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

YES 1.758 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

  ↳ IFR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	_ _ []	=	[]
deleteBy	eq x (y : ys)	=	if x `eq` y then ys else y : deleteBy eq x ys

elem_by :: (a -> a -> Bool) -> a -> [a] -> Bool

elem_by	_ _ []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

nubBy :: (a -> a -> Bool) -> [a] -> [a]

nubBy

eq l

nubBy' l []

where

nubBy'

[] _

[]

nubBy'

(y : ys) xs

elem_by eq y xs

nubBy' ys xs

otherwise

y : nubBy' ys (y : xs)

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

union

unionBy (==)

unionBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

unionBy

eq xs ys

xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs

module Maybe where

import qualified List
import qualified Prelude

If Reductions:
The following If expression

if eq x y then ys else y : deleteBy eq x ys

is transformed to

deleteBy0 ys y eq x True = ys

deleteBy0 ys y eq x False = y : deleteBy eq x ys

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	_ _ []	=	[]
deleteBy	eq x (y : ys)	=	deleteBy0 ys y eq x (x `eq` y)

deleteBy0	ys y eq x True	=	ys
deleteBy0	ys y eq x False	=	y : deleteBy eq x ys

elem_by :: (a -> a -> Bool) -> a -> [a] -> Bool

elem_by	_ _ []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

nubBy :: (a -> a -> Bool) -> [a] -> [a]

nubBy

eq l

nubBy' l []

where

nubBy'

[] _

[]

nubBy'

(y : ys) xs

elem_by eq y xs

nubBy' ys xs

otherwise

y : nubBy' ys (y : xs)

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

union

unionBy (==)

unionBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

unionBy

eq xs ys

xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	vw vx []	=	[]
deleteBy	eq x (y : ys)	=	deleteBy0 ys y eq x (x `eq` y)

deleteBy0	ys y eq x True	=	ys
deleteBy0	ys y eq x False	=	y : deleteBy eq x ys

elem_by :: (a -> a -> Bool) -> a -> [a] -> Bool

elem_by	vz wu []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

nubBy :: (a -> a -> Bool) -> [a] -> [a]

nubBy

eq l

nubBy' l []

where

nubBy'

[] vy

[]

nubBy'

(y : ys) xs

elem_by eq y xs

nubBy' ys xs

otherwise

y : nubBy' ys (y : xs)

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

union

unionBy (==)

unionBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

unionBy

eq xs ys

xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

nubBy' [] vy = []

nubBy' (y : ys) xs

| elem_by eq y xs

= nubBy' ys xs

| otherwise

= y : nubBy' ys (y : xs)

is transformed to

nubBy' [] vy = nubBy'3 [] vy

nubBy' (y : ys) xs = nubBy'2 (y : ys) xs

nubBy'1 y ys xs True = nubBy' ys xs

nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise

nubBy'0 y ys xs True = y : nubBy' ys (y : xs)

nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs)

nubBy'3 [] vy = []

nubBy'3 wz xu = nubBy'2 wz xu

The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	vw vx []	=	[]
deleteBy	eq x (y : ys)	=	deleteBy0 ys y eq x (x `eq` y)

deleteBy0	ys y eq x True	=	ys
deleteBy0	ys y eq x False	=	y : deleteBy eq x ys

elem_by :: (a -> a -> Bool) -> a -> [a] -> Bool

elem_by	vz wu []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

nubBy :: (a -> a -> Bool) -> [a] -> [a]

nubBy

eq l

nubBy' l []

where

nubBy'	[] vy	=	nubBy'3 [] vy
nubBy'	(y : ys) xs	=	nubBy'2 (y : ys) xs

nubBy'0

y ys xs True

y : nubBy' ys (y : xs)

nubBy'1	y ys xs True	=	nubBy' ys xs
nubBy'1	y ys xs False	=	nubBy'0 y ys xs otherwise

nubBy'2

(y : ys) xs

nubBy'1 y ys xs (elem_by eq y xs)

nubBy'3	[] vy	=	[]
nubBy'3	wz xu	=	nubBy'2 wz xu

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

union

unionBy (==)

unionBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

unionBy

eq xs ys

xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs

module Maybe where

import qualified List
import qualified Prelude

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

nubBy' l []

where

nubBy' [] vy = nubBy'3 [] vy

nubBy' (y : ys) xs = nubBy'2 (y : ys) xs

nubBy'0 y ys xs True = y : nubBy' ys (y : xs)

nubBy'1 y ys xs True = nubBy' ys xs

nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise

nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs)

nubBy'3 [] vy = []

nubBy'3 wz xu = nubBy'2 wz xu

are unpacked to the following functions on top level

nubByNubBy'1 xv y ys xs True = nubByNubBy' xv ys xs

nubByNubBy'1 xv y ys xs False = nubByNubBy'0 xv y ys xs otherwise

nubByNubBy' xv [] vy = nubByNubBy'3 xv [] vy

nubByNubBy' xv (y : ys) xs = nubByNubBy'2 xv (y : ys) xs

nubByNubBy'2 xv (y : ys) xs = nubByNubBy'1 xv y ys xs (elem_by xv y xs)

nubByNubBy'3 xv [] vy = []

nubByNubBy'3 xv wz xu = nubByNubBy'2 xv wz xu

nubByNubBy'0 xv y ys xs True = y : nubByNubBy' xv ys (y : xs)

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	vw vx []	=	[]
deleteBy	eq x (y : ys)	=	deleteBy0 ys y eq x (x `eq` y)

deleteBy0	ys y eq x True	=	ys
deleteBy0	ys y eq x False	=	y : deleteBy eq x ys

elem_by :: (a -> a -> Bool) -> a -> [a] -> Bool

elem_by	vz wu []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

nubBy :: (a -> a -> Bool) -> [a] -> [a]

nubBy

eq l

nubByNubBy' eq l []

nubByNubBy'	xv [] vy	=	nubByNubBy'3 xv [] vy
nubByNubBy'	xv (y : ys) xs	=	nubByNubBy'2 xv (y : ys) xs

nubByNubBy'0

xv y ys xs True

y : nubByNubBy' xv ys (y : xs)

nubByNubBy'1	xv y ys xs True	=	nubByNubBy' xv ys xs
nubByNubBy'1	xv y ys xs False	=	nubByNubBy'0 xv y ys xs otherwise

nubByNubBy'2

xv (y : ys) xs

nubByNubBy'1 xv y ys xs (elem_by xv y xs)

nubByNubBy'3	xv [] vy	=	[]
nubByNubBy'3	xv wz xu	=	nubByNubBy'2 xv wz xu

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

union

unionBy (==)

unionBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]

