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

YES 0.842 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

((splitAt :: Int -> [a] -> ([a],[a])) :: Int -> [a] -> ([a],[a]))

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\(xs',_)→xs'

is transformed to

xs'0 (xs',_) = xs'

The following Lambda expression

\(_,xs'')→xs''

is transformed to

xs''0 (_,xs'') = xs''

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

((splitAt :: Int -> [a] -> ([a],[a])) :: Int -> [a] -> ([a],[a]))

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((splitAt :: Int -> [a] -> ([a],[a])) :: Int -> [a] -> ([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

splitAt n xs

| n <= 0

= ([],xs)

splitAt vw [] = ([],[])

splitAt n (x : xs) =

(x : xs',xs'')

where

vu42 = splitAt (n - 1) xs

xs' = xs'0 vu42

xs'' = xs''0 vu42

xs''0 (vx,xs'') = xs''

xs'0 (xs',vy) = xs'

is transformed to

splitAt n xs = splitAt3 n xs

splitAt vw [] = splitAt1 vw []

splitAt n (x : xs) = splitAt0 n (x : xs)

splitAt0 n (x : xs) =

(x : xs',xs'')

where

vu42 = splitAt (n - 1) xs

xs' = xs'0 vu42

xs'' = xs''0 vu42

xs''0 (vx,xs'') = xs''

xs'0 (xs',vy) = xs'

splitAt1 vw [] = ([],[])

splitAt1 ww wx = splitAt0 ww wx

splitAt2 n xs True = ([],xs)

splitAt2 n xs False = splitAt1 n xs

splitAt3 n xs = splitAt2 n xs (n <= 0)

splitAt3 wy wz = splitAt1 wy wz

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

mainModule Main

((splitAt :: Int -> [a] -> ([a],[a])) :: Int -> [a] -> ([a],[a]))

module Main where

import qualified Prelude

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

(x : xs',xs'')

where

vu42 = splitAt (n - 1) xs

xs' = xs'0 vu42

xs'' = xs''0 vu42

xs''0 (vx,xs'') = xs''

xs'0 (xs',vy) = xs'

are unpacked to the following functions on top level

splitAt0Xs' xu xv = splitAt0Xs'0 xu xv (splitAt0Vu42 xu xv)

splitAt0Xs''0 xu xv (vx,xs'') = xs''

splitAt0Xs'' xu xv = splitAt0Xs''0 xu xv (splitAt0Vu42 xu xv)

splitAt0Xs'0 xu xv (xs',vy) = xs'

splitAt0Vu42 xu xv = splitAt (xu - 1) xv

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

mainModule Main

((splitAt :: Int -> [a] -> ([a],[a])) :: Int -> [a] -> ([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

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ NumRed

                    ↳ HASKELL

                      ↳ Narrow

mainModule Main

(splitAt :: Int -> [a] -> ([a],[a]))

