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

YES 2.257 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

((properFraction :: Ratio Int -> (Int,Ratio Int)) :: Ratio Int -> (Int,Ratio Int))

module Main where

import qualified Prelude

Lambda Reductions:
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

((properFraction :: Ratio Int -> (Int,Ratio Int)) :: Ratio Int -> (Int,Ratio 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

((properFraction :: Ratio Int -> (Int,Ratio Int)) :: Ratio Int -> (Int,Ratio Int))

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule Main

((properFraction :: Ratio Int -> (Int,Ratio Int)) :: Ratio Int -> (Int,Ratio 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

((properFraction :: Ratio Int -> (Int,Ratio Int)) :: Ratio Int -> (Int,Ratio Int))

module Main where

import qualified Prelude

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

(fromIntegral q,r :% y)

where

q = q1 vu30

q1 (q,wu) = q

r = r0 vu30

r0 (wv,r) = r

vu30 = quotRem x y

are unpacked to the following functions on top level

properFractionVu30 wx wy = quotRem wx wy

properFractionR0 wx wy (wv,r) = r

properFractionQ wx wy = properFractionQ1 wx wy (properFractionVu30 wx wy)

properFractionQ1 wx wy (q,wu) = q

properFractionR wx wy = properFractionR0 wx wy (properFractionVu30 wx wy)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

mainModule Main

(properFraction :: Ratio Int -> (Int,Ratio Int))

