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

YES 0.769 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

((foldM_ :: (b -> a -> Maybe b) -> b -> [a] -> Maybe ()) :: (b -> a -> Maybe b) -> b -> [a] -> Maybe ())

module Monad where

import qualified Maybe
import qualified Prelude

foldM :: Monad a => (b -> c -> a b) -> b -> [c] -> a b

foldM	_ a []	=	return a
foldM	f a (x : xs)	=	f a x >>= (\fax ->foldM f fax xs)

foldM_ :: Monad a => (c -> b -> a c) -> c -> [b] -> a ()

foldM_

f a xs

foldM f a xs >> return ()

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\fax→foldM f fax xs

is transformed to

foldM0 f xs fax = foldM f fax xs

The following Lambda expression

\_→q

is transformed to

gtGt0 q _ = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

((foldM_ :: (b -> a -> Maybe b) -> b -> [a] -> Maybe ()) :: (b -> a -> Maybe b) -> b -> [a] -> Maybe ())

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

foldM :: Monad a => (b -> c -> a b) -> b -> [c] -> a b

foldM	_ a []	=	return a
foldM	f a (x : xs)	=	f a x >>= foldM0 f xs

foldM0

f xs fax

foldM f fax xs

foldM_ :: Monad b => (c -> a -> b c) -> c -> [a] -> b ()

foldM_

f a xs

foldM f a xs >> return ()

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

((foldM_ :: (a -> b -> Maybe a) -> a -> [b] -> Maybe ()) :: (a -> b -> Maybe a) -> a -> [b] -> Maybe ())

module Monad where

import qualified Maybe
import qualified Prelude

foldM :: Monad b => (a -> c -> b a) -> a -> [c] -> b a

foldM	vw a []	=	return a
foldM	f a (x : xs)	=	f a x >>= foldM0 f xs

foldM0

f xs fax

foldM f fax xs

foldM_ :: Monad a => (c -> b -> a c) -> c -> [b] -> a ()

foldM_

f a xs

foldM f a xs >> return ()

module Maybe where

import qualified Monad
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

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule Monad

(foldM_ :: (b -> a -> Maybe b) -> b -> [a] -> Maybe ())

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

foldM :: Monad b => (c -> a -> b c) -> c -> [a] -> b c

foldM	vw a []	=	return a
foldM	f a (x : xs)	=	f a x >>= foldM0 f xs

foldM0

f xs fax

foldM f fax xs

foldM_ :: Monad b => (c -> a -> b c) -> c -> [a] -> b ()

foldM_

f a xs

foldM f a xs >> return ()

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="foldM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="foldM_ wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="foldM_ wu3 wu4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="foldM_ wu3 wu4 wu5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="foldM wu3 wu4 wu5 >> return ()",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldM wu3 wu4 wu5 >>= gtGt0 (return ())",fontsize=16,color="burlywood",shape="triangle"];30[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];7 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 8[label="",style="solid", color="burlywood", weight=3];
31[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldM wu3 wu4 (wu50 : wu51) >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldM wu3 wu4 [] >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10 -> 12[label="",style="dashed", color="red", weight=0];
10[label="wu3 wu4 wu50 >>= foldM0 wu3 wu51 >>= gtGt0 (return ())",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="return wu4 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
13[label="wu3 wu4 wu50",fontsize=16,color="green",shape="box"];13 -> 19[label="",style="dashed", color="green", weight=3];
13 -> 20[label="",style="dashed", color="green", weight=3];
12[label="wu6 >>= foldM0 wu3 wu51 >>= gtGt0 (return ())",fontsize=16,color="burlywood",shape="triangle"];33[label="wu6/Nothing",fontsize=10,color="white",style="solid",shape="box"];12 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 17[label="",style="solid", color="burlywood", weight=3];
34[label="wu6/Just wu60",fontsize=10,color="white",style="solid",shape="box"];12 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 18[label="",style="solid", color="burlywood", weight=3];
14[label="Just wu4 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];14 -> 21[label="",style="solid", color="black", weight=3];
19[label="wu4",fontsize=16,color="green",shape="box"];20[label="wu50",fontsize=16,color="green",shape="box"];17[label="Nothing >>= foldM0 wu3 wu51 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];17 -> 22[label="",style="solid", color="black", weight=3];
18[label="Just wu60 >>= foldM0 wu3 wu51 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];18 -> 23[label="",style="solid", color="black", weight=3];
21[label="gtGt0 (return ()) wu4",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
22[label="Nothing >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
23[label="foldM0 wu3 wu51 wu60 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="return ()",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
25[label="Nothing",fontsize=16,color="green",shape="box"];26 -> 7[label="",style="dashed", color="red", weight=0];
26[label="foldM wu3 wu60 wu51 >>= gtGt0 (return ())",fontsize=16,color="magenta"];26 -> 28[label="",style="dashed", color="magenta", weight=3];
26 -> 29[label="",style="dashed", color="magenta", weight=3];
27[label="Just ()",fontsize=16,color="green",shape="box"];28[label="wu60",fontsize=16,color="green",shape="box"];29[label="wu51",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_gtGtEs(wu3, wu51, h, ba) → new_gtGtEs0(wu3, wu51, h, ba)
new_gtGtEs0(wu3, :(wu50, wu51), h, ba) → new_gtGtEs(wu3, wu51, h, 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_gtGtEs0(wu3, :(wu50, wu51), h, ba) → new_gtGtEs(wu3, wu51, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
new_gtGtEs(wu3, wu51, h, ba) → new_gtGtEs0(wu3, wu51, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4