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

YES 0.642 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

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

module Main where

import qualified Prelude

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 Main

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

module Main where

import qualified Prelude

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 Main

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

module Main where

import qualified Prelude

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 Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

mainModule Main

((filter :: (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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="filter",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="filter vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="filter vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="concatMap (filter0 vy3) vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concat . map (filter0 vy3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="concat (map (filter0 vy3) vy4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="foldr (++) [] (map (filter0 vy3) vy4)",fontsize=16,color="burlywood",shape="triangle"];31[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];8 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 9[label="",style="solid", color="burlywood", weight=3];
32[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 10[label="",style="solid", color="burlywood", weight=3];
9[label="foldr (++) [] (map (filter0 vy3) (vy40 : vy41))",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldr (++) [] (map (filter0 vy3) [])",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (++) [] (filter0 vy3 vy40 : map (filter0 vy3) vy41)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13 -> 15[label="",style="dashed", color="red", weight=0];
13[label="(++) filter0 vy3 vy40 foldr (++) [] (map (filter0 vy3) vy41)",fontsize=16,color="magenta"];13 -> 16[label="",style="dashed", color="magenta", weight=3];
14[label="[]",fontsize=16,color="green",shape="box"];16 -> 8[label="",style="dashed", color="red", weight=0];
16[label="foldr (++) [] (map (filter0 vy3) vy41)",fontsize=16,color="magenta"];16 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="(++) filter0 vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];15 -> 18[label="",style="solid", color="black", weight=3];
17[label="vy41",fontsize=16,color="green",shape="box"];18[label="(++) filter00 vy3 vy40 vy5",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19 -> 20[label="",style="dashed", color="red", weight=0];
19[label="(++) filter000 vy40 (vy3 vy40) vy5",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
21[label="vy3 vy40",fontsize=16,color="green",shape="box"];21 -> 25[label="",style="dashed", color="green", weight=3];
20[label="(++) filter000 vy40 vy6 vy5",fontsize=16,color="burlywood",shape="triangle"];36[label="vy6/False",fontsize=10,color="white",style="solid",shape="box"];20 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 23[label="",style="solid", color="burlywood", weight=3];
37[label="vy6/True",fontsize=10,color="white",style="solid",shape="box"];20 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 24[label="",style="solid", color="burlywood", weight=3];
25[label="vy40",fontsize=16,color="green",shape="box"];23[label="(++) filter000 vy40 False vy5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="(++) filter000 vy40 True vy5",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
26[label="(++) [] vy5",fontsize=16,color="black",shape="triangle"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="(++) (vy40 : []) vy5",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="vy5",fontsize=16,color="green",shape="box"];29[label="vy40 : [] ++ vy5",fontsize=16,color="green",shape="box"];29 -> 30[label="",style="dashed", color="green", weight=3];
30 -> 26[label="",style="dashed", color="red", weight=0];
30[label="[] ++ vy5",fontsize=16,color="magenta"];}

↳ 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_foldr(vy3, :(vy40, vy41), ba) → new_foldr(vy3, vy41, 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_foldr(vy3, :(vy40, vy41), ba) → new_foldr(vy3, vy41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3