unionBy

eq xs ys

xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
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3[label="union xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="union xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="unionBy (==) xw3 xw4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
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1153 -> 7[label="",style="solid", color="burlywood", weight=3];
1154[label="xw3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 1154[label="",style="solid", color="burlywood", weight=9];
1154 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="(xw30 : xw31) ++ foldl (flip (deleteBy (==))) (nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="[] ++ foldl (flip (deleteBy (==))) (nubBy (==) xw4) []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="xw30 : xw31 ++ foldl (flip (deleteBy (==))) (nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="green",shape="box"];9 -> 11[label="",style="dashed", color="green", weight=3];
10[label="foldl (flip (deleteBy (==))) (nubBy (==) xw4) []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 114[label="",style="dashed", color="red", weight=0];
11[label="xw31 ++ foldl (flip (deleteBy (==))) (nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="magenta"];11 -> 115[label="",style="dashed", color="magenta", weight=3];
11 -> 116[label="",style="dashed", color="magenta", weight=3];
11 -> 117[label="",style="dashed", color="magenta", weight=3];
11 -> 118[label="",style="dashed", color="magenta", weight=3];
12[label="nubBy (==) xw4",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
115[label="xw31",fontsize=16,color="green",shape="box"];116 -> 12[label="",style="dashed", color="red", weight=0];
116[label="nubBy (==) xw4",fontsize=16,color="magenta"];117[label="xw30",fontsize=16,color="green",shape="box"];118[label="xw31",fontsize=16,color="green",shape="box"];114[label="xw8 ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];1157[label="xw8/xw80 : xw81",fontsize=10,color="white",style="solid",shape="box"];114 -> 1157[label="",style="solid", color="burlywood", weight=9];
1157 -> 143[label="",style="solid", color="burlywood", weight=3];
1158[label="xw8/[]",fontsize=10,color="white",style="solid",shape="box"];114 -> 1158[label="",style="solid", color="burlywood", weight=9];
1158 -> 144[label="",style="solid", color="burlywood", weight=3];
15[label="nubByNubBy' (==) xw4 []",fontsize=16,color="burlywood",shape="box"];1159[label="xw4/xw40 : xw41",fontsize=10,color="white",style="solid",shape="box"];15 -> 1159[label="",style="solid", color="burlywood", weight=9];
1159 -> 18[label="",style="solid", color="burlywood", weight=3];
1160[label="xw4/[]",fontsize=10,color="white",style="solid",shape="box"];15 -> 1160[label="",style="solid", color="burlywood", weight=9];
1160 -> 19[label="",style="solid", color="burlywood", weight=3];
143[label="(xw80 : xw81) ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="black",shape="box"];143 -> 149[label="",style="solid", color="black", weight=3];
144[label="[] ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="black",shape="box"];144 -> 150[label="",style="solid", color="black", weight=3];
18[label="nubByNubBy' (==) (xw40 : xw41) []",fontsize=16,color="black",shape="box"];18 -> 23[label="",style="solid", color="black", weight=3];
19[label="nubByNubBy' (==) [] []",fontsize=16,color="black",shape="box"];19 -> 24[label="",style="solid", color="black", weight=3];
149[label="xw80 : xw81 ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="green",shape="box"];149 -> 155[label="",style="dashed", color="green", weight=3];
150[label="foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="black",shape="box"];150 -> 156[label="",style="solid", color="black", weight=3];
23[label="nubByNubBy'2 (==) (xw40 : xw41) []",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
24[label="nubByNubBy'3 (==) [] []",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
155 -> 114[label="",style="dashed", color="red", weight=0];
155[label="xw81 ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="magenta"];155 -> 163[label="",style="dashed", color="magenta", weight=3];
156[label="foldl (flip (deleteBy (==))) (flip (deleteBy (==)) xw9 xw10) xw11\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];1162[label="xw11/xw110 : xw111",fontsize=10,color="white",style="solid",shape="box"];156 -> 1162[label="",style="solid", color="burlywood", weight=9];
1162 -> 164[label="",style="solid", color="burlywood", weight=3];
1163[label="xw11/[]",fontsize=10,color="white",style="solid",shape="box"];156 -> 1163[label="",style="solid", color="burlywood", weight=9];
1163 -> 165[label="",style="solid", color="burlywood", weight=3];
28[label="nubByNubBy'1 (==) xw40 xw41 [] (elem_by (==) xw40 [])",fontsize=16,color="black",shape="box"];28 -> 33[label="",style="solid", color="black", weight=3];
29[label="[]",fontsize=16,color="green",shape="box"];163[label="xw81\n  Eq a",fontsize=16,color="green",shape="box"];164[label="foldl (flip (deleteBy (==))) (flip (deleteBy (==)) xw9 xw10) (xw110 : xw111)\n  Eq a",fontsize=16,color="black",shape="box"];164 -> 168[label="",style="solid", color="black", weight=3];
165[label="foldl (flip (deleteBy (==))) (flip (deleteBy (==)) xw9 xw10) []\n  Eq a",fontsize=16,color="black",shape="box"];165 -> 169[label="",style="solid", color="black", weight=3];
33[label="nubByNubBy'1 (==) xw40 xw41 [] False",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
168 -> 156[label="",style="dashed", color="red", weight=0];
168[label="foldl (flip (deleteBy (==))) (flip (deleteBy (==)) (flip (deleteBy (==)) xw9 xw10) xw110) xw111\n  Eq a",fontsize=16,color="magenta"];168 -> 172[label="",style="dashed", color="magenta", weight=3];
168 -> 173[label="",style="dashed", color="magenta", weight=3];
168 -> 174[label="",style="dashed", color="magenta", weight=3];
169[label="flip (deleteBy (==)) xw9 xw10\n  Eq a",fontsize=16,color="black",shape="triangle"];169 -> 175[label="",style="solid", color="black", weight=3];
37[label="nubByNubBy'0 (==) xw40 xw41 [] otherwise",fontsize=16,color="black",shape="box"];37 -> 42[label="",style="solid", color="black", weight=3];
172[label="xw111\n  Eq a",fontsize=16,color="green",shape="box"];173 -> 169[label="",style="dashed", color="red", weight=0];
173[label="flip (deleteBy (==)) xw9 xw10\n  Eq a",fontsize=16,color="magenta"];174[label="xw110\n  Eq a",fontsize=16,color="green",shape="box"];175[label="deleteBy (==) xw10 xw9\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];1166[label="xw9/xw90 : xw91",fontsize=10,color="white",style="solid",shape="box"];175 -> 1166[label="",style="solid", color="burlywood", weight=9];
1166 -> 178[label="",style="solid", color="burlywood", weight=3];
1167[label="xw9/[]",fontsize=10,color="white",style="solid",shape="box"];175 -> 1167[label="",style="solid", color="burlywood", weight=9];
1167 -> 179[label="",style="solid", color="burlywood", weight=3];
42[label="nubByNubBy'0 (==) xw40 xw41 [] True",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
178[label="deleteBy (==) xw10 (xw90 : xw91)\n  Eq a",fontsize=16,color="black",shape="box"];178 -> 184[label="",style="solid", color="black", weight=3];
179[label="deleteBy (==) xw10 []\n  Eq a",fontsize=16,color="black",shape="box"];179 -> 185[label="",style="solid", color="black", weight=3];
