H-Termination proof of ../tpdb/FP/full_haskell/Maybe_mapMaybe

YES H-Termination proof of ../tpdb/FP/full_haskell/Maybe_mapMaybe_1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ CR

mainModule Main

((Maybe.mapMaybe :: (b -> Maybe a) -> [b] -> [a]) :: (a -> Maybe b) -> [a] -> [b])

module Main where

import qualified Maybe
import qualified Prelude

module Maybe where

import qualified Main
import qualified Prelude

mapMaybe :: (a -> Maybe b) -> [a] -> [b]

mapMaybe

_ []

[]

mapMaybe

f (x : xs)

let

mapMaybe f xs

case	f x of
	Nothing	->	rs
	Just r	->	r : rs

Case Reductions:
The following Case expression

case f x of
Nothing → rs

Just r → r : rs

is transformed to

mapMaybe0 Nothing = rs

mapMaybe0 (Just r) = r : rs

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

mainModule Main

((Maybe.mapMaybe :: (a -> Maybe b) -> [a] -> [b]) :: (a -> Maybe b) -> [a] -> [b])

module Main where

import qualified Maybe
import qualified Prelude

module Maybe where

import qualified Main
import qualified Prelude

mapMaybe :: (b -> Maybe a) -> [b] -> [a]

mapMaybe

_ []

[]

mapMaybe

f (x : xs)

let

mapMaybe0	Nothing	=	rs
mapMaybe0	(Just r)	=	r : rs

mapMaybe f xs

mapMaybe0 (f x)

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ LetRed

mainModule Main

((Maybe.mapMaybe :: (a -> Maybe b) -> [a] -> [b]) :: (a -> Maybe b) -> [a] -> [b])

module Main where

import qualified Maybe
import qualified Prelude

module Maybe where

import qualified Main
import qualified Prelude

mapMaybe :: (a -> Maybe b) -> [a] -> [b]

mapMaybe

vw []

[]

mapMaybe

f (x : xs)

let

mapMaybe0	Nothing	=	rs
mapMaybe0	(Just r)	=	r : rs

mapMaybe f xs

mapMaybe0 (f x)

Let/Where Reductions:
The bindings of the following Let/Where expression

let

mapMaybe0 Nothing = rs

mapMaybe0 (Just r) = r : rs

rs = mapMaybe f xs

in mapMaybe0 (f x)

are unpacked to the following functions on top level

mapMaybeRs vz wu = mapMaybe vz wu

mapMaybeMapMaybe0 vz wu Nothing = mapMaybeRs vz wu

mapMaybeMapMaybe0 vz wu (Just r) = r : mapMaybeRs vz wu

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ NumRed

mainModule Main

((Maybe.mapMaybe :: (b -> Maybe a) -> [b] -> [a]) :: (b -> Maybe a) -> [b] -> [a])

module Main where

import qualified Maybe
import qualified Prelude

module Maybe where

import qualified Main
import qualified Prelude

mapMaybe :: (a -> Maybe b) -> [a] -> [b]

mapMaybe	vw []	=	[]
mapMaybe	f (x : xs)	=	mapMaybeMapMaybe0 f xs (f x)

mapMaybeMapMaybe0	vz wu Nothing	=	mapMaybeRs vz wu
mapMaybeMapMaybe0	vz wu (Just r)	=	r : mapMaybeRs vz wu

mapMaybeRs

vz wu

mapMaybe vz wu

Num Reduction:All numbers are transformed to thier corresponding representation with Succ, Pred and Zero.

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

mainModule Main

(Maybe.mapMaybe :: (b -> Maybe a) -> [b] -> [a])

module Main where

import qualified Maybe
import qualified Prelude

module Maybe where

import qualified Main
import qualified Prelude

mapMaybe :: (a -> Maybe b) -> [a] -> [b]

mapMaybe	vw []	=	[]
mapMaybe	f (x : xs)	=	mapMaybeMapMaybe0 f xs (f x)

mapMaybeMapMaybe0	vz wu Nothing	=	mapMaybeRs vz wu
mapMaybeMapMaybe0	vz wu (Just r)	=	r : mapMaybeRs vz wu

mapMaybeRs

vz wu

mapMaybe vz wu

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Maybe.mapMaybe\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Maybe.mapMaybe wv3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Maybe.mapMaybe wv3 wv4\n",fontsize=16,color="burlywood",shape="triangle"];20[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 5[label="",style="solid", color="burlywood", weight=3];
21[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="Maybe.mapMaybe wv3 (wv40 : wv41)\n",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="Maybe.mapMaybe wv3 []\n",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 (wv3 wv40)\n",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="[]\n",fontsize=16,color="green",shape="box"];10[label="wv3 wv40\n",fontsize=16,color="green",shape="box"];10 -> 14[label="",style="dashed", color="green", weight=3];
9[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 wv5\n",fontsize=16,color="burlywood",shape="triangle"];23[label="wv5/Nothing",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 12[label="",style="solid", color="burlywood", weight=3];
24[label="wv5/Just wv50",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 13[label="",style="solid", color="burlywood", weight=3];
14[label="wv40\n",fontsize=16,color="green",shape="box"];12[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 Nothing\n",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 (Just wv50)\n",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
15[label="Maybe.mapMaybeRs wv3 wv41\n",fontsize=16,color="black",shape="triangle"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="wv50 : Maybe.mapMaybeRs wv3 wv41\n",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
17 -> 4[label="",style="dashed", color="red", weight=0];
17[label="Maybe.mapMaybe wv3 wv41\n",fontsize=16,color="magenta"];17 -> 19[label="",style="dashed", color="magenta", weight=3];
18 -> 15[label="",style="dashed", color="red", weight=0];
18[label="Maybe.mapMaybeRs wv3 wv41\n",fontsize=16,color="magenta"];19[label="wv41\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_mapMaybeMapMaybe0(wv3, wv41, h, ba) → new_mapMaybe(wv3, wv41, h, ba)
new_mapMaybe(wv3, :(wv40, wv41), h, ba) → new_mapMaybeMapMaybe0(wv3, wv41, h, ba)
new_mapMaybeRs(wv3, wv41, h, ba) → new_mapMaybe(wv3, wv41, h, ba)
new_mapMaybeMapMaybe0(wv3, wv41, h, ba) → new_mapMaybeRs(wv3, wv41, 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_mapMaybe(wv3, :(wv40, wv41), h, ba) → new_mapMaybeMapMaybe0(wv3, wv41, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
new_mapMaybeMapMaybe0(wv3, wv41, h, ba) → new_mapMaybe(wv3, wv41, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
new_mapMaybeMapMaybe0(wv3, wv41, h, ba) → new_mapMaybeRs(wv3, wv41, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
new_mapMaybeRs(wv3, wv41, h, ba) → new_mapMaybe(wv3, wv41, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4

mapMaybeMapMaybe0	vz wu Nothing	= mapMaybeRs vz wu
mapMaybeMapMaybe0	vz wu (Just r)	= r : mapMaybeRs vz wu