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

YES 5.673 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 :: [Char] -> [Char] -> [Char]) :: [Char] -> [Char] -> [Char])

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 :: [Char] -> [Char] -> [Char]) :: [Char] -> [Char] -> [Char])

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 :: [Char] -> [Char] -> [Char]) :: [Char] -> [Char] -> [Char])

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 :: [Char] -> [Char] -> [Char]) :: [Char] -> [Char] -> [Char])

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

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

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

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

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

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

mainModule List

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

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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11 -> 108[label="",style="dashed", color="magenta", weight=3];
11 -> 109[label="",style="dashed", color="magenta", weight=3];
11 -> 110[label="",style="dashed", color="magenta", weight=3];
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107[label="xw31",fontsize=16,color="green",shape="box"];108[label="xw31",fontsize=16,color="green",shape="box"];109[label="xw30",fontsize=16,color="green",shape="box"];110 -> 12[label="",style="dashed", color="red", weight=0];
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2445[label="xw8/[]",fontsize=10,color="white",style="solid",shape="box"];106 -> 2445[label="",style="solid", color="burlywood", weight=9];
2445 -> 136[label="",style="solid", color="burlywood", weight=3];
15[label="nubByNubBy' (==) xw4 []",fontsize=16,color="burlywood",shape="box"];2446[label="xw4/xw40 : xw41",fontsize=10,color="white",style="solid",shape="box"];15 -> 2446[label="",style="solid", color="burlywood", weight=9];
2446 -> 18[label="",style="solid", color="burlywood", weight=3];
2447[label="xw4/[]",fontsize=10,color="white",style="solid",shape="box"];15 -> 2447[label="",style="solid", color="burlywood", weight=9];
2447 -> 19[label="",style="solid", color="burlywood", weight=3];
135[label="(xw80 : xw81) ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="black",shape="box"];135 -> 138[label="",style="solid", color="black", weight=3];
136[label="[] ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="black",shape="box"];136 -> 139[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];
138[label="xw80 : xw81 ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="green",shape="box"];138 -> 142[label="",style="dashed", color="green", weight=3];
139[label="foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="black",shape="box"];139 -> 143[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];
142 -> 106[label="",style="dashed", color="red", weight=0];
142[label="xw81 ++ foldl (flip (deleteBy (==))) xw9 (xw10 : xw11)\n  Eq a",fontsize=16,color="magenta"];142 -> 148[label="",style="dashed", color="magenta", weight=3];
143[label="foldl (flip (deleteBy (==))) (flip (deleteBy (==)) xw9 xw10) xw11\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];2449[label="xw11/xw110 : xw111",fontsize=10,color="white",style="solid",shape="box"];143 -> 2449[label="",style="solid", color="burlywood", weight=9];
2449 -> 149[label="",style="solid", color="burlywood", weight=3];
2450[label="xw11/[]",fontsize=10,color="white",style="solid",shape="box"];143 -> 2450[label="",style="solid", color="burlywood", weight=9];
2450 -> 150[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"];148[label="xw81\n  Eq a",fontsize=16,color="green",shape="box"];149[label="foldl (flip (deleteBy (==))) (flip (deleteBy (==)) xw9 xw10) (xw110 : xw111)\n  Eq a",fontsize=16,color="black",shape="box"];149 -> 155[label="",style="solid", color="black", weight=3];
150[label="foldl (flip (deleteBy (==))) (flip (deleteBy (==)) xw9 xw10) []\n  Eq a",fontsize=16,color="black",shape="box"];150 -> 156[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];