49[label="xw40 : nubByNubBy' (==) xw41 (xw40 : [])",fontsize=16,color="green",shape="box"];49 -> 55[label="",style="dashed", color="green", weight=3];
184 -> 190[label="",style="dashed", color="red", weight=0];
184[label="deleteBy0 xw91 xw90 (==) xw10 ((==) xw10 xw90)\n  Eq a",fontsize=16,color="magenta"];184 -> 191[label="",style="dashed", color="magenta", weight=3];
184 -> 192[label="",style="dashed", color="magenta", weight=3];
184 -> 193[label="",style="dashed", color="magenta", weight=3];
184 -> 194[label="",style="dashed", color="magenta", weight=3];
185[label="[]\n  Eq a",fontsize=16,color="green",shape="box"];55 -> 589[label="",style="dashed", color="red", weight=0];
55[label="nubByNubBy' (==) xw41 (xw40 : [])",fontsize=16,color="magenta"];55 -> 590[label="",style="dashed", color="magenta", weight=3];
55 -> 591[label="",style="dashed", color="magenta", weight=3];
55 -> 592[label="",style="dashed", color="magenta", weight=3];
191[label="xw10\n  Eq a",fontsize=16,color="green",shape="box"];192[label="(==) xw10 xw90\n  Eq a",fontsize=16,color="blue",shape="box"];1170[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1170[label="",style="solid", color="blue", weight=9];
1170 -> 195[label="",style="solid", color="blue", weight=3];
1171[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1171[label="",style="solid", color="blue", weight=9];
1171 -> 196[label="",style="solid", color="blue", weight=3];
1172[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1172[label="",style="solid", color="blue", weight=9];
1172 -> 197[label="",style="solid", color="blue", weight=3];
1173[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1173[label="",style="solid", color="blue", weight=9];
1173 -> 198[label="",style="solid", color="blue", weight=3];
1174[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1174[label="",style="solid", color="blue", weight=9];
1174 -> 199[label="",style="solid", color="blue", weight=3];
1175[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1175[label="",style="solid", color="blue", weight=9];
1175 -> 200[label="",style="solid", color="blue", weight=3];
1176[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1176[label="",style="solid", color="blue", weight=9];
1176 -> 201[label="",style="solid", color="blue", weight=3];
1177[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1177[label="",style="solid", color="blue", weight=9];
1177 -> 202[label="",style="solid", color="blue", weight=3];
1178[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1178[label="",style="solid", color="blue", weight=9];
1178 -> 203[label="",style="solid", color="blue", weight=3];
1179[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1179[label="",style="solid", color="blue", weight=9];
1179 -> 204[label="",style="solid", color="blue", weight=3];
1180[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1180[label="",style="solid", color="blue", weight=9];
1180 -> 205[label="",style="solid", color="blue", weight=3];
1181[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1181[label="",style="solid", color="blue", weight=9];
1181 -> 206[label="",style="solid", color="blue", weight=3];
1182[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1182[label="",style="solid", color="blue", weight=9];
1182 -> 207[label="",style="solid", color="blue", weight=3];
1183[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];192 -> 1183[label="",style="solid", color="blue", weight=9];
1183 -> 208[label="",style="solid", color="blue", weight=3];
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1185[label="xw20/True",fontsize=10,color="white",style="solid",shape="box"];190 -> 1185[label="",style="solid", color="burlywood", weight=9];
1185 -> 210[label="",style="solid", color="burlywood", weight=3];
590[label="xw40",fontsize=16,color="green",shape="box"];591[label="xw41",fontsize=16,color="green",shape="box"];592[label="[]",fontsize=16,color="green",shape="box"];589[label="nubByNubBy' (==) xw50 (xw51 : xw52)\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];1186[label="xw50/xw500 : xw501",fontsize=10,color="white",style="solid",shape="box"];589 -> 1186[label="",style="solid", color="burlywood", weight=9];
1186 -> 653[label="",style="solid", color="burlywood", weight=3];
1187[label="xw50/[]",fontsize=10,color="white",style="solid",shape="box"];589 -> 1187[label="",style="solid", color="burlywood", weight=9];
1187 -> 654[label="",style="solid", color="burlywood", weight=3];
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196[label="(==) xw10 xw90",fontsize=16,color="black",shape="triangle"];196 -> 216[label="",style="solid", color="black", weight=3];
197[label="(==) xw10 xw90",fontsize=16,color="black",shape="triangle"];197 -> 217[label="",style="solid", color="black", weight=3];
198[label="(==) xw10 xw90\n  Eq a  Eq b",fontsize=16,color="black",shape="triangle"];198 -> 218[label="",style="solid", color="black", weight=3];
199[label="(==) xw10 xw90",fontsize=16,color="black",shape="triangle"];199 -> 219[label="",style="solid", color="black", weight=3];
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201[label="(==) xw10 xw90\n  Eq a",fontsize=16,color="black",shape="triangle"];201 -> 221[label="",style="solid", color="black", weight=3];
202[label="(==) xw10 xw90\n  Eq a  Eq b  Eq c",fontsize=16,color="black",shape="triangle"];202 -> 222[label="",style="solid", color="black", weight=3];
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204[label="(==) xw10 xw90",fontsize=16,color="black",shape="triangle"];204 -> 224[label="",style="solid", color="black", weight=3];
205[label="(==) xw10 xw90",fontsize=16,color="black",shape="triangle"];205 -> 225[label="",style="solid", color="black", weight=3];
206[label="(==) xw10 xw90",fontsize=16,color="black",shape="triangle"];206 -> 226[label="",style="solid", color="black", weight=3];
207[label="(==) xw10 xw90",fontsize=16,color="black",shape="triangle"];207 -> 227[label="",style="solid", color="black", weight=3];
208[label="(==) xw10 xw90",fontsize=16,color="burlywood",shape="triangle"];1188[label="xw10/False",fontsize=10,color="white",style="solid",shape="box"];208 -> 1188[label="",style="solid", color="burlywood", weight=9];
1188 -> 228[label="",style="solid", color="burlywood", weight=3];
1189[label="xw10/True",fontsize=10,color="white",style="solid",shape="box"];208 -> 1189[label="",style="solid", color="burlywood", weight=9];
1189 -> 229[label="",style="solid", color="burlywood", weight=3];
209[label="deleteBy0 xw17 xw18 (==) xw19 False\n  Eq a",fontsize=16,color="black",shape="box"];209 -> 230[label="",style="solid", color="black", weight=3];
210[label="deleteBy0 xw17 xw18 (==) xw19 True\n  Eq a",fontsize=16,color="black",shape="box"];210 -> 231[label="",style="solid", color="black", weight=3];
653[label="nubByNubBy' (==) (xw500 : xw501) (xw51 : xw52)\n  Eq a",fontsize=16,color="black",shape="box"];653 -> 655[label="",style="solid", color="black", weight=3];
654[label="nubByNubBy' (==) [] (xw51 : xw52)\n  Eq a",fontsize=16,color="black",shape="box"];654 -> 656[label="",style="solid", color="black", weight=3];
