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

YES 1.2530000000000001 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

((delete :: Int -> [Int] -> [Int]) :: Int -> [Int] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

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

delete

deleteBy (==)

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

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

((delete :: Int -> [Int] -> [Int]) :: Int -> [Int] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

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

delete

deleteBy (==)

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

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

((delete :: Int -> [Int] -> [Int]) :: Int -> [Int] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

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

delete

deleteBy (==)

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

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
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

              ↳ Narrow

mainModule List

(delete :: Int -> [Int] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

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

delete

deleteBy (==)

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

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="delete",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="delete ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="delete ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="deleteBy (==) ww3 ww4",fontsize=16,color="burlywood",shape="box"];611[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 611[label="",style="solid", color="burlywood", weight=9];
611 -> 6[label="",style="solid", color="burlywood", weight=3];
612[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 612[label="",style="solid", color="burlywood", weight=9];
612 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="deleteBy (==) ww3 (ww40 : ww41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="deleteBy (==) ww3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="deleteBy0 ww41 ww40 (==) ww3 ((==) ww3 ww40)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="[]",fontsize=16,color="green",shape="box"];10[label="deleteBy0 ww41 ww40 primEqInt ww3 (primEqInt ww3 ww40)",fontsize=16,color="burlywood",shape="triangle"];613[label="ww3/Pos ww30",fontsize=10,color="white",style="solid",shape="box"];10 -> 613[label="",style="solid", color="burlywood", weight=9];
613 -> 11[label="",style="solid", color="burlywood", weight=3];
614[label="ww3/Neg ww30",fontsize=10,color="white",style="solid",shape="box"];10 -> 614[label="",style="solid", color="burlywood", weight=9];
614 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="deleteBy0 ww41 ww40 primEqInt (Pos ww30) (primEqInt (Pos ww30) ww40)",fontsize=16,color="burlywood",shape="box"];615[label="ww30/Succ ww300",fontsize=10,color="white",style="solid",shape="box"];11 -> 615[label="",style="solid", color="burlywood", weight=9];
615 -> 13[label="",style="solid", color="burlywood", weight=3];
616[label="ww30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 616[label="",style="solid", color="burlywood", weight=9];
616 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="deleteBy0 ww41 ww40 primEqInt (Neg ww30) (primEqInt (Neg ww30) ww40)",fontsize=16,color="burlywood",shape="box"];617[label="ww30/Succ ww300",fontsize=10,color="white",style="solid",shape="box"];12 -> 617[label="",style="solid", color="burlywood", weight=9];
617 -> 15[label="",style="solid", color="burlywood", weight=3];
618[label="ww30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 618[label="",style="solid", color="burlywood", weight=9];
618 -> 16[label="",style="solid", color="burlywood", weight=3];
13[label="deleteBy0 ww41 ww40 primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) ww40)",fontsize=16,color="burlywood",shape="box"];619[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];13 -> 619[label="",style="solid", color="burlywood", weight=9];
619 -> 17[label="",style="solid", color="burlywood", weight=3];
620[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];13 -> 620[label="",style="solid", color="burlywood", weight=9];
620 -> 18[label="",style="solid", color="burlywood", weight=3];
14[label="deleteBy0 ww41 ww40 primEqInt (Pos Zero) (primEqInt (Pos Zero) ww40)",fontsize=16,color="burlywood",shape="box"];621[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];14 -> 621[label="",style="solid", color="burlywood", weight=9];
621 -> 19[label="",style="solid", color="burlywood", weight=3];
622[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];14 -> 622[label="",style="solid", color="burlywood", weight=9];
622 -> 20[label="",style="solid", color="burlywood", weight=3];