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
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562 -> 19[label="",style="solid", color="burlywood", weight=3];
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563 -> 20[label="",style="solid", color="burlywood", weight=3];
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568 -> 27[label="",style="solid", color="burlywood", weight=3];
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569 -> 28[label="",style="solid", color="burlywood", weight=3];
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570 -> 29[label="",style="solid", color="burlywood", weight=3];
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571 -> 30[label="",style="solid", color="burlywood", weight=3];
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572 -> 31[label="",style="solid", color="burlywood", weight=3];
573[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 573[label="",style="solid", color="burlywood", weight=9];
573 -> 32[label="",style="solid", color="burlywood", weight=3];
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36 -> 17[label="",style="dashed", color="red", weight=0];
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40 -> 38[label="",style="dashed", color="red", weight=0];
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588 -> 59[label="",style="solid", color="burlywood", weight=3];
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589 -> 60[label="",style="solid", color="burlywood", weight=3];
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55[label="wz3100",fontsize=16,color="green",shape="box"];56[label="wz300",fontsize=16,color="green",shape="box"];57[label="wz300",fontsize=16,color="green",shape="box"];58[label="wz3100",fontsize=16,color="green",shape="box"];59[label="primQuotInt (Pos wz300) (Pos wz310)",fontsize=16,color="burlywood",shape="box"];592[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];59 -> 592[label="",style="solid", color="burlywood", weight=9];
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593[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 593[label="",style="solid", color="burlywood", weight=9];
593 -> 66[label="",style="solid", color="burlywood", weight=3];
60[label="primQuotInt (Pos wz300) (Neg wz310)",fontsize=16,color="burlywood",shape="box"];594[label="wz310/Succ wz3100",fontsize=10,color="white",style="solid",shape="box"];60 -> 594[label="",style="solid", color="burlywood", weight=9];
594 -> 67[label="",style="solid", color="burlywood", weight=3];
595[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 595[label="",style="solid", color="burlywood", weight=9];
595 -> 68[label="",style="solid", color="burlywood", weight=3];
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596 -> 69[label="",style="solid", color="burlywood", weight=3];
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597 -> 70[label="",style="solid", color="burlywood", weight=3];
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598 -> 71[label="",style="solid", color="burlywood", weight=3];
599[label="wz310/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 599[label="",style="solid", color="burlywood", weight=9];
599 -> 72[label="",style="solid", color="burlywood", weight=3];
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600 -> 73[label="",style="solid", color="burlywood", weight=3];
601[label="wz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];63 -> 601[label="",style="solid", color="burlywood", weight=9];
601 -> 74[label="",style="solid", color="burlywood", weight=3];
64[label="Zero",fontsize=16,color="green",shape="box"];65[label="primQuotInt (Pos wz300) (Pos (Succ wz3100))",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
66[label="primQuotInt (Pos wz300) (Pos Zero)",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
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602 -> 83[label="",style="solid", color="burlywood", weight=3];
603[label="wz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];73 -> 603[label="",style="solid", color="burlywood", weight=9];
603 -> 84[label="",style="solid", color="burlywood", weight=3];
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604 -> 85[label="",style="solid", color="burlywood", weight=3];
605[label="wz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];74 -> 605[label="",style="solid", color="burlywood", weight=9];
605 -> 86[label="",style="solid", color="burlywood", weight=3];
75[label="Pos (primDivNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];75 -> 87[label="",style="dashed", color="green", weight=3];
76 -> 38[label="",style="dashed", color="red", weight=0];
76[label="error []",fontsize=16,color="magenta"];77[label="Neg (primDivNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];77 -> 88[label="",style="dashed", color="green", weight=3];
78 -> 38[label="",style="dashed", color="red", weight=0];
78[label="error []",fontsize=16,color="magenta"];79[label="Neg (primDivNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];79 -> 89[label="",style="dashed", color="green", weight=3];
