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

MAYBE 3.099 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could not be shown:

↳ HASKELL

  ↳ BR

mainModule Main

((enumFromThen :: Float -> Float -> [Float]) :: Float -> Float -> [Float])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((enumFromThen :: Float -> Float -> [Float]) :: Float -> Float -> [Float])

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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

          ↳ Narrow

mainModule Main

(enumFromThen :: Float -> Float -> [Float])

module Main where

import qualified Prelude

Haskell To QDPs

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319 -> 71[label="",style="solid", color="burlywood", weight=3];
320[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 320[label="",style="solid", color="burlywood", weight=9];
320 -> 72[label="",style="solid", color="burlywood", weight=3];
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81 -> 45[label="",style="dashed", color="red", weight=0];
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81 -> 96[label="",style="dashed", color="magenta", weight=3];
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326 -> 97[label="",style="solid", color="burlywood", weight=3];
327[label="vx50/Neg vx500",fontsize=10,color="white",style="solid",shape="box"];82 -> 327[label="",style="solid", color="burlywood", weight=9];
327 -> 98[label="",style="solid", color="burlywood", weight=3];
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328 -> 99[label="",style="solid", color="burlywood", weight=3];
329[label="vx50/Neg vx500",fontsize=10,color="white",style="solid",shape="box"];83 -> 329[label="",style="solid", color="burlywood", weight=9];
329 -> 100[label="",style="solid", color="burlywood", weight=3];
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330 -> 101[label="",style="solid", color="burlywood", weight=3];
331[label="vx50/Neg vx500",fontsize=10,color="white",style="solid",shape="box"];84 -> 331[label="",style="solid", color="burlywood", weight=9];
331 -> 102[label="",style="solid", color="burlywood", weight=3];
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332 -> 103[label="",style="solid", color="burlywood", weight=3];
333[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];85 -> 333[label="",style="solid", color="burlywood", weight=9];
333 -> 104[label="",style="solid", color="burlywood", weight=3];
86[label="primMinusNat (primPlusNat (Succ vx4000) vx300) vx500",fontsize=16,color="burlywood",shape="box"];334[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];86 -> 334[label="",style="solid", color="burlywood", weight=9];
334 -> 105[label="",style="solid", color="burlywood", weight=3];
335[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];86 -> 335[label="",style="solid", color="burlywood", weight=9];
335 -> 106[label="",style="solid", color="burlywood", weight=3];
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336 -> 107[label="",style="solid", color="burlywood", weight=3];
337[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];87 -> 337[label="",style="solid", color="burlywood", weight=9];
337 -> 108[label="",style="solid", color="burlywood", weight=3];
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400[label="vx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];189 -> 400[label="",style="solid", color="burlywood", weight=9];
400 -> 217[label="",style="solid", color="burlywood", weight=3];
190[label="Succ vx5000",fontsize=16,color="green",shape="box"];191[label="Succ (primPlusNat vx4000 vx3000)",fontsize=16,color="green",shape="box"];191 -> 218[label="",style="dashed", color="green", weight=3];
192[label="Succ vx5000",fontsize=16,color="green",shape="box"];193[label="vx4000",fontsize=16,color="green",shape="box"];194[label="Succ vx5000",fontsize=16,color="green",shape="box"];195[label="Zero",fontsize=16,color="green",shape="box"];196[label="vx5000",fontsize=16,color="green",shape="box"];197[label="Zero",fontsize=16,color="green",shape="box"];198[label="Succ (primPlusNat vx4000 vx3000)",fontsize=16,color="green",shape="box"];198 -> 219[label="",style="dashed", color="green", weight=3];
199[label="Zero",fontsize=16,color="green",shape="box"];200[label="vx4000",fontsize=16,color="green",shape="box"];201[label="Zero",fontsize=16,color="green",shape="box"];202[label="Zero",fontsize=16,color="green",shape="box"];276 -> 189[label="",style="dashed", color="red", weight=0];
276[label="primPlusNat (primMulNat vx60 (Succ vx5100)) (Succ vx5100)",fontsize=16,color="magenta"];276 -> 282[label="",style="dashed", color="magenta", weight=3];
276 -> 283[label="",style="dashed", color="magenta", weight=3];
277[label="Zero",fontsize=16,color="green",shape="box"];278[label="Zero",fontsize=16,color="green",shape="box"];279[label="Zero",fontsize=16,color="green",shape="box"];207[label="Succ (Succ (primPlusNat vx4000 vx5000))",fontsize=16,color="green",shape="box"];207 -> 224[label="",style="dashed", color="green", weight=3];
208[label="Succ vx4000",fontsize=16,color="green",shape="box"];209[label="primMinusNat (Succ vx40000) vx5000",fontsize=16,color="burlywood",shape="box"];402[label="vx5000/Succ vx50000",fontsize=10,color="white",style="solid",shape="box"];209 -> 402[label="",style="solid", color="burlywood", weight=9];
402 -> 225[label="",style="solid", color="burlywood", weight=3];
403[label="vx5000/Zero",fontsize=10,color="white",style="solid",shape="box"];209 -> 403[label="",style="solid", color="burlywood", weight=9];
403 -> 226[label="",style="solid", color="burlywood", weight=3];