155 -> 143[label="",style="dashed", color="red", weight=0];
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155 -> 163[label="",style="dashed", color="magenta", weight=3];
155 -> 164[label="",style="dashed", color="magenta", weight=3];
156[label="flip (deleteBy (==)) xw9 xw10\n  Eq a",fontsize=16,color="black",shape="triangle"];156 -> 165[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];
162[label="xw111\n  Eq a",fontsize=16,color="green",shape="box"];163[label="xw110\n  Eq a",fontsize=16,color="green",shape="box"];164 -> 156[label="",style="dashed", color="red", weight=0];
164[label="flip (deleteBy (==)) xw9 xw10\n  Eq a",fontsize=16,color="magenta"];165[label="deleteBy (==) xw10 xw9\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];2453[label="xw9/xw90 : xw91",fontsize=10,color="white",style="solid",shape="box"];165 -> 2453[label="",style="solid", color="burlywood", weight=9];
2453 -> 173[label="",style="solid", color="burlywood", weight=3];
2454[label="xw9/[]",fontsize=10,color="white",style="solid",shape="box"];165 -> 2454[label="",style="solid", color="burlywood", weight=9];
2454 -> 174[label="",style="solid", color="burlywood", weight=3];
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173[label="deleteBy (==) xw10 (xw90 : xw91)\n  Eq a",fontsize=16,color="black",shape="box"];173 -> 183[label="",style="solid", color="black", weight=3];
174[label="deleteBy (==) xw10 []\n  Eq a",fontsize=16,color="black",shape="box"];174 -> 184[label="",style="solid", color="black", weight=3];
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183 -> 194[label="",style="dashed", color="red", weight=0];
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183 -> 196[label="",style="dashed", color="magenta", weight=3];
183 -> 197[label="",style="dashed", color="magenta", weight=3];
183 -> 198[label="",style="dashed", color="magenta", weight=3];
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2456 -> 58[label="",style="solid", color="burlywood", weight=3];
2457[label="xw41/[]",fontsize=10,color="white",style="solid",shape="box"];54 -> 2457[label="",style="solid", color="burlywood", weight=9];
2457 -> 59[label="",style="solid", color="burlywood", weight=3];
195[label="(==) xw10 xw90\n  Eq a",fontsize=16,color="blue",shape="box"];2458[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2458[label="",style="solid", color="blue", weight=9];
2458 -> 199[label="",style="solid", color="blue", weight=3];
2459[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2459[label="",style="solid", color="blue", weight=9];
2459 -> 200[label="",style="solid", color="blue", weight=3];
2460[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2460[label="",style="solid", color="blue", weight=9];
2460 -> 201[label="",style="solid", color="blue", weight=3];
2461[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2461[label="",style="solid", color="blue", weight=9];
2461 -> 202[label="",style="solid", color="blue", weight=3];
2462[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2462[label="",style="solid", color="blue", weight=9];
2462 -> 203[label="",style="solid", color="blue", weight=3];
2463[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2463[label="",style="solid", color="blue", weight=9];
2463 -> 204[label="",style="solid", color="blue", weight=3];
2464[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2464[label="",style="solid", color="blue", weight=9];
2464 -> 205[label="",style="solid", color="blue", weight=3];
2465[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2465[label="",style="solid", color="blue", weight=9];
2465 -> 206[label="",style="solid", color="blue", weight=3];
2466[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2466[label="",style="solid", color="blue", weight=9];
2466 -> 207[label="",style="solid", color="blue", weight=3];
2467[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2467[label="",style="solid", color="blue", weight=9];
2467 -> 208[label="",style="solid", color="blue", weight=3];
