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

YES 1.09 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/Monad.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Monad

((replicateM :: Int -> [a] -> [[a]]) :: Int -> [a] -> [[a]])

module Monad where

import qualified Maybe
import qualified Prelude

replicateM :: Monad b => Int -> b a -> b [a]

replicateM

n x

sequence (replicate n x)

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\xs→return (x : xs)

is transformed to

sequence0 x xs = return (x : xs)

The following Lambda expression

\x→sequence cs >>= sequence0 x

is transformed to

sequence1 cs x = sequence cs >>= sequence0 x

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

((replicateM :: Int -> [a] -> [[a]]) :: Int -> [a] -> [[a]])

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

replicateM :: Monad a => Int -> a b -> a [b]

replicateM

n x

sequence (replicate n x)

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

((replicateM :: Int -> [a] -> [[a]]) :: Int -> [a] -> [[a]])

module Monad where

import qualified Maybe
import qualified Prelude

replicateM :: Monad a => Int -> a b -> a [b]

replicateM

n x

sequence (replicate n x)

module Maybe where

import qualified Monad
import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

The following Function with conditions

take n vx

| n <= 0

= []

take vy [] = []

take n (x : xs) = x : take (n - 1) xs

is transformed to

take n vx = take3 n vx

take vy [] = take1 vy []

take n (x : xs) = take0 n (x : xs)

take0 n (x : xs) = x : take (n - 1) xs

take1 vy [] = []

take1 ww wx = take0 ww wx

take2 n vx True = []

take2 n vx False = take1 n vx

take3 n vx = take2 n vx (n <= 0)

take3 wy wz = take1 wy wz

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

mainModule Monad

((replicateM :: Int -> [a] -> [[a]]) :: Int -> [a] -> [[a]])

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

replicateM :: Monad b => Int -> b a -> b [a]

replicateM

n x

sequence (replicate n x)

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

xs

where

xs = x : xs

are unpacked to the following functions on top level

repeatXs xu = xu : repeatXs xu

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

mainModule Monad

((replicateM :: Int -> [a] -> [[a]]) :: Int -> [a] -> [[a]])

module Monad where

import qualified Maybe
import qualified Prelude

replicateM :: Monad a => Int -> a b -> a [b]

replicateM

n x

sequence (replicate n x)

module Maybe where

import qualified Monad
import qualified Prelude

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

                    ↳ HASKELL

                      ↳ Narrow

mainModule Monad

(replicateM :: Int -> [a] -> [[a]])

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

replicateM :: Monad a => Int -> a b -> a [b]

replicateM

n x

sequence (replicate n x)