210[label="primMinusNat Zero vx5000",fontsize=16,color="burlywood",shape="box"];404[label="vx5000/Succ vx50000",fontsize=10,color="white",style="solid",shape="box"];210 -> 404[label="",style="solid", color="burlywood", weight=9];
404 -> 227[label="",style="solid", color="burlywood", weight=3];
405[label="vx5000/Zero",fontsize=10,color="white",style="solid",shape="box"];210 -> 405[label="",style="solid", color="burlywood", weight=9];
405 -> 228[label="",style="solid", color="burlywood", weight=3];
211[label="vx3000",fontsize=16,color="green",shape="box"];212[label="vx5000",fontsize=16,color="green",shape="box"];213[label="Succ vx5000",fontsize=16,color="green",shape="box"];214[label="Zero",fontsize=16,color="green",shape="box"];215 -> 189[label="",style="dashed", color="red", weight=0];
215[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];216[label="primPlusNat (Succ vx40000) vx3000",fontsize=16,color="burlywood",shape="box"];407[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];216 -> 407[label="",style="solid", color="burlywood", weight=9];
407 -> 229[label="",style="solid", color="burlywood", weight=3];
408[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];216 -> 408[label="",style="solid", color="burlywood", weight=9];
408 -> 230[label="",style="solid", color="burlywood", weight=3];
217[label="primPlusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];409[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];217 -> 409[label="",style="solid", color="burlywood", weight=9];
409 -> 231[label="",style="solid", color="burlywood", weight=3];
410[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];217 -> 410[label="",style="solid", color="burlywood", weight=9];
410 -> 232[label="",style="solid", color="burlywood", weight=3];
218 -> 189[label="",style="dashed", color="red", weight=0];
218[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];218 -> 233[label="",style="dashed", color="magenta", weight=3];
218 -> 234[label="",style="dashed", color="magenta", weight=3];
219 -> 189[label="",style="dashed", color="red", weight=0];
219[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];219 -> 235[label="",style="dashed", color="magenta", weight=3];
219 -> 236[label="",style="dashed", color="magenta", weight=3];
282[label="Succ vx5100",fontsize=16,color="green",shape="box"];283 -> 237[label="",style="dashed", color="red", weight=0];
283[label="primMulNat vx60 (Succ vx5100)",fontsize=16,color="magenta"];283 -> 284[label="",style="dashed", color="magenta", weight=3];
283 -> 285[label="",style="dashed", color="magenta", weight=3];
224 -> 189[label="",style="dashed", color="red", weight=0];
224[label="primPlusNat vx4000 vx5000",fontsize=16,color="magenta"];224 -> 260[label="",style="dashed", color="magenta", weight=3];
225[label="primMinusNat (Succ vx40000) (Succ vx50000)",fontsize=16,color="black",shape="box"];225 -> 261[label="",style="solid", color="black", weight=3];
226[label="primMinusNat (Succ vx40000) Zero",fontsize=16,color="black",shape="box"];226 -> 262[label="",style="solid", color="black", weight=3];
227[label="primMinusNat Zero (Succ vx50000)",fontsize=16,color="black",shape="box"];227 -> 263[label="",style="solid", color="black", weight=3];
228[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];228 -> 264[label="",style="solid", color="black", weight=3];
229[label="primPlusNat (Succ vx40000) (Succ vx30000)",fontsize=16,color="black",shape="box"];229 -> 265[label="",style="solid", color="black", weight=3];
230[label="primPlusNat (Succ vx40000) Zero",fontsize=16,color="black",shape="box"];230 -> 266[label="",style="solid", color="black", weight=3];
231[label="primPlusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];231 -> 267[label="",style="solid", color="black", weight=3];
232[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];232 -> 268[label="",style="solid", color="black", weight=3];
233[label="vx3000",fontsize=16,color="green",shape="box"];234[label="vx4000",fontsize=16,color="green",shape="box"];235[label="vx3000",fontsize=16,color="green",shape="box"];236[label="vx4000",fontsize=16,color="green",shape="box"];284[label="vx60",fontsize=16,color="green",shape="box"];285[label="Succ vx5100",fontsize=16,color="green",shape="box"];260[label="vx5000",fontsize=16,color="green",shape="box"];261 -> 177[label="",style="dashed", color="red", weight=0];
261[label="primMinusNat vx40000 vx50000",fontsize=16,color="magenta"];261 -> 273[label="",style="dashed", color="magenta", weight=3];
261 -> 274[label="",style="dashed", color="magenta", weight=3];
262[label="Pos (Succ vx40000)",fontsize=16,color="green",shape="box"];263[label="Neg (Succ vx50000)",fontsize=16,color="green",shape="box"];264[label="Pos Zero",fontsize=16,color="green",shape="box"];265[label="Succ (Succ (primPlusNat vx40000 vx30000))",fontsize=16,color="green",shape="box"];265 -> 275[label="",style="dashed", color="green", weight=3];
266[label="Succ vx40000",fontsize=16,color="green",shape="box"];267[label="Succ vx30000",fontsize=16,color="green",shape="box"];268[label="Zero",fontsize=16,color="green",shape="box"];273[label="vx50000",fontsize=16,color="green",shape="box"];274[label="vx40000",fontsize=16,color="green",shape="box"];275 -> 189[label="",style="dashed", color="red", weight=0];
275[label="primPlusNat vx40000 vx30000",fontsize=16,color="magenta"];275 -> 280[label="",style="dashed", color="magenta", weight=3];
275 -> 281[label="",style="dashed", color="magenta", weight=3];
280[label="vx30000",fontsize=16,color="green",shape="box"];281[label="vx40000",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