80 -> 38[label="",style="dashed", color="red", weight=0];
80[label="error []",fontsize=16,color="magenta"];81[label="Pos (primDivNatS wz300 (Succ wz3100))",fontsize=16,color="green",shape="box"];81 -> 90[label="",style="dashed", color="green", weight=3];
82 -> 38[label="",style="dashed", color="red", weight=0];
82[label="error []",fontsize=16,color="magenta"];83[label="primModNatS0 (Succ wz30000) (Succ wz31000) (primGEqNatS (Succ wz30000) (Succ wz31000))",fontsize=16,color="black",shape="box"];83 -> 91[label="",style="solid", color="black", weight=3];
84[label="primModNatS0 (Succ wz30000) Zero (primGEqNatS (Succ wz30000) Zero)",fontsize=16,color="black",shape="box"];84 -> 92[label="",style="solid", color="black", weight=3];
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86[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];86 -> 94[label="",style="solid", color="black", weight=3];
87[label="primDivNatS wz300 (Succ wz3100)",fontsize=16,color="burlywood",shape="triangle"];610[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];87 -> 610[label="",style="solid", color="burlywood", weight=9];
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92[label="primModNatS0 (Succ wz30000) Zero True",fontsize=16,color="black",shape="box"];92 -> 103[label="",style="solid", color="black", weight=3];
93[label="primModNatS0 Zero (Succ wz31000) False",fontsize=16,color="black",shape="box"];93 -> 104[label="",style="solid", color="black", weight=3];
94[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];94 -> 105[label="",style="solid", color="black", weight=3];
95[label="primDivNatS (Succ wz3000) (Succ wz3100)",fontsize=16,color="black",shape="box"];95 -> 106[label="",style="solid", color="black", weight=3];
96[label="primDivNatS Zero (Succ wz3100)",fontsize=16,color="black",shape="box"];96 -> 107[label="",style="solid", color="black", weight=3];
97[label="wz3100",fontsize=16,color="green",shape="box"];98[label="wz300",fontsize=16,color="green",shape="box"];99[label="wz300",fontsize=16,color="green",shape="box"];100[label="wz3100",fontsize=16,color="green",shape="box"];363[label="wz31000",fontsize=16,color="green",shape="box"];364[label="wz31000",fontsize=16,color="green",shape="box"];365[label="wz30000",fontsize=16,color="green",shape="box"];366[label="wz30000",fontsize=16,color="green",shape="box"];362[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS wz23 wz24)",fontsize=16,color="burlywood",shape="triangle"];616[label="wz23/Succ wz230",fontsize=10,color="white",style="solid",shape="box"];362 -> 616[label="",style="solid", color="burlywood", weight=9];
616 -> 395[label="",style="solid", color="burlywood", weight=3];
617[label="wz23/Zero",fontsize=10,color="white",style="solid",shape="box"];362 -> 617[label="",style="solid", color="burlywood", weight=9];
617 -> 396[label="",style="solid", color="burlywood", weight=3];
103 -> 46[label="",style="dashed", color="red", weight=0];
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103 -> 113[label="",style="dashed", color="magenta", weight=3];
104[label="Succ Zero",fontsize=16,color="green",shape="box"];105 -> 46[label="",style="dashed", color="red", weight=0];
105[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];105 -> 114[label="",style="dashed", color="magenta", weight=3];
105 -> 115[label="",style="dashed", color="magenta", weight=3];
106[label="primDivNatS0 wz3000 wz3100 (primGEqNatS wz3000 wz3100)",fontsize=16,color="burlywood",shape="box"];620[label="wz3000/Succ wz30000",fontsize=10,color="white",style="solid",shape="box"];106 -> 620[label="",style="solid", color="burlywood", weight=9];
620 -> 116[label="",style="solid", color="burlywood", weight=3];
621[label="wz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];106 -> 621[label="",style="solid", color="burlywood", weight=9];
621 -> 117[label="",style="solid", color="burlywood", weight=3];
107[label="Zero",fontsize=16,color="green",shape="box"];395[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS (Succ wz230) wz24)",fontsize=16,color="burlywood",shape="box"];622[label="wz24/Succ wz240",fontsize=10,color="white",style="solid",shape="box"];395 -> 622[label="",style="solid", color="burlywood", weight=9];
622 -> 402[label="",style="solid", color="burlywood", weight=3];
623[label="wz24/Zero",fontsize=10,color="white",style="solid",shape="box"];395 -> 623[label="",style="solid", color="burlywood", weight=9];
623 -> 403[label="",style="solid", color="burlywood", weight=3];
396[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS Zero wz24)",fontsize=16,color="burlywood",shape="box"];624[label="wz24/Succ wz240",fontsize=10,color="white",style="solid",shape="box"];396 -> 624[label="",style="solid", color="burlywood", weight=9];
624 -> 404[label="",style="solid", color="burlywood", weight=3];
625[label="wz24/Zero",fontsize=10,color="white",style="solid",shape="box"];396 -> 625[label="",style="solid", color="burlywood", weight=9];
625 -> 405[label="",style="solid", color="burlywood", weight=3];
112[label="primMinusNatS (Succ wz30000) Zero",fontsize=16,color="black",shape="triangle"];112 -> 122[label="",style="solid", color="black", weight=3];