215[label="error []",fontsize=16,color="red",shape="box"];216[label="error []",fontsize=16,color="red",shape="box"];217[label="error []",fontsize=16,color="red",shape="box"];218[label="error []",fontsize=16,color="red",shape="box"];219[label="error []",fontsize=16,color="red",shape="box"];220[label="error []",fontsize=16,color="red",shape="box"];221[label="error []",fontsize=16,color="red",shape="box"];222[label="error []",fontsize=16,color="red",shape="box"];223[label="error []",fontsize=16,color="red",shape="box"];224[label="error []",fontsize=16,color="red",shape="box"];225[label="error []",fontsize=16,color="red",shape="box"];226[label="error []",fontsize=16,color="red",shape="box"];227[label="error []",fontsize=16,color="red",shape="box"];228[label="(==) False xw90",fontsize=16,color="burlywood",shape="box"];1190[label="xw90/False",fontsize=10,color="white",style="solid",shape="box"];228 -> 1190[label="",style="solid", color="burlywood", weight=9];
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1191[label="xw90/True",fontsize=10,color="white",style="solid",shape="box"];228 -> 1191[label="",style="solid", color="burlywood", weight=9];
1191 -> 235[label="",style="solid", color="burlywood", weight=3];
229[label="(==) True xw90",fontsize=16,color="burlywood",shape="box"];1192[label="xw90/False",fontsize=10,color="white",style="solid",shape="box"];229 -> 1192[label="",style="solid", color="burlywood", weight=9];
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1193[label="xw90/True",fontsize=10,color="white",style="solid",shape="box"];229 -> 1193[label="",style="solid", color="burlywood", weight=9];
1193 -> 237[label="",style="solid", color="burlywood", weight=3];
230[label="xw18 : deleteBy (==) xw19 xw17\n  Eq a",fontsize=16,color="green",shape="box"];230 -> 238[label="",style="dashed", color="green", weight=3];
231[label="xw17\n  Eq a",fontsize=16,color="green",shape="box"];655[label="nubByNubBy'2 (==) (xw500 : xw501) (xw51 : xw52)\n  Eq a",fontsize=16,color="black",shape="box"];655 -> 657[label="",style="solid", color="black", weight=3];
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234[label="(==) False False",fontsize=16,color="black",shape="box"];234 -> 241[label="",style="solid", color="black", weight=3];
235[label="(==) False True",fontsize=16,color="black",shape="box"];235 -> 242[label="",style="solid", color="black", weight=3];
236[label="(==) True False",fontsize=16,color="black",shape="box"];236 -> 243[label="",style="solid", color="black", weight=3];
237[label="(==) True True",fontsize=16,color="black",shape="box"];237 -> 244[label="",style="solid", color="black", weight=3];
238 -> 175[label="",style="dashed", color="red", weight=0];
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238 -> 246[label="",style="dashed", color="magenta", weight=3];
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1196 -> 1048[label="",style="solid", color="blue", weight=3];
1197[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1197[label="",style="solid", color="blue", weight=9];
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1198[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1198[label="",style="solid", color="blue", weight=9];
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1199[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1199[label="",style="solid", color="blue", weight=9];
1199 -> 1051[label="",style="solid", color="blue", weight=3];
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1201[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1201[label="",style="solid", color="blue", weight=9];
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1202[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1202[label="",style="solid", color="blue", weight=9];
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1205[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1205[label="",style="solid", color="blue", weight=9];
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1206[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1206[label="",style="solid", color="blue", weight=9];
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1207[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1207[label="",style="solid", color="blue", weight=9];
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1208[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1208[label="",style="solid", color="blue", weight=9];
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1209[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];1036 -> 1209[label="",style="solid", color="blue", weight=9];
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1048 -> 195[label="",style="dashed", color="red", weight=0];
1048[label="(==) xw51 xw500\n  Integral a",fontsize=16,color="magenta"];1048 -> 1064[label="",style="dashed", color="magenta", weight=3];
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1052[label="(==) xw51 xw500",fontsize=16,color="magenta"];1052 -> 1072[label="",style="dashed", color="magenta", weight=3];
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1057[label="(==) xw51 xw500",fontsize=16,color="magenta"];1057 -> 1082[label="",style="dashed", color="magenta", weight=3];
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1097 -> 1103[label="",style="dashed", color="magenta", weight=3];
1098[label="nubByNubBy'1 (==) xw142 xw143 (xw144 : xw145) False\n  Eq a",fontsize=16,color="black",shape="box"];1098 -> 1104[label="",style="solid", color="black", weight=3];
1099[label="xw144\n  Eq a",fontsize=16,color="green",shape="box"];1100[label="xw143\n  Eq a",fontsize=16,color="green",shape="box"];1101[label="xw145\n  Eq a",fontsize=16,color="green",shape="box"];1102[label="(==) xw1470 xw142\n  Eq a",fontsize=16,color="blue",shape="box"];1230[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1230[label="",style="solid", color="blue", weight=9];
1230 -> 1105[label="",style="solid", color="blue", weight=3];
1231[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1231[label="",style="solid", color="blue", weight=9];
1231 -> 1106[label="",style="solid", color="blue", weight=3];
1232[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1232[label="",style="solid", color="blue", weight=9];
1232 -> 1107[label="",style="solid", color="blue", weight=3];
1233[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1233[label="",style="solid", color="blue", weight=9];
1233 -> 1108[label="",style="solid", color="blue", weight=3];
1234[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1234[label="",style="solid", color="blue", weight=9];
1234 -> 1109[label="",style="solid", color="blue", weight=3];
1235[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1235[label="",style="solid", color="blue", weight=9];
1235 -> 1110[label="",style="solid", color="blue", weight=3];
1236[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1236[label="",style="solid", color="blue", weight=9];
1236 -> 1111[label="",style="solid", color="blue", weight=3];
1237[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1237[label="",style="solid", color="blue", weight=9];
1237 -> 1112[label="",style="solid", color="blue", weight=3];
1238[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1238[label="",style="solid", color="blue", weight=9];
1238 -> 1113[label="",style="solid", color="blue", weight=3];
1239[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1239[label="",style="solid", color="blue", weight=9];