2468[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2468[label="",style="solid", color="blue", weight=9];
2468 -> 209[label="",style="solid", color="blue", weight=3];
2469[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2469[label="",style="solid", color="blue", weight=9];
2469 -> 210[label="",style="solid", color="blue", weight=3];
2470[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2470[label="",style="solid", color="blue", weight=9];
2470 -> 211[label="",style="solid", color="blue", weight=3];
2471[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];195 -> 2471[label="",style="solid", color="blue", weight=9];
2471 -> 212[label="",style="solid", color="blue", weight=3];
196[label="xw10\n  Eq a",fontsize=16,color="green",shape="box"];197[label="xw90\n  Eq a",fontsize=16,color="green",shape="box"];198[label="xw91\n  Eq a",fontsize=16,color="green",shape="box"];194[label="deleteBy0 xw17 xw18 (==) xw19 xw20\n  Eq a",fontsize=16,color="burlywood",shape="triangle"];2472[label="xw20/False",fontsize=10,color="white",style="solid",shape="box"];194 -> 2472[label="",style="solid", color="burlywood", weight=9];
2472 -> 213[label="",style="solid", color="burlywood", weight=3];
2473[label="xw20/True",fontsize=10,color="white",style="solid",shape="box"];194 -> 2473[label="",style="solid", color="burlywood", weight=9];
2473 -> 214[label="",style="solid", color="burlywood", weight=3];
58[label="nubByNubBy' (==) (xw410 : xw411) (xw40 : [])",fontsize=16,color="black",shape="box"];58 -> 66[label="",style="solid", color="black", weight=3];
59[label="nubByNubBy' (==) [] (xw40 : [])",fontsize=16,color="black",shape="box"];59 -> 67[label="",style="solid", color="black", weight=3];
199[label="(==) xw10 xw90\n  Eq a",fontsize=16,color="black",shape="box"];199 -> 226[label="",style="solid", color="black", weight=3];
200[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];200 -> 227[label="",style="solid", color="black", weight=3];
201[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];201 -> 228[label="",style="solid", color="black", weight=3];
202[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];202 -> 229[label="",style="solid", color="black", weight=3];
203[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];203 -> 230[label="",style="solid", color="black", weight=3];
204[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];204 -> 231[label="",style="solid", color="black", weight=3];
205[label="(==) xw10 xw90\n  Eq a",fontsize=16,color="black",shape="box"];205 -> 232[label="",style="solid", color="black", weight=3];
206[label="(==) xw10 xw90\n  Eq a  Eq b",fontsize=16,color="black",shape="box"];206 -> 233[label="",style="solid", color="black", weight=3];
207[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];207 -> 234[label="",style="solid", color="black", weight=3];
208[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];208 -> 235[label="",style="solid", color="black", weight=3];
209[label="(==) xw10 xw90\n  Eq a  Eq b  Eq c",fontsize=16,color="black",shape="box"];209 -> 236[label="",style="solid", color="black", weight=3];
210[label="(==) xw10 xw90\n  Eq a  Eq b",fontsize=16,color="black",shape="box"];210 -> 237[label="",style="solid", color="black", weight=3];
211[label="(==) xw10 xw90\n  Integral a",fontsize=16,color="black",shape="box"];211 -> 238[label="",style="solid", color="black", weight=3];
212[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];212 -> 239[label="",style="solid", color="black", weight=3];
213[label="deleteBy0 xw17 xw18 (==) xw19 False\n  Eq a",fontsize=16,color="black",shape="box"];213 -> 240[label="",style="solid", color="black", weight=3];
214[label="deleteBy0 xw17 xw18 (==) xw19 True\n  Eq a",fontsize=16,color="black",shape="box"];214 -> 241[label="",style="solid", color="black", weight=3];
66[label="nubByNubBy'2 (==) (xw410 : xw411) (xw40 : [])",fontsize=16,color="black",shape="box"];66 -> 72[label="",style="solid", color="black", weight=3];
67[label="nubByNubBy'3 (==) [] (xw40 : [])",fontsize=16,color="black",shape="box"];67 -> 73[label="",style="solid", color="black", weight=3];
