashed", color="red", weight=0];
275[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];275 -> 279[label="",style="dashed", color="magenta", weight=3];
275 -> 280[label="",style="dashed", color="magenta", weight=3];
276[label="primDivNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="black",shape="triangle"];276 -> 281[label="",style="solid", color="black", weight=3];
277[label="primDivNatS0 (Succ vz21) (Succ vz22) False",fontsize=16,color="black",shape="box"];277 -> 282[label="",style="solid", color="black", weight=3];
278 -> 276[label="",style="dashed", color="red", weight=0];
278[label="primDivNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="magenta"];93[label="Succ vz3000",fontsize=16,color="green",shape="box"];94[label="Zero",fontsize=16,color="green",shape="box"];95[label="Zero",fontsize=16,color="green",shape="box"];96[label="Zero",fontsize=16,color="green",shape="box"];279[label="vz240",fontsize=16,color="green",shape="box"];280[label="vz230",fontsize=16,color="green",shape="box"];281[label="Succ (primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22)))",fontsize=16,color="green",shape="box"];281 -> 283[label="",style="dashed", color="green", weight=3];
282[label="Zero",fontsize=16,color="green",shape="box"];283 -> 28[label="",style="dashed", color="red", weight=0];
283[label="primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];283 -> 284[label="",style="dashed", color="magenta", weight=3];
283 -> 285[label="",style="dashed", color="magenta", weight=3];
284[label="Succ vz22",fontsize=16,color="green",shape="box"];285 -> 52[label="",style="dashed", color="red", weight=0];
285[label="primMinusNatS (Succ vz21) (Succ vz22)",fontsize=16,color="magenta"];285 -> 286[label="",style="dashed", color="magenta", weight=3];
285 -> 287[label="",style="dashed", color="magenta", weight=3];
286[label="Succ vz21",fontsize=16,color="green",shape="box"];287[label="Succ vz22",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_primMinusNatS(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS(vz3000, vz4000)

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_primMinusNatS(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS(vz3000, vz4000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                    ↳ QDPOrderProof

                  ↳ QDP

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

new_primDivNatP(Succ(vz300), vz400) → new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

new_primDivNatP(Succ(vz300), vz400) → new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25]:

POL(Succ(x₁)) = 1 + x₁
POL(Zero) = 0
POL(new_primDivNatP(x₁, x₂)) = x₁
POL(new_primMinusNatS0(x₁, x₂)) = x₁

The following usable rules [17] were oriented:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)
new_primMinusNatS0(Zero, Zero) → Zero

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                    ↳ QDPOrderProof

                  ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

new_primDivNatP(Succ(vz300), vz400) → new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( new_primMinusNatS0(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( Succ(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( Zero ) =

/	0	\
\	0	/

Tuple symbols:

M( new_primDivNatP(x₁, x₂) ) =

[

]

x₁

[

]

x₂

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)
new_primMinusNatS0(Zero, Zero) → Zero

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                    ↳ QDPOrderProof

                      ↳ QDP

                  ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

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

new_primDivNatS0(vz21, vz22, Zero, Zero) → new_primDivNatS00(vz21, vz22)
new_primDivNatS00(vz21, vz22) → new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))
new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)
new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))
new_primDivNatS(Succ(Succ(vz3000)), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vz3000), Zero), Zero)
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS0(Zero, Zero), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ UsableRulesProof

                        ↳ QDP

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

new_primDivNatS(Succ(Succ(vz3000)), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vz3000), Zero), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ RuleRemovalProof

                        ↳ QDP

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

new_primDivNatS(Succ(Succ(vz3000)), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vz3000), Zero), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

new_primDivNatS(Succ(Succ(vz3000)), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vz3000), Zero), Zero)

Strictly oriented rules of the TRS R:

new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

Used ordering: POLO with Polynomial interpretation [25]:

POL(Succ(x₁)) = 2 + 2·x₁
POL(Zero) = 0
POL(new_primDivNatS(x₁, x₂)) = x₁ + x₂
POL(new_primMinusNatS0(x₁, x₂)) = 1 + 2·x₁ + x₂

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ UsableRulesProof

                            ↳ QDP

                              ↳ RuleRemovalProof

                                ↳ QDP

                                  ↳ PisEmptyProof

                        ↳ QDP

Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ Rewriting

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

new_primDivNatS00(vz21, vz22) → new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))
new_primDivNatS0(vz21, vz22, Zero, Zero) → new_primDivNatS00(vz21, vz22)
new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)
new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS00(vz21, vz22) → new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22)) at position [0] we obtained the following new rules:

new_primDivNatS00(vz21, vz22) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ Rewriting

                            ↳ QDP

                              ↳ Rewriting

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

new_primDivNatS0(vz21, vz22, Zero, Zero) → new_primDivNatS00(vz21, vz22)
new_primDivNatS00(vz21, vz22) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)
new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vz21), Succ(vz22)), Succ(vz22)) at position [0] we obtained the following new rules:

new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ Rewriting

                            ↳ QDP

                              ↳ Rewriting

                                ↳ QDP

                                  ↳ QDPOrderProof

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

new_primDivNatS0(vz21, vz22, Zero, Zero) → new_primDivNatS00(vz21, vz22)
new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)
new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)
new_primDivNatS00(vz21, vz22) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primDivNatS0(vz3000, vz4000, vz3000, vz4000)

The remaining pairs can at least be oriented weakly.

new_primDivNatS0(vz21, vz22, Zero, Zero) → new_primDivNatS00(vz21, vz22)
new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)
new_primDivNatS00(vz21, vz22) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))

Used ordering: Polynomial interpretation [25]:

POL(Succ(x₁)) = 1 + x₁
POL(Zero) = 0
POL(new_primDivNatS(x₁, x₂)) = x₁
POL(new_primDivNatS0(x₁, x₂, x₃, x₄)) = x₁
POL(new_primDivNatS00(x₁, x₂)) = x₁
POL(new_primMinusNatS0(x₁, x₂)) = x₁

The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ Rewriting

                            ↳ QDP

                              ↳ Rewriting

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

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

new_primDivNatS0(vz21, vz22, Zero, Zero) → new_primDivNatS00(vz21, vz22)
new_primDivNatS00(vz21, vz22) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)
new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) → new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ Rewriting

                            ↳ QDP

                              ↳ Rewriting

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ UsableRulesProof

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

new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz4000)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz3000), Succ(vz4000)) → new_primMinusNatS0(vz3000, vz4000)
new_primMinusNatS0(Succ(vz3000), Zero) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ Rewriting

                            ↳ QDP

                              ↳ Rewriting

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ UsableRulesProof

                                            ↳ QDP

                                              ↳ QReductionProof

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

new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)

R is empty.
The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ Rewriting

                            ↳ QDP

                              ↳ Rewriting

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ UsableRulesProof

                                            ↳ QDP

                                              ↳ QReductionProof

                                                ↳ QDP

                                                  ↳ QDPSizeChangeProof

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

new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)

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

new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primDivNatS0(vz21, vz22, vz230, vz240)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4

primDivNatS0	x y True	= Succ (primDivNatS (primMinusNatS x y) (Succ y))
primDivNatS0	x y False	= Zero