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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_rem_1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ IFR

mainModule Main

((rem :: Int -> Int -> Int) :: Int -> Int -> Int)

module Main where

import qualified Prelude

If Reductions:
The following If expression

if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x

is transformed to

primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y)

primModNatS0 x y False = Succ x

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

mainModule Main

((rem :: 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

((rem :: 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

              ↳ NumRed

mainModule Main

((rem :: Int -> Int -> 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

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

mainModule Main

(rem :: Int -> Int -> Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="rem\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="rem vz3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="rem vz3 vz4\n",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primRemInt vz3 vz4\n",fontsize=16,color="burlywood",shape="box"];264[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 264[label="",style="solid", color="burlywood", weight=9];
264 -> 6[label="",style="solid", color="burlywood", weight=3];
265[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 265[label="",style="solid", color="burlywood", weight=9];
265 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="primRemInt (Pos vz30) vz4\n",fontsize=16,color="burlywood",shape="box"];266[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 266[label="",style="solid", color="burlywood", weight=9];
266 -> 8[label="",style="solid", color="burlywood", weight=3];
267[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 267[label="",style="solid", color="burlywood", weight=9];
267 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="primRemInt (Neg vz30) vz4\n",fontsize=16,color="burlywood",shape="box"];268[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 268[label="",style="solid", color="burlywood", weight=9];
268 -> 10[label="",style="solid", color="burlywood", weight=3];
269[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 269[label="",style="solid", color="burlywood", weight=9];
269 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="primRemInt (Pos vz30) (Pos vz40)\n",fontsize=16,color="burlywood",shape="box"];270[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 270[label="",style="solid", color="burlywood", weight=9];
270 -> 12[label="",style="solid", color="burlywood", weight=3];
271[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 271[label="",style="solid", color="burlywood", weight=9];
271 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="primRemInt (Pos vz30) (Neg vz40)\n",fontsize=16,color="burlywood",shape="box"];272[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 272[label="",style="solid", color="burlywood", weight=9];
272 -> 14[label="",style="solid", color="burlywood", weight=3];
273[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 273[label="",style="solid", color="burlywood", weight=9];
273 -> 15[label="",style="solid", color="burlywood", weight=3];
10[label="primRemInt (Neg vz30) (Pos vz40)\n",fontsize=16,color="burlywood",shape="box"];274[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 274[label="",style="solid", color="burlywood", weight=9];
274 -> 16[label="",style="solid", color="burlywood", weight=3];
275[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 275[label="",style="solid", color="burlywood", weight=9];
275 -> 17[label="",style="solid", color="burlywood", weight=3];
11[label="primRemInt (Neg vz30) (Neg vz40)\n",fontsize=16,color="burlywood",shape="box"];276[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 276[label="",style="solid", color="burlywood", weight=9];
276 -> 18[label="",style="solid", color="burlywood", weight=3];
277[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 277[label="",style="solid", color="burlywood", weight=9];
277 -> 19[label="",style="solid", color="burlywood", weight=3];
12[label="primRemInt (Pos vz30) (Pos (Succ vz400))\n",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
13[label="primRemInt (Pos vz30) (Pos Zero)\n",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="primRemInt (Pos vz30) (Neg (Succ vz400))\n",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="primRemInt (Pos vz30) (Neg Zero)\n",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16[label="primRemInt (Neg vz30) (Pos (Succ vz400))\n",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="primRemInt (Neg vz30) (Pos Zero)\n",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primRemInt (Neg vz30) (Neg (Succ vz400))\n",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
19[label="primRemInt (Neg vz30) (Neg Zero)\n",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="Pos (primModNatS vz30 (Succ vz400))\n",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3];
21[label="error []\n",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="Pos (primModNatS vz30 (Succ vz400))\n",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 []\n",fontsize=16,color="magenta"];24[label="Neg (primModNatS vz30 (Succ vz400))\n",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 []\n",fontsize=16,color="magenta"];26[label="Neg (primModNatS vz30 (Succ vz400))\n",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 []\n",fontsize=16,color="magenta"];28[label="primModNatS vz30 (Succ vz400)\n",fontsize=16,color="burlywood",shape="triangle"];281[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 281[label="",style="solid", color="burlywood", weight=9];
