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

YES 0.451 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

((liftM2 :: (c -> a -> b) -> IO c -> IO a -> IO b) :: (c -> a -> b) -> IO c -> IO a -> IO b)

module Monad where

import qualified Maybe
import qualified Prelude

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

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

((liftM2 :: (b -> c -> a) -> IO b -> IO c -> IO a) :: (b -> c -> a) -> IO b -> IO c -> IO a)

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

((liftM2 :: (a -> b -> c) -> IO a -> IO b -> IO c) :: (a -> b -> c) -> IO a -> IO b -> IO c)

module Monad where

import qualified Maybe
import qualified Prelude

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

(liftM2 :: (c -> a -> b) -> IO c -> IO a -> IO b)

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

Haskell To QDPs

digraph dp_graph {
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3[label="liftM2 vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="liftM2 vy3 vy4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="liftM2 vy3 vy4 vy5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="vy4 >>= liftM21 vy5 vy3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="primbindIO vy4 (liftM21 vy5 vy3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="terminator vy4 (liftM21 vy5 vy3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
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9 -> 11[label="",style="dashed", color="green", weight=3];
10[label="vy4",fontsize=16,color="green",shape="box"];11[label="liftM21 vy5 vy3",fontsize=16,color="grey",shape="box"];11 -> 12[label="",style="dashed", color="grey", weight=3];
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15[label="terminator vy5 (liftM20 vy3 vy6)",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
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16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="vy5",fontsize=16,color="green",shape="box"];18[label="liftM20 vy3 vy6",fontsize=16,color="grey",shape="box"];18 -> 19[label="",style="dashed", color="grey", weight=3];
19[label="liftM20 vy3 vy6 vy7",fontsize=16,color="black",shape="triangle"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="return (vy3 vy6 vy7)",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="primretIO (vy3 vy6 vy7)",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="terminator (vy3 vy6 vy7)",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="ter2m (vy3 vy6 vy7)",fontsize=16,color="green",shape="box"];23 -> 24[label="",style="dashed", color="green", weight=3];
24[label="vy3 vy6 vy7",fontsize=16,color="green",shape="box"];24 -> 25[label="",style="dashed", color="green", weight=3];
24 -> 26[label="",style="dashed", color="green", weight=3];
25[label="vy6",fontsize=16,color="green",shape="box"];26[label="vy7",fontsize=16,color="green",shape="box"];}