          ↳ Narrow

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

new_primMinusNat(Succ(vx40000), Succ(vx50000)) → new_primMinusNat(vx40000, vx50000)

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_primMinusNat(Succ(vx40000), Succ(vx50000)) → new_primMinusNat(vx40000, vx50000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

          ↳ Narrow

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

new_primPlusNat(Succ(vx40000), Succ(vx30000)) → new_primPlusNat(vx40000, vx30000)

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

new_primPlusNat(Succ(vx40000), Succ(vx30000)) → new_primPlusNat(vx40000, vx30000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

          ↳ Narrow

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

new_primMulNat(Succ(vx60), Succ(vx5100)) → new_primMulNat(vx60, Succ(vx5100))

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

new_primMulNat(Succ(vx60), Succ(vx5100)) → new_primMulNat(vx60, Succ(vx5100))
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

          ↳ Narrow

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

new_primPlusInt(Succ(vx4000), Succ(vx3000), vx50) → new_primPlusInt(vx4000, vx3000, vx50)

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

new_primPlusInt(Succ(vx4000), Succ(vx3000), vx50) → new_primPlusInt(vx4000, vx3000, vx50)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ MNOCProof

          ↳ Narrow

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

new_iterate(vx4, vx3, vx5) → new_iterate(vx4, vx3, new_ps(vx4, vx3, vx5))

The TRS R consists of the following rules:

