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

YES 0.743 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

((mapAndUnzipM :: (a -> IO (b,c)) -> [a] -> IO ([b],[c])) :: (a -> IO (b,c)) -> [a] -> IO ([b],[c]))

module Monad where

import qualified Maybe
import qualified Prelude

mapAndUnzipM :: Monad c => (d -> c (b,a)) -> [d] -> c ([b],[a])

mapAndUnzipM

f xs

sequence (map f xs) >>= return . unzip

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\xs→return (x : xs)

is transformed to

sequence0 x xs = return (x : xs)

The following Lambda expression

\x→sequence cs >>= sequence0 x

is transformed to

sequence1 cs x = sequence cs >>= sequence0 x

The following Lambda expression

\(a,b)~(as,bs)→(a : as,b : bs)

is transformed to

unzip0 (a,b) ~(as,bs) = (a : as,b : bs)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

mainModule Monad

((mapAndUnzipM :: (b -> IO (c,a)) -> [b] -> IO ([c],[a])) :: (b -> IO (c,a)) -> [b] -> IO ([c],[a]))

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

mapAndUnzipM :: Monad d => (b -> d (c,a)) -> [b] -> d ([c],[a])

mapAndUnzipM

f xs

sequence (map f xs) >>= return . unzip

IrrPat Reductions:
The variables of the following irrefutable Pattern

~(as,bs)

are replaced by calls to these functions

unzip00 (as,bs) = as

unzip01 (as,bs) = bs

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

        ↳ HASKELL

          ↳ BR

mainModule Monad

((mapAndUnzipM :: (b -> IO (a,c)) -> [b] -> IO ([a],[c])) :: (b -> IO (a,c)) -> [b] -> IO ([a],[c]))

module Monad where

import qualified Maybe
import qualified Prelude

mapAndUnzipM :: Monad c => (a -> c (b,d)) -> [a] -> c ([b],[d])

mapAndUnzipM

f xs

sequence (map f xs) >>= return . unzip

module Maybe where

import qualified Monad
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule Monad

((mapAndUnzipM :: (c -> IO (b,a)) -> [c] -> IO ([b],[a])) :: (c -> IO (b,a)) -> [c] -> IO ([b],[a]))

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

mapAndUnzipM :: Monad b => (d -> b (a,c)) -> [d] -> b ([a],[c])

mapAndUnzipM

f xs

sequence (map f xs) >>= return . unzip

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

mainModule Monad

(mapAndUnzipM :: (a -> IO (b,c)) -> [a] -> IO ([b],[c]))

module Monad where

import qualified Maybe
import qualified Prelude

mapAndUnzipM :: Monad c => (a -> c (b,d)) -> [a] -> c ([b],[d])