1239 -> 1114[label="",style="solid", color="blue", weight=3];
1240[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1240[label="",style="solid", color="blue", weight=9];
1240 -> 1115[label="",style="solid", color="blue", weight=3];
1241[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1241[label="",style="solid", color="blue", weight=9];
1241 -> 1116[label="",style="solid", color="blue", weight=3];
1242[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1242[label="",style="solid", color="blue", weight=9];
1242 -> 1117[label="",style="solid", color="blue", weight=3];
1243[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];1102 -> 1243[label="",style="solid", color="blue", weight=9];
1243 -> 1118[label="",style="solid", color="blue", weight=3];
1103[label="xw1471\n  Eq a",fontsize=16,color="green",shape="box"];1104[label="nubByNubBy'0 (==) xw142 xw143 (xw144 : xw145) otherwise\n  Eq a",fontsize=16,color="black",shape="box"];1104 -> 1119[label="",style="solid", color="black", weight=3];
1105 -> 195[label="",style="dashed", color="red", weight=0];
1105[label="(==) xw1470 xw142\n  Integral a",fontsize=16,color="magenta"];1105 -> 1120[label="",style="dashed", color="magenta", weight=3];
1105 -> 1121[label="",style="dashed", color="magenta", weight=3];
1106 -> 196[label="",style="dashed", color="red", weight=0];
1106[label="(==) xw1470 xw142",fontsize=16,color="magenta"];1106 -> 1122[label="",style="dashed", color="magenta", weight=3];
1106 -> 1123[label="",style="dashed", color="magenta", weight=3];
1107 -> 197[label="",style="dashed", color="red", weight=0];
1107[label="(==) xw1470 xw142",fontsize=16,color="magenta"];1107 -> 1124[label="",style="dashed", color="magenta", weight=3];
1107 -> 1125[label="",style="dashed", color="magenta", weight=3];
1108 -> 198[label="",style="dashed", color="red", weight=0];
1108[label="(==) xw1470 xw142\n  Eq a  Eq b",fontsize=16,color="magenta"];1108 -> 1126[label="",style="dashed", color="magenta", weight=3];
1108 -> 1127[label="",style="dashed", color="magenta", weight=3];
1109 -> 199[label="",style="dashed", color="red", weight=0];
1109[label="(==) xw1470 xw142",fontsize=16,color="magenta"];1109 -> 1128[label="",style="dashed", color="magenta", weight=3];
1109 -> 1129[label="",style="dashed", color="magenta", weight=3];
1110 -> 200[label="",style="dashed", color="red", weight=0];
1110[label="(==) xw1470 xw142\n  Eq a",fontsize=16,color="magenta"];1110 -> 1130[label="",style="dashed", color="magenta", weight=3];
1110 -> 1131[label="",style="dashed", color="magenta", weight=3];
1111 -> 201[label="",style="dashed", color="red", weight=0];
1111[label="(==) xw1470 xw142\n  Eq a",fontsize=16,color="magenta"];1111 -> 1132[label="",style="dashed", color="magenta", weight=3];
1111 -> 1133[label="",style="dashed", color="magenta", weight=3];
1112 -> 202[label="",style="dashed", color="red", weight=0];
1112[label="(==) xw1470 xw142\n  Eq a  Eq b  Eq c",fontsize=16,color="magenta"];1112 -> 1134[label="",style="dashed", color="magenta", weight=3];
1112 -> 1135[label="",style="dashed", color="magenta", weight=3];
1113 -> 203[label="",style="dashed", color="red", weight=0];
1113[label="(==) xw1470 xw142\n  Eq a  Eq b",fontsize=16,color="magenta"];1113 -> 1136[label="",style="dashed", color="magenta", weight=3];
1113 -> 1137[label="",style="dashed", color="magenta", weight=3];
1114 -> 204[label="",style="dashed", color="red", weight=0];
1114[label="(==) xw1470 xw142",fontsize=16,color="magenta"];1114 -> 1138[label="",style="dashed", color="magenta", weight=3];
1114 -> 1139[label="",style="dashed", color="magenta", weight=3];
1115 -> 205[label="",style="dashed", color="red", weight=0];
1115[label="(==) xw1470 xw142",fontsize=16,color="magenta"];1115 -> 1140[label="",style="dashed", color="magenta", weight=3];
1115 -> 1141[label="",style="dashed", color="magenta", weight=3];
1116 -> 206[label="",style="dashed", color="red", weight=0];
1116[label="(==) xw1470 xw142",fontsize=16,color="magenta"];1116 -> 1142[label="",style="dashed", color="magenta", weight=3];
1116 -> 1143[label="",style="dashed", color="magenta", weight=3];
1117 -> 207[label="",style="dashed", color="red", weight=0];
1117[label="(==) xw1470 xw142",fontsize=16,color="magenta"];1117 -> 1144[label="",style="dashed", color="magenta", weight=3];
1117 -> 1145[label="",style="dashed", color="magenta", weight=3];
1118 -> 208[label="",style="dashed", color="red", weight=0];
1118[label="(==) xw1470 xw142",fontsize=16,color="magenta"];1118 -> 1146[label="",style="dashed", color="magenta", weight=3];
1118 -> 1147[label="",style="dashed", color="magenta", weight=3];
1119[label="nubByNubBy'0 (==) xw142 xw143 (xw144 : xw145) True\n  Eq a",fontsize=16,color="black",shape="box"];1119 -> 1148[label="",style="solid", color="black", weight=3];
1120[label="xw142\n  Integral a",fontsize=16,color="green",shape="box"];1121[label="xw1470\n  Integral a",fontsize=16,color="green",shape="box"];1122[label="xw142",fontsize=16,color="green",shape="box"];1123[label="xw1470",fontsize=16,color="green",shape="box"];1124[label="xw142",fontsize=16,color="green",shape="box"];1125[label="xw1470",fontsize=16,color="green",shape="box"];1126[label="xw142\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];1127[label="xw1470\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];1128[label="xw142",fontsize=16,color="green",shape="box"];1129[label="xw1470",fontsize=16,color="green",shape="box"];1130[label="xw142\n  Eq a",fontsize=16,color="green",shape="box"];1131[label="xw1470\n  Eq a",fontsize=16,color="green",shape="box"];1132[label="xw142\n  Eq a",fontsize=16,color="green",shape="box"];1133[label="xw1470\n  Eq a",fontsize=16,color="green",shape="box"];1134[label="xw142\n  Eq a  Eq b  Eq c",fontsize=16,color="green",shape="box"];1135[label="xw1470\n  Eq a  Eq b  Eq c",fontsize=16,color="green",shape="box"];1136[label="xw142\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];1137[label="xw1470\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];1138[label="xw142",fontsize=16,color="green",shape="box"];1139[label="xw1470",fontsize=16,color="green",shape="box"];1140[label="xw142",fontsize=16,color="green",shape="box"];1141[label="xw1470",fontsize=16,color="green",shape="box"];1142[label="xw142",fontsize=16,color="green",shape="box"];1143[label="xw1470",fontsize=16,color="green",shape="box"];1144[label="xw142",fontsize=16,color="green",shape="box"];1145[label="xw1470",fontsize=16,color="green",shape="box"];1146[label="xw142",fontsize=16,color="green",shape="box"];1147[label="xw1470",fontsize=16,color="green",shape="box"];1148[label="xw142 : nubByNubBy' (==) xw143 (xw142 : xw144 : xw145)\n  Eq a",fontsize=16,color="green",shape="box"];1148 -> 1149[label="",style="dashed", color="green", weight=3];
1149 -> 589[label="",style="dashed", color="red", weight=0];
1149[label="nubByNubBy' (==) xw143 (xw142 : xw144 : xw145)\n  Eq a",fontsize=16,color="magenta"];1149 -> 1150[label="",style="dashed", color="magenta", weight=3];
1149 -> 1151[label="",style="dashed", color="magenta", weight=3];
1149 -> 1152[label="",style="dashed", color="magenta", weight=3];
1150[label="xw142\n  Eq a",fontsize=16,color="green",shape="box"];1151[label="xw143\n  Eq a",fontsize=16,color="green",shape="box"];1152[label="xw144 : xw145\n  Eq a",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