Haskell To QDPs

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315 -> 12[label="",style="solid", color="burlywood", weight=3];
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317 -> 14[label="",style="solid", color="burlywood", weight=3];
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318 -> 15[label="",style="solid", color="burlywood", weight=3];
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319 -> 16[label="",style="solid", color="burlywood", weight=3];
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16[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not (primCmpInt (Neg (Succ xv300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="sequence (take2 (Neg Zero) (repeat xv4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
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23[label="sequence (take2 (Pos Zero) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="sequence (take2 (Neg Zero) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not True))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="sequence (take2 (Pos Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) True)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="sequence (take2 (Neg Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
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323 -> 42[label="",style="solid", color="burlywood", weight=3];
324[label="xv4/[]",fontsize=10,color="white",style="solid",shape="box"];41 -> 324[label="",style="solid", color="burlywood", weight=9];
324 -> 43[label="",style="solid", color="burlywood", weight=3];
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278[label="xv41",fontsize=16,color="green",shape="box"];279[label="xv40",fontsize=16,color="green",shape="box"];280[label="xv300",fontsize=16,color="green",shape="box"];281[label="xv41",fontsize=16,color="green",shape="box"];277[label="xv13 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="burlywood",shape="triangle"];329[label="xv13/xv130 : xv131",fontsize=10,color="white",style="solid",shape="box"];277 -> 329[label="",style="solid", color="burlywood", weight=9];
329 -> 302[label="",style="solid", color="burlywood", weight=3];
330[label="xv13/[]",fontsize=10,color="white",style="solid",shape="box"];277 -> 330[label="",style="solid", color="burlywood", weight=9];
330 -> 303[label="",style="solid", color="burlywood", weight=3];
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306 -> 277[label="",style="dashed", color="red", weight=0];
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221[label="xv11 ++ xv8",fontsize=16,color="burlywood",shape="triangle"];333[label="xv11/xv110 : xv111",fontsize=10,color="white",style="solid",shape="box"];221 -> 333[label="",style="solid", color="burlywood", weight=9];
333 -> 229[label="",style="solid", color="burlywood", weight=3];
334[label="xv11/[]",fontsize=10,color="white",style="solid",shape="box"];221 -> 334[label="",style="solid", color="burlywood", weight=9];
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201[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];201 -> 206[label="",style="solid", color="black", weight=3];
308[label="xv131",fontsize=16,color="green",shape="box"];309 -> 213[label="",style="dashed", color="red", weight=0];
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309 -> 311[label="",style="dashed", color="magenta", weight=3];
229[label="(xv110 : xv111) ++ xv8",fontsize=16,color="black",shape="box"];229 -> 237[label="",style="solid", color="black", weight=3];
230[label="[] ++ xv8",fontsize=16,color="black",shape="box"];230 -> 238[label="",style="solid", color="black", weight=3];
205 -> 209[label="",style="dashed", color="red", weight=0];
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206[label="sequence (take2 (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];206 -> 211[label="",style="solid", color="black", weight=3];
310 -> 112[label="",style="dashed", color="red", weight=0];
310[label="sequence (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="magenta"];310 -> 312[label="",style="dashed", color="magenta", weight=3];
310 -> 313[label="",style="dashed", color="magenta", weight=3];
310 -> 314[label="",style="dashed", color="magenta", weight=3];
311[label="xv130",fontsize=16,color="green",shape="box"];213[label="xv101 >>= sequence0 xv410",fontsize=16,color="burlywood",shape="triangle"];338[label="xv101/xv1010 : xv1011",fontsize=10,color="white",style="solid",shape="box"];213 -> 338[label="",style="solid", color="burlywood", weight=9];
338 -> 219[label="",style="solid", color="burlywood", weight=3];
339[label="xv101/[]",fontsize=10,color="white",style="solid",shape="box"];213 -> 339[label="",style="solid", color="burlywood", weight=9];
339 -> 220[label="",style="solid", color="burlywood", weight=3];
237[label="xv110 : xv111 ++ xv8",fontsize=16,color="green",shape="box"];237 -> 245[label="",style="dashed", color="green", weight=3];
238[label="xv8",fontsize=16,color="green",shape="box"];210 -> 192[label="",style="dashed", color="red", weight=0];
210[label="[] ++ (xv101 >>= sequence0 xv410)",fontsize=16,color="magenta"];210 -> 213[label="",style="dashed", color="magenta", weight=3];
209[label="((xv410 : xv100) : xv11) ++ xv8",fontsize=16,color="black",shape="triangle"];209 -> 214[label="",style="solid", color="black", weight=3];
211[label="sequence (take2 (primMinusNat (Succ xv300) (Succ Zero)) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat (Succ xv300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];211 -> 215[label="",style="solid", color="black", weight=3];
312[label="xv14",fontsize=16,color="green",shape="box"];313[label="xv15",fontsize=16,color="green",shape="box"];314[label="xv16",fontsize=16,color="green",shape="box"];219[label="xv1010 : xv1011 >>= sequence0 xv410",fontsize=16,color="black",shape="box"];219 -> 227[label="",style="solid", color="black", weight=3];
220[label="[] >>= sequence0 xv410",fontsize=16,color="black",shape="box"];220 -> 228[label="",style="solid", color="black", weight=3];
245 -> 221[label="",style="dashed", color="red", weight=0];
245[label="xv111 ++ xv8",fontsize=16,color="magenta"];245 -> 252[label="",style="dashed", color="magenta", weight=3];
214[label="(xv410 : xv100) : xv11 ++ xv8",fontsize=16,color="green",shape="box"];214 -> 221[label="",style="dashed", color="green", weight=3];
215[label="sequence (take2 (primMinusNat xv300 Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat xv300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];342[label="xv300/Succ xv3000",fontsize=10,color="white",style="solid",shape="box"];215 -> 342[label="",style="solid", color="burlywood", weight=9];
342 -> 222[label="",style="solid", color="burlywood", weight=3];
343[label="xv300/Zero",fontsize=10,color="white",style="solid",shape="box"];215 -> 343[label="",style="solid", color="burlywood", weight=9];
343 -> 223[label="",style="solid", color="burlywood", weight=3];
227 -> 221[label="",style="dashed", color="red", weight=0];
227[label="sequence0 xv410 xv1010 ++ (xv1011 >>= sequence0 xv410)",fontsize=16,color="magenta"];227 -> 235[label="",style="dashed", color="magenta", weight=3];
227 -> 236[label="",style="dashed", color="magenta", weight=3];
228[label="[]",fontsize=16,color="green",shape="box"];252[label="xv111",fontsize=16,color="green",shape="box"];222[label="sequence (take2 (primMinusNat (Succ xv3000) Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat (Succ xv3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];222 -> 231[label="",style="solid", color="black", weight=3];
223[label="sequence (take2 (primMinusNat Zero Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];223 -> 232[label="",style="solid", color="black", weight=3];
235 -> 213[label="",style="dashed", color="red", weight=0];
235[label="xv1011 >>= sequence0 xv410",fontsize=16,color="magenta"];235 -> 243[label="",style="dashed", color="magenta", weight=3];
236[label="sequence0 xv410 xv1010",fontsize=16,color="black",shape="box"];236 -> 244[label="",style="solid", color="black", weight=3];
231[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos (Succ xv3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];231 -> 239[label="",style="solid", color="black", weight=3];
232[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];232 -> 240[label="",style="solid", color="black", weight=3];
243[label="xv1011",fontsize=16,color="green",shape="box"];244[label="return (xv410 : xv1010)",fontsize=16,color="black",shape="box"];244 -> 251[label="",style="solid", color="black", weight=3];
239[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (primCmpNat (Succ xv3000) Zero == GT)))",fontsize=16,color="black",shape="box"];239 -> 246[label="",style="solid", color="black", weight=3];
240[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];240 -> 247[label="",style="solid", color="black", weight=3];
251[label="(xv410 : xv1010) : []",fontsize=16,color="green",shape="box"];246[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];246 -> 253[label="",style="solid", color="black", weight=3];
247[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not False))",fontsize=16,color="black",shape="box"];247 -> 254[label="",style="solid", color="black", weight=3];
253[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not True))",fontsize=16,color="black",shape="box"];253 -> 259[label="",style="solid", color="black", weight=3];
254[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) True)",fontsize=16,color="black",shape="box"];254 -> 260[label="",style="solid", color="black", weight=3];
259[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) False)",fontsize=16,color="black",shape="box"];259 -> 264[label="",style="solid", color="black", weight=3];
260 -> 31[label="",style="dashed", color="red", weight=0];
260[label="sequence []",fontsize=16,color="magenta"];264 -> 36[label="",style="dashed", color="red", weight=0];
264[label="sequence (take1 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)))",fontsize=16,color="magenta"];264 -> 268[label="",style="dashed", color="magenta", weight=3];
264 -> 269[label="",style="dashed", color="magenta", weight=3];
268[label="xv3000",fontsize=16,color="green",shape="box"];269[label="xv40 : xv41",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                          ↳ QDP