226[label="error []",fontsize=16,color="red",shape="box"];227[label="error []",fontsize=16,color="red",shape="box"];228[label="error []",fontsize=16,color="red",shape="box"];229[label="error []",fontsize=16,color="red",shape="box"];230[label="error []",fontsize=16,color="red",shape="box"];231[label="error []",fontsize=16,color="red",shape="box"];232[label="error []",fontsize=16,color="red",shape="box"];233[label="error []",fontsize=16,color="red",shape="box"];234[label="error []",fontsize=16,color="red",shape="box"];235[label="error []",fontsize=16,color="red",shape="box"];236[label="error []",fontsize=16,color="red",shape="box"];237[label="error []",fontsize=16,color="red",shape="box"];238[label="error []",fontsize=16,color="red",shape="box"];239[label="primEqChar xw10 xw90",fontsize=16,color="burlywood",shape="triangle"];2474[label="xw10/Char xw100",fontsize=10,color="white",style="solid",shape="box"];239 -> 2474[label="",style="solid", color="burlywood", weight=9];
2474 -> 253[label="",style="solid", color="burlywood", weight=3];
240[label="xw18 : deleteBy (==) xw19 xw17\n  Eq a",fontsize=16,color="green",shape="box"];240 -> 254[label="",style="dashed", color="green", weight=3];
241[label="xw17\n  Eq a",fontsize=16,color="green",shape="box"];72[label="nubByNubBy'1 (==) xw410 xw411 (xw40 : []) (elem_by (==) xw410 (xw40 : []))",fontsize=16,color="black",shape="box"];72 -> 78[label="",style="solid", color="black", weight=3];
73[label="[]",fontsize=16,color="green",shape="box"];253[label="primEqChar (Char xw100) xw90",fontsize=16,color="burlywood",shape="box"];2475[label="xw90/Char xw900",fontsize=10,color="white",style="solid",shape="box"];253 -> 2475[label="",style="solid", color="burlywood", weight=9];
2475 -> 271[label="",style="solid", color="burlywood", weight=3];
254 -> 165[label="",style="dashed", color="red", weight=0];
254[label="deleteBy (==) xw19 xw17\n  Eq a",fontsize=16,color="magenta"];254 -> 272[label="",style="dashed", color="magenta", weight=3];
254 -> 273[label="",style="dashed", color="magenta", weight=3];
78[label="nubByNubBy'1 (==) xw410 xw411 (xw40 : []) ((==) xw40 xw410 || elem_by (==) xw410 [])",fontsize=16,color="black",shape="box"];78 -> 89[label="",style="solid", color="black", weight=3];
271[label="primEqChar (Char xw100) (Char xw900)",fontsize=16,color="black",shape="box"];271 -> 289[label="",style="solid", color="black", weight=3];
272[label="xw17\n  Eq a",fontsize=16,color="green",shape="box"];273[label="xw19\n  Eq a",fontsize=16,color="green",shape="box"];89 -> 2402[label="",style="dashed", color="red", weight=0];
89[label="nubByNubBy'1 primEqChar xw410 xw411 (xw40 : []) (primEqChar xw40 xw410 || elem_by primEqChar xw410 [])",fontsize=16,color="magenta"];89 -> 2403[label="",style="dashed", color="magenta", weight=3];
89 -> 2404[label="",style="dashed", color="magenta", weight=3];
89 -> 2405[label="",style="dashed", color="magenta", weight=3];
89 -> 2406[label="",style="dashed", color="magenta", weight=3];
89 -> 2407[label="",style="dashed", color="magenta", weight=3];
89 -> 2408[label="",style="dashed", color="magenta", weight=3];
289[label="primEqNat xw100 xw900",fontsize=16,color="burlywood",shape="triangle"];2478[label="xw100/Succ xw1000",fontsize=10,color="white",style="solid",shape="box"];289 -> 2478[label="",style="solid", color="burlywood", weight=9];
2478 -> 305[label="",style="solid", color="burlywood", weight=3];
2479[label="xw100/Zero",fontsize=10,color="white",style="solid",shape="box"];289 -> 2479[label="",style="solid", color="burlywood", weight=9];
2479 -> 306[label="",style="solid", color="burlywood", weight=3];
2403[label="xw40",fontsize=16,color="green",shape="box"];2404[label="xw411",fontsize=16,color="green",shape="box"];2405[label="[]",fontsize=16,color="green",shape="box"];2406 -> 239[label="",style="dashed", color="red", weight=0];
2406[label="primEqChar xw40 xw410",fontsize=16,color="magenta"];2406 -> 2410[label="",style="dashed", color="magenta", weight=3];
2406 -> 2411[label="",style="dashed", color="magenta", weight=3];
