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

YES 0.71 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 :: [a -> b] -> [a] -> [b]) :: [a -> b] -> [a] -> [b])

module Monad where

import qualified Maybe
import qualified Prelude

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

liftM2 id

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

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 :: [b -> a] -> [b] -> [a]) :: [b -> a] -> [b] -> [a])

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 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

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

((ap :: [b -> a] -> [b] -> [a]) :: [b -> a] -> [b] -> [a])

module Monad where

import qualified Maybe
import qualified Prelude

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

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

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 :: [b -> a] -> [b] -> [a])

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

liftM2 id

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

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="triangle"];49[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 7[label="",style="solid", color="burlywood", weight=3];
50[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="vy30 : vy31 >>= liftM21 vy4 id",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="[] >>= liftM21 vy4 id",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="liftM21 vy4 id vy30 ++ (vy31 >>= liftM21 vy4 id)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="[]",fontsize=16,color="green",shape="box"];12 -> 6[label="",style="dashed", color="red", weight=0];
12[label="vy31 >>= liftM21 vy4 id",fontsize=16,color="magenta"];12 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="liftM21 vy4 id vy30 ++ vy5",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
13[label="vy31",fontsize=16,color="green",shape="box"];14[label="(vy4 >>= liftM20 id vy30) ++ vy5",fontsize=16,color="burlywood",shape="box"];53[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];14 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 15[label="",style="solid", color="burlywood", weight=3];
54[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];14 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 16[label="",style="solid", color="burlywood", weight=3];
15[label="(vy40 : vy41 >>= liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="([] >>= liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="(liftM20 id vy30 vy40 ++ (vy41 >>= liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="[] ++ vy5",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="(return (id vy30 vy40) ++ (vy41 >>= liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="vy5",fontsize=16,color="green",shape="box"];21[label="((id vy30 vy40 : []) ++ (vy41 >>= liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22 -> 23[label="",style="dashed", color="red", weight=0];
22[label="(id vy30 vy40 : [] ++ (vy41 >>= liftM20 id vy30)) ++ vy5",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
24 -> 18[label="",style="dashed", color="red", weight=0];
24[label="[] ++ (vy41 >>= liftM20 id vy30)",fontsize=16,color="magenta"];24 -> 25[label="",style="dashed", color="magenta", weight=3];
23[label="(id vy30 vy40 : vy6) ++ vy5",fontsize=16,color="black",shape="triangle"];23 -> 26[label="",style="solid", color="black", weight=3];
25[label="vy41 >>= liftM20 id vy30",fontsize=16,color="burlywood",shape="triangle"];57[label="vy41/vy410 : vy411",fontsize=10,color="white",style="solid",shape="box"];25 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 27[label="",style="solid", color="burlywood", weight=3];
58[label="vy41/[]",fontsize=10,color="white",style="solid",shape="box"];25 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 28[label="",style="solid", color="burlywood", weight=3];
26[label="id vy30 vy40 : vy6 ++ vy5",fontsize=16,color="green",shape="box"];26 -> 29[label="",style="dashed", color="green", weight=3];
26 -> 30[label="",style="dashed", color="green", weight=3];
27[label="vy410 : vy411 >>= liftM20 id vy30",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="[] >>= liftM20 id vy30",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="id vy30 vy40",fontsize=16,color="black",shape="triangle"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="vy6 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];59[label="vy6/vy60 : vy61",fontsize=10,color="white",style="solid",shape="box"];30 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 34[label="",style="solid", color="burlywood", weight=3];
60[label="vy6/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 35[label="",style="solid", color="burlywood", weight=3];
31 -> 30[label="",style="dashed", color="red", weight=0];
31[label="liftM20 id vy30 vy410 ++ (vy411 >>= liftM20 id vy30)",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3];
31 -> 37[label="",style="dashed", color="magenta", weight=3];
32[label="[]",fontsize=16,color="green",shape="box"];33[label="vy30 vy40",fontsize=16,color="green",shape="box"];33 -> 38[label="",style="dashed", color="green", weight=3];
34[label="(vy60 : vy61) ++ vy5",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
35[label="[] ++ vy5",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
36[label="liftM20 id vy30 vy410",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
37 -> 25[label="",style="dashed", color="red", weight=0];
37[label="vy411 >>= liftM20 id vy30",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3];
38[label="vy40",fontsize=16,color="green",shape="box"];39[label="vy60 : vy61 ++ vy5",fontsize=16,color="green",shape="box"];39 -> 43[label="",style="dashed", color="green", weight=3];
40[label="vy5",fontsize=16,color="green",shape="box"];41 -> 44[label="",style="dashed", color="red", weight=0];
41[label="return (id vy30 vy410)",fontsize=16,color="magenta"];41 -> 45[label="",style="dashed", color="magenta", weight=3];
42[label="vy411",fontsize=16,color="green",shape="box"];43 -> 30[label="",style="dashed", color="red", weight=0];
43[label="vy61 ++ vy5",fontsize=16,color="magenta"];43 -> 46[label="",style="dashed", color="magenta", weight=3];
45 -> 29[label="",style="dashed", color="red", weight=0];
45[label="id vy30 vy410",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3];
44[label="return vy7",fontsize=16,color="black",shape="triangle"];44 -> 48[label="",style="solid", color="black", weight=3];
46[label="vy61",fontsize=16,color="green",shape="box"];47[label="vy410",fontsize=16,color="green",shape="box"];48[label="vy7 : []",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_psPs(:(vy60, vy61), vy5, h) → new_psPs(vy61, vy5, 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_psPs(:(vy60, vy61), vy5, h) → new_psPs(vy61, vy5, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_gtGtEs(:(vy410, vy411), vy30, h, ba) → new_gtGtEs(vy411, vy30, h, ba)

From the DPs we obtained the following set of size-change graphs:

new_gtGtEs(:(vy410, vy411), vy30, h, ba) → new_gtGtEs(vy411, vy30, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_gtGtEs0(:(vy30, vy31), vy4, h, ba) → new_gtGtEs0(vy31, vy4, h, ba)

From the DPs we obtained the following set of size-change graphs:

new_gtGtEs0(:(vy30, vy31), vy4, h, ba) → new_gtGtEs0(vy31, vy4, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4