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

YES 0.964 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 :: [()] -> [()]) :: [()] -> [()])

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 :: [()] -> [()]) :: [()] -> [()])

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'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

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 :: [()] -> [()]) :: [()] -> [()])

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

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

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

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

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

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

nubNub'3 [] vw = []

nubNub'3 wv ww = nubNub'2 wv ww

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

mainModule List

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

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"];42[label="wx3/wx30 : wx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 5[label="",style="solid", color="burlywood", weight=3];
43[label="wx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 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"];44[label="wx31/wx310 : wx311",fontsize=10,color="white",style="solid",shape="box"];21 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 22[label="",style="solid", color="burlywood", weight=3];
45[label="wx31/[]",fontsize=10,color="white",style="solid",shape="box"];21 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 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"];46[label="wx310/()",fontsize=10,color="white",style="solid",shape="box"];34 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 35[label="",style="solid", color="burlywood", weight=3];
35[label="nubNub'1 () wx311 (wx30 : []) ((||) (==) () wx30 foldr (||) False (map ((==) ()) []))",fontsize=16,color="burlywood",shape="box"];47[label="wx30/()",fontsize=10,color="white",style="solid",shape="box"];35 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 36[label="",style="solid", color="burlywood", weight=3];
36[label="nubNub'1 () wx311 (() : []) ((||) (==) () () foldr (||) False (map ((==) ()) []))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="nubNub'1 () wx311 (() : []) ((||) True foldr (||) False (map ((==) ()) []))",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
38[label="nubNub'1 () wx311 (() : []) True",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
39 -> 21[label="",style="dashed", color="red", weight=0];
39[label="nubNub' wx311 (() : [])",fontsize=16,color="magenta"];39 -> 40[label="",style="dashed", color="magenta", weight=3];
39 -> 41[label="",style="dashed", color="magenta", weight=3];
40[label="wx311",fontsize=16,color="green",shape="box"];41[label="()",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_nubNub'(:(@0, wx311), @0) → new_nubNub'(wx311, @0)

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'(:(@0, wx311), @0) → new_nubNub'(wx311, @0)
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'	[] vw	= nubNub'3 [] vw
nubNub'	(x : xs) ls	= nubNub'2 (x : xs) ls

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