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

  ↳ BR

mainModule Main

((drop :: 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

((drop :: 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

drop n xs

| n <= 0

= xs

drop vw [] = []

drop n (vx : xs) = drop (n - 1) xs

is transformed to

drop n xs = drop3 n xs

drop vw [] = drop1 vw []

drop n (vx : xs) = drop0 n (vx : xs)

drop0 n (vx : xs) = drop (n - 1) xs

drop1 vw [] = []

drop1 wv ww = drop0 wv ww

drop2 n xs True = xs

drop2 n xs False = drop1 n xs

drop3 n xs = drop2 n xs (n <= 0)

drop3 wx wy = drop1 wx wy

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

((drop :: 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

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="drop",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="drop wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="drop wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="drop3 wz3 wz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="drop2 wz3 wz4 (wz3 <= Pos Zero)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="drop2 wz3 wz4 (compare wz3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="drop2 wz3 wz4 (not (compare wz3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="drop2 wz3 wz4 (not (primCmpInt wz3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];47[label="wz3/Pos wz30",fontsize=10,color="white",style="solid",shape="box"];9 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 10[label="",style="solid", color="burlywood", weight=3];
48[label="wz3/Neg wz30",fontsize=10,color="white",style="solid",shape="box"];9 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="drop2 (Pos wz30) wz4 (not (primCmpInt (Pos wz30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];49[label="wz30/Succ wz300",fontsize=10,color="white",style="solid",shape="box"];10 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 12[label="",style="solid", color="burlywood", weight=3];
50[label="wz30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="drop2 (Neg wz30) wz4 (not (primCmpInt (Neg wz30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];51[label="wz30/Succ wz300",fontsize=10,color="white",style="solid",shape="box"];11 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 14[label="",style="solid", color="burlywood", weight=3];
52[label="wz30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 15[label="",style="solid", color="burlywood", weight=3];
12[label="drop2 (Pos (Succ wz300)) wz4 (not (primCmpInt (Pos (Succ wz300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="drop2 (Pos Zero) wz4 (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="drop2 (Neg (Succ wz300)) wz4 (not (primCmpInt (Neg (Succ wz300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="drop2 (Neg Zero) wz4 (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="drop2 (Pos (Succ wz300)) wz4 (not (primCmpNat (Succ wz300) Zero == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="drop2 (Pos Zero) wz4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="drop2 (Neg (Succ wz300)) wz4 (not (LT == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="drop2 (Neg Zero) wz4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="drop2 (Pos (Succ wz300)) wz4 (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="drop2 (Pos Zero) wz4 (not False)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="drop2 (Neg (Succ wz300)) wz4 (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="drop2 (Neg Zero) wz4 (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="drop2 (Pos (Succ wz300)) wz4 (not True)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="drop2 (Pos Zero) wz4 True",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="drop2 (Neg (Succ wz300)) wz4 True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="drop2 (Neg Zero) wz4 True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="drop2 (Pos (Succ wz300)) wz4 False",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="wz4",fontsize=16,color="green",shape="box"];30[label="wz4",fontsize=16,color="green",shape="box"];31[label="wz4",fontsize=16,color="green",shape="box"];32[label="drop1 (Pos (Succ wz300)) wz4",fontsize=16,color="burlywood",shape="box"];53[label="wz4/wz40 : wz41",fontsize=10,color="white",style="solid",shape="box"];32 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 33[label="",style="solid", color="burlywood", weight=3];
54[label="wz4/[]",fontsize=10,color="white",style="solid",shape="box"];32 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 34[label="",style="solid", color="burlywood", weight=3];
33[label="drop1 (Pos (Succ wz300)) (wz40 : wz41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="drop1 (Pos (Succ wz300)) []",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="drop0 (Pos (Succ wz300)) (wz40 : wz41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="[]",fontsize=16,color="green",shape="box"];37 -> 4[label="",style="dashed", color="red", weight=0];
37[label="drop (Pos (Succ wz300) - Pos (Succ Zero)) wz41",fontsize=16,color="magenta"];37 -> 38[label="",style="dashed", color="magenta", weight=3];
37 -> 39[label="",style="dashed", color="magenta", weight=3];
38[label="Pos (Succ wz300) - Pos (Succ Zero)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="wz41",fontsize=16,color="green",shape="box"];40[label="primMinusInt (Pos (Succ wz300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
41[label="primMinusNat (Succ wz300) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42[label="primMinusNat wz300 Zero",fontsize=16,color="burlywood",shape="box"];56[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];42 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 43[label="",style="solid", color="burlywood", weight=3];
57[label="wz300/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 44[label="",style="solid", color="burlywood", weight=3];
43[label="primMinusNat (Succ wz3000) Zero",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="Pos (Succ wz3000)",fontsize=16,color="green",shape="box"];46[label="Pos Zero",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_drop(Pos(Succ(wz300)), :(wz40, wz41), ba) → new_drop(new_primMinusNat(wz300), wz41, ba)

The TRS R consists of the following rules:

new_primMinusNat(Succ(wz3000)) → Pos(Succ(wz3000))
new_primMinusNat(Zero) → Pos(Zero)

The set Q consists of the following terms:

new_primMinusNat(Succ(x0))
new_primMinusNat(Zero)

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_drop(Pos(Succ(wz300)), :(wz40, wz41), ba) → new_drop(new_primMinusNat(wz300), wz41, ba)
The graph contains the following edges 2 > 2, 3 >= 3

drop	n xs	= drop3 n xs
drop	vw []	= drop1 vw []
drop	n (vx : xs)	= drop0 n (vx : xs)