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

new_nubByNubBy'1(xw142, xw143, xw144, xw145, False, [], ba) → new_nubByNubBy'(xw143, xw142, :(xw144, xw145), ba)
new_nubByNubBy'1(xw142, xw143, xw144, xw145, True, xw147, ba) → new_nubByNubBy'(xw143, xw144, xw145, ba)
new_nubByNubBy'1(xw142, xw143, xw144, xw145, False, :(xw1470, xw1471), ba) → new_nubByNubBy'1(xw142, xw143, xw144, xw145, new_esEs(xw1470, xw142, ba), xw1471, ba)
new_nubByNubBy'(:(xw500, xw501), xw51, xw52, bb) → new_nubByNubBy'1(xw500, xw501, xw51, xw52, new_esEs0(xw51, xw500, bb), xw52, bb)

The TRS R consists of the following rules:

new_esEs0(xw51, xw500, ty_Integer) → new_esEs4(xw51, xw500)
new_esEs11(xw10, xw90, dc, dd) → error([])
new_esEs14(True, False) → False
new_esEs14(False, True) → False
new_esEs0(xw51, xw500, app(ty_Ratio, bc)) → new_esEs3(xw51, xw500, bc)
new_esEs5(xw10, xw90) → error([])
new_esEs0(xw51, xw500, ty_Int) → new_esEs12(xw51, xw500)
new_esEs14(False, False) → True
new_esEs7(xw10, xw90) → error([])
new_esEs(xw1470, xw142, ty_Integer) → new_esEs4(xw1470, xw142)
new_esEs2(xw10, xw90) → error([])
new_esEs0(xw51, xw500, app(ty_[], bf)) → new_esEs8(xw51, xw500, bf)
new_esEs(xw1470, xw142, app(ty_Ratio, de)) → new_esEs3(xw1470, xw142, de)
new_esEs14(True, True) → True
new_esEs0(xw51, xw500, app(app(app(ty_@3, bh), ca), cb)) → new_esEs10(xw51, xw500, bh, ca, cb)
new_esEs0(xw51, xw500, app(app(ty_@2, cc), cd)) → new_esEs11(xw51, xw500, cc, cd)
new_esEs3(xw10, xw90, ce) → error([])
new_esEs8(xw10, xw90, cg) → error([])
new_esEs0(xw51, xw500, ty_Double) → new_esEs2(xw51, xw500)
new_esEs(xw1470, xw142, app(ty_Maybe, ea)) → new_esEs9(xw1470, xw142, ea)
new_esEs(xw1470, xw142, ty_Double) → new_esEs2(xw1470, xw142)
new_esEs4(xw10, xw90) → error([])
new_esEs13(xw10, xw90) → error([])
new_esEs0(xw51, xw500, app(app(ty_Either, bd), be)) → new_esEs6(xw51, xw500, bd, be)
new_esEs(xw1470, xw142, ty_Float) → new_esEs13(xw1470, xw142)
new_esEs0(xw51, xw500, ty_Char) → new_esEs1(xw51, xw500)
new_esEs(xw1470, xw142, app(ty_[], dh)) → new_esEs8(xw1470, xw142, dh)
new_esEs12(xw10, xw90) → error([])
new_esEs6(xw10, xw90, da, db) → error([])
new_esEs(xw1470, xw142, app(app(ty_Either, df), dg)) → new_esEs6(xw1470, xw142, df, dg)
new_esEs(xw1470, xw142, app(app(app(ty_@3, eb), ec), ed)) → new_esEs10(xw1470, xw142, eb, ec, ed)
new_esEs(xw1470, xw142, app(app(ty_@2, ee), ef)) → new_esEs11(xw1470, xw142, ee, ef)
new_esEs9(xw10, xw90, cf) → error([])
new_esEs1(xw10, xw90) → error([])
new_esEs(xw1470, xw142, ty_@0) → new_esEs5(xw1470, xw142)
new_esEs10(xw10, xw90, eg, eh, fa) → error([])
new_esEs(xw1470, xw142, ty_Char) → new_esEs1(xw1470, xw142)
new_esEs(xw1470, xw142, ty_Int) → new_esEs12(xw1470, xw142)
new_esEs0(xw51, xw500, ty_Bool) → new_esEs14(xw51, xw500)
new_esEs0(xw51, xw500, ty_Ordering) → new_esEs7(xw51, xw500)
new_esEs(xw1470, xw142, ty_Bool) → new_esEs14(xw1470, xw142)
new_esEs0(xw51, xw500, ty_@0) → new_esEs5(xw51, xw500)
new_esEs0(xw51, xw500, ty_Float) → new_esEs13(xw51, xw500)
new_esEs(xw1470, xw142, ty_Ordering) → new_esEs7(xw1470, xw142)
new_esEs0(xw51, xw500, app(ty_Maybe, bg)) → new_esEs9(xw51, xw500, bg)

