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

YES 0.982 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule List

((nub :: [Bool] -> [Bool]) :: [Bool] -> [Bool])

module List where

import qualified Maybe
import qualified Prelude

nub :: Eq a => [a] -> [a]

nub

nub' l []

where

nub'

[] _

[]

nub'

(x : xs) ls

x `elem` ls

nub' xs ls

otherwise

x : nub' xs (x : ls)

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule List

((nub :: [Bool] -> [Bool]) :: [Bool] -> [Bool])

module List where

import qualified Maybe
import qualified Prelude

nub :: Eq a => [a] -> [a]

nub

nub' l []

where

nub'

[] vw

[]

nub'

(x : xs) ls

x `elem` ls

nub' xs ls

otherwise

x : nub' xs (x : ls)

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

nub' [] vw = []

nub' (x : xs) ls

| x `elem` ls

= nub' xs ls

| otherwise

= x : nub' xs (x : ls)

is transformed to

nub' [] vw = nub'3 [] vw

nub' (x : xs) ls = nub'2 (x : xs) ls

nub'1 x xs ls True = nub' xs ls

nub'1 x xs ls False = nub'0 x xs ls otherwise

nub'0 x xs ls True = x : nub' xs (x : ls)

nub'2 (x : xs) ls = nub'1 x xs ls (x `elem` ls)

nub'3 [] vw = []

nub'3 wv ww = nub'2 wv ww

The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

mainModule List

((nub :: [Bool] -> [Bool]) :: [Bool] -> [Bool])

module List where

import qualified Maybe
import qualified Prelude

nub :: Eq a => [a] -> [a]

nub

nub' l []

where

nub'	[] vw	=	nub'3 [] vw
nub'	(x : xs) ls	=	nub'2 (x : xs) ls

nub'0

x xs ls True

x : nub' xs (x : ls)

nub'1	x xs ls True	=	nub' xs ls
nub'1	x xs ls False	=	nub'0 x xs ls otherwise

nub'2

(x : xs) ls

nub'1 x xs ls (x `elem` ls)

nub'3	[] vw	=	[]
nub'3	wv ww	=	nub'2 wv ww

module Maybe where

import qualified List
import qualified Prelude

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

nub' l []

where

nub' [] vw = nub'3 [] vw

nub' (x : xs) ls = nub'2 (x : xs) ls

nub'0 x xs ls True = x : nub' xs (x : ls)

nub'1 x xs ls True = nub' xs ls

nub'1 x xs ls False = nub'0 x xs ls otherwise

nub'2 (x : xs) ls = nub'1 x xs ls (x `elem` ls)

nub'3 [] vw = []

nub'3 wv ww = nub'2 wv ww

are unpacked to the following functions on top level

nubNub'1 x xs ls True = nubNub' xs ls

nubNub'1 x xs ls False = nubNub'0 x xs ls otherwise

nubNub' [] vw = nubNub'3 [] vw

nubNub' (x : xs) ls = nubNub'2 (x : xs) ls

nubNub'2 (x : xs) ls = nubNub'1 x xs ls (x `elem` ls)

nubNub'3 [] vw = []

nubNub'3 wv ww = nubNub'2 wv ww

nubNub'0 x xs ls True = x : nubNub' xs (x : ls)

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

mainModule List

(nub :: [Bool] -> [Bool])

module List where

import qualified Maybe
import qualified Prelude

nub :: Eq a => [a] -> [a]

nub

nubNub' l []

nubNub'	[] vw	=	nubNub'3 [] vw
nubNub'	(x : xs) ls	=	nubNub'2 (x : xs) ls

nubNub'0

x xs ls True

x : nubNub' xs (x : ls)

nubNub'1	x xs ls True	=	nubNub' xs ls
nubNub'1	x xs ls False	=	nubNub'0 x xs ls otherwise

nubNub'2

(x : xs) ls

nubNub'1 x xs ls (x `elem` ls)