                          ↳ QDP

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

new_psPs(:(xv110, xv111), xv8, h) → new_psPs(xv111, xv8, h)

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_psPs(:(xv110, xv111), xv8, h) → new_psPs(xv111, xv8, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                          ↳ QDP

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

new_gtGtEs(:(xv1010, xv1011), xv410, h) → new_gtGtEs(xv1011, xv410, h)

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

new_gtGtEs(:(xv1010, xv1011), xv410, h) → new_gtGtEs(xv1011, xv410, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPSizeChangeProof

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

new_sequence0(xv300, :(xv40, xv41), h) → new_gtGtEs0(xv41, xv300, xv40, xv41, h)
new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, ba) → new_sequence(xv14, xv15, xv16, ba)
new_sequence(Succ(xv3000), xv40, xv41, h) → new_sequence0(xv3000, :(xv40, xv41), h)
new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, ba) → new_gtGtEs0(xv131, xv14, xv15, xv16, ba)
new_sequence0(Succ(xv3000), :(xv40, xv41), h) → new_sequence0(xv3000, :(xv40, xv41), h)

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

new_sequence(Succ(xv3000), xv40, xv41, h) → new_sequence0(xv3000, :(xv40, xv41), h)
The graph contains the following edges 1 > 1, 4 >= 3
new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, ba) → new_gtGtEs0(xv131, xv14, xv15, xv16, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5
new_sequence0(xv300, :(xv40, xv41), h) → new_gtGtEs0(xv41, xv300, xv40, xv41, h)
The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3, 2 > 4, 3 >= 5
new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, ba) → new_sequence(xv14, xv15, xv16, ba)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4
new_sequence0(Succ(xv3000), :(xv40, xv41), h) → new_sequence0(xv3000, :(xv40, xv41), h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

take	n vx	= take3 n vx
take	vy []	= take1 vy []
take	n (x : xs)	= take0 n (x : xs)