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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_truncate_2.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

((truncate :: Float -> Int) :: Float -> Int)

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\(m,_)→m

is transformed to

m0 (m,_) = m

The following Lambda expression

\(_,r)→r

is transformed to

r0 (_,r) = r

The following Lambda expression

\(q,_)→q

is transformed to

q1 (q,_) = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

mainModule Main

((truncate :: Float -> Int) :: Float -> 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

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

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

mainModule Main

((truncate :: Float -> Int) :: Float -> Int)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.
Binding Reductions:
The bind variable of the following binding Pattern

frac@(Double vx vy)

is replaced by the following term

Double vx vy

The bind variable of the following binding Pattern

frac@(Float wv ww)

is replaced by the following term

Float wv ww

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule Main

((truncate :: Float -> Int) :: Float -> 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

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

mainModule Main

((truncate :: Float -> Int) :: Float -> Int)

module Main where

import qualified Prelude

Let/Where Reductions:
The bindings of the following Let/Where expression

m

where

m = m0 vu6

m0 (m,wy) = m

vu6 = properFraction x

are unpacked to the following functions on top level

truncateM0 xw (m,wy) = m

truncateM xw = truncateM0 xw (truncateVu6 xw)

truncateVu6 xw = properFraction xw

The bindings of the following Let/Where expression

(fromIntegral q,r :% y)

where

q = q1 vu30

q1 (q,wz) = q

r = r0 vu30

r0 (xu,r) = r

vu30 = quotRem x y

are unpacked to the following functions on top level

properFractionQ xx xy = properFractionQ1 xx xy (properFractionVu30 xx xy)

properFractionVu30 xx xy = quotRem xx xy

properFractionR0 xx xy (xu,r) = r

properFractionQ1 xx xy (q,wz) = q

properFractionR xx xy = properFractionR0 xx xy (properFractionVu30 xx xy)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ NumRed

mainModule Main

((truncate :: Float -> Int) :: Float -> Int)

