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

YES 0.841 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule List

((find :: (a -> Bool) -> [a] -> Maybe a) :: (a -> Bool) -> [a] -> Maybe a)

module List where

import qualified Maybe
import qualified Prelude

find :: (a -> Bool) -> [a] -> Maybe a

find

Maybe.listToMaybe . filter p

module Maybe where

import qualified List
import qualified Prelude

listToMaybe :: [a] -> Maybe a

listToMaybe	[]	=	Nothing
listToMaybe	(a : _)	=	Just a

Lambda Reductions:
The following Lambda expression

\vu39→

case vu39 of
x → if p x then x : [] else []

_ → []

is transformed to

filter0 p vu39 =

case vu39 of
x → if p x then x : [] else []

_ → []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

mainModule List

((find :: (a -> Bool) -> [a] -> Maybe a) :: (a -> Bool) -> [a] -> Maybe a)

module List where

import qualified Maybe
import qualified Prelude

find :: (a -> Bool) -> [a] -> Maybe a

find

Maybe.listToMaybe . filter p

module Maybe where

import qualified List
import qualified Prelude

listToMaybe :: [a] -> Maybe a

listToMaybe	[]	=	Nothing
listToMaybe	(a : _)	=	Just a

Case Reductions:
The following Case expression

case vu39 of
x → if p x then x : [] else []

_ → []

is transformed to

filter00 p x = if p x then x : [] else []

filter00 p _ = []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

mainModule List

((find :: (a -> Bool) -> [a] -> Maybe a) :: (a -> Bool) -> [a] -> Maybe a)

module List where

import qualified Maybe
import qualified Prelude

find :: (a -> Bool) -> [a] -> Maybe a

find

Maybe.listToMaybe . filter p

module Maybe where

import qualified List
import qualified Prelude

listToMaybe :: [a] -> Maybe a

listToMaybe	[]	=	Nothing
listToMaybe	(a : _)	=	Just a

If Reductions:
The following If expression

if p x then x : [] else []

is transformed to

filter000 x True = x : []

filter000 x False = []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

mainModule List

((find :: (a -> Bool) -> [a] -> Maybe a) :: (a -> Bool) -> [a] -> Maybe a)

module List where

import qualified Maybe
import qualified Prelude

find :: (a -> Bool) -> [a] -> Maybe a

find

Maybe.listToMaybe . filter p

module Maybe where

import qualified List
import qualified Prelude

listToMaybe :: [a] -> Maybe a

listToMaybe	[]	=	Nothing
listToMaybe	(a : _)	=	Just a

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

mainModule List

((find :: (a -> Bool) -> [a] -> Maybe a) :: (a -> Bool) -> [a] -> Maybe a)

module List where

import qualified Maybe
import qualified Prelude

find :: (a -> Bool) -> [a] -> Maybe a

find

Maybe.listToMaybe . filter p

module Maybe where

import qualified List
import qualified Prelude

listToMaybe :: [a] -> Maybe a

listToMaybe	[]	=	Nothing
listToMaybe	(a : vw)	=	Just a

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ Narrow

mainModule List

(find :: (a -> Bool) -> [a] -> Maybe a)

module List where

import qualified Maybe
import qualified Prelude

find :: (a -> Bool) -> [a] -> Maybe a

find

Maybe.listToMaybe . filter p

module Maybe where

import qualified List
import qualified Prelude

listToMaybe :: [a] -> Maybe a

listToMaybe	[]	=	Nothing
listToMaybe	(a : vw)	=	Just a

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="find",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="find wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="find wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="Maybe.listToMaybe . filter wu3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="Maybe.listToMaybe (filter wu3 wu4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="Maybe.listToMaybe (concatMap (filter0 wu3) wu4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="Maybe.listToMaybe (concat . map (filter0 wu3))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="Maybe.listToMaybe (concat (map (filter0 wu3) wu4))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) wu4))",fontsize=16,color="burlywood",shape="triangle"];32[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];10 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 11[label="",style="solid", color="burlywood", weight=3];
33[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) (wu40 : wu41)))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) []))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="Maybe.listToMaybe (foldr (++) [] (filter0 wu3 wu40 : map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="Maybe.listToMaybe (foldr (++) [] [])",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="Maybe.listToMaybe ((++) filter0 wu3 wu40 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="Maybe.listToMaybe []",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="Maybe.listToMaybe ((++) filter00 wu3 wu40 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="Nothing",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="red", weight=0];
19[label="Maybe.listToMaybe ((++) filter000 wu40 (wu3 wu40) foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
21[label="wu3 wu40",fontsize=16,color="green",shape="box"];21 -> 25[label="",style="dashed", color="green", weight=3];
20[label="Maybe.listToMaybe ((++) filter000 wu40 wu5 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="burlywood",shape="triangle"];35[label="wu5/False",fontsize=10,color="white",style="solid",shape="box"];20 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 23[label="",style="solid", color="burlywood", weight=3];
36[label="wu5/True",fontsize=10,color="white",style="solid",shape="box"];20 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 24[label="",style="solid", color="burlywood", weight=3];
25[label="wu40",fontsize=16,color="green",shape="box"];23[label="Maybe.listToMaybe ((++) filter000 wu40 False foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="Maybe.listToMaybe ((++) filter000 wu40 True foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
26[label="Maybe.listToMaybe ((++) [] foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="Maybe.listToMaybe ((++) (wu40 : []) foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28 -> 10[label="",style="dashed", color="red", weight=0];
28[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="magenta"];28 -> 30[label="",style="dashed", color="magenta", weight=3];
29[label="Maybe.listToMaybe (wu40 : [] ++ foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="wu41",fontsize=16,color="green",shape="box"];31[label="Just wu40",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ QDP

                          ↳ QDPSizeChangeProof

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

new_listToMaybe0(wu3, :(wu40, wu41), ba) → new_listToMaybe(wu40, wu3, wu41, ba)
new_listToMaybe(wu40, wu3, wu41, ba) → new_listToMaybe0(wu3, wu41, 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_listToMaybe(wu40, wu3, wu41, ba) → new_listToMaybe0(wu3, wu41, ba)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3
new_listToMaybe0(wu3, :(wu40, wu41), ba) → new_listToMaybe(wu40, wu3, wu41, ba)
The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3, 3 >= 4