The set Q consists of the following terms:

new_esEs0(x0, x1, ty_Char)
new_esEs10(x0, x1, x2, x3, x4)
new_esEs0(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs14(False, False)
new_esEs0(x0, x1, ty_Int)
new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs(x0, x1, ty_Bool)
new_esEs(x0, x1, ty_@0)
new_esEs(x0, x1, app(ty_[], x2))
new_esEs7(x0, x1)
new_esEs(x0, x1, ty_Ordering)
new_esEs1(x0, x1)
new_esEs14(True, True)
new_esEs4(x0, x1)
new_esEs(x0, x1, ty_Float)
new_esEs2(x0, x1)
new_esEs(x0, x1, ty_Double)
new_esEs(x0, x1, ty_Integer)
new_esEs0(x0, x1, ty_Integer)
new_esEs0(x0, x1, app(app(ty_Either, x2), x3))
new_esEs3(x0, x1, x2)
new_esEs(x0, x1, app(app(ty_@2, x2), x3))
new_esEs0(x0, x1, ty_Double)
new_esEs0(x0, x1, app(ty_Ratio, x2))
new_esEs0(x0, x1, app(ty_Maybe, x2))
new_esEs5(x0, x1)
new_esEs14(True, False)
new_esEs14(False, True)
new_esEs0(x0, x1, ty_@0)
new_esEs13(x0, x1)
new_esEs11(x0, x1, x2, x3)
new_esEs(x0, x1, ty_Char)
new_esEs0(x0, x1, app(app(ty_@2, x2), x3))
new_esEs(x0, x1, app(app(ty_Either, x2), x3))
new_esEs9(x0, x1, x2)
new_esEs0(x0, x1, ty_Float)
new_esEs(x0, x1, app(ty_Ratio, x2))
new_esEs0(x0, x1, app(ty_[], x2))
new_esEs8(x0, x1, x2)
new_esEs(x0, x1, app(ty_Maybe, x2))
new_esEs0(x0, x1, ty_Ordering)
new_esEs12(x0, x1)
new_esEs6(x0, x1, x2, x3)
new_esEs0(x0, x1, ty_Bool)
new_esEs(x0, x1, ty_Int)

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_nubByNubBy'(:(xw500, xw501), xw51, xw52, bb) → new_nubByNubBy'1(xw500, xw501, xw51, xw52, new_esEs0(xw51, xw500, bb), xw52, bb)
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3, 3 >= 4, 3 >= 6, 4 >= 7
new_nubByNubBy'1(xw142, xw143, xw144, xw145, False, :(xw1470, xw1471), ba) → new_nubByNubBy'1(xw142, xw143, xw144, xw145, new_esEs(xw1470, xw142, ba), xw1471, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 6 > 6, 7 >= 7
new_nubByNubBy'1(xw142, xw143, xw144, xw145, False, [], ba) → new_nubByNubBy'(xw143, xw142, :(xw144, xw145), ba)
The graph contains the following edges 2 >= 1, 1 >= 2, 7 >= 4
new_nubByNubBy'1(xw142, xw143, xw144, xw145, True, xw147, ba) → new_nubByNubBy'(xw143, xw144, xw145, ba)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 7 >= 4

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

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

new_deleteBy0(xw17, xw18, xw19, False, ba) → new_deleteBy(xw19, xw17, ba)
new_deleteBy(xw10, :(xw90, xw91), bb) → new_deleteBy0(xw91, xw90, xw10, new_esEs15(xw10, xw90, bb), bb)

The TRS R consists of the following rules:

new_esEs15(xw10, xw90, ty_Integer) → new_esEs4(xw10, xw90)
new_esEs15(xw10, xw90, ty_Bool) → new_esEs14(xw10, xw90)
new_esEs11(xw10, xw90, cc, cd) → error([])
new_esEs14(True, False) → False
new_esEs14(False, True) → False
new_esEs5(xw10, xw90) → error([])
new_esEs15(xw10, xw90, app(app(ty_@2, cc), cd)) → new_esEs11(xw10, xw90, cc, cd)
new_esEs14(False, False) → True
new_esEs7(xw10, xw90) → error([])
new_esEs2(xw10, xw90) → error([])
new_esEs14(True, True) → True
new_esEs3(xw10, xw90, bc) → error([])
new_esEs8(xw10, xw90, bf) → error([])
new_esEs15(xw10, xw90, ty_Float) → new_esEs13(xw10, xw90)
new_esEs4(xw10, xw90) → error([])
new_esEs13(xw10, xw90) → error([])
new_esEs12(xw10, xw90) → error([])
new_esEs15(xw10, xw90, ty_Int) → new_esEs12(xw10, xw90)
new_esEs6(xw10, xw90, bd, be) → error([])
new_esEs15(xw10, xw90, ty_@0) → new_esEs5(xw10, xw90)
new_esEs15(xw10, xw90, app(app(app(ty_@3, bh), ca), cb)) → new_esEs10(xw10, xw90, bh, ca, cb)
new_esEs15(xw10, xw90, ty_Ordering) → new_esEs7(xw10, xw90)
new_esEs15(xw10, xw90, app(ty_Maybe, bg)) → new_esEs9(xw10, xw90, bg)
new_esEs15(xw10, xw90, ty_Double) → new_esEs2(xw10, xw90)
new_esEs9(xw10, xw90, bg) → error([])
new_esEs1(xw10, xw90) → error([])
new_esEs10(xw10, xw90, bh, ca, cb) → error([])
new_esEs15(xw10, xw90, ty_Char) → new_esEs1(xw10, xw90)
new_esEs15(xw10, xw90, app(ty_[], bf)) → new_esEs8(xw10, xw90, bf)
new_esEs15(xw10, xw90, app(ty_Ratio, bc)) → new_esEs3(xw10, xw90, bc)
new_esEs15(xw10, xw90, app(app(ty_Either, bd), be)) → new_esEs6(xw10, xw90, bd, be)

The set Q consists of the following terms:

new_esEs14(False, False)
new_esEs15(x0, x1, ty_Ordering)
new_esEs3(x0, x1, x2)
new_esEs8(x0, x1, x2)
new_esEs15(x0, x1, ty_Double)
new_esEs6(x0, x1, x2, x3)
new_esEs7(x0, x1)
new_esEs14(True, True)
new_esEs4(x0, x1)
new_esEs1(x0, x1)
new_esEs2(x0, x1)
new_esEs15(x0, x1, ty_@0)
new_esEs9(x0, x1, x2)
new_esEs15(x0, x1, app(ty_Ratio, x2))
new_esEs15(x0, x1, ty_Float)
new_esEs15(x0, x1, app(ty_Maybe, x2))
new_esEs15(x0, x1, ty_Char)
new_esEs15(x0, x1, ty_Int)
new_esEs5(x0, x1)
new_esEs14(True, False)
new_esEs14(False, True)
new_esEs15(x0, x1, app(app(ty_@2, x2), x3))
new_esEs15(x0, x1, app(app(ty_Either, x2), x3))
new_esEs13(x0, x1)
new_esEs10(x0, x1, x2, x3, x4)
new_esEs15(x0, x1, ty_Bool)
new_esEs12(x0, x1)
new_esEs11(x0, x1, x2, x3)
new_esEs15(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs15(x0, x1, app(ty_[], x2))
new_esEs15(x0, x1, ty_Integer)