module Main where

import qualified Prelude

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

mainModule Main

(truncate :: Float -> Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="truncate\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="truncate xz3\n",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="truncateM xz3\n",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="truncateM0 xz3 (truncateVu6 xz3)\n",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="truncateM0 xz3 (properFraction xz3)\n",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="truncateM0 xz3 (floatProperFractionFloat xz3)\n",fontsize=16,color="burlywood",shape="box"];281[label="xz3/Float xz30 xz31",fontsize=10,color="white",style="solid",shape="box"];7 -> 281[label="",style="solid", color="burlywood", weight=9];
281 -> 8[label="",style="solid", color="burlywood", weight=3];
8[label="truncateM0 (Float xz30 xz31) (floatProperFractionFloat (Float xz30 xz31))\n",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="truncateM0 (Float xz30 xz31) (fromInt (xz30 `quot` xz31),Float xz30 xz31 - fromInt (xz30 `quot` xz31))\n",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="fromInt (xz30 `quot` xz31)\n",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="xz30 `quot` xz31\n",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="primQuotInt xz30 xz31\n",fontsize=16,color="burlywood",shape="box"];282[label="xz30/Pos xz300",fontsize=10,color="white",style="solid",shape="box"];12 -> 282[label="",style="solid", color="burlywood", weight=9];
282 -> 13[label="",style="solid", color="burlywood", weight=3];
283[label="xz30/Neg xz300",fontsize=10,color="white",style="solid",shape="box"];12 -> 283[label="",style="solid", color="burlywood", weight=9];
283 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="primQuotInt (Pos xz300) xz31\n",fontsize=16,color="burlywood",shape="box"];284[label="xz31/Pos xz310",fontsize=10,color="white",style="solid",shape="box"];13 -> 284[label="",style="solid", color="burlywood", weight=9];
284 -> 15[label="",style="solid", color="burlywood", weight=3];
285[label="xz31/Neg xz310",fontsize=10,color="white",style="solid",shape="box"];13 -> 285[label="",style="solid", color="burlywood", weight=9];
285 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="primQuotInt (Neg xz300) xz31\n",fontsize=16,color="burlywood",shape="box"];286[label="xz31/Pos xz310",fontsize=10,color="white",style="solid",shape="box"];14 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 17[label="",style="solid", color="burlywood", weight=3];
287[label="xz31/Neg xz310",fontsize=10,color="white",style="solid",shape="box"];14 -> 287[label="",style="solid", color="burlywood", weight=9];
287 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="primQuotInt (Pos xz300) (Pos xz310)\n",fontsize=16,color="burlywood",shape="box"];288[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];15 -> 288[label="",style="solid", color="burlywood", weight=9];
288 -> 19[label="",style="solid", color="burlywood", weight=3];
289[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 20[label="",style="solid", color="burlywood", weight=3];
16[label="primQuotInt (Pos xz300) (Neg xz310)\n",fontsize=16,color="burlywood",shape="box"];290[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];16 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 21[label="",style="solid", color="burlywood", weight=3];
291[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 22[label="",style="solid", color="burlywood", weight=3];
17[label="primQuotInt (Neg xz300) (Pos xz310)\n",fontsize=16,color="burlywood",shape="box"];292[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];17 -> 292[label="",style="solid", color="burlywood", weight=9];
292 -> 23[label="",style="solid", color="burlywood", weight=3];
293[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 293[label="",style="solid", color="burlywood", weight=9];
293 -> 24[label="",style="solid", color="burlywood", weight=3];
18[label="primQuotInt (Neg xz300) (Neg xz310)\n",fontsize=16,color="burlywood",shape="box"];294[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];18 -> 294[label="",style="solid", color="burlywood", weight=9];
294 -> 25[label="",style="solid", color="burlywood", weight=3];
295[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 295[label="",style="solid", color="burlywood", weight=9];
295 -> 26[label="",style="solid", color="burlywood", weight=3];
19[label="primQuotInt (Pos xz300) (Pos (Succ xz3100))\n",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="primQuotInt (Pos xz300) (Pos Zero)\n",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3];
