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

YES 2.008 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

  ↳ LR

mainModule Main

((truncate :: RealFrac a => a -> Int) :: RealFrac a => a -> 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 :: RealFrac a => a -> Int) :: RealFrac a => a -> 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 :: RealFrac a => a -> Int) :: RealFrac a => a -> 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 vy vz)

is replaced by the following term

Double vy vz

The bind variable of the following binding Pattern

frac@(Float ww wx)

is replaced by the following term

Float ww wx

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule Main

((truncate :: RealFrac a => a -> Int) :: RealFrac a => a -> 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 :: RealFrac a => a -> Int) :: RealFrac a => a -> 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,vv) = m

vu6 = properFraction x

are unpacked to the following functions on top level

truncateM0 xw (m,vv) = 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

properFractionVu30 xx xy = quotRem xx xy

properFractionQ1 xx xy (q,wz) = q

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

properFractionR0 xx xy (xu,r) = r

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ NumRed

mainModule Main

((truncate :: RealFrac a => a -> Int) :: RealFrac a => a -> Int)

module Main where

import qualified Prelude

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

mainModule Main

(truncate :: RealFrac a => a -> Int)

module Main where

import qualified Prelude

Haskell To QDPs

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354 -> 74[label="",style="solid", color="burlywood", weight=3];
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355 -> 75[label="",style="solid", color="burlywood", weight=3];
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81 -> 87[label="",style="dashed", color="magenta", weight=3];
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83 -> 34[label="",style="dashed", color="red", weight=0];
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83 -> 90[label="",style="dashed", color="magenta", weight=3];
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84[label="primDivNatS0 xz3000 xz3100 (primGEqNatS xz3000 xz3100)",fontsize=16,color="burlywood",shape="box"];363[label="xz3000/Succ xz30000",fontsize=10,color="white",style="solid",shape="box"];84 -> 363[label="",style="solid", color="burlywood", weight=9];
363 -> 92[label="",style="solid", color="burlywood", weight=3];
364[label="xz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];84 -> 364[label="",style="solid", color="burlywood", weight=9];
364 -> 93[label="",style="solid", color="burlywood", weight=3];
85[label="Zero",fontsize=16,color="green",shape="box"];86[label="xz300",fontsize=16,color="green",shape="box"];87[label="xz310",fontsize=16,color="green",shape="box"];88[label="fromInteger (Integer xz6)",fontsize=16,color="black",shape="box"];88 -> 94[label="",style="solid", color="black", weight=3];
89[label="xz30",fontsize=16,color="green",shape="box"];90[label="xz31",fontsize=16,color="green",shape="box"];91[label="xz7",fontsize=16,color="green",shape="box"];92[label="primDivNatS0 (Succ xz30000) xz3100 (primGEqNatS (Succ xz30000) xz3100)",fontsize=16,color="burlywood",shape="box"];365[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];92 -> 365[label="",style="solid", color="burlywood", weight=9];
365 -> 95[label="",style="solid", color="burlywood", weight=3];
366[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];92 -> 366[label="",style="solid", color="burlywood", weight=9];
366 -> 96[label="",style="solid", color="burlywood", weight=3];
93[label="primDivNatS0 Zero xz3100 (primGEqNatS Zero xz3100)",fontsize=16,color="burlywood",shape="box"];367[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];93 -> 367[label="",style="solid", color="burlywood", weight=9];
367 -> 97[label="",style="solid", color="burlywood", weight=3];
368[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];93 -> 368[label="",style="solid", color="burlywood", weight=9];
368 -> 98[label="",style="solid", color="burlywood", weight=3];
94[label="xz6",fontsize=16,color="green",shape="box"];95[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS (Succ xz30000) (Succ xz31000))",fontsize=16,color="black",shape="box"];95 -> 99[label="",style="solid", color="black", weight=3];
96[label="primDivNatS0 (Succ xz30000) Zero (primGEqNatS (Succ xz30000) Zero)",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3];
97[label="primDivNatS0 Zero (Succ xz31000) (primGEqNatS Zero (Succ xz31000))",fontsize=16,color="black",shape="box"];97 -> 101[label="",style="solid", color="black", weight=3];
98[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];98 -> 102[label="",style="solid", color="black", weight=3];
99 -> 263[label="",style="dashed", color="red", weight=0];
99[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS xz30000 xz31000)",fontsize=16,color="magenta"];99 -> 264[label="",style="dashed", color="magenta", weight=3];
99 -> 265[label="",style="dashed", color="magenta", weight=3];
99 -> 266[label="",style="dashed", color="magenta", weight=3];
99 -> 267[label="",style="dashed", color="magenta", weight=3];
100[label="primDivNatS0 (Succ xz30000) Zero True",fontsize=16,color="black",shape="box"];100 -> 105[label="",style="solid", color="black", weight=3];
101[label="primDivNatS0 Zero (Succ xz31000) False",fontsize=16,color="black",shape="box"];101 -> 106[label="",style="solid", color="black", weight=3];
102[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];102 -> 107[label="",style="solid", color="black", weight=3];