15[label="deleteBy0 ww41 ww40 primEqInt (Neg (Succ ww300)) (primEqInt (Neg (Succ ww300)) ww40)",fontsize=16,color="burlywood",shape="box"];623[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];15 -> 623[label="",style="solid", color="burlywood", weight=9];
623 -> 21[label="",style="solid", color="burlywood", weight=3];
624[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];15 -> 624[label="",style="solid", color="burlywood", weight=9];
624 -> 22[label="",style="solid", color="burlywood", weight=3];
16[label="deleteBy0 ww41 ww40 primEqInt (Neg Zero) (primEqInt (Neg Zero) ww40)",fontsize=16,color="burlywood",shape="box"];625[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];16 -> 625[label="",style="solid", color="burlywood", weight=9];
625 -> 23[label="",style="solid", color="burlywood", weight=3];
626[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];16 -> 626[label="",style="solid", color="burlywood", weight=9];
626 -> 24[label="",style="solid", color="burlywood", weight=3];
17[label="deleteBy0 ww41 (Pos ww400) primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) (Pos ww400))",fontsize=16,color="burlywood",shape="box"];627[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];17 -> 627[label="",style="solid", color="burlywood", weight=9];
627 -> 25[label="",style="solid", color="burlywood", weight=3];
628[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 628[label="",style="solid", color="burlywood", weight=9];
628 -> 26[label="",style="solid", color="burlywood", weight=3];
18[label="deleteBy0 ww41 (Neg ww400) primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) (Neg ww400))",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
19[label="deleteBy0 ww41 (Pos ww400) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Pos ww400))",fontsize=16,color="burlywood",shape="box"];629[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];19 -> 629[label="",style="solid", color="burlywood", weight=9];
629 -> 28[label="",style="solid", color="burlywood", weight=3];
630[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 630[label="",style="solid", color="burlywood", weight=9];
630 -> 29[label="",style="solid", color="burlywood", weight=3];
20[label="deleteBy0 ww41 (Neg ww400) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Neg ww400))",fontsize=16,color="burlywood",shape="box"];631[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];20 -> 631[label="",style="solid", color="burlywood", weight=9];
631 -> 30[label="",style="solid", color="burlywood", weight=3];
632[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 632[label="",style="solid", color="burlywood", weight=9];
632 -> 31[label="",style="solid", color="burlywood", weight=3];
21[label="deleteBy0 ww41 (Pos ww400) primEqInt (Neg (Succ ww300)) (primEqInt (Neg (Succ ww300)) (Pos ww400))",fontsize=16,color="black",shape="box"];21 -> 32[label="",style="solid", color="black", weight=3];
22[label="deleteBy0 ww41 (Neg ww400) primEqInt (Neg (Succ ww300)) (primEqInt (Neg (Succ ww300)) (Neg ww400))",fontsize=16,color="burlywood",shape="box"];633[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];22 -> 633[label="",style="solid", color="burlywood", weight=9];
633 -> 33[label="",style="solid", color="burlywood", weight=3];
634[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 634[label="",style="solid", color="burlywood", weight=9];
634 -> 34[label="",style="solid", color="burlywood", weight=3];
23[label="deleteBy0 ww41 (Pos ww400) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Pos ww400))",fontsize=16,color="burlywood",shape="box"];635[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];23 -> 635[label="",style="solid", color="burlywood", weight=9];
635 -> 35[label="",style="solid", color="burlywood", weight=3];
636[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 636[label="",style="solid", color="burlywood", weight=9];
636 -> 36[label="",style="solid", color="burlywood", weight=3];
24[label="deleteBy0 ww41 (Neg ww400) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Neg ww400))",fontsize=16,color="burlywood",shape="box"];637[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];24 -> 637[label="",style="solid", color="burlywood", weight=9];
637 -> 37[label="",style="solid", color="burlywood", weight=3];
638[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 638[label="",style="solid", color="burlywood", weight=9];
638 -> 38[label="",style="solid", color="burlywood", weight=3];
25[label="deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) (Pos (Succ ww4000)))",fontsize=16,color="black",shape="box"];25 -> 39[label="",style="solid", color="black", weight=3];
26[label="deleteBy0 ww41 (Pos Zero) primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) (Pos Zero))",fontsize=16,color="black",shape="box"];26 -> 40[label="",style="solid", color="black", weight=3];
27[label="deleteBy0 ww41 (Neg ww400) primEqInt (Pos (Succ ww300)) False",fontsize=16,color="black",shape="box"];27 -> 41[label="",style="solid", color="black", weight=3];