2407[label="[]",fontsize=16,color="green",shape="box"];2408[label="xw410",fontsize=16,color="green",shape="box"];2402[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (xw161 || elem_by primEqChar xw155 xw160)",fontsize=16,color="burlywood",shape="triangle"];2481[label="xw161/False",fontsize=10,color="white",style="solid",shape="box"];2402 -> 2481[label="",style="solid", color="burlywood", weight=9];
2481 -> 2412[label="",style="solid", color="burlywood", weight=3];
2482[label="xw161/True",fontsize=10,color="white",style="solid",shape="box"];2402 -> 2482[label="",style="solid", color="burlywood", weight=9];
2482 -> 2413[label="",style="solid", color="burlywood", weight=3];
305[label="primEqNat (Succ xw1000) xw900",fontsize=16,color="burlywood",shape="box"];2483[label="xw900/Succ xw9000",fontsize=10,color="white",style="solid",shape="box"];305 -> 2483[label="",style="solid", color="burlywood", weight=9];
2483 -> 309[label="",style="solid", color="burlywood", weight=3];
2484[label="xw900/Zero",fontsize=10,color="white",style="solid",shape="box"];305 -> 2484[label="",style="solid", color="burlywood", weight=9];
2484 -> 310[label="",style="solid", color="burlywood", weight=3];
306[label="primEqNat Zero xw900",fontsize=16,color="burlywood",shape="box"];2485[label="xw900/Succ xw9000",fontsize=10,color="white",style="solid",shape="box"];306 -> 2485[label="",style="solid", color="burlywood", weight=9];
2485 -> 311[label="",style="solid", color="burlywood", weight=3];
2486[label="xw900/Zero",fontsize=10,color="white",style="solid",shape="box"];306 -> 2486[label="",style="solid", color="burlywood", weight=9];
2486 -> 312[label="",style="solid", color="burlywood", weight=3];
2410[label="xw410",fontsize=16,color="green",shape="box"];2411[label="xw40",fontsize=16,color="green",shape="box"];2412[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (False || elem_by primEqChar xw155 xw160)",fontsize=16,color="black",shape="box"];2412 -> 2414[label="",style="solid", color="black", weight=3];
2413[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (True || elem_by primEqChar xw155 xw160)",fontsize=16,color="black",shape="box"];2413 -> 2415[label="",style="solid", color="black", weight=3];
309[label="primEqNat (Succ xw1000) (Succ xw9000)",fontsize=16,color="black",shape="box"];309 -> 333[label="",style="solid", color="black", weight=3];
310[label="primEqNat (Succ xw1000) Zero",fontsize=16,color="black",shape="box"];310 -> 334[label="",style="solid", color="black", weight=3];
311[label="primEqNat Zero (Succ xw9000)",fontsize=16,color="black",shape="box"];311 -> 335[label="",style="solid", color="black", weight=3];
312[label="primEqNat Zero Zero",fontsize=16,color="black",shape="box"];312 -> 336[label="",style="solid", color="black", weight=3];
2414[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (elem_by primEqChar xw155 xw160)",fontsize=16,color="burlywood",shape="triangle"];2487[label="xw160/xw1600 : xw1601",fontsize=10,color="white",style="solid",shape="box"];2414 -> 2487[label="",style="solid", color="burlywood", weight=9];
2487 -> 2416[label="",style="solid", color="burlywood", weight=3];
2488[label="xw160/[]",fontsize=10,color="white",style="solid",shape="box"];2414 -> 2488[label="",style="solid", color="burlywood", weight=9];
2488 -> 2417[label="",style="solid", color="burlywood", weight=3];
2415[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) True",fontsize=16,color="black",shape="box"];2415 -> 2418[label="",style="solid", color="black", weight=3];
333 -> 289[label="",style="dashed", color="red", weight=0];
333[label="primEqNat xw1000 xw9000",fontsize=16,color="magenta"];333 -> 350[label="",style="dashed", color="magenta", weight=3];
333 -> 351[label="",style="dashed", color="magenta", weight=3];
334[label="False",fontsize=16,color="green",shape="box"];335[label="False",fontsize=16,color="green",shape="box"];336[label="True",fontsize=16,color="green",shape="box"];2416[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (elem_by primEqChar xw155 (xw1600 : xw1601))",fontsize=16,color="black",shape="box"];2416 -> 2419[label="",style="solid", color="black", weight=3];