281 -> 33[label="",style="solid", color="burlywood", weight=3];
282[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 282[label="",style="solid", color="burlywood", weight=9];
282 -> 34[label="",style="solid", color="burlywood", weight=3];
29[label="error []\n",fontsize=16,color="red",shape="box"];30 -> 28[label="",style="dashed", color="red", weight=0];
30[label="primModNatS vz30 (Succ vz400)\n",fontsize=16,color="magenta"];30 -> 35[label="",style="dashed", color="magenta", weight=3];
31 -> 28[label="",style="dashed", color="red", weight=0];
31[label="primModNatS vz30 (Succ vz400)\n",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3];
32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="primModNatS vz30 (Succ vz400)\n",fontsize=16,color="magenta"];32 -> 37[label="",style="dashed", color="magenta", weight=3];
32 -> 38[label="",style="dashed", color="magenta", weight=3];
33[label="primModNatS (Succ vz300) (Succ vz400)\n",fontsize=16,color="black",shape="box"];33 -> 39[label="",style="solid", color="black", weight=3];
34[label="primModNatS Zero (Succ vz400)\n",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3];
35[label="vz400\n",fontsize=16,color="green",shape="box"];36[label="vz30\n",fontsize=16,color="green",shape="box"];37[label="vz30\n",fontsize=16,color="green",shape="box"];38[label="vz400\n",fontsize=16,color="green",shape="box"];39[label="primModNatS0 vz300 vz400 (primGEqNatS vz300 vz400)\n",fontsize=16,color="burlywood",shape="box"];286[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];39 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 41[label="",style="solid", color="burlywood", weight=3];
287[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 287[label="",style="solid", color="burlywood", weight=9];
287 -> 42[label="",style="solid", color="burlywood", weight=3];
40[label="Zero\n",fontsize=16,color="green",shape="box"];41[label="primModNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)\n",fontsize=16,color="burlywood",shape="box"];288[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];41 -> 288[label="",style="solid", color="burlywood", weight=9];
288 -> 43[label="",style="solid", color="burlywood", weight=3];
289[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 44[label="",style="solid", color="burlywood", weight=3];
42[label="primModNatS0 Zero vz400 (primGEqNatS Zero vz400)\n",fontsize=16,color="burlywood",shape="box"];290[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];42 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 45[label="",style="solid", color="burlywood", weight=3];
291[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 46[label="",style="solid", color="burlywood", weight=3];
43[label="primModNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))\n",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
44[label="primModNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)\n",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
45[label="primModNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))\n",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
46[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)\n",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
47 -> 202[label="",style="dashed", color="red", weight=0];
47[label="primModNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)\n",fontsize=16,color="magenta"];47 -> 203[label="",style="dashed", color="magenta", weight=3];
47 -> 204[label="",style="dashed", color="magenta", weight=3];
47 -> 205[label="",style="dashed", color="magenta", weight=3];
47 -> 206[label="",style="dashed", color="magenta", weight=3];
48[label="primModNatS0 (Succ vz3000) Zero True\n",fontsize=16,color="black",shape="box"];48 -> 53[label="",style="solid", color="black", weight=3];
49[label="primModNatS0 Zero (Succ vz4000) False\n",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3];
50[label="primModNatS0 Zero Zero True\n",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
203[label="vz4000\n",fontsize=16,color="green",shape="box"];204[label="vz4000\n",fontsize=16,color="green",shape="box"];205[label="vz3000\n",fontsize=16,color="green",shape="box"];206[label="vz3000\n",fontsize=16,color="green",shape="box"];202[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)\n",fontsize=16,color="burlywood",shape="triangle"];293[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];202 -> 293[label="",style="solid", color="burlywood", weight=9];
293 -> 235[label="",style="solid", color="burlywood", weight=3];
294[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];202 -> 294[label="",style="solid", color="burlywood", weight=9];
294 -> 236[label="",style="solid", color="burlywood", weight=3];
53 -> 28[label="",style="dashed", color="red", weight=0];
53[label="primModNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)\n",fontsize=16,color="magenta"];53 -> 60[label="",style="dashed", color="magenta", weight=3];
53 -> 61[label="",style="dashed", color="magenta", weight=3];
54[label="Succ Zero\n",fontsize=16,color="green",shape="box"];55 -> 28[label="",style="dashed", color="red", weight=0];
55[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)\n",fontsize=16,color="magenta"];55 -> 62[label="",style="dashed", color="magenta", weight=3];
55 -> 63[label="",style="dashed", color="magenta", weight=3];
235[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)\n",fontsize=16,color="burlywood",shape="box"];297[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];235 -> 297[label="",style="solid", color="burlywood", weight=9];
297 -> 237[label="",style="solid", color="burlywood", weight=3];
298[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];235 -> 298[label="",style="solid", color="burlywood", weight=9];