mapAndUnzipM

f xs

sequence (map f xs) >>= return . unzip

module Maybe where

import qualified Monad
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="mapAndUnzipM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
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6[label="primbindIO (sequence (map vz3 vz4)) (return . unzip)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="terminator (sequence (map vz3 vz4)) (return . unzip)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="ter0m (sequence (map vz3 vz4)) (return . unzip)",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="green", weight=3];
8 -> 10[label="",style="dashed", color="green", weight=3];
9[label="sequence (map vz3 vz4)",fontsize=16,color="burlywood",shape="triangle"];65[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 11[label="",style="solid", color="burlywood", weight=3];
66[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 12[label="",style="solid", color="burlywood", weight=3];
10[label="return . unzip",fontsize=16,color="grey",shape="box"];10 -> 13[label="",style="dashed", color="grey", weight=3];
11[label="sequence (map vz3 (vz40 : vz41))",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="sequence (map vz3 [])",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13[label="return . unzip",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="sequence (vz3 vz40 : map vz3 vz41)",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="sequence []",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
16[label="return (unzip vz5)",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
17[label="vz3 vz40 >>= sequence1 (map vz3 vz41)",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
18[label="return []",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
19[label="primretIO (unzip vz5)",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
20[label="primbindIO (vz3 vz40) (sequence1 (map vz3 vz41))",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3];
21[label="primretIO []",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
22[label="terminator (unzip vz5)",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
23[label="terminator (vz3 vz40) (sequence1 (map vz3 vz41))",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="terminator []",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
25[label="ter1m (unzip vz5)",fontsize=16,color="green",shape="box"];25 -> 28[label="",style="dashed", color="green", weight=3];
26[label="ter2m (vz3 vz40) (sequence1 (map vz3 vz41))",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="ter3m []",fontsize=16,color="green",shape="box"];27 -> 31[label="",style="dashed", color="green", weight=3];
28[label="unzip vz5",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="vz3 vz40",fontsize=16,color="green",shape="box"];29 -> 33[label="",style="dashed", color="green", weight=3];
30[label="sequence1 (map vz3 vz41)",fontsize=16,color="grey",shape="box"];30 -> 34[label="",style="dashed", color="grey", weight=3];
31[label="[]",fontsize=16,color="green",shape="box"];32[label="foldr unzip0 ([],[]) vz5",fontsize=16,color="burlywood",shape="triangle"];67[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];32 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 35[label="",style="solid", color="burlywood", weight=3];
68[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];32 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 36[label="",style="solid", color="burlywood", weight=3];
33[label="vz40",fontsize=16,color="green",shape="box"];34[label="sequence1 (map vz3 vz41) vz6",fontsize=16,color="black",shape="triangle"];34 -> 37[label="",style="solid", color="black", weight=3];
35[label="foldr unzip0 ([],[]) (vz50 : vz51)",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
36[label="foldr unzip0 ([],[]) []",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
37 -> 40[label="",style="dashed", color="red", weight=0];
37[label="sequence (map vz3 vz41) >>= sequence0 vz6",fontsize=16,color="magenta"];37 -> 41[label="",style="dashed", color="magenta", weight=3];
38 -> 42[label="",style="dashed", color="red", weight=0];
38[label="unzip0 vz50 (foldr unzip0 ([],[]) vz51)",fontsize=16,color="magenta"];38 -> 43[label="",style="dashed", color="magenta", weight=3];
39[label="([],[])",fontsize=16,color="green",shape="box"];41 -> 9[label="",style="dashed", color="red", weight=0];
41[label="sequence (map vz3 vz41)",fontsize=16,color="magenta"];41 -> 44[label="",style="dashed", color="magenta", weight=3];
40[label="vz7 >>= sequence0 vz6",fontsize=16,color="black",shape="triangle"];40 -> 45[label="",style="solid", color="black", weight=3];
43 -> 32[label="",style="dashed", color="red", weight=0];
43[label="foldr unzip0 ([],[]) vz51",fontsize=16,color="magenta"];43 -> 46[label="",style="dashed", color="magenta", weight=3];
42[label="unzip0 vz50 vz8",fontsize=16,color="burlywood",shape="triangle"];73[label="vz50/(vz500,vz501)",fontsize=10,color="white",style="solid",shape="box"];42 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 47[label="",style="solid", color="burlywood", weight=3];
44[label="vz41",fontsize=16,color="green",shape="box"];45[label="primbindIO vz7 (sequence0 vz6)",fontsize=16,color="black",shape="box"];45 -> 48[label="",style="solid", color="black", weight=3];
46[label="vz51",fontsize=16,color="green",shape="box"];47[label="unzip0 (vz500,vz501) vz8",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48[label="terminator vz7 (sequence0 vz6)",fontsize=16,color="black",shape="box"];48 -> 50[label="",style="solid", color="black", weight=3];
49[label="(vz500 : unzip00 vz8,vz501 : unzip01 vz8)",fontsize=16,color="green",shape="box"];49 -> 51[label="",style="dashed", color="green", weight=3];
49 -> 52[label="",style="dashed", color="green", weight=3];
50[label="ter4m vz7 (sequence0 vz6)",fontsize=16,color="green",shape="box"];50 -> 53[label="",style="dashed", color="green", weight=3];
50 -> 54[label="",style="dashed", color="green", weight=3];
51[label="unzip00 vz8",fontsize=16,color="burlywood",shape="box"];74[label="vz8/(vz80,vz81)",fontsize=10,color="white",style="solid",shape="box"];51 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 55[label="",style="solid", color="burlywood", weight=3];
52[label="unzip01 vz8",fontsize=16,color="burlywood",shape="box"];75[label="vz8/(vz80,vz81)",fontsize=10,color="white",style="solid",shape="box"];52 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 56[label="",style="solid", color="burlywood", weight=3];
53[label="vz7",fontsize=16,color="green",shape="box"];54[label="sequence0 vz6",fontsize=16,color="grey",shape="box"];54 -> 57[label="",style="dashed", color="grey", weight=3];
55[label="unzip00 (vz80,vz81)",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
56[label="unzip01 (vz80,vz81)",fontsize=16,color="black",shape="box"];56 -> 59[label="",style="solid", color="black", weight=3];
57[label="sequence0 vz6 vz9",fontsize=16,color="black",shape="triangle"];57 -> 60[label="",style="solid", color="black", weight=3];
58[label="vz80",fontsize=16,color="green",shape="box"];59[label="vz81",fontsize=16,color="green",shape="box"];60[label="return (vz6 : vz9)",fontsize=16,color="black",shape="box"];60 -> 61[label="",style="solid", color="black", weight=3];
61[label="primretIO (vz6 : vz9)",fontsize=16,color="black",shape="box"];61 -> 62[label="",style="solid", color="black", weight=3];
62[label="terminator (vz6 : vz9)",fontsize=16,color="black",shape="box"];62 -> 63[label="",style="solid", color="black", weight=3];
63[label="ter5m (vz6 : vz9)",fontsize=16,color="green",shape="box"];63 -> 64[label="",style="dashed", color="green", weight=3];
64[label="vz6 : vz9",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

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

new_foldr(:(vz50, vz51), h, ba) → new_foldr(vz51, 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_foldr(:(vz50, vz51), h, ba) → new_foldr(vz51, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

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

new_sequence1(vz3, vz41, vz6, h, ba, bb) → new_sequence(vz3, vz41, h, ba, bb)
new_sequence(vz3, :(vz40, vz41), h, ba, bb) → new_sequence(vz3, vz41, h, ba, bb)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

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

new_sequence(vz3, :(vz40, vz41), h, ba, bb) → new_sequence(vz3, vz41, h, ba, bb)

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

new_sequence(vz3, :(vz40, vz41), h, ba, bb) → new_sequence(vz3, vz41, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5