new_primMulInt0(vx410, vx310, Pos(vx510)) → Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primPlusInt0(Neg(Zero), Pos(Zero), Pos(Zero)) → new_primMinusNat3(Zero)
new_primMulNat0(Succ(vx60), Succ(vx5100)) → new_primPlusNat2(new_primMulNat0(vx60, Succ(vx5100)), Succ(vx5100))
new_primPlusInt1(Zero, Zero, Pos(vx500)) → Pos(new_primPlusNat3(vx500))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Zero), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), vx4000)
new_primPlusInt0(Neg(Zero), Pos(Succ(vx3000)), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), vx3000)
new_primPlusNat0(Zero, Succ(vx3000), vx500) → new_primPlusNat1(vx3000, vx500)
new_primPlusNat0(Succ(vx4000), Zero, vx500) → new_primPlusNat1(vx4000, vx500)
new_primMinusNat0(vx4000, Zero) → Pos(Succ(vx4000))
new_primMulInt(Neg(vx410), Neg(vx310), vx51) → new_primMulInt0(vx410, vx310, vx51)
new_primMinusNat1(Succ(vx40000), Zero) → Pos(Succ(vx40000))
new_primPlusNat1(vx4000, Zero) → Succ(vx4000)
new_primPlusInt0(Pos(vx400), Pos(vx300), vx50) → new_primPlusInt1(vx400, vx300, vx50)
new_primMulNat0(Succ(vx60), Zero) → Zero
new_primMulNat0(Zero, Succ(vx5100)) → Zero
new_primMinusNat2(Zero, vx3000) → Neg(Succ(vx3000))
new_primPlusInt1(Succ(vx4000), Zero, Neg(vx500)) → new_primMinusNat0(vx4000, vx500)
new_primPlusInt0(Neg(vx400), Pos(vx300), Neg(vx500)) → Neg(new_primPlusNat0(vx400, vx300, vx500))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Succ(vx3000)), Pos(Zero)) → new_primMinusNat2(Zero, Succ(new_primPlusNat2(vx4000, vx3000)))
new_primPlusInt1(Zero, Succ(vx3000), Pos(vx500)) → new_primMinusNat2(vx500, vx3000)
new_primPlusInt1(Zero, Succ(vx3000), Neg(vx500)) → Neg(new_primPlusNat1(vx3000, vx500))
new_primPlusInt0(Neg(Zero), Pos(Zero), Pos(Succ(vx5000))) → new_primMinusNat0(vx5000, Zero)
new_primMulInt0(vx410, vx310, Neg(vx510)) → Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Succ(vx3000)), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), Succ(new_primPlusNat2(vx4000, vx3000)))
new_primMinusNat1(Zero, Succ(vx50000)) → Neg(Succ(vx50000))
new_primPlusNat3(Zero) → Zero
new_primPlusInt0(Pos(vx400), Neg(vx300), Pos(vx500)) → Pos(new_primPlusNat0(vx400, vx300, vx500))
new_primMinusNat1(Zero, Zero) → Pos(Zero)
new_primMulNat0(Zero, Zero) → Zero
new_primMinusNat3(Succ(vx5000)) → Neg(Succ(vx5000))
new_primMinusNat3(Zero) → Pos(Zero)
new_primMinusNat0(vx4000, Succ(vx5000)) → new_primMinusNat1(vx4000, vx5000)
new_primPlusInt0(Neg(Zero), Pos(Succ(vx3000)), Pos(Zero)) → new_primMinusNat2(Zero, vx3000)
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Zero), Pos(Zero)) → new_primMinusNat2(Zero, vx4000)
new_primPlusNat2(Succ(vx40000), Zero) → Succ(vx40000)
new_primPlusNat2(Zero, Succ(vx30000)) → Succ(vx30000)
new_primPlusNat3(Succ(vx5000)) → Succ(vx5000)
new_primPlusInt0(Neg(vx400), Neg(vx300), vx50) → new_primPlusInt1(vx300, vx400, vx50)
new_primPlusInt0(Pos(Succ(vx4000)), Neg(Succ(vx3000)), Neg(vx500)) → new_primMinusNat0(Succ(new_primPlusNat2(vx4000, vx3000)), vx500)
new_primPlusInt0(Pos(Zero), Neg(Zero), Neg(vx500)) → new_primMinusNat3(vx500)
new_primPlusInt1(Succ(vx4000), Zero, Pos(vx500)) → Pos(new_primPlusNat1(vx4000, vx500))
new_primPlusNat1(vx4000, Succ(vx5000)) → Succ(Succ(new_primPlusNat2(vx4000, vx5000)))
new_primMulInt(Pos(vx410), Pos(vx310), vx51) → new_primMulInt0(vx410, vx310, vx51)
new_primMinusNat2(Succ(vx5000), vx3000) → new_primMinusNat1(vx5000, vx3000)
new_primPlusNat2(Zero, Zero) → Zero
new_primPlusNat2(Succ(vx40000), Succ(vx30000)) → Succ(Succ(new_primPlusNat2(vx40000, vx30000)))
new_primMinusNat1(Succ(vx40000), Succ(vx50000)) → new_primMinusNat1(vx40000, vx50000)
new_primPlusNat0(Succ(vx4000), Succ(vx3000), vx500) → new_primPlusNat1(Succ(new_primPlusNat2(vx4000, vx3000)), vx500)
new_primPlusInt0(Pos(Succ(vx4000)), Neg(Zero), Neg(vx500)) → new_primMinusNat0(vx4000, vx500)
new_primPlusInt0(Pos(Zero), Neg(Succ(vx3000)), Neg(vx500)) → new_primMinusNat0(vx3000, vx500)
new_primPlusInt1(Zero, Zero, Neg(vx500)) → new_primMinusNat3(vx500)
new_primMulInt1(vx410, vx310, Neg(vx510)) → Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_ps(Float(vx40, vx41), Float(vx30, vx31), Float(vx50, vx51)) → Float(new_primPlusInt0(vx40, vx30, vx50), new_primMulInt(vx41, vx31, vx51))
new_primPlusInt1(Succ(vx4000), Succ(vx3000), vx50) → new_primPlusInt1(vx4000, vx3000, vx50)
new_primPlusNat0(Zero, Zero, vx500) → new_primPlusNat3(vx500)
new_primMulInt1(vx410, vx310, Pos(vx510)) → Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primMulInt(Neg(vx410), Pos(vx310), vx51) → new_primMulInt1(vx410, vx310, vx51)
new_primMulInt(Pos(vx410), Neg(vx310), vx51) → new_primMulInt1(vx410, vx310, vx51)

