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

YES 0.856 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 -> Maybe a -> Maybe ()) :: Int -> Maybe a -> Maybe ())

module Monad where

import qualified Maybe
import qualified Prelude

replicateM_ :: Monad a => Int -> 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

\_→q

is transformed to

gtGt0 q _ = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

((replicateM_ :: Int -> Maybe a -> Maybe ()) :: Int -> Maybe a -> Maybe ())

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 ()

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 -> Maybe a -> Maybe ()) :: Int -> Maybe a -> Maybe ())

module Monad where

import qualified Maybe
import qualified Prelude

replicateM_ :: Monad b => Int -> 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 wx wy = take0 wx wy

take2 n vx True = []

take2 n vx False = take1 n vx

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

take3 wz xu = take1 wz xu

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

mainModule Monad

((replicateM_ :: Int -> Maybe a -> Maybe ()) :: Int -> Maybe a -> Maybe ())

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 ()

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 xv = xv : repeatXs xv

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

mainModule Monad

((replicateM_ :: Int -> Maybe a -> Maybe ()) :: Int -> Maybe a -> Maybe ())

module Monad where

import qualified Maybe
import qualified Prelude

replicateM_ :: Monad b => Int -> 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 -> Maybe a -> Maybe ())

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 ()

replicateM_

n x

sequence_ (replicate n x)