2417[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (elem_by primEqChar xw155 [])",fontsize=16,color="black",shape="box"];2417 -> 2420[label="",style="solid", color="black", weight=3];
2418[label="nubByNubBy' primEqChar xw156 (xw157 : xw158)",fontsize=16,color="burlywood",shape="triangle"];2490[label="xw156/xw1560 : xw1561",fontsize=10,color="white",style="solid",shape="box"];2418 -> 2490[label="",style="solid", color="burlywood", weight=9];
2490 -> 2421[label="",style="solid", color="burlywood", weight=3];
2491[label="xw156/[]",fontsize=10,color="white",style="solid",shape="box"];2418 -> 2491[label="",style="solid", color="burlywood", weight=9];
2491 -> 2422[label="",style="solid", color="burlywood", weight=3];
350[label="xw9000",fontsize=16,color="green",shape="box"];351[label="xw1000",fontsize=16,color="green",shape="box"];2419 -> 2402[label="",style="dashed", color="red", weight=0];
2419[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (primEqChar xw1600 xw155 || elem_by primEqChar xw155 xw1601)",fontsize=16,color="magenta"];2419 -> 2423[label="",style="dashed", color="magenta", weight=3];
2419 -> 2424[label="",style="dashed", color="magenta", weight=3];
2420[label="nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) False",fontsize=16,color="black",shape="box"];2420 -> 2425[label="",style="solid", color="black", weight=3];
2421[label="nubByNubBy' primEqChar (xw1560 : xw1561) (xw157 : xw158)",fontsize=16,color="black",shape="box"];2421 -> 2426[label="",style="solid", color="black", weight=3];
2422[label="nubByNubBy' primEqChar [] (xw157 : xw158)",fontsize=16,color="black",shape="box"];2422 -> 2427[label="",style="solid", color="black", weight=3];
2423[label="xw1601",fontsize=16,color="green",shape="box"];2424 -> 239[label="",style="dashed", color="red", weight=0];
2424[label="primEqChar xw1600 xw155",fontsize=16,color="magenta"];2424 -> 2428[label="",style="dashed", color="magenta", weight=3];
2424 -> 2429[label="",style="dashed", color="magenta", weight=3];
2425[label="nubByNubBy'0 primEqChar xw155 xw156 (xw157 : xw158) otherwise",fontsize=16,color="black",shape="box"];2425 -> 2430[label="",style="solid", color="black", weight=3];
2426[label="nubByNubBy'2 primEqChar (xw1560 : xw1561) (xw157 : xw158)",fontsize=16,color="black",shape="box"];2426 -> 2431[label="",style="solid", color="black", weight=3];
2427[label="nubByNubBy'3 primEqChar [] (xw157 : xw158)",fontsize=16,color="black",shape="box"];2427 -> 2432[label="",style="solid", color="black", weight=3];
2428[label="xw155",fontsize=16,color="green",shape="box"];2429[label="xw1600",fontsize=16,color="green",shape="box"];2430[label="nubByNubBy'0 primEqChar xw155 xw156 (xw157 : xw158) True",fontsize=16,color="black",shape="box"];2430 -> 2433[label="",style="solid", color="black", weight=3];
2431 -> 2414[label="",style="dashed", color="red", weight=0];
2431[label="nubByNubBy'1 primEqChar xw1560 xw1561 (xw157 : xw158) (elem_by primEqChar xw1560 (xw157 : xw158))",fontsize=16,color="magenta"];2431 -> 2434[label="",style="dashed", color="magenta", weight=3];
2431 -> 2435[label="",style="dashed", color="magenta", weight=3];
2431 -> 2436[label="",style="dashed", color="magenta", weight=3];
2432[label="[]",fontsize=16,color="green",shape="box"];2433[label="xw155 : nubByNubBy' primEqChar xw156 (xw155 : xw157 : xw158)",fontsize=16,color="green",shape="box"];2433 -> 2437[label="",style="dashed", color="green", weight=3];
2434[label="xw1561",fontsize=16,color="green",shape="box"];2435[label="xw157 : xw158",fontsize=16,color="green",shape="box"];2436[label="xw1560",fontsize=16,color="green",shape="box"];2437 -> 2418[label="",style="dashed", color="red", weight=0];
2437[label="nubByNubBy' primEqChar xw156 (xw155 : xw157 : xw158)",fontsize=16,color="magenta"];2437 -> 2438[label="",style="dashed", color="magenta", weight=3];
2437 -> 2439[label="",style="dashed", color="magenta", weight=3];
2438[label="xw155",fontsize=16,color="green",shape="box"];2439[label="xw157 : xw158",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