nubNub'3	[] vw	=	[]
nubNub'3	wv ww	=	nubNub'2 wv ww

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="nub",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="nub wx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="nubNub' wx3 []",fontsize=16,color="burlywood",shape="box"];121[label="wx3/wx30 : wx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 5[label="",style="solid", color="burlywood", weight=3];
122[label="wx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="nubNub' (wx30 : wx31) []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="nubNub' [] []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="nubNub'2 (wx30 : wx31) []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="nubNub'3 [] []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="nubNub'1 wx30 wx31 [] (wx30 `elem` [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="[]",fontsize=16,color="green",shape="box"];11[label="nubNub'1 wx30 wx31 [] (any . (==))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="nubNub'1 wx30 wx31 [] (any ((==) wx30) [])",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="nubNub'1 wx30 wx31 [] (or . map ((==) wx30))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="nubNub'1 wx30 wx31 [] (or (map ((==) wx30) []))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="nubNub'1 wx30 wx31 [] (foldr (||) False (map ((==) wx30) []))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="nubNub'1 wx30 wx31 [] (foldr (||) False [])",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="nubNub'1 wx30 wx31 [] False",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="nubNub'0 wx30 wx31 [] otherwise",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="nubNub'0 wx30 wx31 [] True",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="wx30 : nubNub' wx31 (wx30 : [])",fontsize=16,color="green",shape="box"];20 -> 21[label="",style="dashed", color="green", weight=3];
21[label="nubNub' wx31 (wx30 : [])",fontsize=16,color="burlywood",shape="triangle"];123[label="wx31/wx310 : wx311",fontsize=10,color="white",style="solid",shape="box"];21 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 22[label="",style="solid", color="burlywood", weight=3];
124[label="wx31/[]",fontsize=10,color="white",style="solid",shape="box"];21 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 23[label="",style="solid", color="burlywood", weight=3];
22[label="nubNub' (wx310 : wx311) (wx30 : [])",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="nubNub' [] (wx30 : [])",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="nubNub'2 (wx310 : wx311) (wx30 : [])",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="nubNub'3 [] (wx30 : [])",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="nubNub'1 wx310 wx311 (wx30 : []) (wx310 `elem` wx30 : [])",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="[]",fontsize=16,color="green",shape="box"];28[label="nubNub'1 wx310 wx311 (wx30 : []) (any . (==))",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3];
29[label="nubNub'1 wx310 wx311 (wx30 : []) (any ((==) wx310) (wx30 : []))",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
30[label="nubNub'1 wx310 wx311 (wx30 : []) (or . map ((==) wx310))",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
31[label="nubNub'1 wx310 wx311 (wx30 : []) (or (map ((==) wx310) (wx30 : [])))",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32[label="nubNub'1 wx310 wx311 (wx30 : []) (foldr (||) False (map ((==) wx310) (wx30 : [])))",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3];
33[label="nubNub'1 wx310 wx311 (wx30 : []) (foldr (||) False (((==) wx310 wx30) : map ((==) wx310) []))",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
34[label="nubNub'1 wx310 wx311 (wx30 : []) ((||) (==) wx310 wx30 foldr (||) False (map ((==) wx310) []))",fontsize=16,color="burlywood",shape="box"];125[label="wx310/False",fontsize=10,color="white",style="solid",shape="box"];34 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 35[label="",style="solid", color="burlywood", weight=3];
126[label="wx310/True",fontsize=10,color="white",style="solid",shape="box"];34 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 36[label="",style="solid", color="burlywood", weight=3];
35[label="nubNub'1 False wx311 (wx30 : []) ((||) (==) False wx30 foldr (||) False (map ((==) False) []))",fontsize=16,color="burlywood",shape="box"];127[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];35 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 37[label="",style="solid", color="burlywood", weight=3];
128[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];35 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 38[label="",style="solid", color="burlywood", weight=3];