113[label="Zero",fontsize=16,color="green",shape="box"];114[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];114 -> 123[label="",style="solid", color="black", weight=3];
115[label="Zero",fontsize=16,color="green",shape="box"];116[label="primDivNatS0 (Succ wz30000) wz3100 (primGEqNatS (Succ wz30000) wz3100)",fontsize=16,color="burlywood",shape="box"];626[label="wz3100/Succ wz31000",fontsize=10,color="white",style="solid",shape="box"];116 -> 626[label="",style="solid", color="burlywood", weight=9];
626 -> 124[label="",style="solid", color="burlywood", weight=3];
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627 -> 125[label="",style="solid", color="burlywood", weight=3];
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122[label="Succ wz30000",fontsize=16,color="green",shape="box"];123[label="Zero",fontsize=16,color="green",shape="box"];124[label="primDivNatS0 (Succ wz30000) (Succ wz31000) (primGEqNatS (Succ wz30000) (Succ wz31000))",fontsize=16,color="black",shape="box"];124 -> 133[label="",style="solid", color="black", weight=3];
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127[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];127 -> 136[label="",style="solid", color="black", weight=3];
408 -> 362[label="",style="dashed", color="red", weight=0];
408[label="primModNatS0 (Succ wz21) (Succ wz22) (primGEqNatS wz230 wz240)",fontsize=16,color="magenta"];408 -> 420[label="",style="dashed", color="magenta", weight=3];
408 -> 421[label="",style="dashed", color="magenta", weight=3];
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410[label="primModNatS0 (Succ wz21) (Succ wz22) False",fontsize=16,color="black",shape="box"];410 -> 423[label="",style="solid", color="black", weight=3];
411 -> 409[label="",style="dashed", color="red", weight=0];
411[label="primModNatS0 (Succ wz21) (Succ wz22) True",fontsize=16,color="magenta"];133 -> 505[label="",style="dashed", color="red", weight=0];
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133 -> 508[label="",style="dashed", color="magenta", weight=3];
133 -> 509[label="",style="dashed", color="magenta", weight=3];
134[label="primDivNatS0 (Succ wz30000) Zero True",fontsize=16,color="black",shape="box"];134 -> 147[label="",style="solid", color="black", weight=3];
135[label="primDivNatS0 Zero (Succ wz31000) False",fontsize=16,color="black",shape="box"];135 -> 148[label="",style="solid", color="black", weight=3];
136[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];136 -> 149[label="",style="solid", color="black", weight=3];
420[label="wz240",fontsize=16,color="green",shape="box"];421[label="wz230",fontsize=16,color="green",shape="box"];422 -> 46[label="",style="dashed", color="red", weight=0];
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422 -> 431[label="",style="dashed", color="magenta", weight=3];
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634 -> 542[label="",style="solid", color="burlywood", weight=3];
635[label="wz47/Zero",fontsize=10,color="white",style="solid",shape="box"];505 -> 635[label="",style="solid", color="burlywood", weight=9];
635 -> 543[label="",style="solid", color="burlywood", weight=3];
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148[label="Zero",fontsize=16,color="green",shape="box"];149[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];149 -> 161[label="",style="dashed", color="green", weight=3];
430[label="primMinusNatS (Succ wz21) (Succ wz22)",fontsize=16,color="black",shape="box"];430 -> 436[label="",style="solid", color="black", weight=3];
431[label="Succ wz22",fontsize=16,color="green",shape="box"];542[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS (Succ wz470) wz48)",fontsize=16,color="burlywood",shape="box"];636[label="wz48/Succ wz480",fontsize=10,color="white",style="solid",shape="box"];542 -> 636[label="",style="solid", color="burlywood", weight=9];
636 -> 544[label="",style="solid", color="burlywood", weight=3];
637[label="wz48/Zero",fontsize=10,color="white",style="solid",shape="box"];542 -> 637[label="",style="solid", color="burlywood", weight=9];
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639[label="wz48/Zero",fontsize=10,color="white",style="solid",shape="box"];543 -> 639[label="",style="solid", color="burlywood", weight=9];
639 -> 547[label="",style="solid", color="burlywood", weight=3];
160 -> 87[label="",style="dashed", color="red", weight=0];
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160 -> 173[label="",style="dashed", color="magenta", weight=3];
161 -> 87[label="",style="dashed", color="red", weight=0];
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161 -> 175[label="",style="dashed", color="magenta", weight=3];
436[label="primMinusNatS wz21 wz22",fontsize=16,color="burlywood",shape="triangle"];642[label="wz21/Succ wz210",fontsize=10,color="white",style="solid",shape="box"];436 -> 642[label="",style="solid", color="burlywood", weight=9];
642 -> 440[label="",style="solid", color="burlywood", weight=3];
643[label="wz21/Zero",fontsize=10,color="white",style="solid",shape="box"];436 -> 643[label="",style="solid", color="burlywood", weight=9];