298 -> 238[label="",style="solid", color="burlywood", weight=3];
236[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)\n",fontsize=16,color="burlywood",shape="box"];299[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];236 -> 299[label="",style="solid", color="burlywood", weight=9];
299 -> 239[label="",style="solid", color="burlywood", weight=3];
300[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];236 -> 300[label="",style="solid", color="burlywood", weight=9];
300 -> 240[label="",style="solid", color="burlywood", weight=3];
60[label="primMinusNatS (Succ vz3000) Zero\n",fontsize=16,color="black",shape="triangle"];60 -> 68[label="",style="solid", color="black", weight=3];
61[label="Zero\n",fontsize=16,color="green",shape="box"];62[label="primMinusNatS Zero Zero\n",fontsize=16,color="black",shape="triangle"];62 -> 69[label="",style="solid", color="black", weight=3];
63[label="Zero\n",fontsize=16,color="green",shape="box"];237[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))\n",fontsize=16,color="black",shape="box"];237 -> 241[label="",style="solid", color="black", weight=3];
238[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)\n",fontsize=16,color="black",shape="box"];238 -> 242[label="",style="solid", color="black", weight=3];
239[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))\n",fontsize=16,color="black",shape="box"];239 -> 243[label="",style="solid", color="black", weight=3];
240[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)\n",fontsize=16,color="black",shape="box"];240 -> 244[label="",style="solid", color="black", weight=3];
68[label="Succ vz3000\n",fontsize=16,color="green",shape="box"];69[label="Zero\n",fontsize=16,color="green",shape="box"];241 -> 202[label="",style="dashed", color="red", weight=0];
241[label="primModNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)\n",fontsize=16,color="magenta"];241 -> 245[label="",style="dashed", color="magenta", weight=3];
241 -> 246[label="",style="dashed", color="magenta", weight=3];
242[label="primModNatS0 (Succ vz21) (Succ vz22) True\n",fontsize=16,color="black",shape="triangle"];242 -> 247[label="",style="solid", color="black", weight=3];
243[label="primModNatS0 (Succ vz21) (Succ vz22) False\n",fontsize=16,color="black",shape="box"];243 -> 248[label="",style="solid", color="black", weight=3];
244 -> 242[label="",style="dashed", color="red", weight=0];
244[label="primModNatS0 (Succ vz21) (Succ vz22) True\n",fontsize=16,color="magenta"];245[label="vz240\n",fontsize=16,color="green",shape="box"];246[label="vz230\n",fontsize=16,color="green",shape="box"];247 -> 28[label="",style="dashed", color="red", weight=0];
247[label="primModNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))\n",fontsize=16,color="magenta"];247 -> 249[label="",style="dashed", color="magenta", weight=3];
247 -> 250[label="",style="dashed", color="magenta", weight=3];
248[label="Succ (Succ vz21)\n",fontsize=16,color="green",shape="box"];249[label="primMinusNatS (Succ vz21) (Succ vz22)\n",fontsize=16,color="black",shape="box"];249 -> 251[label="",style="solid", color="black", weight=3];
250[label="Succ vz22\n",fontsize=16,color="green",shape="box"];251[label="primMinusNatS vz21 vz22\n",fontsize=16,color="burlywood",shape="triangle"];304[label="vz21/Succ vz210",fontsize=10,color="white",style="solid",shape="box"];251 -> 304[label="",style="solid", color="burlywood", weight=9];
304 -> 252[label="",style="solid", color="burlywood", weight=3];
305[label="vz21/Zero",fontsize=10,color="white",style="solid",shape="box"];251 -> 305[label="",style="solid", color="burlywood", weight=9];
305 -> 253[label="",style="solid", color="burlywood", weight=3];
252[label="primMinusNatS (Succ vz210) vz22\n",fontsize=16,color="burlywood",shape="box"];306[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];252 -> 306[label="",style="solid", color="burlywood", weight=9];
306 -> 254[label="",style="solid", color="burlywood", weight=3];
307[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];252 -> 307[label="",style="solid", color="burlywood", weight=9];
307 -> 255[label="",style="solid", color="burlywood", weight=3];
253[label="primMinusNatS Zero vz22\n",fontsize=16,color="burlywood",shape="box"];308[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];253 -> 308[label="",style="solid", color="burlywood", weight=9];
308 -> 256[label="",style="solid", color="burlywood", weight=3];
309[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];253 -> 309[label="",style="solid", color="burlywood", weight=9];
309 -> 257[label="",style="solid", color="burlywood", weight=3];
254[label="primMinusNatS (Succ vz210) (Succ vz220)\n",fontsize=16,color="black",shape="box"];254 -> 258[label="",style="solid", color="black", weight=3];
255[label="primMinusNatS (Succ vz210) Zero\n",fontsize=16,color="black",shape="box"];255 -> 259[label="",style="solid", color="black", weight=3];
256[label="primMinusNatS Zero (Succ vz220)\n",fontsize=16,color="black",shape="box"];256 -> 260[label="",style="solid", color="black", weight=3];
257[label="primMinusNatS Zero Zero\n",fontsize=16,color="black",shape="box"];257 -> 261[label="",style="solid", color="black", weight=3];
258 -> 251[label="",style="dashed", color="red", weight=0];
258[label="primMinusNatS vz210 vz220\n",fontsize=16,color="magenta"];258 -> 262[label="",style="dashed", color="magenta", weight=3];
258 -> 263[label="",style="dashed", color="magenta", weight=3];
259[label="Succ vz210\n",fontsize=16,color="green",shape="box"];260[label="Zero\n",fontsize=16,color="green",shape="box"];261[label="Zero\n",fontsize=16,color="green",shape="box"];262[label="vz220\n",fontsize=16,color="green",shape="box"];263[label="vz210\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

