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

YES 0.709 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

  ↳ LR

mainModule Monad

((filterM :: (a -> IO Bool) -> [a] -> IO [a]) :: (a -> IO Bool) -> [a] -> IO [a])

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad b => (a -> b Bool) -> [a] -> b [a]

filterM	_ []	=	return []
filterM	p (x : xs)	=	p x >>= (\flg ->filterM p xs >>= (\ys ->return ( if flg then x : ys else ys)))

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\ys→return (if flg then x : ys else ys)

is transformed to

filterM0 flg x ys = return (if flg then x : ys else ys)

The following Lambda expression

\flg→filterM p xs >>= filterM0 flg x

is transformed to

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

mainModule Monad

((filterM :: (a -> IO Bool) -> [a] -> IO [a]) :: (a -> IO Bool) -> [a] -> IO [a])

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad a => (b -> a Bool) -> [b] -> a [b]

filterM	_ []	=	return []
filterM	p (x : xs)	=	p x >>= filterM1 p xs x

filterM0

flg x ys

return ( if flg then x : ys else ys)

filterM1

p xs x flg

filterM p xs >>= filterM0 flg x

If Reductions:
The following If expression

if flg then x : ys else ys

is transformed to

filterM00 x ys True = x : ys

filterM00 x ys False = ys

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

mainModule Monad

((filterM :: (a -> IO Bool) -> [a] -> IO [a]) :: (a -> IO Bool) -> [a] -> IO [a])

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad a => (b -> a Bool) -> [b] -> a [b]

filterM	_ []	=	return []
filterM	p (x : xs)	=	p x >>= filterM1 p xs x

filterM0

flg x ys

return (filterM00 x ys flg)

filterM00	x ys True	=	x : ys
filterM00	x ys False	=	ys

filterM1

p xs x flg

filterM p xs >>= filterM0 flg x

module Maybe where

import qualified Monad
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule Monad

((filterM :: (a -> IO Bool) -> [a] -> IO [a]) :: (a -> IO Bool) -> [a] -> IO [a])

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad a => (b -> a Bool) -> [b] -> a [b]

filterM	vw []	=	return []
filterM	p (x : xs)	=	p x >>= filterM1 p xs x

filterM0

flg x ys

return (filterM00 x ys flg)

filterM00	x ys True	=	x : ys
filterM00	x ys False	=	ys

filterM1

p xs x flg

filterM p xs >>= filterM0 flg x

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

mainModule Monad

(filterM :: (a -> IO Bool) -> [a] -> IO [a])

module Monad where

import qualified Maybe
import qualified Prelude

filterM :: Monad b => (a -> b Bool) -> [a] -> b [a]

filterM	vw []	=	return []
filterM	p (x : xs)	=	p x >>= filterM1 p xs x

filterM0

flg x ys

return (filterM00 x ys flg)

filterM00	x ys True	=	x : ys
filterM00	x ys False	=	ys

filterM1

p xs x flg

filterM p xs >>= filterM0 flg x

module Maybe where

import qualified Monad
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="filterM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="filterM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="filterM vz3 vz4",fontsize=16,color="burlywood",shape="triangle"];39[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 5[label="",style="solid", color="burlywood", weight=3];
40[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="filterM vz3 (vz40 : vz41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="filterM vz3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="vz3 vz40 >>= filterM1 vz3 vz41 vz40",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="return []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="primbindIO (vz3 vz40) (filterM1 vz3 vz41 vz40)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="primretIO []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="terminator (vz3 vz40) (filterM1 vz3 vz41 vz40)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="terminator []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="ter0m (vz3 vz40) (filterM1 vz3 vz41 vz40)",fontsize=16,color="green",shape="box"];13 -> 15[label="",style="dashed", color="green", weight=3];
13 -> 16[label="",style="dashed", color="green", weight=3];
14[label="ter1m []",fontsize=16,color="green",shape="box"];14 -> 17[label="",style="dashed", color="green", weight=3];
15[label="vz3 vz40",fontsize=16,color="green",shape="box"];15 -> 18[label="",style="dashed", color="green", weight=3];
16[label="filterM1 vz3 vz41 vz40",fontsize=16,color="grey",shape="box"];16 -> 19[label="",style="dashed", color="grey", weight=3];
17[label="[]",fontsize=16,color="green",shape="box"];18[label="vz40",fontsize=16,color="green",shape="box"];19[label="filterM1 vz3 vz41 vz40 vz5",fontsize=16,color="black",shape="triangle"];19 -> 20[label="",style="solid", color="black", weight=3];
20 -> 21[label="",style="dashed", color="red", weight=0];
20[label="filterM vz3 vz41 >>= filterM0 vz5 vz40",fontsize=16,color="magenta"];20 -> 22[label="",style="dashed", color="magenta", weight=3];
22 -> 4[label="",style="dashed", color="red", weight=0];
22[label="filterM vz3 vz41",fontsize=16,color="magenta"];22 -> 23[label="",style="dashed", color="magenta", weight=3];
21[label="vz6 >>= filterM0 vz5 vz40",fontsize=16,color="black",shape="triangle"];21 -> 24[label="",style="solid", color="black", weight=3];
23[label="vz41",fontsize=16,color="green",shape="box"];24[label="primbindIO vz6 (filterM0 vz5 vz40)",fontsize=16,color="black",shape="box"];24 -> 25[label="",style="solid", color="black", weight=3];
25[label="terminator vz6 (filterM0 vz5 vz40)",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3];
26[label="ter2m vz6 (filterM0 vz5 vz40)",fontsize=16,color="green",shape="box"];26 -> 27[label="",style="dashed", color="green", weight=3];
26 -> 28[label="",style="dashed", color="green", weight=3];
27[label="vz6",fontsize=16,color="green",shape="box"];28[label="filterM0 vz5 vz40",fontsize=16,color="grey",shape="box"];28 -> 29[label="",style="dashed", color="grey", weight=3];
29[label="filterM0 vz5 vz40 vz7",fontsize=16,color="black",shape="triangle"];29 -> 30[label="",style="solid", color="black", weight=3];
30[label="return (filterM00 vz40 vz7 vz5)",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
31[label="primretIO (filterM00 vz40 vz7 vz5)",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32[label="terminator (filterM00 vz40 vz7 vz5)",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3];
33[label="ter3m (filterM00 vz40 vz7 vz5)",fontsize=16,color="green",shape="box"];33 -> 34[label="",style="dashed", color="green", weight=3];
34[label="filterM00 vz40 vz7 vz5",fontsize=16,color="burlywood",shape="box"];43[label="vz5/False",fontsize=10,color="white",style="solid",shape="box"];34 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 35[label="",style="solid", color="burlywood", weight=3];
44[label="vz5/True",fontsize=10,color="white",style="solid",shape="box"];34 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 36[label="",style="solid", color="burlywood", weight=3];
35[label="filterM00 vz40 vz7 False",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="filterM00 vz40 vz7 True",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="vz7",fontsize=16,color="green",shape="box"];38[label="vz40 : vz7",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ DependencyGraphProof

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

new_filterM1(vz3, vz41, vz40, vz5, h) → new_filterM(vz3, vz41, h)
new_filterM(vz3, :(vz40, vz41), h) → new_filterM(vz3, vz41, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IFR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPSizeChangeProof

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

new_filterM(vz3, :(vz40, vz41), h) → new_filterM(vz3, vz41, 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_filterM(vz3, :(vz40, vz41), h) → new_filterM(vz3, vz41, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3