643 -> 441[label="",style="solid", color="burlywood", weight=3];
544[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS (Succ wz470) (Succ wz480))",fontsize=16,color="black",shape="box"];544 -> 548[label="",style="solid", color="black", weight=3];
545[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS (Succ wz470) Zero)",fontsize=16,color="black",shape="box"];545 -> 549[label="",style="solid", color="black", weight=3];
546[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS Zero (Succ wz480))",fontsize=16,color="black",shape="box"];546 -> 550[label="",style="solid", color="black", weight=3];
547[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];547 -> 551[label="",style="solid", color="black", weight=3];
172 -> 112[label="",style="dashed", color="red", weight=0];
172[label="primMinusNatS (Succ wz30000) Zero",fontsize=16,color="magenta"];173[label="Zero",fontsize=16,color="green",shape="box"];174 -> 114[label="",style="dashed", color="red", weight=0];
174[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];175[label="Zero",fontsize=16,color="green",shape="box"];440[label="primMinusNatS (Succ wz210) wz22",fontsize=16,color="burlywood",shape="box"];646[label="wz22/Succ wz220",fontsize=10,color="white",style="solid",shape="box"];440 -> 646[label="",style="solid", color="burlywood", weight=9];
646 -> 445[label="",style="solid", color="burlywood", weight=3];
647[label="wz22/Zero",fontsize=10,color="white",style="solid",shape="box"];440 -> 647[label="",style="solid", color="burlywood", weight=9];
647 -> 446[label="",style="solid", color="burlywood", weight=3];
441[label="primMinusNatS Zero wz22",fontsize=16,color="burlywood",shape="box"];648[label="wz22/Succ wz220",fontsize=10,color="white",style="solid",shape="box"];441 -> 648[label="",style="solid", color="burlywood", weight=9];
648 -> 447[label="",style="solid", color="burlywood", weight=3];
649[label="wz22/Zero",fontsize=10,color="white",style="solid",shape="box"];441 -> 649[label="",style="solid", color="burlywood", weight=9];
649 -> 448[label="",style="solid", color="burlywood", weight=3];
548 -> 505[label="",style="dashed", color="red", weight=0];
548[label="primDivNatS0 (Succ wz45) (Succ wz46) (primGEqNatS wz470 wz480)",fontsize=16,color="magenta"];548 -> 552[label="",style="dashed", color="magenta", weight=3];
548 -> 553[label="",style="dashed", color="magenta", weight=3];
549[label="primDivNatS0 (Succ wz45) (Succ wz46) True",fontsize=16,color="black",shape="triangle"];549 -> 554[label="",style="solid", color="black", weight=3];
550[label="primDivNatS0 (Succ wz45) (Succ wz46) False",fontsize=16,color="black",shape="box"];550 -> 555[label="",style="solid", color="black", weight=3];
551 -> 549[label="",style="dashed", color="red", weight=0];
551[label="primDivNatS0 (Succ wz45) (Succ wz46) True",fontsize=16,color="magenta"];445[label="primMinusNatS (Succ wz210) (Succ wz220)",fontsize=16,color="black",shape="box"];445 -> 451[label="",style="solid", color="black", weight=3];
446[label="primMinusNatS (Succ wz210) Zero",fontsize=16,color="black",shape="box"];446 -> 452[label="",style="solid", color="black", weight=3];
447[label="primMinusNatS Zero (Succ wz220)",fontsize=16,color="black",shape="box"];447 -> 453[label="",style="solid", color="black", weight=3];
448[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];448 -> 454[label="",style="solid", color="black", weight=3];
552[label="wz480",fontsize=16,color="green",shape="box"];553[label="wz470",fontsize=16,color="green",shape="box"];554[label="Succ (primDivNatS (primMinusNatS (Succ wz45) (Succ wz46)) (Succ (Succ wz46)))",fontsize=16,color="green",shape="box"];554 -> 556[label="",style="dashed", color="green", weight=3];
555[label="Zero",fontsize=16,color="green",shape="box"];451 -> 436[label="",style="dashed", color="red", weight=0];
451[label="primMinusNatS wz210 wz220",fontsize=16,color="magenta"];451 -> 491[label="",style="dashed", color="magenta", weight=3];
451 -> 492[label="",style="dashed", color="magenta", weight=3];
452[label="Succ wz210",fontsize=16,color="green",shape="box"];453[label="Zero",fontsize=16,color="green",shape="box"];454[label="Zero",fontsize=16,color="green",shape="box"];556 -> 87[label="",style="dashed", color="red", weight=0];
556[label="primDivNatS (primMinusNatS (Succ wz45) (Succ wz46)) (Succ (Succ wz46))",fontsize=16,color="magenta"];556 -> 557[label="",style="dashed", color="magenta", weight=3];
556 -> 558[label="",style="dashed", color="magenta", weight=3];
491[label="wz220",fontsize=16,color="green",shape="box"];492[label="wz210",fontsize=16,color="green",shape="box"];557 -> 436[label="",style="dashed", color="red", weight=0];
557[label="primMinusNatS (Succ wz45) (Succ wz46)",fontsize=16,color="magenta"];557 -> 559[label="",style="dashed", color="magenta", weight=3];
557 -> 560[label="",style="dashed", color="magenta", weight=3];
558[label="Succ wz46",fontsize=16,color="green",shape="box"];559[label="Succ wz46",fontsize=16,color="green",shape="box"];560[label="Succ wz45",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                          ↳ QDP