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

new_primMinusNatS(Succ(vz210), Succ(vz220)) → new_primMinusNatS(vz210, vz220)

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(vz210), Succ(vz220)) → new_primMinusNatS(vz210, vz220)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

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

new_primModNatS00(vz21, vz22) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS(Succ(Zero), Zero) → new_primModNatS(new_primMinusNatS2, Zero)
new_primModNatS0(vz21, vz22, Succ(vz230), Zero) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS1(vz3000), Zero)
new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)
new_primModNatS0(vz21, vz22, Zero, Zero) → new_primModNatS00(vz21, vz22)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz220)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vz3000) → Succ(vz3000)
new_primMinusNatS0(Succ(vz210), Succ(vz220)) → new_primMinusNatS0(vz210, vz220)
new_primMinusNatS0(Succ(vz210), Zero) → Succ(vz210)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS1(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

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ UsableRulesProof

                            ↳ QDP

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

new_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS1(vz3000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz220)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vz3000) → Succ(vz3000)
new_primMinusNatS0(Succ(vz210), Succ(vz220)) → new_primMinusNatS0(vz210, vz220)
new_primMinusNatS0(Succ(vz210), Zero) → Succ(vz210)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS1(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

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

                            ↳ QDP

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

new_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS1(vz3000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS1(vz3000) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS1(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_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

                                    ↳ QDP

                                      ↳ RuleRemovalProof

                            ↳ QDP

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

new_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS1(vz3000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS1(vz3000) → Succ(vz3000)

The set Q consists of the following terms:

new_primMinusNatS1(x0)