28[label="deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Pos (Succ ww4000)))",fontsize=16,color="black",shape="box"];28 -> 42[label="",style="solid", color="black", weight=3];
29[label="deleteBy0 ww41 (Pos Zero) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];29 -> 43[label="",style="solid", color="black", weight=3];
30[label="deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Neg (Succ ww4000)))",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
31[label="deleteBy0 ww41 (Neg Zero) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
32[label="deleteBy0 ww41 (Pos ww400) primEqInt (Neg (Succ ww300)) False",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
33[label="deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Neg (Succ ww300)) (primEqInt (Neg (Succ ww300)) (Neg (Succ ww4000)))",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
34[label="deleteBy0 ww41 (Neg Zero) primEqInt (Neg (Succ ww300)) (primEqInt (Neg (Succ ww300)) (Neg Zero))",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
35[label="deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Pos (Succ ww4000)))",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
36[label="deleteBy0 ww41 (Pos Zero) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
37[label="deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Neg (Succ ww4000)))",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
38[label="deleteBy0 ww41 (Neg Zero) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
39 -> 475[label="",style="dashed", color="red", weight=0];
39[label="deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Pos (Succ ww300)) (primEqNat ww300 ww4000)",fontsize=16,color="magenta"];39 -> 476[label="",style="dashed", color="magenta", weight=3];
39 -> 477[label="",style="dashed", color="magenta", weight=3];
39 -> 478[label="",style="dashed", color="magenta", weight=3];
39 -> 479[label="",style="dashed", color="magenta", weight=3];
39 -> 480[label="",style="dashed", color="magenta", weight=3];
40[label="deleteBy0 ww41 (Pos Zero) primEqInt (Pos (Succ ww300)) False",fontsize=16,color="black",shape="box"];40 -> 55[label="",style="solid", color="black", weight=3];
41[label="Neg ww400 : deleteBy primEqInt (Pos (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];41 -> 56[label="",style="dashed", color="green", weight=3];
42[label="deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Pos Zero) False",fontsize=16,color="black",shape="box"];42 -> 57[label="",style="solid", color="black", weight=3];
43[label="deleteBy0 ww41 (Pos Zero) primEqInt (Pos Zero) True",fontsize=16,color="black",shape="box"];43 -> 58[label="",style="solid", color="black", weight=3];
44[label="deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Pos Zero) False",fontsize=16,color="black",shape="box"];44 -> 59[label="",style="solid", color="black", weight=3];
45[label="deleteBy0 ww41 (Neg Zero) primEqInt (Pos Zero) True",fontsize=16,color="black",shape="box"];45 -> 60[label="",style="solid", color="black", weight=3];
46[label="Pos ww400 : deleteBy primEqInt (Neg (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];46 -> 61[label="",style="dashed", color="green", weight=3];
47 -> 528[label="",style="dashed", color="red", weight=0];
47[label="deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Neg (Succ ww300)) (primEqNat ww300 ww4000)",fontsize=16,color="magenta"];47 -> 529[label="",style="dashed", color="magenta", weight=3];
47 -> 530[label="",style="dashed", color="magenta", weight=3];
47 -> 531[label="",style="dashed", color="magenta", weight=3];
47 -> 532[label="",style="dashed", color="magenta", weight=3];
47 -> 533[label="",style="dashed", color="magenta", weight=3];
48[label="deleteBy0 ww41 (Neg Zero) primEqInt (Neg (Succ ww300)) False",fontsize=16,color="black",shape="box"];48 -> 64[label="",style="solid", color="black", weight=3];
49[label="deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Neg Zero) False",fontsize=16,color="black",shape="box"];49 -> 65[label="",style="solid", color="black", weight=3];
50[label="deleteBy0 ww41 (Pos Zero) primEqInt (Neg Zero) True",fontsize=16,color="black",shape="box"];50 -> 66[label="",style="solid", color="black", weight=3];
51[label="deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Neg Zero) False",fontsize=16,color="black",shape="box"];51 -> 67[label="",style="solid", color="black", weight=3];
52[label="deleteBy0 ww41 (Neg Zero) primEqInt (Neg Zero) True",fontsize=16,color="black",shape="box"];52 -> 68[label="",style="solid", color="black", weight=3];
476[label="ww300",fontsize=16,color="green",shape="box"];477[label="ww4000",fontsize=16,color="green",shape="box"];478[label="ww300",fontsize=16,color="green",shape="box"];479[label="ww4000",fontsize=16,color="green",shape="box"];480[label="ww41",fontsize=16,color="green",shape="box"];475[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat ww48 ww49)",fontsize=16,color="burlywood",shape="triangle"];641[label="ww48/Succ ww480",fontsize=10,color="white",style="solid",shape="box"];475 -> 641[label="",style="solid", color="burlywood", weight=9];