36[label="nubNub'1 True wx311 (wx30 : []) ((||) (==) True wx30 foldr (||) False (map ((==) True) []))",fontsize=16,color="burlywood",shape="box"];129[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];36 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 39[label="",style="solid", color="burlywood", weight=3];
130[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];36 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 40[label="",style="solid", color="burlywood", weight=3];
37[label="nubNub'1 False wx311 (False : []) ((||) (==) False False foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="nubNub'1 False wx311 (True : []) ((||) (==) False True foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="nubNub'1 True wx311 (False : []) ((||) (==) True False foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
40[label="nubNub'1 True wx311 (True : []) ((||) (==) True True foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="nubNub'1 False wx311 (False : []) ((||) True foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="nubNub'1 False wx311 (True : []) ((||) False foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
43[label="nubNub'1 True wx311 (False : []) ((||) False foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
44[label="nubNub'1 True wx311 (True : []) ((||) True foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
45[label="nubNub'1 False wx311 (False : []) True",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
46[label="nubNub'1 False wx311 (True : []) (foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
47[label="nubNub'1 True wx311 (False : []) (foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3];
48[label="nubNub'1 True wx311 (True : []) True",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3];
49 -> 21[label="",style="dashed", color="red", weight=0];
49[label="nubNub' wx311 (False : [])",fontsize=16,color="magenta"];49 -> 53[label="",style="dashed", color="magenta", weight=3];
49 -> 54[label="",style="dashed", color="magenta", weight=3];
50[label="nubNub'1 False wx311 (True : []) (foldr (||) False [])",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3];
51[label="nubNub'1 True wx311 (False : []) (foldr (||) False [])",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3];
52 -> 21[label="",style="dashed", color="red", weight=0];
52[label="nubNub' wx311 (True : [])",fontsize=16,color="magenta"];52 -> 57[label="",style="dashed", color="magenta", weight=3];
52 -> 58[label="",style="dashed", color="magenta", weight=3];
53[label="wx311",fontsize=16,color="green",shape="box"];54[label="False",fontsize=16,color="green",shape="box"];55[label="nubNub'1 False wx311 (True : []) False",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="nubNub'1 True wx311 (False : []) False",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3];
57[label="wx311",fontsize=16,color="green",shape="box"];58[label="True",fontsize=16,color="green",shape="box"];59[label="nubNub'0 False wx311 (True : []) otherwise",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3];
60[label="nubNub'0 True wx311 (False : []) otherwise",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
61[label="nubNub'0 False wx311 (True : []) True",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="nubNub'0 True wx311 (False : []) True",fontsize=16,color="black",shape="box"];62 -> 64[label="",style="solid", color="black", weight=3];
63[label="False : nubNub' wx311 (False : True : [])",fontsize=16,color="green",shape="box"];63 -> 65[label="",style="dashed", color="green", weight=3];
64[label="True : nubNub' wx311 (True : False : [])",fontsize=16,color="green",shape="box"];64 -> 66[label="",style="dashed", color="green", weight=3];
65[label="nubNub' wx311 (False : True : [])",fontsize=16,color="burlywood",shape="triangle"];133[label="wx311/wx3110 : wx3111",fontsize=10,color="white",style="solid",shape="box"];65 -> 133[label="",style="solid", color="burlywood", weight=9];
133 -> 67[label="",style="solid", color="burlywood", weight=3];
134[label="wx311/[]",fontsize=10,color="white",style="solid",shape="box"];65 -> 134[label="",style="solid", color="burlywood", weight=9];
134 -> 68[label="",style="solid", color="burlywood", weight=3];
66[label="nubNub' wx311 (True : False : [])",fontsize=16,color="burlywood",shape="triangle"];135[label="wx311/wx3110 : wx3111",fontsize=10,color="white",style="solid",shape="box"];66 -> 135[label="",style="solid", color="burlywood", weight=9];
135 -> 69[label="",style="solid", color="burlywood", weight=3];
136[label="wx311/[]",fontsize=10,color="white",style="solid",shape="box"];66 -> 136[label="",style="solid", color="burlywood", weight=9];
136 -> 70[label="",style="solid", color="burlywood", weight=3];
67[label="nubNub' (wx3110 : wx3111) (False : True : [])",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
68[label="nubNub' [] (False : True : [])",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="nubNub' (wx3110 : wx3111) (True : False : [])",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