                          ↳ QDP

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

new_primMinusNatS(Succ(wz210), Succ(wz220)) → new_primMinusNatS(wz210, wz220)

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(wz210), Succ(wz220)) → new_primMinusNatS(wz210, wz220)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                          ↳ QDP

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

new_primModNatS0(wz21, wz22, Succ(wz230), Zero) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS00(wz21, wz22) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS(Succ(Zero), Zero) → new_primModNatS(new_primMinusNatS1, Zero)
new_primModNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primModNatS0(wz30000, wz31000, wz30000, wz31000)
new_primModNatS(Succ(Succ(wz30000)), Zero) → new_primModNatS(new_primMinusNatS0(wz30000), Zero)
new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)
new_primModNatS0(wz21, wz22, Zero, Zero) → new_primModNatS00(wz21, wz22)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)
new_primMinusNatS1 → Zero
new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

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

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ UsableRulesProof

                                ↳ QDP

                          ↳ QDP

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

new_primModNatS(Succ(Succ(wz30000)), Zero) → new_primModNatS(new_primMinusNatS0(wz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)
new_primMinusNatS1 → Zero
new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

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

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                ↳ QDP

                          ↳ QDP

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

new_primModNatS(Succ(Succ(wz30000)), Zero) → new_primModNatS(new_primMinusNatS0(wz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ RuleRemovalProof

                                ↳ QDP

                          ↳ QDP

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

new_primModNatS(Succ(Succ(wz30000)), Zero) → new_primModNatS(new_primMinusNatS0(wz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

new_primMinusNatS0(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(wz30000)), Zero) → new_primModNatS(new_primMinusNatS0(wz30000), Zero)

Strictly oriented rules of the TRS R:

new_primMinusNatS0(wz30000) → Succ(wz30000)

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ RuleRemovalProof

                                            ↳ QDP

                                              ↳ PisEmptyProof

                                ↳ QDP

                          ↳ QDP

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

new_primMinusNatS0(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

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                          ↳ QDP

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

new_primModNatS0(wz21, wz22, Succ(wz230), Zero) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS00(wz21, wz22) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primModNatS0(wz30000, wz31000, wz30000, wz31000)
new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)
new_primModNatS0(wz21, wz22, Zero, Zero) → new_primModNatS00(wz21, wz22)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)
new_primMinusNatS1 → Zero
new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                          ↳ QDP

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

new_primModNatS0(wz21, wz22, Succ(wz230), Zero) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS00(wz21, wz22) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primModNatS0(wz30000, wz31000, wz30000, wz31000)
new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)
new_primModNatS0(wz21, wz22, Zero, Zero) → new_primModNatS00(wz21, wz22)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                          ↳ QDP

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

new_primModNatS0(wz21, wz22, Succ(wz230), Zero) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS00(wz21, wz22) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primModNatS0(wz30000, wz31000, wz30000, wz31000)
new_primModNatS0(wz21, wz22, Zero, Zero) → new_primModNatS00(wz21, wz22)
new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(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_primModNatS0(wz21, wz22, Succ(wz230), Zero) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primModNatS0(wz30000, wz31000, wz30000, wz31000)
new_primModNatS0(wz21, wz22, Zero, Zero) → new_primModNatS00(wz21, wz22)

The remaining pairs can at least be oriented weakly.

new_primModNatS00(wz21, wz22) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)

Used ordering: Polynomial interpretation [25]:

POL(Succ(x₁)) = 1 + x₁
POL(Zero) = 0
POL(new_primMinusNatS2(x₁, x₂)) = x₁
POL(new_primModNatS(x₁, x₂)) = x₁
POL(new_primModNatS0(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(new_primModNatS00(x₁, x₂)) = x₁

The following usable rules [17] were oriented:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                          ↳ QDP

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

new_primModNatS00(wz21, wz22) → new_primModNatS(new_primMinusNatS2(wz21, wz22), Succ(wz22))
new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

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

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ UsableRulesProof

                          ↳ QDP

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

new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ UsableRulesProof

                                                    ↳ QDP

                                                      ↳ QReductionProof

                          ↳ QDP

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

new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)

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

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ UsableRulesProof

                                                    ↳ QDP

                                                      ↳ QReductionProof

                                                        ↳ QDP

                                                          ↳ QDPSizeChangeProof

                          ↳ QDP

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

new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)

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

new_primModNatS0(wz21, wz22, Succ(wz230), Succ(wz240)) → new_primModNatS0(wz21, wz22, wz230, wz240)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

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

new_primDivNatS0(wz45, wz46, Zero, Zero) → new_primDivNatS00(wz45, wz46)
new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) → new_primDivNatS0(wz45, wz46, wz470, wz480)
new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
new_primDivNatS(Succ(Succ(wz30000)), Zero) → new_primDivNatS(new_primMinusNatS0(wz30000), Zero)
new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS1, Zero)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)
new_primMinusNatS1 → Zero
new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

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

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ UsableRulesProof

                                ↳ QDP

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

new_primDivNatS(Succ(Succ(wz30000)), Zero) → new_primDivNatS(new_primMinusNatS0(wz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)
new_primMinusNatS1 → Zero
new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                ↳ QDP

