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

YES 0.651 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

((dropWhile :: (a -> Bool) -> [a] -> [a]) :: (a -> Bool) -> [a] -> [a])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.
Binding Reductions:
The bind variable of the following binding Pattern

xs@(vv : vw)

is replaced by the following term

vv : vw

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((dropWhile :: (a -> Bool) -> [a] -> [a]) :: (a -> Bool) -> [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

dropWhile p [] = []

dropWhile p (vv : vw)

| p vv

= dropWhile p vw

| otherwise

= vv : vw

is transformed to

dropWhile p [] = dropWhile3 p []

dropWhile p (vv : vw) = dropWhile2 p (vv : vw)

dropWhile0 p vv vw True = vv : vw

dropWhile1 p vv vw True = dropWhile p vw

dropWhile1 p vv vw False = dropWhile0 p vv vw otherwise

dropWhile2 p (vv : vw) = dropWhile1 p vv vw (p vv)

dropWhile3 p [] = []

dropWhile3 wv ww = dropWhile2 wv ww

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(dropWhile :: (a -> Bool) -> [a] -> [a])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="dropWhile",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="dropWhile wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="dropWhile wx3 wx4",fontsize=16,color="burlywood",shape="triangle"];22[label="wx4/wx40 : wx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 5[label="",style="solid", color="burlywood", weight=3];
23[label="wx4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="dropWhile wx3 (wx40 : wx41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="dropWhile wx3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="dropWhile2 wx3 (wx40 : wx41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="dropWhile3 wx3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="dropWhile1 wx3 wx40 wx41 (wx3 wx40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="[]",fontsize=16,color="green",shape="box"];12[label="wx3 wx40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3];
11[label="dropWhile1 wx3 wx40 wx41 wx5",fontsize=16,color="burlywood",shape="triangle"];25[label="wx5/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 14[label="",style="solid", color="burlywood", weight=3];
26[label="wx5/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 15[label="",style="solid", color="burlywood", weight=3];
16[label="wx40",fontsize=16,color="green",shape="box"];14[label="dropWhile1 wx3 wx40 wx41 False",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="dropWhile1 wx3 wx40 wx41 True",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
17[label="dropWhile0 wx3 wx40 wx41 otherwise",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18 -> 4[label="",style="dashed", color="red", weight=0];
18[label="dropWhile wx3 wx41",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
19[label="dropWhile0 wx3 wx40 wx41 True",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="wx41",fontsize=16,color="green",shape="box"];21[label="wx40 : wx41",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

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

new_dropWhile1(wx3, wx40, wx41, ba) → new_dropWhile(wx3, wx41, ba)
new_dropWhile(wx3, :(wx40, wx41), ba) → new_dropWhile1(wx3, wx40, wx41, 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_dropWhile(wx3, :(wx40, wx41), ba) → new_dropWhile1(wx3, wx40, wx41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4
new_dropWhile1(wx3, wx40, wx41, ba) → new_dropWhile(wx3, wx41, ba)
The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3

dropWhile	p []	= dropWhile3 p []
dropWhile	p (vv : vw)	= dropWhile2 p (vv : vw)

dropWhile1	p vv vw True	= dropWhile p vw
dropWhile1	p vv vw False	= dropWhile0 p vv vw otherwise