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

YES 0.6890000000000001 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/Monad.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Monad

((msum :: [Maybe a] -> Maybe a) :: [Maybe a] -> Maybe a)

module Monad where

import qualified Maybe
import qualified Prelude

class Monad a => MonadPlus a where

mplus :: MonadPlus a => a b -> a b -> a b

mzero :: MonadPlus a => a b

instance MonadPlus Maybe where

mplus	Nothing ys	=	ys
mplus	xs _ys	=	xs

mzero

Nothing

instance MonadPlus [] where

msum :: MonadPlus b => [b a] -> b a

msum

foldr mplus mzero

module Maybe where

import qualified Monad
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Monad

((msum :: [Maybe a] -> Maybe a) :: [Maybe a] -> Maybe a)

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

class Monad a => MonadPlus a where

mplus :: MonadPlus a => a b -> a b -> a b

mzero :: MonadPlus a => a b

instance MonadPlus Maybe where

mplus	Nothing ys	=	ys
mplus	xs _ys	=	xs

mzero

Nothing

instance MonadPlus [] where

msum :: MonadPlus a => [a b] -> a b

msum

foldr mplus mzero

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Monad

(msum :: [Maybe a] -> Maybe a)

module Monad where

import qualified Maybe
import qualified Prelude

class Monad a => MonadPlus a where

mplus :: MonadPlus a => a b -> a b -> a b

mzero :: MonadPlus a => a b

instance MonadPlus Maybe where

mplus	Nothing ys	=	ys
mplus	xs _ys	=	xs

mzero

Nothing

instance MonadPlus [] where

msum :: MonadPlus b => [b a] -> b a

msum

foldr mplus mzero

module Maybe where

import qualified Monad
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="msum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="msum vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr mplus mzero vy3",fontsize=16,color="burlywood",shape="triangle"];17[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 5[label="",style="solid", color="burlywood", weight=3];
18[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr mplus mzero (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr mplus mzero []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="mplus vy30 (foldr mplus mzero vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="mzero",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="foldr mplus mzero vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
9[label="mplus vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];21[label="vy30/Nothing",fontsize=10,color="white",style="solid",shape="box"];9 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 13[label="",style="solid", color="burlywood", weight=3];
22[label="vy30/Just vy300",fontsize=10,color="white",style="solid",shape="box"];9 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 14[label="",style="solid", color="burlywood", weight=3];
11[label="Nothing",fontsize=16,color="green",shape="box"];12[label="vy31",fontsize=16,color="green",shape="box"];13[label="mplus Nothing vy4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="mplus (Just vy300) vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="vy4",fontsize=16,color="green",shape="box"];16[label="Just vy300",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_foldr(:(vy30, vy31), h) → new_foldr(vy31, h)

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(:(vy30, vy31), h) → new_foldr(vy31, h)
The graph contains the following edges 1 > 1, 2 >= 2