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_deleteBy(xw10, :(xw90, xw91), bb) → new_deleteBy0(xw91, xw90, xw10, new_esEs15(xw10, xw90, bb), bb)
The graph contains the following edges 2 > 1, 2 > 2, 1 >= 3, 3 >= 5
new_deleteBy0(xw17, xw18, xw19, False, ba) → new_deleteBy(xw19, xw17, ba)
The graph contains the following edges 3 >= 1, 1 >= 2, 5 >= 3

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

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

new_foldl(xw9, xw10, :(xw110, xw111), ba) → new_foldl(new_flip(xw9, xw10, ba), xw110, xw111, ba)

The TRS R consists of the following rules:

new_esEs15(xw10, xw90, ty_Integer) → new_esEs4(xw10, xw90)
new_esEs15(xw10, xw90, ty_Bool) → new_esEs14(xw10, xw90)
new_esEs11(xw10, xw90, bg, bh) → error([])
new_esEs14(False, True) → False
new_esEs14(True, False) → False
new_esEs5(xw10, xw90) → error([])
new_esEs15(xw10, xw90, app(app(ty_@2, bg), bh)) → new_esEs11(xw10, xw90, bg, bh)
new_esEs14(False, False) → True
new_esEs7(xw10, xw90) → error([])
new_deleteBy00(xw17, xw18, xw19, True, ca) → xw17
new_esEs2(xw10, xw90) → error([])
new_esEs14(True, True) → True
new_esEs3(xw10, xw90, bb) → error([])
new_esEs8(xw10, xw90, bd) → error([])
new_esEs15(xw10, xw90, ty_Float) → new_esEs13(xw10, xw90)
new_esEs4(xw10, xw90) → error([])
new_esEs13(xw10, xw90) → error([])
new_deleteBy1(xw10, [], ba) → []
new_esEs12(xw10, xw90) → error([])
new_esEs15(xw10, xw90, ty_Int) → new_esEs12(xw10, xw90)
new_esEs15(xw10, xw90, app(app(app(ty_@3, cb), cc), cd)) → new_esEs10(xw10, xw90, cb, cc, cd)
new_esEs15(xw10, xw90, ty_@0) → new_esEs5(xw10, xw90)
new_esEs6(xw10, xw90, be, bf) → error([])
new_esEs15(xw10, xw90, app(ty_Maybe, bc)) → new_esEs9(xw10, xw90, bc)
new_esEs15(xw10, xw90, ty_Ordering) → new_esEs7(xw10, xw90)
new_deleteBy1(xw10, :(xw90, xw91), ba) → new_deleteBy00(xw91, xw90, xw10, new_esEs15(xw10, xw90, ba), ba)
new_deleteBy00(xw17, xw18, xw19, False, ca) → :(xw18, new_deleteBy1(xw19, xw17, ca))
new_esEs15(xw10, xw90, ty_Double) → new_esEs2(xw10, xw90)
new_esEs1(xw10, xw90) → error([])
new_esEs9(xw10, xw90, bc) → error([])
new_esEs10(xw10, xw90, cb, cc, cd) → error([])
new_flip(xw9, xw10, ba) → new_deleteBy1(xw10, xw9, ba)
new_esEs15(xw10, xw90, ty_Char) → new_esEs1(xw10, xw90)
new_esEs15(xw10, xw90, app(ty_[], bd)) → new_esEs8(xw10, xw90, bd)
new_esEs15(xw10, xw90, app(ty_Ratio, bb)) → new_esEs3(xw10, xw90, bb)
new_esEs15(xw10, xw90, app(app(ty_Either, be), bf)) → new_esEs6(xw10, xw90, be, bf)

The set Q consists of the following terms:

new_esEs14(False, False)
new_deleteBy1(x0, :(x1, x2), x3)
new_esEs15(x0, x1, ty_Ordering)
new_flip(x0, x1, x2)
new_esEs15(x0, x1, ty_Double)
new_esEs7(x0, x1)
new_esEs11(x0, x1, x2, x3)
new_esEs1(x0, x1)
new_esEs14(True, True)
new_esEs4(x0, x1)
new_esEs2(x0, x1)
new_esEs15(x0, x1, ty_@0)
new_deleteBy00(x0, x1, x2, False, x3)
new_esEs15(x0, x1, ty_Float)
new_esEs15(x0, x1, ty_Char)
new_deleteBy00(x0, x1, x2, True, x3)
new_esEs15(x0, x1, ty_Int)
new_deleteBy1(x0, [], x1)
new_esEs5(x0, x1)
new_esEs14(False, True)
new_esEs14(True, False)
new_esEs15(x0, x1, app(ty_Maybe, x2))
new_esEs10(x0, x1, x2, x3, x4)
new_esEs13(x0, x1)
new_esEs15(x0, x1, app(app(ty_Either, x2), x3))
new_esEs15(x0, x1, app(app(ty_@2, x2), x3))
new_esEs15(x0, x1, ty_Bool)
new_esEs15(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs15(x0, x1, app(ty_Ratio, x2))
new_esEs9(x0, x1, x2)
new_esEs15(x0, x1, app(ty_[], x2))
new_esEs3(x0, x1, x2)
new_esEs6(x0, x1, x2, x3)
new_esEs12(x0, x1)
new_esEs15(x0, x1, ty_Integer)
new_esEs8(x0, x1, x2)

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_foldl(xw9, xw10, :(xw110, xw111), ba) → new_foldl(new_flip(xw9, xw10, ba), xw110, xw111, ba)
The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

new_psPs(:(xw80, xw81), xw9, xw10, xw11, ba) → new_psPs(xw81, xw9, xw10, xw11, 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_psPs(:(xw80, xw81), xw9, xw10, xw11, ba) → new_psPs(xw81, xw9, xw10, xw11, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5

deleteBy0	ys y eq x True	= ys
deleteBy0	ys y eq x False	= y : deleteBy eq x ys

nubBy'	[] vy	= nubBy'3 [] vy
nubBy'	(y : ys) xs	= nubBy'2 (y : ys) xs

nubBy'1	y ys xs True	= nubBy' ys xs
nubBy'1	y ys xs False	= nubBy'0 y ys xs otherwise

nubByNubBy'1	xv y ys xs True	= nubByNubBy' xv ys xs
nubByNubBy'1	xv y ys xs False	= nubByNubBy'0 xv y ys xs otherwise

nubByNubBy'	xv [] vy	= nubByNubBy'3 xv [] vy
nubByNubBy'	xv (y : ys) xs	= nubByNubBy'2 xv (y : ys) xs

nubByNubBy'3	xv [] vy	= []
nubByNubBy'3	xv wz xu	= nubByNubBy'2 xv wz xu