The set Q consists of the following terms:

new_primPlusInt0(Pos(Succ(x0)), Neg(Succ(x1)), Neg(x2))
new_primPlusInt0(Neg(Succ(x0)), Pos(Succ(x1)), Pos(Zero))
new_primPlusNat3(Succ(x0))
new_primMulInt(Pos(x0), Pos(x1), x2)
new_primMulNat0(Zero, Succ(x0))
new_primPlusInt1(Zero, Succ(x0), Pos(x1))
new_primPlusInt1(Zero, Zero, Pos(x0))
new_primMulNat0(Zero, Zero)
new_primPlusInt0(Pos(Zero), Neg(Succ(x0)), Neg(x1))
new_primPlusInt1(Succ(x0), Zero, Pos(x1))
new_primPlusInt0(Neg(x0), Pos(x1), Neg(x2))
new_primMinusNat1(Succ(x0), Zero)
new_primMulInt(Neg(x0), Pos(x1), x2)
new_primMulInt(Pos(x0), Neg(x1), x2)
new_primMulInt1(x0, x1, Pos(x2))
new_primPlusInt0(Pos(x0), Neg(x1), Pos(x2))
new_primMinusNat0(x0, Zero)
new_primPlusInt1(Zero, Zero, Neg(x0))
new_primPlusInt0(Pos(Succ(x0)), Neg(Zero), Neg(x1))
new_primMulInt0(x0, x1, Neg(x2))
new_primPlusNat0(Succ(x0), Succ(x1), x2)
new_primPlusInt0(Neg(Succ(x0)), Pos(Zero), Pos(Succ(x1)))
new_primPlusNat2(Zero, Zero)
new_primPlusInt1(Succ(x0), Succ(x1), x2)
new_primMulInt1(x0, x1, Neg(x2))
new_primPlusNat2(Zero, Succ(x0))
new_primPlusInt0(Neg(x0), Neg(x1), x2)
new_primPlusNat1(x0, Succ(x1))
new_primPlusNat2(Succ(x0), Zero)
new_primMinusNat2(Zero, x0)
new_primPlusNat2(Succ(x0), Succ(x1))
new_primMinusNat1(Succ(x0), Succ(x1))
new_primPlusNat0(Zero, Succ(x0), x1)
new_primPlusNat1(x0, Zero)
new_primMinusNat1(Zero, Zero)
new_primPlusInt0(Neg(Zero), Pos(Zero), Pos(Zero))
new_primPlusInt0(Neg(Zero), Pos(Succ(x0)), Pos(Zero))
new_primMulInt0(x0, x1, Pos(x2))
new_primMinusNat1(Zero, Succ(x0))
new_primMinusNat0(x0, Succ(x1))
new_ps(Float(x0, x1), Float(x2, x3), Float(x4, x5))
new_primPlusInt0(Neg(Zero), Pos(Zero), Pos(Succ(x0)))
new_primPlusInt1(Succ(x0), Zero, Neg(x1))
new_primMinusNat2(Succ(x0), x1)
new_primPlusInt0(Neg(Succ(x0)), Pos(Succ(x1)), Pos(Succ(x2)))
new_primPlusInt0(Neg(Zero), Pos(Succ(x0)), Pos(Succ(x1)))
new_primPlusInt1(Zero, Succ(x0), Neg(x1))
new_primMulNat0(Succ(x0), Zero)
new_primPlusNat3(Zero)
new_primMulNat0(Succ(x0), Succ(x1))
new_primMinusNat3(Succ(x0))
new_primMulInt(Neg(x0), Neg(x1), x2)
new_primPlusNat0(Zero, Zero, x0)
new_primPlusInt0(Pos(x0), Pos(x1), x2)
new_primPlusInt0(Pos(Zero), Neg(Zero), Neg(x0))
new_primPlusInt0(Neg(Succ(x0)), Pos(Zero), Pos(Zero))
new_primMinusNat3(Zero)
new_primPlusNat0(Succ(x0), Zero, x1)

We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ MNOCProof

                  ↳ QDP

                    ↳ NonTerminationProof

          ↳ Narrow

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

new_iterate(vx4, vx3, vx5) → new_iterate(vx4, vx3, new_ps(vx4, vx3, vx5))