70[label="nubNub' [] (True : False : [])",fontsize=16,color="black",shape="box"];70 -> 74[label="",style="solid", color="black", weight=3];
71[label="nubNub'2 (wx3110 : wx3111) (False : True : [])",fontsize=16,color="black",shape="box"];71 -> 75[label="",style="solid", color="black", weight=3];
72[label="nubNub'3 [] (False : True : [])",fontsize=16,color="black",shape="box"];72 -> 76[label="",style="solid", color="black", weight=3];
73[label="nubNub'2 (wx3110 : wx3111) (True : False : [])",fontsize=16,color="black",shape="box"];73 -> 77[label="",style="solid", color="black", weight=3];
74[label="nubNub'3 [] (True : False : [])",fontsize=16,color="black",shape="box"];74 -> 78[label="",style="solid", color="black", weight=3];
75[label="nubNub'1 wx3110 wx3111 (False : True : []) (wx3110 `elem` False : True : [])",fontsize=16,color="black",shape="box"];75 -> 79[label="",style="solid", color="black", weight=3];
76[label="[]",fontsize=16,color="green",shape="box"];77[label="nubNub'1 wx3110 wx3111 (True : False : []) (wx3110 `elem` True : False : [])",fontsize=16,color="black",shape="box"];77 -> 80[label="",style="solid", color="black", weight=3];
78[label="[]",fontsize=16,color="green",shape="box"];79[label="nubNub'1 wx3110 wx3111 (False : True : []) (any . (==))",fontsize=16,color="black",shape="box"];79 -> 81[label="",style="solid", color="black", weight=3];
80[label="nubNub'1 wx3110 wx3111 (True : False : []) (any . (==))",fontsize=16,color="black",shape="box"];80 -> 82[label="",style="solid", color="black", weight=3];
81[label="nubNub'1 wx3110 wx3111 (False : True : []) (any ((==) wx3110) (False : True : []))",fontsize=16,color="black",shape="box"];81 -> 83[label="",style="solid", color="black", weight=3];
82[label="nubNub'1 wx3110 wx3111 (True : False : []) (any ((==) wx3110) (True : False : []))",fontsize=16,color="black",shape="box"];82 -> 84[label="",style="solid", color="black", weight=3];
83[label="nubNub'1 wx3110 wx3111 (False : True : []) (or . map ((==) wx3110))",fontsize=16,color="black",shape="box"];83 -> 85[label="",style="solid", color="black", weight=3];
84[label="nubNub'1 wx3110 wx3111 (True : False : []) (or . map ((==) wx3110))",fontsize=16,color="black",shape="box"];84 -> 86[label="",style="solid", color="black", weight=3];
85[label="nubNub'1 wx3110 wx3111 (False : True : []) (or (map ((==) wx3110) (False : True : [])))",fontsize=16,color="black",shape="box"];85 -> 87[label="",style="solid", color="black", weight=3];
86[label="nubNub'1 wx3110 wx3111 (True : False : []) (or (map ((==) wx3110) (True : False : [])))",fontsize=16,color="black",shape="box"];86 -> 88[label="",style="solid", color="black", weight=3];
87[label="nubNub'1 wx3110 wx3111 (False : True : []) (foldr (||) False (map ((==) wx3110) (False : True : [])))",fontsize=16,color="black",shape="box"];87 -> 89[label="",style="solid", color="black", weight=3];
88[label="nubNub'1 wx3110 wx3111 (True : False : []) (foldr (||) False (map ((==) wx3110) (True : False : [])))",fontsize=16,color="black",shape="box"];88 -> 90[label="",style="solid", color="black", weight=3];
89[label="nubNub'1 wx3110 wx3111 (False : True : []) (foldr (||) False (((==) wx3110 False) : map ((==) wx3110) (True : [])))",fontsize=16,color="black",shape="box"];89 -> 91[label="",style="solid", color="black", weight=3];
90[label="nubNub'1 wx3110 wx3111 (True : False : []) (foldr (||) False (((==) wx3110 True) : map ((==) wx3110) (False : [])))",fontsize=16,color="black",shape="box"];90 -> 92[label="",style="solid", color="black", weight=3];
91[label="nubNub'1 wx3110 wx3111 (False : True : []) ((||) (==) wx3110 False foldr (||) False (map ((==) wx3110) (True : [])))",fontsize=16,color="burlywood",shape="box"];137[label="wx3110/False",fontsize=10,color="white",style="solid",shape="box"];91 -> 137[label="",style="solid", color="burlywood", weight=9];
137 -> 93[label="",style="solid", color="burlywood", weight=3];
138[label="wx3110/True",fontsize=10,color="white",style="solid",shape="box"];91 -> 138[label="",style="solid", color="burlywood", weight=9];
138 -> 94[label="",style="solid", color="burlywood", weight=3];
92[label="nubNub'1 wx3110 wx3111 (True : False : []) ((||) (==) wx3110 True foldr (||) False (map ((==) wx3110) (False : [])))",fontsize=16,color="burlywood",shape="box"];139[label="wx3110/False",fontsize=10,color="white",style="solid",shape="box"];92 -> 139[label="",style="solid", color="burlywood", weight=9];
139 -> 95[label="",style="solid", color="burlywood", weight=3];
140[label="wx3110/True",fontsize=10,color="white",style="solid",shape="box"];92 -> 140[label="",style="solid", color="burlywood", weight=9];
140 -> 96[label="",style="solid", color="burlywood", weight=3];
93[label="nubNub'1 False wx3111 (False : True : []) ((||) (==) False False foldr (||) False (map ((==) False) (True : [])))",fontsize=16,color="black",shape="box"];93 -> 97[label="",style="solid", color="black", weight=3];