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

new_primDivNatS(Succ(Succ(wz30000)), Zero) → new_primDivNatS(new_primMinusNatS0(wz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ 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(wz30000)), Zero) → new_primDivNatS(new_primMinusNatS0(wz30000), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

new_primMinusNatS0(x0)

new_primDivNatS(Succ(Succ(wz30000)), Zero) → new_primDivNatS(new_primMinusNatS0(wz30000), Zero)

Strictly oriented rules of the TRS R:

new_primMinusNatS0(wz30000) → Succ(wz30000)

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_primMinusNatS0(x₁)) = 2 + 2·x₁

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ 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_primMinusNatS0(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

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

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

new_primDivNatS0(wz45, wz46, Zero, Zero) → new_primDivNatS00(wz45, wz46)
new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) → new_primDivNatS0(wz45, wz46, wz470, wz480)
new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)
new_primMinusNatS1 → Zero
new_primMinusNatS0(wz30000) → Succ(wz30000)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

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

new_primDivNatS0(wz45, wz46, Zero, Zero) → new_primDivNatS00(wz45, wz46)
new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) → new_primDivNatS0(wz45, wz46, wz470, wz480)
new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ Rewriting

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

new_primDivNatS0(wz45, wz46, Zero, Zero) → new_primDivNatS00(wz45, wz46)
new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) → new_primDivNatS0(wz45, wz46, wz470, wz480)
new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))
new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46)) at position [0] we obtained the following new rules:

new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(wz45, wz46), Succ(wz46))

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ Rewriting

                                            ↳ QDP

                                              ↳ Rewriting

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

new_primDivNatS0(wz45, wz46, Zero, Zero) → new_primDivNatS00(wz45, wz46)
new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) → new_primDivNatS0(wz45, wz46, wz470, wz480)
new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(wz45, wz46), Succ(wz46))
new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46))

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(wz45), Succ(wz46)), Succ(wz46)) at position [0] we obtained the following new rules:

new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(wz45, wz46), Succ(wz46))

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ Rewriting

                                            ↳ QDP

                                              ↳ Rewriting

                                                ↳ QDP

                                                  ↳ QDPOrderProof

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

new_primDivNatS0(wz45, wz46, Zero, Zero) → new_primDivNatS00(wz45, wz46)
new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(wz45, wz46), Succ(wz46))
new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) → new_primDivNatS0(wz45, wz46, wz470, wz480)
new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(wz45, wz46), Succ(wz46))

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(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(wz45, wz46, Zero, Zero) → new_primDivNatS00(wz45, wz46)
new_primDivNatS0(wz45, wz46, Succ(wz470), Zero) → new_primDivNatS(new_primMinusNatS2(wz45, wz46), Succ(wz46))
new_primDivNatS(Succ(Succ(wz30000)), Succ(wz31000)) → new_primDivNatS0(wz30000, wz31000, wz30000, wz31000)
new_primDivNatS0(wz45, wz46, Succ(wz470), Succ(wz480)) → new_primDivNatS0(wz45, wz46, wz470, wz480)

The remaining pairs can at least be oriented weakly.

new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(wz45, wz46), Succ(wz46))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( Succ(x₁) ) =

/	1	\
\	1	/

/	1	1	\
\	1	0	/

x₁

M( new_primMinusNatS2(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

/	0	0	\
\	1	1	/

x₂

M( Zero ) =

/	0	\
\	0	/

Tuple symbols:

M( new_primDivNatS00(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( new_primDivNatS(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( new_primDivNatS0(x₁, ..., x₄) ) =

[

]

x₁

[

]

x₂

[

]

x₃

[

]

x₄

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

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

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ UsableRulesProof

                                    ↳ QDP

                                      ↳ QReductionProof

                                        ↳ QDP

                                          ↳ Rewriting

                                            ↳ QDP

                                              ↳ Rewriting

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

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

new_primDivNatS00(wz45, wz46) → new_primDivNatS(new_primMinusNatS2(wz45, wz46), Succ(wz46))

The TRS R consists of the following rules:

new_primMinusNatS2(Succ(wz210), Succ(wz220)) → new_primMinusNatS2(wz210, wz220)
new_primMinusNatS2(Zero, Succ(wz220)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(wz210), Zero) → Succ(wz210)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

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