The TRS R consists of the following rules:

new_primMulInt0(vx410, vx310, Pos(vx510)) → Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primPlusInt0(Neg(Zero), Pos(Zero), Pos(Zero)) → new_primMinusNat3(Zero)
new_primMulNat0(Succ(vx60), Succ(vx5100)) → new_primPlusNat2(new_primMulNat0(vx60, Succ(vx5100)), Succ(vx5100))
new_primPlusInt1(Zero, Zero, Pos(vx500)) → Pos(new_primPlusNat3(vx500))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Zero), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), vx4000)
new_primPlusInt0(Neg(Zero), Pos(Succ(vx3000)), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), vx3000)
new_primPlusNat0(Zero, Succ(vx3000), vx500) → new_primPlusNat1(vx3000, vx500)
new_primPlusNat0(Succ(vx4000), Zero, vx500) → new_primPlusNat1(vx4000, vx500)
new_primMinusNat0(vx4000, Zero) → Pos(Succ(vx4000))
new_primMulInt(Neg(vx410), Neg(vx310), vx51) → new_primMulInt0(vx410, vx310, vx51)
new_primMinusNat1(Succ(vx40000), Zero) → Pos(Succ(vx40000))
new_primPlusNat1(vx4000, Zero) → Succ(vx4000)
new_primPlusInt0(Pos(vx400), Pos(vx300), vx50) → new_primPlusInt1(vx400, vx300, vx50)
new_primMulNat0(Succ(vx60), Zero) → Zero
new_primMulNat0(Zero, Succ(vx5100)) → Zero
new_primMinusNat2(Zero, vx3000) → Neg(Succ(vx3000))
new_primPlusInt1(Succ(vx4000), Zero, Neg(vx500)) → new_primMinusNat0(vx4000, vx500)
new_primPlusInt0(Neg(vx400), Pos(vx300), Neg(vx500)) → Neg(new_primPlusNat0(vx400, vx300, vx500))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Succ(vx3000)), Pos(Zero)) → new_primMinusNat2(Zero, Succ(new_primPlusNat2(vx4000, vx3000)))
new_primPlusInt1(Zero, Succ(vx3000), Pos(vx500)) → new_primMinusNat2(vx500, vx3000)
new_primPlusInt1(Zero, Succ(vx3000), Neg(vx500)) → Neg(new_primPlusNat1(vx3000, vx500))
new_primPlusInt0(Neg(Zero), Pos(Zero), Pos(Succ(vx5000))) → new_primMinusNat0(vx5000, Zero)
new_primMulInt0(vx410, vx310, Neg(vx510)) → Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Succ(vx3000)), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), Succ(new_primPlusNat2(vx4000, vx3000)))
new_primMinusNat1(Zero, Succ(vx50000)) → Neg(Succ(vx50000))
new_primPlusNat3(Zero) → Zero
new_primPlusInt0(Pos(vx400), Neg(vx300), Pos(vx500)) → Pos(new_primPlusNat0(vx400, vx300, vx500))
new_primMinusNat1(Zero, Zero) → Pos(Zero)
new_primMulNat0(Zero, Zero) → Zero
new_primMinusNat3(Succ(vx5000)) → Neg(Succ(vx5000))
new_primMinusNat3(Zero) → Pos(Zero)
new_primMinusNat0(vx4000, Succ(vx5000)) → new_primMinusNat1(vx4000, vx5000)
new_primPlusInt0(Neg(Zero), Pos(Succ(vx3000)), Pos(Zero)) → new_primMinusNat2(Zero, vx3000)
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Zero), Pos(Zero)) → new_primMinusNat2(Zero, vx4000)
new_primPlusNat2(Succ(vx40000), Zero) → Succ(vx40000)
new_primPlusNat2(Zero, Succ(vx30000)) → Succ(vx30000)
new_primPlusNat3(Succ(vx5000)) → Succ(vx5000)
new_primPlusInt0(Neg(vx400), Neg(vx300), vx50) → new_primPlusInt1(vx300, vx400, vx50)
new_primPlusInt0(Pos(Succ(vx4000)), Neg(Succ(vx3000)), Neg(vx500)) → new_primMinusNat0(Succ(new_primPlusNat2(vx4000, vx3000)), vx500)
new_primPlusInt0(Pos(Zero), Neg(Zero), Neg(vx500)) → new_primMinusNat3(vx500)
new_primPlusInt1(Succ(vx4000), Zero, Pos(vx500)) → Pos(new_primPlusNat1(vx4000, vx500))
new_primPlusNat1(vx4000, Succ(vx5000)) → Succ(Succ(new_primPlusNat2(vx4000, vx5000)))
new_primMulInt(Pos(vx410), Pos(vx310), vx51) → new_primMulInt0(vx410, vx310, vx51)
new_primMinusNat2(Succ(vx5000), vx3000) → new_primMinusNat1(vx5000, vx3000)
new_primPlusNat2(Zero, Zero) → Zero
new_primPlusNat2(Succ(vx40000), Succ(vx30000)) → Succ(Succ(new_primPlusNat2(vx40000, vx30000)))
new_primMinusNat1(Succ(vx40000), Succ(vx50000)) → new_primMinusNat1(vx40000, vx50000)
new_primPlusNat0(Succ(vx4000), Succ(vx3000), vx500) → new_primPlusNat1(Succ(new_primPlusNat2(vx4000, vx3000)), vx500)
new_primPlusInt0(Pos(Succ(vx4000)), Neg(Zero), Neg(vx500)) → new_primMinusNat0(vx4000, vx500)
new_primPlusInt0(Pos(Zero), Neg(Succ(vx3000)), Neg(vx500)) → new_primMinusNat0(vx3000, vx500)
new_primPlusInt1(Zero, Zero, Neg(vx500)) → new_primMinusNat3(vx500)
new_primMulInt1(vx410, vx310, Neg(vx510)) → Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_ps(Float(vx40, vx41), Float(vx30, vx31), Float(vx50, vx51)) → Float(new_primPlusInt0(vx40, vx30, vx50), new_primMulInt(vx41, vx31, vx51))
new_primPlusInt1(Succ(vx4000), Succ(vx3000), vx50) → new_primPlusInt1(vx4000, vx3000, vx50)
new_primPlusNat0(Zero, Zero, vx500) → new_primPlusNat3(vx500)
new_primMulInt1(vx410, vx310, Pos(vx510)) → Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primMulInt(Neg(vx410), Pos(vx310), vx51) → new_primMulInt1(vx410, vx310, vx51)
new_primMulInt(Pos(vx410), Neg(vx310), vx51) → new_primMulInt1(vx410, vx310, vx51)

Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_iterate(vx4, vx3, vx5) → new_iterate(vx4, vx3, new_ps(vx4, vx3, vx5))

The TRS R consists of the following rules:

new_primMulInt0(vx410, vx310, Pos(vx510)) → Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primPlusInt0(Neg(Zero), Pos(Zero), Pos(Zero)) → new_primMinusNat3(Zero)
new_primMulNat0(Succ(vx60), Succ(vx5100)) → new_primPlusNat2(new_primMulNat0(vx60, Succ(vx5100)), Succ(vx5100))
new_primPlusInt1(Zero, Zero, Pos(vx500)) → Pos(new_primPlusNat3(vx500))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Zero), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), vx4000)
new_primPlusInt0(Neg(Zero), Pos(Succ(vx3000)), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), vx3000)
new_primPlusNat0(Zero, Succ(vx3000), vx500) → new_primPlusNat1(vx3000, vx500)
new_primPlusNat0(Succ(vx4000), Zero, vx500) → new_primPlusNat1(vx4000, vx500)
new_primMinusNat0(vx4000, Zero) → Pos(Succ(vx4000))
new_primMulInt(Neg(vx410), Neg(vx310), vx51) → new_primMulInt0(vx410, vx310, vx51)
new_primMinusNat1(Succ(vx40000), Zero) → Pos(Succ(vx40000))
new_primPlusNat1(vx4000, Zero) → Succ(vx4000)
new_primPlusInt0(Pos(vx400), Pos(vx300), vx50) → new_primPlusInt1(vx400, vx300, vx50)
new_primMulNat0(Succ(vx60), Zero) → Zero
new_primMulNat0(Zero, Succ(vx5100)) → Zero
new_primMinusNat2(Zero, vx3000) → Neg(Succ(vx3000))
new_primPlusInt1(Succ(vx4000), Zero, Neg(vx500)) → new_primMinusNat0(vx4000, vx500)
new_primPlusInt0(Neg(vx400), Pos(vx300), Neg(vx500)) → Neg(new_primPlusNat0(vx400, vx300, vx500))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Succ(vx3000)), Pos(Zero)) → new_primMinusNat2(Zero, Succ(new_primPlusNat2(vx4000, vx3000)))
new_primPlusInt1(Zero, Succ(vx3000), Pos(vx500)) → new_primMinusNat2(vx500, vx3000)
new_primPlusInt1(Zero, Succ(vx3000), Neg(vx500)) → Neg(new_primPlusNat1(vx3000, vx500))
new_primPlusInt0(Neg(Zero), Pos(Zero), Pos(Succ(vx5000))) → new_primMinusNat0(vx5000, Zero)
new_primMulInt0(vx410, vx310, Neg(vx510)) → Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Succ(vx3000)), Pos(Succ(vx5000))) → new_primMinusNat2(Succ(vx5000), Succ(new_primPlusNat2(vx4000, vx3000)))
new_primMinusNat1(Zero, Succ(vx50000)) → Neg(Succ(vx50000))
new_primPlusNat3(Zero) → Zero
new_primPlusInt0(Pos(vx400), Neg(vx300), Pos(vx500)) → Pos(new_primPlusNat0(vx400, vx300, vx500))
new_primMinusNat1(Zero, Zero) → Pos(Zero)
new_primMulNat0(Zero, Zero) → Zero
new_primMinusNat3(Succ(vx5000)) → Neg(Succ(vx5000))
new_primMinusNat3(Zero) → Pos(Zero)
new_primMinusNat0(vx4000, Succ(vx5000)) → new_primMinusNat1(vx4000, vx5000)