94[label="nubNub'1 True wx3111 (False : True : []) ((||) (==) True False foldr (||) False (map ((==) True) (True : [])))",fontsize=16,color="black",shape="box"];94 -> 98[label="",style="solid", color="black", weight=3];
95[label="nubNub'1 False wx3111 (True : False : []) ((||) (==) False True foldr (||) False (map ((==) False) (False : [])))",fontsize=16,color="black",shape="box"];95 -> 99[label="",style="solid", color="black", weight=3];
96[label="nubNub'1 True wx3111 (True : False : []) ((||) (==) True True foldr (||) False (map ((==) True) (False : [])))",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3];
97[label="nubNub'1 False wx3111 (False : True : []) ((||) True foldr (||) False (map ((==) False) (True : [])))",fontsize=16,color="black",shape="box"];97 -> 101[label="",style="solid", color="black", weight=3];
98[label="nubNub'1 True wx3111 (False : True : []) ((||) False foldr (||) False (map ((==) True) (True : [])))",fontsize=16,color="black",shape="box"];98 -> 102[label="",style="solid", color="black", weight=3];
99[label="nubNub'1 False wx3111 (True : False : []) ((||) False foldr (||) False (map ((==) False) (False : [])))",fontsize=16,color="black",shape="box"];99 -> 103[label="",style="solid", color="black", weight=3];
100[label="nubNub'1 True wx3111 (True : False : []) ((||) True foldr (||) False (map ((==) True) (False : [])))",fontsize=16,color="black",shape="box"];100 -> 104[label="",style="solid", color="black", weight=3];
101[label="nubNub'1 False wx3111 (False : True : []) True",fontsize=16,color="black",shape="box"];101 -> 105[label="",style="solid", color="black", weight=3];
102[label="nubNub'1 True wx3111 (False : True : []) (foldr (||) False (map ((==) True) (True : [])))",fontsize=16,color="black",shape="box"];102 -> 106[label="",style="solid", color="black", weight=3];
103[label="nubNub'1 False wx3111 (True : False : []) (foldr (||) False (map ((==) False) (False : [])))",fontsize=16,color="black",shape="box"];103 -> 107[label="",style="solid", color="black", weight=3];
104[label="nubNub'1 True wx3111 (True : False : []) True",fontsize=16,color="black",shape="box"];104 -> 108[label="",style="solid", color="black", weight=3];
105 -> 65[label="",style="dashed", color="red", weight=0];
105[label="nubNub' wx3111 (False : True : [])",fontsize=16,color="magenta"];105 -> 109[label="",style="dashed", color="magenta", weight=3];
106[label="nubNub'1 True wx3111 (False : True : []) (foldr (||) False (((==) True True) : map ((==) True) []))",fontsize=16,color="black",shape="box"];106 -> 110[label="",style="solid", color="black", weight=3];
107[label="nubNub'1 False wx3111 (True : False : []) (foldr (||) False (((==) False False) : map ((==) False) []))",fontsize=16,color="black",shape="box"];107 -> 111[label="",style="solid", color="black", weight=3];
108 -> 66[label="",style="dashed", color="red", weight=0];
108[label="nubNub' wx3111 (True : False : [])",fontsize=16,color="magenta"];108 -> 112[label="",style="dashed", color="magenta", weight=3];
109[label="wx3111",fontsize=16,color="green",shape="box"];110[label="nubNub'1 True wx3111 (False : True : []) ((||) (==) True True foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];110 -> 113[label="",style="solid", color="black", weight=3];
111[label="nubNub'1 False wx3111 (True : False : []) ((||) (==) False False foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];111 -> 114[label="",style="solid", color="black", weight=3];
112[label="wx3111",fontsize=16,color="green",shape="box"];113[label="nubNub'1 True wx3111 (False : True : []) ((||) True foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];113 -> 115[label="",style="solid", color="black", weight=3];
114[label="nubNub'1 False wx3111 (True : False : []) ((||) True foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];114 -> 116[label="",style="solid", color="black", weight=3];
115[label="nubNub'1 True wx3111 (False : True : []) True",fontsize=16,color="black",shape="box"];115 -> 117[label="",style="solid", color="black", weight=3];
116[label="nubNub'1 False wx3111 (True : False : []) True",fontsize=16,color="black",shape="box"];116 -> 118[label="",style="solid", color="black", weight=3];
117 -> 65[label="",style="dashed", color="red", weight=0];
117[label="nubNub' wx3111 (False : True : [])",fontsize=16,color="magenta"];117 -> 119[label="",style="dashed", color="magenta", weight=3];
118 -> 66[label="",style="dashed", color="red", weight=0];
118[label="nubNub' wx3111 (True : False : [])",fontsize=16,color="magenta"];118 -> 120[label="",style="dashed", color="magenta", weight=3];
119[label="wx3111",fontsize=16,color="green",shape="box"];120[label="wx3111",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_nubNub'(:(True, wx3111)) → new_nubNub'(wx3111)
new_nubNub'(:(False, wx3111)) → new_nubNub'(wx3111)

