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

YES 0.459 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

((ap :: Maybe (a -> b) -> Maybe a -> Maybe b) :: Maybe (a -> b) -> Maybe a -> Maybe b)

module Monad where

import qualified Maybe
import qualified Prelude

ap :: Monad c => c (a -> b) -> c a -> c b

liftM2 id

liftM2 :: Monad a => (c -> b -> d) -> a c -> a b -> a d

liftM2

f m1 m2

m1 >>= (\x1 ->m2 >>= (\x2 ->return (f x1 x2)))

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\x2→return (f x1 x2)

is transformed to

liftM20 f x1 x2 = return (f x1 x2)

The following Lambda expression

\x1→m2 >>= liftM20 f x1

is transformed to

liftM21 m2 f x1 = m2 >>= liftM20 f x1

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

((ap :: Maybe (a -> b) -> Maybe a -> Maybe b) :: Maybe (a -> b) -> Maybe a -> Maybe b)

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

ap :: Monad a => a (b -> c) -> a b -> a c

liftM2 id

liftM2 :: Monad c => (a -> d -> b) -> c a -> c d -> c b

liftM2

f m1 m2

m1 >>= liftM21 m2 f

liftM20

f x1 x2

return (f x1 x2)

liftM21

m2 f x1

m2 >>= liftM20 f x1

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

((ap :: Maybe (a -> b) -> Maybe a -> Maybe b) :: Maybe (a -> b) -> Maybe a -> Maybe b)

module Monad where

import qualified Maybe
import qualified Prelude

ap :: Monad b => b (c -> a) -> b c -> b a

liftM2 id

liftM2 :: Monad c => (a -> b -> d) -> c a -> c b -> c d

liftM2

f m1 m2

m1 >>= liftM21 m2 f

liftM20

f x1 x2

return (f x1 x2)

liftM21

m2 f x1

m2 >>= liftM20 f x1

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

(ap :: Maybe (a -> b) -> Maybe a -> Maybe b)

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

ap :: Monad a => a (b -> c) -> a b -> a c

liftM2 id

liftM2 :: Monad c => (d -> b -> a) -> c d -> c b -> c a

liftM2

f m1 m2

m1 >>= liftM21 m2 f

liftM20

f x1 x2

return (f x1 x2)

liftM21

m2 f x1

m2 >>= liftM20 f x1

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="ap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="ap vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="ap vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="liftM2 id vy3 vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="vy3 >>= liftM21 vy4 id",fontsize=16,color="burlywood",shape="box"];21[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 7[label="",style="solid", color="burlywood", weight=3];
22[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];6 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="Nothing >>= liftM21 vy4 id",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="Just vy30 >>= liftM21 vy4 id",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="Nothing",fontsize=16,color="green",shape="box"];10[label="liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="vy4 >>= liftM20 id vy30",fontsize=16,color="burlywood",shape="box"];23[label="vy4/Nothing",fontsize=10,color="white",style="solid",shape="box"];11 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 12[label="",style="solid", color="burlywood", weight=3];
24[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 13[label="",style="solid", color="burlywood", weight=3];
12[label="Nothing >>= liftM20 id vy30",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="Just vy40 >>= liftM20 id vy30",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="Nothing",fontsize=16,color="green",shape="box"];15[label="liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="return (id vy30 vy40)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="Just (id vy30 vy40)",fontsize=16,color="green",shape="box"];17 -> 18[label="",style="dashed", color="green", weight=3];
18[label="id vy30 vy40",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="vy30 vy40",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="green", weight=3];
20[label="vy40",fontsize=16,color="green",shape="box"];}