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_primModNatS(Succ(Succ(vz3000)), Zero) → new_primModNatS(new_primMinusNatS1(vz3000), Zero)

Strictly oriented rules of the TRS R:

new_primMinusNatS1(vz3000) → Succ(vz3000)

Used ordering: POLO with Polynomial interpretation [25]:

POL(Succ(x₁)) = 1 + 2·x₁
POL(Zero) = 0
POL(new_primMinusNatS1(x₁)) = 2 + 2·x₁
POL(new_primModNatS(x₁, x₂)) = x₁ + x₂

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

                                    ↳ QDP

                                      ↳ RuleRemovalProof

                                        ↳ QDP

                                          ↳ PisEmptyProof

                            ↳ QDP

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

new_primMinusNatS1(x0)

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

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ UsableRulesProof

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

new_primModNatS00(vz21, vz22) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS0(vz21, vz22, Succ(vz230), Zero) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)
new_primModNatS0(vz21, vz22, Zero, Zero) → new_primModNatS00(vz21, vz22)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz220)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vz3000) → Succ(vz3000)
new_primMinusNatS0(Succ(vz210), Succ(vz220)) → new_primMinusNatS0(vz210, vz220)
new_primMinusNatS0(Succ(vz210), Zero) → Succ(vz210)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS1(x0)
new_primMinusNatS0(Zero, Zero)

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

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

new_primModNatS00(vz21, vz22) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS0(vz21, vz22, Succ(vz230), Zero) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)
new_primModNatS0(vz21, vz22, Zero, Zero) → new_primModNatS00(vz21, vz22)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz220)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz210), Succ(vz220)) → new_primMinusNatS0(vz210, vz220)
new_primMinusNatS0(Succ(vz210), Zero) → Succ(vz210)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS1(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_primMinusNatS2
new_primMinusNatS1(x0)

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

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

new_primModNatS00(vz21, vz22) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS0(vz21, vz22, Succ(vz230), Zero) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS0(vz21, vz22, Zero, Zero) → new_primModNatS00(vz21, vz22)
new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz220)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz210), Succ(vz220)) → new_primMinusNatS0(vz210, vz220)
new_primMinusNatS0(Succ(vz210), Zero) → Succ(vz210)

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_primModNatS00(vz21, vz22) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))
new_primModNatS(Succ(Succ(vz3000)), Succ(vz4000)) → new_primModNatS0(vz3000, vz4000, vz3000, vz4000)
new_primModNatS0(vz21, vz22, Succ(vz230), Zero) → new_primModNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22))

The remaining pairs can at least be oriented weakly.

new_primModNatS0(vz21, vz22, Zero, Zero) → new_primModNatS00(vz21, vz22)
new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)

Used ordering: Polynomial interpretation [25]:

POL(Succ(x₁)) = 1 + x₁
POL(Zero) = 0
POL(new_primMinusNatS0(x₁, x₂)) = x₁
POL(new_primModNatS(x₁, x₂)) = x₁
POL(new_primModNatS0(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(new_primModNatS00(x₁, x₂)) = 1 + x₁

The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vz210), Zero) → Succ(vz210)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vz220)) → Zero
new_primMinusNatS0(Succ(vz210), Succ(vz220)) → new_primMinusNatS0(vz210, vz220)

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

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

new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)
new_primModNatS0(vz21, vz22, Zero, Zero) → new_primModNatS00(vz21, vz22)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz220)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz210), Succ(vz220)) → new_primMinusNatS0(vz210, vz220)
new_primMinusNatS0(Succ(vz210), Zero) → Succ(vz210)

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 1 less node.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ UsableRulesProof

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

new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vz220)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vz210), Succ(vz220)) → new_primMinusNatS0(vz210, vz220)
new_primMinusNatS0(Succ(vz210), Zero) → Succ(vz210)

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

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

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

new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(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

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QReductionProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ QDPSizeChangeProof

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

new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)

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

new_primModNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) → new_primModNatS0(vz21, vz22, vz230, vz240)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4

primModNatS0	x y True	= primModNatS (primMinusNatS x y) (Succ y)
primModNatS0	x y False	= Succ x