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="splitAt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="splitAt xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="splitAt xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="splitAt3 xw3 xw4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="splitAt2 xw3 xw4 (xw3 <= Pos Zero)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="splitAt2 xw3 xw4 (compare xw3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="splitAt2 xw3 xw4 (not (compare xw3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="splitAt2 xw3 xw4 (not (primCmpInt xw3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];64[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];9 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 10[label="",style="solid", color="burlywood", weight=3];
65[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="splitAt2 (Pos xw30) xw4 (not (primCmpInt (Pos xw30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];66[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];10 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 12[label="",style="solid", color="burlywood", weight=3];
67[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="splitAt2 (Neg xw30) xw4 (not (primCmpInt (Neg xw30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];68[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];11 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 14[label="",style="solid", color="burlywood", weight=3];
69[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="splitAt2 (Pos (Succ xw300)) xw4 (not (primCmpInt (Pos (Succ xw300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="splitAt2 (Pos Zero) xw4 (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="splitAt2 (Neg (Succ xw300)) xw4 (not (primCmpInt (Neg (Succ xw300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="splitAt2 (Neg Zero) xw4 (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="splitAt2 (Pos (Succ xw300)) xw4 (not (primCmpNat (Succ xw300) Zero == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="splitAt2 (Pos Zero) xw4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="splitAt2 (Neg (Succ xw300)) xw4 (not (LT == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="splitAt2 (Neg Zero) xw4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="splitAt2 (Pos (Succ xw300)) xw4 (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="splitAt2 (Pos Zero) xw4 (not False)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="splitAt2 (Neg (Succ xw300)) xw4 (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="splitAt2 (Neg Zero) xw4 (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="splitAt2 (Pos (Succ xw300)) xw4 (not True)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="splitAt2 (Pos Zero) xw4 True",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="splitAt2 (Neg (Succ xw300)) xw4 True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="splitAt2 (Neg Zero) xw4 True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="splitAt2 (Pos (Succ xw300)) xw4 False",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="([],xw4)",fontsize=16,color="green",shape="box"];30[label="([],xw4)",fontsize=16,color="green",shape="box"];31[label="([],xw4)",fontsize=16,color="green",shape="box"];32[label="splitAt1 (Pos (Succ xw300)) xw4",fontsize=16,color="burlywood",shape="box"];70[label="xw4/xw40 : xw41",fontsize=10,color="white",style="solid",shape="box"];32 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 33[label="",style="solid", color="burlywood", weight=3];
71[label="xw4/[]",fontsize=10,color="white",style="solid",shape="box"];32 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 34[label="",style="solid", color="burlywood", weight=3];
33[label="splitAt1 (Pos (Succ xw300)) (xw40 : xw41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="splitAt1 (Pos (Succ xw300)) []",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="splitAt0 (Pos (Succ xw300)) (xw40 : xw41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="([],[])",fontsize=16,color="green",shape="box"];37[label="(xw40 : splitAt0Xs' (Pos (Succ xw300)) xw41,splitAt0Xs'' (Pos (Succ xw300)) xw41)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
37 -> 39[label="",style="dashed", color="green", weight=3];
38[label="splitAt0Xs' (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="splitAt0Xs'' (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40 -> 44[label="",style="dashed", color="red", weight=0];
40[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 (splitAt0Vu42 (Pos (Succ xw300)) xw41)",fontsize=16,color="magenta"];40 -> 45[label="",style="dashed", color="magenta", weight=3];
41 -> 49[label="",style="dashed", color="red", weight=0];
41[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 (splitAt0Vu42 (Pos (Succ xw300)) xw41)",fontsize=16,color="magenta"];41 -> 50[label="",style="dashed", color="magenta", weight=3];
45[label="splitAt0Vu42 (Pos (Succ xw300)) xw41",fontsize=16,color="black",shape="triangle"];45 -> 47[label="",style="solid", color="black", weight=3];
44[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 xw5",fontsize=16,color="burlywood",shape="triangle"];74[label="xw5/(xw50,xw51)",fontsize=10,color="white",style="solid",shape="box"];44 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 48[label="",style="solid", color="burlywood", weight=3];
50 -> 45[label="",style="dashed", color="red", weight=0];
50[label="splitAt0Vu42 (Pos (Succ xw300)) xw41",fontsize=16,color="magenta"];49[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 xw6",fontsize=16,color="burlywood",shape="triangle"];76[label="xw6/(xw60,xw61)",fontsize=10,color="white",style="solid",shape="box"];49 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 52[label="",style="solid", color="burlywood", weight=3];
47 -> 4[label="",style="dashed", color="red", weight=0];
47[label="splitAt (Pos (Succ xw300) - Pos (Succ Zero)) xw41",fontsize=16,color="magenta"];47 -> 53[label="",style="dashed", color="magenta", weight=3];
47 -> 54[label="",style="dashed", color="magenta", weight=3];
48[label="splitAt0Xs'0 (Pos (Succ xw300)) xw41 (xw50,xw51)",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
52[label="splitAt0Xs''0 (Pos (Succ xw300)) xw41 (xw60,xw61)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="xw41",fontsize=16,color="green",shape="box"];54[label="Pos (Succ xw300) - Pos (Succ Zero)",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="xw50",fontsize=16,color="green",shape="box"];56[label="xw61",fontsize=16,color="green",shape="box"];57[label="primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];57 -> 58[label="",style="solid", color="black", weight=3];
58[label="primMinusNat (Succ xw300) (Succ Zero)",fontsize=16,color="black",shape="box"];58 -> 59[label="",style="solid", color="black", weight=3];
59[label="primMinusNat xw300 Zero",fontsize=16,color="burlywood",shape="box"];78[label="xw300/Succ xw3000",fontsize=10,color="white",style="solid",shape="box"];59 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 60[label="",style="solid", color="burlywood", weight=3];
79[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 61[label="",style="solid", color="burlywood", weight=3];
60[label="primMinusNat (Succ xw3000) Zero",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
61[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="Pos (Succ xw3000)",fontsize=16,color="green",shape="box"];63[label="Pos Zero",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_splitAt0Vu42(xw300, xw41, ba) → new_splitAt(new_primMinusNat(xw300), xw41, ba)
new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), ba) → new_splitAt(new_primMinusNat(xw300), xw41, ba)
new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), ba) → new_splitAt0Vu42(xw300, xw41, ba)

The TRS R consists of the following rules:

new_primMinusNat(Succ(xw3000)) → Pos(Succ(xw3000))
new_primMinusNat(Zero) → Pos(Zero)

The set Q consists of the following terms:

new_primMinusNat(Zero)
new_primMinusNat(Succ(x0))

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_splitAt(Pos(Succ(xw300)), :(xw40, xw41), ba) → new_splitAt0Vu42(xw300, xw41, ba)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
new_splitAt(Pos(Succ(xw300)), :(xw40, xw41), ba) → new_splitAt(new_primMinusNat(xw300), xw41, ba)
The graph contains the following edges 2 > 2, 3 >= 3
new_splitAt0Vu42(xw300, xw41, ba) → new_splitAt(new_primMinusNat(xw300), xw41, ba)
The graph contains the following edges 2 >= 2, 3 >= 3

splitAt	n xs	= splitAt3 n xs
splitAt	vw []	= splitAt1 vw []
splitAt	n (x : xs)	= splitAt0 n (x : xs)

splitAt3	n xs	= splitAt2 n xs (n <= 0)
splitAt3	wy wz	= splitAt1 wy wz