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_nubNub'(:(True, wx3111)) → new_nubNub'(wx3111)
The graph contains the following edges 1 > 1
new_nubNub'(:(False, wx3111)) → new_nubNub'(wx3111)
The graph contains the following edges 1 > 1

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_nubNub'0(:(True, wx3111)) → new_nubNub'0(wx3111)
new_nubNub'0(:(False, wx3111)) → new_nubNub'0(wx3111)

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

new_nubNub'0(:(True, wx3111)) → new_nubNub'0(wx3111)
The graph contains the following edges 1 > 1
new_nubNub'0(:(False, wx3111)) → new_nubNub'0(wx3111)
The graph contains the following edges 1 > 1

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

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

new_nubNub'1(:(True, wx311), True) → new_nubNub'1(wx311, True)
new_nubNub'1(:(False, wx311), False) → new_nubNub'1(wx311, False)

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 2 SCCs.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ QDPSizeChangeProof

                        ↳ QDP

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

new_nubNub'1(:(False, wx311), False) → new_nubNub'1(wx311, False)

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

new_nubNub'1(:(False, wx311), False) → new_nubNub'1(wx311, False)
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ QDPSizeChangeProof

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

new_nubNub'1(:(True, wx311), True) → new_nubNub'1(wx311, True)

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

new_nubNub'1(:(True, wx311), True) → new_nubNub'1(wx311, True)
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

nub'1	x xs ls True	= nub' xs ls
nub'1	x xs ls False	= nub'0 x xs ls otherwise

nubNub'1	x xs ls True	= nubNub' xs ls
nubNub'1	x xs ls False	= nubNub'0 x xs ls otherwise

nubNub'	[] vw	= nubNub'3 [] vw
nubNub'	(x : xs) ls	= nubNub'2 (x : xs) ls