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="replicateM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="replicateM_ xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="replicateM_ xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="sequence_ (replicate xw3 xw4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="foldr (>>) (return ()) (replicate xw3 xw4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr (>>) (return ()) (take xw3 (repeat xw4))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="foldr (>>) (return ()) (take3 xw3 (repeat xw4))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (xw3 <= Pos Zero))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (compare xw3 (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (compare xw3 (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (primCmpInt xw3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];75[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 13[label="",style="solid", color="burlywood", weight=3];
76[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="foldr (>>) (return ()) (take2 (Pos xw30) (repeat xw4) (not (primCmpInt (Pos xw30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];77[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 15[label="",style="solid", color="burlywood", weight=3];
78[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="foldr (>>) (return ()) (take2 (Neg xw30) (repeat xw4) (not (primCmpInt (Neg xw30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];79[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];14 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 17[label="",style="solid", color="burlywood", weight=3];
80[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (primCmpInt (Pos (Succ xw300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not (primCmpInt (Neg (Succ xw300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (primCmpNat (Succ xw300) Zero == GT)))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not (LT == GT)))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (GT == GT)))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not True))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) True)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) True)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) False)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];32 -> 36[label="",style="solid", color="black", weight=3];
33 -> 32[label="",style="dashed", color="red", weight=0];
33[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];34 -> 32[label="",style="dashed", color="red", weight=0];
34[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];35[label="foldr (>>) (return ()) (take1 (Pos (Succ xw300)) (repeat xw4))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="return ()",fontsize=16,color="black",shape="triangle"];36 -> 38[label="",style="solid", color="black", weight=3];
37 -> 39[label="",style="dashed", color="red", weight=0];
37[label="foldr (>>) (return ()) (take1 (Pos (Succ xw300)) (repeatXs xw4))",fontsize=16,color="magenta"];37 -> 40[label="",style="dashed", color="magenta", weight=3];
38[label="Just ()",fontsize=16,color="green",shape="box"];40 -> 36[label="",style="dashed", color="red", weight=0];
40[label="return ()",fontsize=16,color="magenta"];39[label="foldr (>>) xw5 (take1 (Pos (Succ xw300)) (repeatXs xw4))",fontsize=16,color="black",shape="triangle"];39 -> 41[label="",style="solid", color="black", weight=3];
41[label="foldr (>>) xw5 (take1 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42[label="foldr (>>) xw5 (take0 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
43[label="foldr (>>) xw5 (xw4 : take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
44[label="(>>) xw4 foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3];
45[label="xw4 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4)))",fontsize=16,color="burlywood",shape="box"];85[label="xw4/Nothing",fontsize=10,color="white",style="solid",shape="box"];45 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 46[label="",style="solid", color="burlywood", weight=3];
86[label="xw4/Just xw40",fontsize=10,color="white",style="solid",shape="box"];45 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 47[label="",style="solid", color="burlywood", weight=3];
46[label="Nothing >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs Nothing)))",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="Just xw40 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40))))",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48[label="Nothing",fontsize=16,color="green",shape="box"];49[label="gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)))) xw40",fontsize=16,color="black",shape="box"];49 -> 50[label="",style="solid", color="black", weight=3];
50[label="foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)))",fontsize=16,color="black",shape="box"];50 -> 51[label="",style="solid", color="black", weight=3];
51[label="foldr (>>) xw5 (take3 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)))",fontsize=16,color="black",shape="box"];51 -> 52[label="",style="solid", color="black", weight=3];
52[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)) (Pos (Succ xw300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];52 -> 53[label="",style="solid", color="black", weight=3];
53[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)) (compare (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];53 -> 54[label="",style="solid", color="black", weight=3];
54[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)) (not (compare (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];54 -> 55[label="",style="solid", color="black", weight=3];
55[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (Just xw40)) (not (primCmpInt (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];55 -> 56[label="",style="solid", color="black", weight=3];
56[label="foldr (>>) xw5 (take2 (primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))) (repeatXs (Just xw40)) (not (primCmpInt (primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];56 -> 57[label="",style="solid", color="black", weight=3];
57[label="foldr (>>) xw5 (take2 (primMinusNat (Succ xw300) (Succ Zero)) (repeatXs (Just xw40)) (not (primCmpInt (primMinusNat (Succ xw300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];57 -> 58[label="",style="solid", color="black", weight=3];
58[label="foldr (>>) xw5 (take2 (primMinusNat xw300 Zero) (repeatXs (Just xw40)) (not (primCmpInt (primMinusNat xw300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];87[label="xw300/Succ xw3000",fontsize=10,color="white",style="solid",shape="box"];58 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 59[label="",style="solid", color="burlywood", weight=3];
88[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 60[label="",style="solid", color="burlywood", weight=3];
59[label="foldr (>>) xw5 (take2 (primMinusNat (Succ xw3000) Zero) (repeatXs (Just xw40)) (not (primCmpInt (primMinusNat (Succ xw3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3];
60[label="foldr (>>) xw5 (take2 (primMinusNat Zero Zero) (repeatXs (Just xw40)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
61[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) (not (primCmpInt (Pos (Succ xw3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (Just xw40)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];62 -> 64[label="",style="solid", color="black", weight=3];
63[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) (not (primCmpNat (Succ xw3000) Zero == GT)))",fontsize=16,color="black",shape="box"];63 -> 65[label="",style="solid", color="black", weight=3];
64[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (Just xw40)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];64 -> 66[label="",style="solid", color="black", weight=3];
65[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];65 -> 67[label="",style="solid", color="black", weight=3];
66[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (Just xw40)) (not False))",fontsize=16,color="black",shape="box"];66 -> 68[label="",style="solid", color="black", weight=3];
67[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) (not True))",fontsize=16,color="black",shape="box"];67 -> 69[label="",style="solid", color="black", weight=3];
68[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (Just xw40)) True)",fontsize=16,color="black",shape="box"];68 -> 70[label="",style="solid", color="black", weight=3];
69[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (Just xw40)) False)",fontsize=16,color="black",shape="box"];69 -> 71[label="",style="solid", color="black", weight=3];
70[label="foldr (>>) xw5 []",fontsize=16,color="black",shape="box"];70 -> 72[label="",style="solid", color="black", weight=3];
71 -> 39[label="",style="dashed", color="red", weight=0];
71[label="foldr (>>) xw5 (take1 (Pos (Succ xw3000)) (repeatXs (Just xw40)))",fontsize=16,color="magenta"];71 -> 73[label="",style="dashed", color="magenta", weight=3];
71 -> 74[label="",style="dashed", color="magenta", weight=3];
72[label="xw5",fontsize=16,color="green",shape="box"];73[label="xw3000",fontsize=16,color="green",shape="box"];74[label="Just xw40",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ QDP

                          ↳ QDPSizeChangeProof

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

new_foldr(xw5, Succ(xw3000), Just(xw40), h) → new_foldr(xw5, xw3000, Just(xw40), 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_foldr(xw5, Succ(xw3000), Just(xw40), h) → new_foldr(xw5, xw3000, Just(xw40), h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4

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