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

new_primEqNat(Succ(xw1000), Succ(xw9000)) → new_primEqNat(xw1000, xw9000)

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_primEqNat(Succ(xw1000), Succ(xw9000)) → new_primEqNat(xw1000, xw9000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

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

new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) → new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) → new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
new_nubByNubBy'10(xw155, xw156, xw157, xw158, []) → new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) → new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) → new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) → new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))

The TRS R consists of the following rules:

new_primEqNat0(Succ(xw1000), Zero) → False
new_primEqNat0(Zero, Succ(xw9000)) → False
new_primEqChar(Char(xw100), Char(xw900)) → new_primEqNat0(xw100, xw900)
new_primEqNat0(Succ(xw1000), Succ(xw9000)) → new_primEqNat0(xw1000, xw9000)
new_primEqNat0(Zero, Zero) → True

The set Q consists of the following terms:

new_primEqChar(Char(x0), Char(x1))
new_primEqNat0(Zero, Zero)
new_primEqNat0(Succ(x0), Succ(x1))
new_primEqNat0(Succ(x0), Zero)
new_primEqNat0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

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

new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) → new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) → new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) → new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) → new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) → new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))

The TRS R consists of the following rules:

new_primEqNat0(Succ(xw1000), Zero) → False
new_primEqNat0(Zero, Succ(xw9000)) → False
new_primEqChar(Char(xw100), Char(xw900)) → new_primEqNat0(xw100, xw900)
new_primEqNat0(Succ(xw1000), Succ(xw9000)) → new_primEqNat0(xw1000, xw9000)
new_primEqNat0(Zero, Zero) → True

The set Q consists of the following terms:

new_primEqChar(Char(x0), Char(x1))
new_primEqNat0(Zero, Zero)
new_primEqNat0(Succ(x0), Succ(x1))
new_primEqNat0(Succ(x0), Zero)
new_primEqNat0(Zero, Succ(x0))

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'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) → new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 6 > 6
new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) → new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 6
new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) → new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3, 3 >= 4
new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) → new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
The graph contains the following edges 2 >= 1, 1 >= 2
new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) → new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
The graph contains the following edges 2 > 1, 2 > 2, 3 >= 3, 4 >= 4

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

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

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

The TRS R consists of the following rules:

new_esEs(xw10, xw90, app(app(app(ty_@3, bg), bh), ca)) → error([])
new_esEs(xw10, xw90, ty_Double) → error([])
new_primEqNat0(Succ(xw1000), Zero) → False
new_primEqNat0(Zero, Succ(xw9000)) → False
new_esEs(xw10, xw90, ty_Integer) → error([])
new_primEqChar(Char(xw100), Char(xw900)) → new_primEqNat0(xw100, xw900)
new_esEs(xw10, xw90, ty_Ordering) → error([])
new_esEs(xw10, xw90, app(app(ty_@2, be), bf)) → error([])
new_esEs(xw10, xw90, app(ty_[], bd)) → error([])
new_esEs(xw10, xw90, app(ty_Ratio, cd)) → error([])
new_esEs(xw10, xw90, ty_@0) → error([])
new_esEs(xw10, xw90, ty_Bool) → error([])
new_primEqNat0(Succ(xw1000), Succ(xw9000)) → new_primEqNat0(xw1000, xw9000)
new_primEqNat0(Zero, Zero) → True
new_esEs(xw10, xw90, ty_Char) → new_primEqChar(xw10, xw90)
new_esEs(xw10, xw90, app(app(ty_Either, cb), cc)) → error([])
new_esEs(xw10, xw90, ty_Float) → error([])
new_esEs(xw10, xw90, app(ty_Maybe, bc)) → error([])
new_esEs(xw10, xw90, ty_Int) → error([])

