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

YES 0.6930000000000001 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

((zipWithM_ :: (c -> a -> Maybe b) -> [c] -> [a] -> Maybe ()) :: (c -> a -> Maybe b) -> [c] -> [a] -> Maybe ())

module Monad where

import qualified Maybe
import qualified Prelude

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

zipWithM_

f xs ys

sequence_ (zipWith f xs ys)

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\_→q

is transformed to

gtGt0 q _ = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

((zipWithM_ :: (a -> c -> Maybe b) -> [a] -> [c] -> Maybe ()) :: (a -> c -> Maybe b) -> [a] -> [c] -> Maybe ())

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

zipWithM_

f xs ys

sequence_ (zipWith f xs ys)

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

((zipWithM_ :: (c -> a -> Maybe b) -> [c] -> [a] -> Maybe ()) :: (c -> a -> Maybe b) -> [c] -> [a] -> Maybe ())

module Monad where

import qualified Maybe
import qualified Prelude

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

zipWithM_

f xs ys

sequence_ (zipWith f xs ys)

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

(zipWithM_ :: (c -> a -> Maybe b) -> [c] -> [a] -> Maybe ())

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

zipWithM_

f xs ys

sequence_ (zipWith f xs ys)

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="zipWithM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="zipWithM_ ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="zipWithM_ ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="zipWithM_ ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence_ (zipWith ww3 ww4 ww5)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr (>>) (return ()) (zipWith ww3 ww4 ww5)",fontsize=16,color="burlywood",shape="triangle"];34[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];7 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 8[label="",style="solid", color="burlywood", weight=3];
35[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) ww5)",fontsize=16,color="burlywood",shape="box"];36[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];8 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 10[label="",style="solid", color="burlywood", weight=3];
37[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 11[label="",style="solid", color="burlywood", weight=3];
9[label="foldr (>>) (return ()) (zipWith ww3 [] ww5)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) (ww50 : ww51))",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) [])",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
13[label="foldr (>>) (return ()) (ww3 ww40 ww50 : zipWith ww3 ww41 ww51)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14 -> 12[label="",style="dashed", color="red", weight=0];
14[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];15[label="return ()",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16 -> 18[label="",style="dashed", color="red", weight=0];
16[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="Just ()",fontsize=16,color="green",shape="box"];19 -> 7[label="",style="dashed", color="red", weight=0];
19[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
19 -> 21[label="",style="dashed", color="magenta", weight=3];
18[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];18 -> 22[label="",style="solid", color="black", weight=3];
20[label="ww41",fontsize=16,color="green",shape="box"];21[label="ww51",fontsize=16,color="green",shape="box"];22 -> 23[label="",style="dashed", color="red", weight=0];
22[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
24[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];24 -> 29[label="",style="dashed", color="green", weight=3];
24 -> 30[label="",style="dashed", color="green", weight=3];
23[label="ww8 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];42[label="ww8/Nothing",fontsize=10,color="white",style="solid",shape="box"];23 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 27[label="",style="solid", color="burlywood", weight=3];
43[label="ww8/Just ww80",fontsize=10,color="white",style="solid",shape="box"];23 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 28[label="",style="solid", color="burlywood", weight=3];
29[label="ww40",fontsize=16,color="green",shape="box"];30[label="ww50",fontsize=16,color="green",shape="box"];27[label="Nothing >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="Just ww80 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
31[label="Nothing",fontsize=16,color="green",shape="box"];32[label="gtGt0 ww6 ww80",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3];
33[label="ww6",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_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) → new_foldr(ww3, ww41, ww51, h, ba, bb)

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(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) → new_foldr(ww3, ww41, ww51, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6