21[label="primQuotInt (Pos xz300) (Neg (Succ xz3100))\n",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="primQuotInt (Pos xz300) (Neg Zero)\n",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
23[label="primQuotInt (Neg xz300) (Pos (Succ xz3100))\n",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
24[label="primQuotInt (Neg xz300) (Pos Zero)\n",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
25[label="primQuotInt (Neg xz300) (Neg (Succ xz3100))\n",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
26[label="primQuotInt (Neg xz300) (Neg Zero)\n",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
27[label="Pos (primDivNatS xz300 (Succ xz3100))\n",fontsize=16,color="green",shape="box"];27 -> 35[label="",style="dashed", color="green", weight=3];
28[label="error []\n",fontsize=16,color="black",shape="triangle"];28 -> 36[label="",style="solid", color="black", weight=3];
29[label="Neg (primDivNatS xz300 (Succ xz3100))\n",fontsize=16,color="green",shape="box"];29 -> 37[label="",style="dashed", color="green", weight=3];
30 -> 28[label="",style="dashed", color="red", weight=0];
30[label="error []\n",fontsize=16,color="magenta"];31[label="Neg (primDivNatS xz300 (Succ xz3100))\n",fontsize=16,color="green",shape="box"];31 -> 38[label="",style="dashed", color="green", weight=3];
32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="error []\n",fontsize=16,color="magenta"];33[label="Pos (primDivNatS xz300 (Succ xz3100))\n",fontsize=16,color="green",shape="box"];33 -> 39[label="",style="dashed", color="green", weight=3];
34 -> 28[label="",style="dashed", color="red", weight=0];
34[label="error []\n",fontsize=16,color="magenta"];35[label="primDivNatS xz300 (Succ xz3100)\n",fontsize=16,color="burlywood",shape="triangle"];299[label="xz300/Succ xz3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 299[label="",style="solid", color="burlywood", weight=9];
299 -> 40[label="",style="solid", color="burlywood", weight=3];
300[label="xz300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 300[label="",style="solid", color="burlywood", weight=9];
300 -> 41[label="",style="solid", color="burlywood", weight=3];
36[label="error []\n",fontsize=16,color="red",shape="box"];37 -> 35[label="",style="dashed", color="red", weight=0];
37[label="primDivNatS xz300 (Succ xz3100)\n",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3];
38 -> 35[label="",style="dashed", color="red", weight=0];
38[label="primDivNatS xz300 (Succ xz3100)\n",fontsize=16,color="magenta"];38 -> 43[label="",style="dashed", color="magenta", weight=3];
39 -> 35[label="",style="dashed", color="red", weight=0];
39[label="primDivNatS xz300 (Succ xz3100)\n",fontsize=16,color="magenta"];39 -> 44[label="",style="dashed", color="magenta", weight=3];
39 -> 45[label="",style="dashed", color="magenta", weight=3];
40[label="primDivNatS (Succ xz3000) (Succ xz3100)\n",fontsize=16,color="black",shape="box"];40 -> 46[label="",style="solid", color="black", weight=3];
41[label="primDivNatS Zero (Succ xz3100)\n",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
42[label="xz3100\n",fontsize=16,color="green",shape="box"];43[label="xz300\n",fontsize=16,color="green",shape="box"];44[label="xz300\n",fontsize=16,color="green",shape="box"];45[label="xz3100\n",fontsize=16,color="green",shape="box"];46[label="primDivNatS0 xz3000 xz3100 (primGEqNatS xz3000 xz3100)\n",fontsize=16,color="burlywood",shape="box"];304[label="xz3000/Succ xz30000",fontsize=10,color="white",style="solid",shape="box"];46 -> 304[label="",style="solid", color="burlywood", weight=9];
304 -> 48[label="",style="solid", color="burlywood", weight=3];
305[label="xz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 305[label="",style="solid", color="burlywood", weight=9];
305 -> 49[label="",style="solid", color="burlywood", weight=3];
47[label="Zero\n",fontsize=16,color="green",shape="box"];48[label="primDivNatS0 (Succ xz30000) xz3100 (primGEqNatS (Succ xz30000) xz3100)\n",fontsize=16,color="burlywood",shape="box"];306[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];48 -> 306[label="",style="solid", color="burlywood", weight=9];
306 -> 50[label="",style="solid", color="burlywood", weight=3];
307[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 307[label="",style="solid", color="burlywood", weight=9];
307 -> 51[label="",style="solid", color="burlywood", weight=3];
49[label="primDivNatS0 Zero xz3100 (primGEqNatS Zero xz3100)\n",fontsize=16,color="burlywood",shape="box"];308[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];49 -> 308[label="",style="solid", color="burlywood", weight=9];
308 -> 52[label="",style="solid", color="burlywood", weight=3];
309[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 309[label="",style="solid", color="burlywood", weight=9];