264[label="xz31000",fontsize=16,color="green",shape="box"];265[label="xz30000",fontsize=16,color="green",shape="box"];266[label="xz30000",fontsize=16,color="green",shape="box"];267[label="xz31000",fontsize=16,color="green",shape="box"];263[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS xz26 xz27)",fontsize=16,color="burlywood",shape="triangle"];370[label="xz26/Succ xz260",fontsize=10,color="white",style="solid",shape="box"];263 -> 370[label="",style="solid", color="burlywood", weight=9];
370 -> 296[label="",style="solid", color="burlywood", weight=3];
371[label="xz26/Zero",fontsize=10,color="white",style="solid",shape="box"];263 -> 371[label="",style="solid", color="burlywood", weight=9];
371 -> 297[label="",style="solid", color="burlywood", weight=3];
105[label="Succ (primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];105 -> 112[label="",style="dashed", color="green", weight=3];
106[label="Zero",fontsize=16,color="green",shape="box"];107[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];107 -> 113[label="",style="dashed", color="green", weight=3];
296[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) xz27)",fontsize=16,color="burlywood",shape="box"];372[label="xz27/Succ xz270",fontsize=10,color="white",style="solid",shape="box"];296 -> 372[label="",style="solid", color="burlywood", weight=9];
372 -> 298[label="",style="solid", color="burlywood", weight=3];
373[label="xz27/Zero",fontsize=10,color="white",style="solid",shape="box"];296 -> 373[label="",style="solid", color="burlywood", weight=9];
373 -> 299[label="",style="solid", color="burlywood", weight=3];
297[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero xz27)",fontsize=16,color="burlywood",shape="box"];374[label="xz27/Succ xz270",fontsize=10,color="white",style="solid",shape="box"];297 -> 374[label="",style="solid", color="burlywood", weight=9];
374 -> 300[label="",style="solid", color="burlywood", weight=3];
375[label="xz27/Zero",fontsize=10,color="white",style="solid",shape="box"];297 -> 375[label="",style="solid", color="burlywood", weight=9];
375 -> 301[label="",style="solid", color="burlywood", weight=3];
112 -> 67[label="",style="dashed", color="red", weight=0];
112[label="primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];112 -> 118[label="",style="dashed", color="magenta", weight=3];
112 -> 119[label="",style="dashed", color="magenta", weight=3];
113 -> 67[label="",style="dashed", color="red", weight=0];
113[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];113 -> 120[label="",style="dashed", color="magenta", weight=3];
113 -> 121[label="",style="dashed", color="magenta", weight=3];
298[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) (Succ xz270))",fontsize=16,color="black",shape="box"];298 -> 302[label="",style="solid", color="black", weight=3];
299[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) Zero)",fontsize=16,color="black",shape="box"];299 -> 303[label="",style="solid", color="black", weight=3];
300[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero (Succ xz270))",fontsize=16,color="black",shape="box"];300 -> 304[label="",style="solid", color="black", weight=3];
301[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];301 -> 305[label="",style="solid", color="black", weight=3];
118[label="primMinusNatS (Succ xz30000) Zero",fontsize=16,color="black",shape="triangle"];118 -> 127[label="",style="solid", color="black", weight=3];
119[label="Zero",fontsize=16,color="green",shape="box"];120[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];120 -> 128[label="",style="solid", color="black", weight=3];
121[label="Zero",fontsize=16,color="green",shape="box"];302 -> 263[label="",style="dashed", color="red", weight=0];
302[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS xz260 xz270)",fontsize=16,color="magenta"];302 -> 306[label="",style="dashed", color="magenta", weight=3];
302 -> 307[label="",style="dashed", color="magenta", weight=3];
303[label="primDivNatS0 (Succ xz24) (Succ xz25) True",fontsize=16,color="black",shape="triangle"];303 -> 308[label="",style="solid", color="black", weight=3];
304[label="primDivNatS0 (Succ xz24) (Succ xz25) False",fontsize=16,color="black",shape="box"];304 -> 309[label="",style="solid", color="black", weight=3];
305 -> 303[label="",style="dashed", color="red", weight=0];
305[label="primDivNatS0 (Succ xz24) (Succ xz25) True",fontsize=16,color="magenta"];127[label="Succ xz30000",fontsize=16,color="green",shape="box"];128[label="Zero",fontsize=16,color="green",shape="box"];306[label="xz270",fontsize=16,color="green",shape="box"];307[label="xz260",fontsize=16,color="green",shape="box"];308[label="Succ (primDivNatS (primMinusNatS (Succ xz24) (Succ xz25)) (Succ (Succ xz25)))",fontsize=16,color="green",shape="box"];308 -> 310[label="",style="dashed", color="green", weight=3];
309[label="Zero",fontsize=16,color="green",shape="box"];310 -> 67[label="",style="dashed", color="red", weight=0];
310[label="primDivNatS (primMinusNatS (Succ xz24) (Succ xz25)) (Succ (Succ xz25))",fontsize=16,color="magenta"];310 -> 311[label="",style="dashed", color="magenta", weight=3];
310 -> 312[label="",style="dashed", color="magenta", weight=3];
311[label="primMinusNatS (Succ xz24) (Succ xz25)",fontsize=16,color="black",shape="box"];311 -> 313[label="",style="solid", color="black", weight=3];
312[label="Succ xz25",fontsize=16,color="green",shape="box"];313[label="primMinusNatS xz24 xz25",fontsize=16,color="burlywood",shape="triangle"];381[label="xz24/Succ xz240",fontsize=10,color="white",style="solid",shape="box"];313 -> 381[label="",style="solid", color="burlywood", weight=9];
381 -> 314[label="",style="solid", color="burlywood", weight=3];
382[label="xz24/Zero",fontsize=10,color="white",style="solid",shape="box"];313 -> 382[label="",style="solid", color="burlywood", weight=9];
