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

YES 0.8270000000000001 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

  ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

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

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

The following Function with conditions

take n vv

| n <= 0

= []

take vw [] = []

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

is transformed to

take n vv = take3 n vv

take vw [] = take1 vw []

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

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

take1 vw [] = []

take1 wv ww = take0 wv ww

take2 n vv True = []

take2 n vv False = take1 n vv

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

take3 wx wy = take1 wx wy

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

mainModule Main

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

module Main where

import qualified Prelude

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 wz = wz : repeatXs wz

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ NumRed

mainModule Main

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

module Main where

import qualified Prelude

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="replicate",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="replicate xu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="replicate xu3 xu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="take xu3 (repeat xu4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="take3 xu3 (repeat xu4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="take2 xu3 (repeat xu4) (xu3 <= Pos Zero)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="take2 xu3 (repeat xu4) (compare xu3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="take2 xu3 (repeat xu4) (not (compare xu3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="take2 xu3 (repeat xu4) (not (primCmpInt xu3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];61[label="xu3/Pos xu30",fontsize=10,color="white",style="solid",shape="box"];10 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 11[label="",style="solid", color="burlywood", weight=3];
62[label="xu3/Neg xu30",fontsize=10,color="white",style="solid",shape="box"];10 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="take2 (Pos xu30) (repeat xu4) (not (primCmpInt (Pos xu30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];63[label="xu30/Succ xu300",fontsize=10,color="white",style="solid",shape="box"];11 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 13[label="",style="solid", color="burlywood", weight=3];
64[label="xu30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="take2 (Neg xu30) (repeat xu4) (not (primCmpInt (Neg xu30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];65[label="xu30/Succ xu300",fontsize=10,color="white",style="solid",shape="box"];12 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 15[label="",style="solid", color="burlywood", weight=3];
66[label="xu30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 16[label="",style="solid", color="burlywood", weight=3];
13[label="take2 (Pos (Succ xu300)) (repeat xu4) (not (primCmpInt (Pos (Succ xu300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="take2 (Pos Zero) (repeat xu4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="take2 (Neg (Succ xu300)) (repeat xu4) (not (primCmpInt (Neg (Succ xu300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="take2 (Neg Zero) (repeat xu4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="take2 (Pos (Succ xu300)) (repeat xu4) (not (primCmpNat (Succ xu300) Zero == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="take2 (Pos Zero) (repeat xu4) (not (EQ == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="take2 (Neg (Succ xu300)) (repeat xu4) (not (LT == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="take2 (Neg Zero) (repeat xu4) (not (EQ == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="take2 (Pos (Succ xu300)) (repeat xu4) (not (GT == GT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="take2 (Pos Zero) (repeat xu4) (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="take2 (Neg (Succ xu300)) (repeat xu4) (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="take2 (Neg Zero) (repeat xu4) (not False)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="take2 (Pos (Succ xu300)) (repeat xu4) (not True)",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="take2 (Pos Zero) (repeat xu4) True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="take2 (Neg (Succ xu300)) (repeat xu4) True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="take2 (Neg Zero) (repeat xu4) True",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="take2 (Pos (Succ xu300)) (repeat xu4) False",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="[]",fontsize=16,color="green",shape="box"];31[label="[]",fontsize=16,color="green",shape="box"];32[label="[]",fontsize=16,color="green",shape="box"];33[label="take1 (Pos (Succ xu300)) (repeat xu4)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
34[label="take1 (Pos (Succ xu300)) (repeatXs xu4)",fontsize=16,color="black",shape="triangle"];34 -> 35[label="",style="solid", color="black", weight=3];
35[label="take1 (Pos (Succ xu300)) (xu4 : repeatXs xu4)",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
36[label="take0 (Pos (Succ xu300)) (xu4 : repeatXs xu4)",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="xu4 : take (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
38[label="take (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4)",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
39[label="take3 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4)",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3];
40[label="take2 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4) (Pos (Succ xu300) - Pos (Succ Zero) <= Pos Zero)",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
41[label="take2 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4) (compare (Pos (Succ xu300) - Pos (Succ Zero)) (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42[label="take2 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4) (not (compare (Pos (Succ xu300) - Pos (Succ Zero)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
43[label="take2 (Pos (Succ xu300) - Pos (Succ Zero)) (repeatXs xu4) (not (primCmpInt (Pos (Succ xu300) - Pos (Succ Zero)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
44[label="take2 (primMinusInt (Pos (Succ xu300)) (Pos (Succ Zero))) (repeatXs xu4) (not (primCmpInt (primMinusInt (Pos (Succ xu300)) (Pos (Succ Zero))) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3];
45[label="take2 (primMinusNat (Succ xu300) (Succ Zero)) (repeatXs xu4) (not (primCmpInt (primMinusNat (Succ xu300) (Succ Zero)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];45 -> 46[label="",style="solid", color="black", weight=3];
46[label="take2 (primMinusNat xu300 Zero) (repeatXs xu4) (not (primCmpInt (primMinusNat xu300 Zero) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];67[label="xu300/Succ xu3000",fontsize=10,color="white",style="solid",shape="box"];46 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 47[label="",style="solid", color="burlywood", weight=3];
68[label="xu300/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 48[label="",style="solid", color="burlywood", weight=3];
47[label="take2 (primMinusNat (Succ xu3000) Zero) (repeatXs xu4) (not (primCmpInt (primMinusNat (Succ xu3000) Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48[label="take2 (primMinusNat Zero Zero) (repeatXs xu4) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];48 -> 50[label="",style="solid", color="black", weight=3];
49[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) (not (primCmpInt (Pos (Succ xu3000)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];49 -> 51[label="",style="solid", color="black", weight=3];
50[label="take2 (Pos Zero) (repeatXs xu4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];50 -> 52[label="",style="solid", color="black", weight=3];
51[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) (not (primCmpNat (Succ xu3000) Zero == GT))",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3];
52[label="take2 (Pos Zero) (repeatXs xu4) (not (EQ == GT))",fontsize=16,color="black",shape="box"];52 -> 54[label="",style="solid", color="black", weight=3];
53[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) (not (GT == GT))",fontsize=16,color="black",shape="box"];53 -> 55[label="",style="solid", color="black", weight=3];
54[label="take2 (Pos Zero) (repeatXs xu4) (not False)",fontsize=16,color="black",shape="box"];54 -> 56[label="",style="solid", color="black", weight=3];
55[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) (not True)",fontsize=16,color="black",shape="box"];55 -> 57[label="",style="solid", color="black", weight=3];
56[label="take2 (Pos Zero) (repeatXs xu4) True",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
57[label="take2 (Pos (Succ xu3000)) (repeatXs xu4) False",fontsize=16,color="black",shape="box"];57 -> 59[label="",style="solid", color="black", weight=3];
58[label="[]",fontsize=16,color="green",shape="box"];59 -> 34[label="",style="dashed", color="red", weight=0];
59[label="take1 (Pos (Succ xu3000)) (repeatXs xu4)",fontsize=16,color="magenta"];59 -> 60[label="",style="dashed", color="magenta", weight=3];
60[label="xu3000",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_take1(Succ(xu3000), xu4, ba) → new_take1(xu3000, xu4, ba)

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_take1(Succ(xu3000), xu4, ba) → new_take1(xu3000, xu4, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

take	n vv	= take3 n vv
take	vw []	= take1 vw []
take	n (x : xs)	= take0 n (x : xs)