309 -> 53[label="",style="solid", color="burlywood", weight=3];
50[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS (Succ xz30000) (Succ xz31000))\n",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3];
51[label="primDivNatS0 (Succ xz30000) Zero (primGEqNatS (Succ xz30000) Zero)\n",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
52[label="primDivNatS0 Zero (Succ xz31000) (primGEqNatS Zero (Succ xz31000))\n",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)\n",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54 -> 218[label="",style="dashed", color="red", weight=0];
54[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS xz30000 xz31000)\n",fontsize=16,color="magenta"];54 -> 219[label="",style="dashed", color="magenta", weight=3];
54 -> 220[label="",style="dashed", color="magenta", weight=3];
54 -> 221[label="",style="dashed", color="magenta", weight=3];
54 -> 222[label="",style="dashed", color="magenta", weight=3];
55[label="primDivNatS0 (Succ xz30000) Zero True\n",fontsize=16,color="black",shape="box"];55 -> 60[label="",style="solid", color="black", weight=3];
56[label="primDivNatS0 Zero (Succ xz31000) False\n",fontsize=16,color="black",shape="box"];56 -> 61[label="",style="solid", color="black", weight=3];
57[label="primDivNatS0 Zero Zero True\n",fontsize=16,color="black",shape="box"];57 -> 62[label="",style="solid", color="black", weight=3];
219[label="xz31000\n",fontsize=16,color="green",shape="box"];220[label="xz31000\n",fontsize=16,color="green",shape="box"];221[label="xz30000\n",fontsize=16,color="green",shape="box"];222[label="xz30000\n",fontsize=16,color="green",shape="box"];218[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz22 xz23)\n",fontsize=16,color="burlywood",shape="triangle"];311[label="xz22/Succ xz220",fontsize=10,color="white",style="solid",shape="box"];218 -> 311[label="",style="solid", color="burlywood", weight=9];
311 -> 251[label="",style="solid", color="burlywood", weight=3];
312[label="xz22/Zero",fontsize=10,color="white",style="solid",shape="box"];218 -> 312[label="",style="solid", color="burlywood", weight=9];
312 -> 252[label="",style="solid", color="burlywood", weight=3];
60[label="Succ (primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero))\n",fontsize=16,color="green",shape="box"];60 -> 67[label="",style="dashed", color="green", weight=3];
61[label="Zero\n",fontsize=16,color="green",shape="box"];62[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))\n",fontsize=16,color="green",shape="box"];62 -> 68[label="",style="dashed", color="green", weight=3];
251[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) xz23)\n",fontsize=16,color="burlywood",shape="box"];313[label="xz23/Succ xz230",fontsize=10,color="white",style="solid",shape="box"];251 -> 313[label="",style="solid", color="burlywood", weight=9];
313 -> 253[label="",style="solid", color="burlywood", weight=3];
314[label="xz23/Zero",fontsize=10,color="white",style="solid",shape="box"];251 -> 314[label="",style="solid", color="burlywood", weight=9];
314 -> 254[label="",style="solid", color="burlywood", weight=3];
252[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero xz23)\n",fontsize=16,color="burlywood",shape="box"];315[label="xz23/Succ xz230",fontsize=10,color="white",style="solid",shape="box"];252 -> 315[label="",style="solid", color="burlywood", weight=9];
315 -> 255[label="",style="solid", color="burlywood", weight=3];
316[label="xz23/Zero",fontsize=10,color="white",style="solid",shape="box"];252 -> 316[label="",style="solid", color="burlywood", weight=9];
316 -> 256[label="",style="solid", color="burlywood", weight=3];
67 -> 35[label="",style="dashed", color="red", weight=0];
67[label="primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero)\n",fontsize=16,color="magenta"];67 -> 73[label="",style="dashed", color="magenta", weight=3];
67 -> 74[label="",style="dashed", color="magenta", weight=3];
68 -> 35[label="",style="dashed", color="red", weight=0];
68[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)\n",fontsize=16,color="magenta"];68 -> 75[label="",style="dashed", color="magenta", weight=3];
68 -> 76[label="",style="dashed", color="magenta", weight=3];
253[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) (Succ xz230))\n",fontsize=16,color="black",shape="box"];253 -> 257[label="",style="solid", color="black", weight=3];
254[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) Zero)\n",fontsize=16,color="black",shape="box"];254 -> 258[label="",style="solid", color="black", weight=3];
255[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero (Succ xz230))\n",fontsize=16,color="black",shape="box"];255 -> 259[label="",style="solid", color="black", weight=3];
