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

YES 0.731 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

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

module Monad where

import qualified Maybe
import qualified Prelude

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

liftM

f m1

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

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\x1→return (f x1)

is transformed to

liftM0 f x1 = return (f x1)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

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

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

liftM

f m1

m1 >>= liftM0 f

liftM0

f x1

return (f x1)

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

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

module Monad where

import qualified Maybe
import qualified Prelude

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

liftM

f m1

m1 >>= liftM0 f

liftM0

f x1

return (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

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

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

liftM

f m1

m1 >>= liftM0 f

liftM0

f x1

return (f x1)

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="liftM\n  Monad a",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="liftM vy3\n  Monad a",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="liftM vy3 vy4\n  Monad a",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="vy4 >>= liftM0 vy3\n  Monad a",fontsize=16,color="blue",shape="box"];43[label=">>= :: (IO a) -> (a -> IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];5 -> 43[label="",style="solid", color="blue", weight=9];
43 -> 6[label="",style="solid", color="blue", weight=3];
44[label=">>= :: ([] a) -> (a -> [] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];5 -> 44[label="",style="solid", color="blue", weight=9];
44 -> 7[label="",style="solid", color="blue", weight=3];
45[label=">>= :: (Maybe a) -> (a -> Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];5 -> 45[label="",style="solid", color="blue", weight=9];
45 -> 8[label="",style="solid", color="blue", weight=3];
6[label="vy4 >>= liftM0 vy3",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="vy4 >>= liftM0 vy3",fontsize=16,color="burlywood",shape="triangle"];46[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 10[label="",style="solid", color="burlywood", weight=3];
47[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="vy4 >>= liftM0 vy3",fontsize=16,color="burlywood",shape="box"];48[label="vy4/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 12[label="",style="solid", color="burlywood", weight=3];
49[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];8 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="primbindIO vy4 (liftM0 vy3)",fontsize=16,color="black",shape="box"];9 -> 14[label="",style="solid", color="black", weight=3];
10[label="vy40 : vy41 >>= liftM0 vy3",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3];
11[label="[] >>= liftM0 vy3",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3];
12[label="Nothing >>= liftM0 vy3",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3];
13[label="Just vy40 >>= liftM0 vy3",fontsize=16,color="black",shape="box"];13 -> 18[label="",style="solid", color="black", weight=3];
14[label="terminator vy4 (liftM0 vy3)",fontsize=16,color="black",shape="box"];14 -> 19[label="",style="solid", color="black", weight=3];
15 -> 20[label="",style="dashed", color="red", weight=0];
15[label="liftM0 vy3 vy40 ++ (vy41 >>= liftM0 vy3)",fontsize=16,color="magenta"];15 -> 21[label="",style="dashed", color="magenta", weight=3];
16[label="[]",fontsize=16,color="green",shape="box"];17[label="Nothing",fontsize=16,color="green",shape="box"];18[label="liftM0 vy3 vy40",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="ter0m vy4 (liftM0 vy3)",fontsize=16,color="green",shape="box"];19 -> 23[label="",style="dashed", color="green", weight=3];
19 -> 24[label="",style="dashed", color="green", weight=3];
21 -> 7[label="",style="dashed", color="red", weight=0];
21[label="vy41 >>= liftM0 vy3",fontsize=16,color="magenta"];21 -> 25[label="",style="dashed", color="magenta", weight=3];
20[label="liftM0 vy3 vy40 ++ vy5",fontsize=16,color="black",shape="triangle"];20 -> 26[label="",style="solid", color="black", weight=3];
22[label="return (vy3 vy40)",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3];
23[label="vy4",fontsize=16,color="green",shape="box"];24[label="liftM0 vy3",fontsize=16,color="grey",shape="box"];24 -> 28[label="",style="dashed", color="grey", weight=3];
25[label="vy41",fontsize=16,color="green",shape="box"];26[label="return (vy3 vy40) ++ vy5",fontsize=16,color="black",shape="box"];26 -> 29[label="",style="solid", color="black", weight=3];
27[label="Just (vy3 vy40)",fontsize=16,color="green",shape="box"];27 -> 30[label="",style="dashed", color="green", weight=3];
28[label="liftM0 vy3 vy6",fontsize=16,color="black",shape="triangle"];28 -> 31[label="",style="solid", color="black", weight=3];
29[label="(vy3 vy40 : []) ++ vy5",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
30[label="vy3 vy40",fontsize=16,color="green",shape="box"];30 -> 33[label="",style="dashed", color="green", weight=3];
31[label="return (vy3 vy6)",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3];
32[label="vy3 vy40 : [] ++ vy5",fontsize=16,color="green",shape="box"];32 -> 35[label="",style="dashed", color="green", weight=3];
32 -> 36[label="",style="dashed", color="green", weight=3];
33[label="vy40",fontsize=16,color="green",shape="box"];34[label="primretIO (vy3 vy6)",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3];
35[label="vy3 vy40",fontsize=16,color="green",shape="box"];35 -> 38[label="",style="dashed", color="green", weight=3];
36[label="[] ++ vy5",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
37[label="terminator (vy3 vy6)",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
38[label="vy40",fontsize=16,color="green",shape="box"];39[label="vy5",fontsize=16,color="green",shape="box"];40[label="ter1m (vy3 vy6)",fontsize=16,color="green",shape="box"];40 -> 41[label="",style="dashed", color="green", weight=3];
41[label="vy3 vy6",fontsize=16,color="green",shape="box"];41 -> 42[label="",style="dashed", color="green", weight=3];
42[label="vy6",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(:(vy40, vy41), vy3, h, ba) → new_gtGtEs(vy41, vy3, 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_gtGtEs(:(vy40, vy41), vy3, h, ba) → new_gtGtEs(vy41, vy3, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4