The set Q consists of the following terms:

new_esEs(x0, x1, ty_Double)
new_esEs(x0, x1, app(app(ty_Either, x2), x3))
new_esEs(x0, x1, ty_@0)
new_primEqNat0(Zero, Zero)
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs(x0, x1, ty_Ordering)
new_esEs(x0, x1, ty_Int)
new_primEqNat0(Zero, Succ(x0))
new_esEs(x0, x1, ty_Integer)
new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs(x0, x1, ty_Char)
new_primEqChar(Char(x0), Char(x1))
new_esEs(x0, x1, app(app(ty_@2, x2), x3))
new_esEs(x0, x1, ty_Bool)
new_esEs(x0, x1, app(ty_Ratio, x2))
new_esEs(x0, x1, ty_Float)
new_esEs(x0, x1, app(ty_Maybe, x2))
new_primEqNat0(Succ(x0), Zero)
new_esEs(x0, x1, app(ty_[], 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_deleteBy0(xw17, xw18, xw19, False, ba) → new_deleteBy(xw19, xw17, ba)
The graph contains the following edges 3 >= 1, 1 >= 2, 5 >= 3
new_deleteBy(xw10, :(xw90, xw91), bb) → new_deleteBy0(xw91, xw90, xw10, new_esEs(xw10, xw90, bb), bb)
The graph contains the following edges 2 > 1, 2 > 2, 1 >= 3, 3 >= 5

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ AND

                      ↳ QDP

                      ↳ 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_primEqChar(Char(xw100), Char(xw900)) → new_primEqNat0(xw100, xw900)
new_esEs(xw10, xw90, ty_Ordering) → error([])
new_esEs(xw10, xw90, ty_@0) → error([])
new_esEs(xw10, xw90, app(ty_Ratio, cc)) → error([])
new_deleteBy00(xw17, xw18, xw19, True, cd) → xw17
new_esEs(xw10, xw90, ty_Bool) → error([])
new_primEqNat0(Succ(xw1000), Succ(xw9000)) → new_primEqNat0(xw1000, xw9000)
new_primEqNat0(Zero, Zero) → True
new_esEs(xw10, xw90, ty_Char) → new_primEqChar(xw10, xw90)
new_esEs(xw10, xw90, app(app(ty_Either, ca), cb)) → error([])
new_deleteBy1(xw10, [], ba) → []
new_esEs(xw10, xw90, app(app(app(ty_@3, bf), bg), bh)) → error([])
new_esEs(xw10, xw90, ty_Double) → error([])
new_primEqNat0(Succ(xw1000), Zero) → False
new_primEqNat0(Zero, Succ(xw9000)) → False
new_esEs(xw10, xw90, ty_Integer) → error([])
new_esEs(xw10, xw90, app(app(ty_@2, bd), be)) → error([])
new_esEs(xw10, xw90, app(ty_[], bc)) → error([])
new_deleteBy00(xw17, xw18, xw19, False, cd) → :(xw18, new_deleteBy1(xw19, xw17, cd))
new_deleteBy1(xw10, :(xw90, xw91), ba) → new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
new_flip(xw9, xw10, ba) → new_deleteBy1(xw10, xw9, ba)
new_esEs(xw10, xw90, ty_Float) → error([])
new_esEs(xw10, xw90, app(ty_Maybe, bb)) → error([])
new_esEs(xw10, xw90, ty_Int) → error([])

The set Q consists of the following terms:

new_esEs(x0, x1, ty_Double)
new_deleteBy00(x0, x1, x2, True, x3)
new_esEs(x0, x1, ty_@0)
new_deleteBy1(x0, [], x1)
new_primEqNat0(Zero, Zero)
new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs(x0, x1, ty_Ordering)
new_deleteBy1(x0, :(x1, x2), x3)
new_esEs(x0, x1, app(ty_[], x2))
new_esEs(x0, x1, app(app(ty_@2, x2), x3))
new_esEs(x0, x1, ty_Int)
new_flip(x0, x1, x2)
new_primEqNat0(Zero, Succ(x0))
new_esEs(x0, x1, ty_Integer)
new_esEs(x0, x1, app(ty_Ratio, x2))
new_esEs(x0, x1, app(ty_Maybe, x2))
new_esEs(x0, x1, ty_Char)
new_primEqChar(Char(x0), Char(x1))
new_esEs(x0, x1, ty_Bool)
new_esEs(x0, x1, app(app(ty_Either, x2), x3))
new_esEs(x0, x1, ty_Float)
new_primEqNat0(Succ(x0), Zero)
new_deleteBy00(x0, x1, x2, False, x3)

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

                      ↳ 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)

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