new_primPlusInt0(Neg(Zero), Pos(Succ(vx3000)), Pos(Zero)) → new_primMinusNat2(Zero, vx3000)
new_primPlusInt0(Neg(Succ(vx4000)), Pos(Zero), Pos(Zero)) → new_primMinusNat2(Zero, vx4000)
new_primPlusNat2(Succ(vx40000), Zero) → Succ(vx40000)
new_primPlusNat2(Zero, Succ(vx30000)) → Succ(vx30000)
new_primPlusNat3(Succ(vx5000)) → Succ(vx5000)
new_primPlusInt0(Neg(vx400), Neg(vx300), vx50) → new_primPlusInt1(vx300, vx400, vx50)
new_primPlusInt0(Pos(Succ(vx4000)), Neg(Succ(vx3000)), Neg(vx500)) → new_primMinusNat0(Succ(new_primPlusNat2(vx4000, vx3000)), vx500)
new_primPlusInt0(Pos(Zero), Neg(Zero), Neg(vx500)) → new_primMinusNat3(vx500)
new_primPlusInt1(Succ(vx4000), Zero, Pos(vx500)) → Pos(new_primPlusNat1(vx4000, vx500))
new_primPlusNat1(vx4000, Succ(vx5000)) → Succ(Succ(new_primPlusNat2(vx4000, vx5000)))
new_primMulInt(Pos(vx410), Pos(vx310), vx51) → new_primMulInt0(vx410, vx310, vx51)
new_primMinusNat2(Succ(vx5000), vx3000) → new_primMinusNat1(vx5000, vx3000)
new_primPlusNat2(Zero, Zero) → Zero
new_primPlusNat2(Succ(vx40000), Succ(vx30000)) → Succ(Succ(new_primPlusNat2(vx40000, vx30000)))
new_primMinusNat1(Succ(vx40000), Succ(vx50000)) → new_primMinusNat1(vx40000, vx50000)
new_primPlusNat0(Succ(vx4000), Succ(vx3000), vx500) → new_primPlusNat1(Succ(new_primPlusNat2(vx4000, vx3000)), vx500)
new_primPlusInt0(Pos(Succ(vx4000)), Neg(Zero), Neg(vx500)) → new_primMinusNat0(vx4000, vx500)
new_primPlusInt0(Pos(Zero), Neg(Succ(vx3000)), Neg(vx500)) → new_primMinusNat0(vx3000, vx500)
new_primPlusInt1(Zero, Zero, Neg(vx500)) → new_primMinusNat3(vx500)
new_primMulInt1(vx410, vx310, Neg(vx510)) → Pos(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_ps(Float(vx40, vx41), Float(vx30, vx31), Float(vx50, vx51)) → Float(new_primPlusInt0(vx40, vx30, vx50), new_primMulInt(vx41, vx31, vx51))
new_primPlusInt1(Succ(vx4000), Succ(vx3000), vx50) → new_primPlusInt1(vx4000, vx3000, vx50)
new_primPlusNat0(Zero, Zero, vx500) → new_primPlusNat3(vx500)
new_primMulInt1(vx410, vx310, Pos(vx510)) → Neg(new_primMulNat0(new_primMulNat0(vx410, vx310), vx510))
new_primMulInt(Neg(vx410), Pos(vx310), vx51) → new_primMulInt1(vx410, vx310, vx51)
new_primMulInt(Pos(vx410), Neg(vx310), vx51) → new_primMulInt1(vx410, vx310, vx51)

s = new_iterate(vx4, vx3, vx5) evaluates to t =new_iterate(vx4, vx3, new_ps(vx4, vx3, vx5))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [vx5 / new_ps(vx4, vx3, vx5)]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_iterate(vx4, vx3, vx5) to new_iterate(vx4, vx3, new_ps(vx4, vx3, vx5)).

Haskell To QDPs