256[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero Zero)\n",fontsize=16,color="black",shape="box"];256 -> 260[label="",style="solid", color="black", weight=3];
73[label="primMinusNatS (Succ xz30000) Zero\n",fontsize=16,color="black",shape="triangle"];73 -> 82[label="",style="solid", color="black", weight=3];
74[label="Zero\n",fontsize=16,color="green",shape="box"];75[label="primMinusNatS Zero Zero\n",fontsize=16,color="black",shape="triangle"];75 -> 83[label="",style="solid", color="black", weight=3];
76[label="Zero\n",fontsize=16,color="green",shape="box"];257 -> 218[label="",style="dashed", color="red", weight=0];
257[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz220 xz230)\n",fontsize=16,color="magenta"];257 -> 261[label="",style="dashed", color="magenta", weight=3];
257 -> 262[label="",style="dashed", color="magenta", weight=3];
258[label="primDivNatS0 (Succ xz20) (Succ xz21) True\n",fontsize=16,color="black",shape="triangle"];258 -> 263[label="",style="solid", color="black", weight=3];
259[label="primDivNatS0 (Succ xz20) (Succ xz21) False\n",fontsize=16,color="black",shape="box"];259 -> 264[label="",style="solid", color="black", weight=3];
260 -> 258[label="",style="dashed", color="red", weight=0];
260[label="primDivNatS0 (Succ xz20) (Succ xz21) True\n",fontsize=16,color="magenta"];82[label="Succ xz30000\n",fontsize=16,color="green",shape="box"];83[label="Zero\n",fontsize=16,color="green",shape="box"];261[label="xz230\n",fontsize=16,color="green",shape="box"];262[label="xz220\n",fontsize=16,color="green",shape="box"];263[label="Succ (primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21)))\n",fontsize=16,color="green",shape="box"];263 -> 265[label="",style="dashed", color="green", weight=3];
264[label="Zero\n",fontsize=16,color="green",shape="box"];265 -> 35[label="",style="dashed", color="red", weight=0];
265[label="primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21))\n",fontsize=16,color="magenta"];265 -> 266[label="",style="dashed", color="magenta", weight=3];
265 -> 267[label="",style="dashed", color="magenta", weight=3];
266[label="primMinusNatS (Succ xz20) (Succ xz21)\n",fontsize=16,color="black",shape="box"];266 -> 268[label="",style="solid", color="black", weight=3];
267[label="Succ xz21\n",fontsize=16,color="green",shape="box"];268[label="primMinusNatS xz20 xz21\n",fontsize=16,color="burlywood",shape="triangle"];322[label="xz20/Succ xz200",fontsize=10,color="white",style="solid",shape="box"];268 -> 322[label="",style="solid", color="burlywood", weight=9];
322 -> 269[label="",style="solid", color="burlywood", weight=3];
323[label="xz20/Zero",fontsize=10,color="white",style="solid",shape="box"];268 -> 323[label="",style="solid", color="burlywood", weight=9];
323 -> 270[label="",style="solid", color="burlywood", weight=3];
269[label="primMinusNatS (Succ xz200) xz21\n",fontsize=16,color="burlywood",shape="box"];324[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];269 -> 324[label="",style="solid", color="burlywood", weight=9];
324 -> 271[label="",style="solid", color="burlywood", weight=3];
325[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];269 -> 325[label="",style="solid", color="burlywood", weight=9];
325 -> 272[label="",style="solid", color="burlywood", weight=3];
270[label="primMinusNatS Zero xz21\n",fontsize=16,color="burlywood",shape="box"];326[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];270 -> 326[label="",style="solid", color="burlywood", weight=9];
326 -> 273[label="",style="solid", color="burlywood", weight=3];
327[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 327[label="",style="solid", color="burlywood", weight=9];
327 -> 274[label="",style="solid", color="burlywood", weight=3];
271[label="primMinusNatS (Succ xz200) (Succ xz210)\n",fontsize=16,color="black",shape="box"];271 -> 275[label="",style="solid", color="black", weight=3];
272[label="primMinusNatS (Succ xz200) Zero\n",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3];
273[label="primMinusNatS Zero (Succ xz210)\n",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3];
274[label="primMinusNatS Zero Zero\n",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3];
275 -> 268[label="",style="dashed", color="red", weight=0];
275[label="primMinusNatS xz200 xz210\n",fontsize=16,color="magenta"];275 -> 279[label="",style="dashed", color="magenta", weight=3];
275 -> 280[label="",style="dashed", color="magenta", weight=3];
276[label="Succ xz200\n",fontsize=16,color="green",shape="box"];277[label="Zero\n",fontsize=16,color="green",shape="box"];278[label="Zero\n",fontsize=16,color="green",shape="box"];279[label="xz210\n",fontsize=16,color="green",shape="box"];280[label="xz200\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