382 -> 315[label="",style="solid", color="burlywood", weight=3];
314[label="primMinusNatS (Succ xz240) xz25",fontsize=16,color="burlywood",shape="box"];383[label="xz25/Succ xz250",fontsize=10,color="white",style="solid",shape="box"];314 -> 383[label="",style="solid", color="burlywood", weight=9];
383 -> 316[label="",style="solid", color="burlywood", weight=3];
384[label="xz25/Zero",fontsize=10,color="white",style="solid",shape="box"];314 -> 384[label="",style="solid", color="burlywood", weight=9];
384 -> 317[label="",style="solid", color="burlywood", weight=3];
315[label="primMinusNatS Zero xz25",fontsize=16,color="burlywood",shape="box"];385[label="xz25/Succ xz250",fontsize=10,color="white",style="solid",shape="box"];315 -> 385[label="",style="solid", color="burlywood", weight=9];
385 -> 318[label="",style="solid", color="burlywood", weight=3];
386[label="xz25/Zero",fontsize=10,color="white",style="solid",shape="box"];315 -> 386[label="",style="solid", color="burlywood", weight=9];
386 -> 319[label="",style="solid", color="burlywood", weight=3];
316[label="primMinusNatS (Succ xz240) (Succ xz250)",fontsize=16,color="black",shape="box"];316 -> 320[label="",style="solid", color="black", weight=3];
317[label="primMinusNatS (Succ xz240) Zero",fontsize=16,color="black",shape="box"];317 -> 321[label="",style="solid", color="black", weight=3];
318[label="primMinusNatS Zero (Succ xz250)",fontsize=16,color="black",shape="box"];318 -> 322[label="",style="solid", color="black", weight=3];
319[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];319 -> 323[label="",style="solid", color="black", weight=3];
320 -> 313[label="",style="dashed", color="red", weight=0];
320[label="primMinusNatS xz240 xz250",fontsize=16,color="magenta"];320 -> 324[label="",style="dashed", color="magenta", weight=3];
320 -> 325[label="",style="dashed", color="magenta", weight=3];
321[label="Succ xz240",fontsize=16,color="green",shape="box"];322[label="Zero",fontsize=16,color="green",shape="box"];323[label="Zero",fontsize=16,color="green",shape="box"];324[label="xz240",fontsize=16,color="green",shape="box"];325[label="xz250",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(xz240), Succ(xz250)) → new_primMinusNatS(xz240, xz250)

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(xz240), Succ(xz250)) → new_primMinusNatS(xz240, xz250)
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(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
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(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

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

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(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

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

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

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_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Succ(x0), Succ(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

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(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(xz30000) → Succ(xz30000)
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
new_primMinusNatS2 → Zero

The set Q consists of the following terms:

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

↳ 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(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)

The set Q consists of the following terms:

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

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(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)

The set Q consists of the following terms:

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

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_primDivNatS00(xz24, xz25) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))
new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) → new_primDivNatS0(xz30000, xz31000, xz30000, xz31000)
new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) → new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25))

The remaining pairs can at least be oriented weakly.

new_primDivNatS0(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)

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

The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)
new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Zero, Zero) → 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(xz24, xz25, Zero, Zero) → new_primDivNatS00(xz24, xz25)
new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)

The set Q consists of the following terms:

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

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(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(xz250)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(xz240), Succ(xz250)) → new_primMinusNatS0(xz240, xz250)
new_primMinusNatS0(Succ(xz240), Zero) → Succ(xz240)

The set Q consists of the following terms:

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

↳ 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(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)

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

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

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

↳ 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(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)

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

new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) → new_primDivNatS0(xz24, xz25, xz260, xz270)
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