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

YES 0.41300000000000003 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

((lookup :: () -> [((),a)] -> Maybe a) :: () -> [((),a)] -> Maybe a)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((lookup :: () -> [((),a)] -> Maybe a) :: () -> [((),a)] -> Maybe a)

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

The following Function with conditions

lookup k [] = Nothing

lookup k ((x,y) : xys)

| k == x

= Just y

| otherwise

= lookup k xys

is transformed to

lookup k [] = lookup3 k []

lookup k ((x,y) : xys) = lookup2 k ((x,y) : xys)

lookup1 k x y xys True = Just y

lookup1 k x y xys False = lookup0 k x y xys otherwise

lookup0 k x y xys True = lookup k xys

lookup2 k ((x,y) : xys) = lookup1 k x y xys (k == x)

lookup3 k [] = Nothing

lookup3 wu wv = lookup2 wu wv

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(lookup :: () -> [((),a)] -> Maybe a)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="lookup ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="lookup ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];16[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];4 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 5[label="",style="solid", color="burlywood", weight=3];
17[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="lookup ww3 (ww40 : ww41)",fontsize=16,color="burlywood",shape="box"];18[label="ww40/(ww400,ww401)",fontsize=10,color="white",style="solid",shape="box"];5 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="lookup ww3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="lookup ww3 ((ww400,ww401) : ww41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="lookup3 ww3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="lookup2 ww3 ((ww400,ww401) : ww41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 ww3 ww400 ww401 ww41 (ww3 == ww400)",fontsize=16,color="burlywood",shape="box"];19[label="ww3/()",fontsize=10,color="white",style="solid",shape="box"];11 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 12[label="",style="solid", color="burlywood", weight=3];
12[label="lookup1 () ww400 ww401 ww41 (() == ww400)",fontsize=16,color="burlywood",shape="box"];20[label="ww400/()",fontsize=10,color="white",style="solid",shape="box"];12 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 13[label="",style="solid", color="burlywood", weight=3];
13[label="lookup1 () () ww401 ww41 (() == ())",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="lookup1 () () ww401 ww41 True",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="Just ww401",fontsize=16,color="green",shape="box"];}

lookup	k []	= lookup3 k []
lookup	k ((x,y) : xys)	= lookup2 k ((x,y) : xys)

lookup1	k x y xys True	= Just y
lookup1	k x y xys False	= lookup0 k x y xys otherwise