                            ↳ AND

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                              ↳ QDP

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

new_primMinusNatS(Succ(xz200), Succ(xz210)) → new_primMinusNatS(xz200, xz210)

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(xz200), Succ(xz210)) → new_primMinusNatS(xz200, xz210)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                                ↳ DependencyGraphProof

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

new_primDivNatS0(xz20, xz21, Zero, Zero) → new_primDivNatS00(xz20, xz21)
new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)
new_primDivNatS00(xz20, xz21) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS2, Zero)
new_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz210)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz200), Succ(xz210)) → new_primMinusNatS0(xz200, xz210)
new_primMinusNatS0(Succ(xz200), Zero) → Succ(xz200)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

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

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

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                      ↳ UsableRulesProof

                                    ↳ QDP

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

new_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz210)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz200), Succ(xz210)) → new_primMinusNatS0(xz200, xz210)
new_primMinusNatS0(Succ(xz200), Zero) → Succ(xz200)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

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

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

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS1(xz30000) → Succ(xz30000)

The set Q consists of the following terms:

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

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS1(xz30000) → Succ(xz30000)

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_primDivNatS(Succ(Succ(xz30000)), Zero) → new_primDivNatS(new_primMinusNatS1(xz30000), Zero)

Strictly oriented rules of the TRS R:

new_primMinusNatS1(xz30000) → Succ(xz30000)

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                    ↳ QDP

                                      ↳ UsableRulesProof

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

new_primDivNatS0(xz20, xz21, Zero, Zero) → new_primDivNatS00(xz20, xz21)
new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS00(xz20, xz21) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz210)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz200), Succ(xz210)) → new_primMinusNatS0(xz200, xz210)
new_primMinusNatS0(Succ(xz200), Zero) → Succ(xz200)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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_primDivNatS0(xz20, xz21, Zero, Zero) → new_primDivNatS00(xz20, xz21)
new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS00(xz20, xz21) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz210)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz200), Succ(xz210)) → new_primMinusNatS0(xz200, xz210)
new_primMinusNatS0(Succ(xz200), Zero) → Succ(xz200)

The set Q consists of the following terms:

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

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

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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_primDivNatS0(xz20, xz21, Zero, Zero) → new_primDivNatS00(xz20, xz21)
new_primDivNatS00(xz20, xz21) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)
new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz210)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz200), Succ(xz210)) → new_primMinusNatS0(xz200, xz210)
new_primMinusNatS0(Succ(xz200), Zero) → Succ(xz200)

The set Q consists of the following terms:

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

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_primDivNatS0(xz20, xz21, Zero, Zero) → new_primDivNatS00(xz20, xz21)
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))

The remaining pairs can at least be oriented weakly.

new_primDivNatS00(xz20, xz21) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))
new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)

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)) = 1 + x₁
POL(new_primDivNatS00(x₁, x₂)) = x₁
POL(new_primMinusNatS0(x₁, x₂)) = x₁

The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(xz200), Succ(xz210)) → new_primMinusNatS0(xz200, xz210)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz200), Zero) → Succ(xz200)
new_primMinusNatS0(Zero, Succ(xz210)) → Zero

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)
new_primDivNatS00(xz20, xz21) → new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz210)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz200), Succ(xz210)) → new_primMinusNatS0(xz200, xz210)
new_primMinusNatS0(Succ(xz200), Zero) → Succ(xz200)

The set Q consists of the following terms:

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

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

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz210)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz200), Succ(xz210)) → new_primMinusNatS0(xz200, xz210)
new_primMinusNatS0(Succ(xz200), Zero) → Succ(xz200)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)

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

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

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ 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_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)

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

new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) → new_primDivNatS0(xz20, xz21, xz220, xz230)
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

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