641 -> 526[label="",style="solid", color="burlywood", weight=3];
642[label="ww48/Zero",fontsize=10,color="white",style="solid",shape="box"];475 -> 642[label="",style="solid", color="burlywood", weight=9];
642 -> 527[label="",style="solid", color="burlywood", weight=3];
55[label="Pos Zero : deleteBy primEqInt (Pos (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];55 -> 73[label="",style="dashed", color="green", weight=3];
56[label="deleteBy primEqInt (Pos (Succ ww300)) ww41",fontsize=16,color="burlywood",shape="triangle"];643[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];56 -> 643[label="",style="solid", color="burlywood", weight=9];
643 -> 74[label="",style="solid", color="burlywood", weight=3];
644[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];56 -> 644[label="",style="solid", color="burlywood", weight=9];
644 -> 75[label="",style="solid", color="burlywood", weight=3];
57[label="Pos (Succ ww4000) : deleteBy primEqInt (Pos Zero) ww41",fontsize=16,color="green",shape="box"];57 -> 76[label="",style="dashed", color="green", weight=3];
58[label="ww41",fontsize=16,color="green",shape="box"];59[label="Neg (Succ ww4000) : deleteBy primEqInt (Pos Zero) ww41",fontsize=16,color="green",shape="box"];59 -> 77[label="",style="dashed", color="green", weight=3];
60[label="ww41",fontsize=16,color="green",shape="box"];61[label="deleteBy primEqInt (Neg (Succ ww300)) ww41",fontsize=16,color="burlywood",shape="triangle"];645[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];61 -> 645[label="",style="solid", color="burlywood", weight=9];
645 -> 78[label="",style="solid", color="burlywood", weight=3];
646[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 646[label="",style="solid", color="burlywood", weight=9];
646 -> 79[label="",style="solid", color="burlywood", weight=3];
529[label="ww300",fontsize=16,color="green",shape="box"];530[label="ww300",fontsize=16,color="green",shape="box"];531[label="ww4000",fontsize=16,color="green",shape="box"];532[label="ww41",fontsize=16,color="green",shape="box"];533[label="ww4000",fontsize=16,color="green",shape="box"];528[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat ww54 ww55)",fontsize=16,color="burlywood",shape="triangle"];647[label="ww54/Succ ww540",fontsize=10,color="white",style="solid",shape="box"];528 -> 647[label="",style="solid", color="burlywood", weight=9];
647 -> 579[label="",style="solid", color="burlywood", weight=3];
648[label="ww54/Zero",fontsize=10,color="white",style="solid",shape="box"];528 -> 648[label="",style="solid", color="burlywood", weight=9];
648 -> 580[label="",style="solid", color="burlywood", weight=3];
64[label="Neg Zero : deleteBy primEqInt (Neg (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];64 -> 84[label="",style="dashed", color="green", weight=3];
65[label="Pos (Succ ww4000) : deleteBy primEqInt (Neg Zero) ww41",fontsize=16,color="green",shape="box"];65 -> 85[label="",style="dashed", color="green", weight=3];
66[label="ww41",fontsize=16,color="green",shape="box"];67[label="Neg (Succ ww4000) : deleteBy primEqInt (Neg Zero) ww41",fontsize=16,color="green",shape="box"];67 -> 86[label="",style="dashed", color="green", weight=3];
68[label="ww41",fontsize=16,color="green",shape="box"];526[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) ww49)",fontsize=16,color="burlywood",shape="box"];649[label="ww49/Succ ww490",fontsize=10,color="white",style="solid",shape="box"];526 -> 649[label="",style="solid", color="burlywood", weight=9];
649 -> 581[label="",style="solid", color="burlywood", weight=3];
650[label="ww49/Zero",fontsize=10,color="white",style="solid",shape="box"];526 -> 650[label="",style="solid", color="burlywood", weight=9];
650 -> 582[label="",style="solid", color="burlywood", weight=3];
527[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero ww49)",fontsize=16,color="burlywood",shape="box"];651[label="ww49/Succ ww490",fontsize=10,color="white",style="solid",shape="box"];527 -> 651[label="",style="solid", color="burlywood", weight=9];
651 -> 583[label="",style="solid", color="burlywood", weight=3];
652[label="ww49/Zero",fontsize=10,color="white",style="solid",shape="box"];527 -> 652[label="",style="solid", color="burlywood", weight=9];
652 -> 584[label="",style="solid", color="burlywood", weight=3];
73 -> 56[label="",style="dashed", color="red", weight=0];
73[label="deleteBy primEqInt (Pos (Succ ww300)) ww41",fontsize=16,color="magenta"];74[label="deleteBy primEqInt (Pos (Succ ww300)) (ww410 : ww411)",fontsize=16,color="black",shape="box"];74 -> 91[label="",style="solid", color="black", weight=3];
75[label="deleteBy primEqInt (Pos (Succ ww300)) []",fontsize=16,color="black",shape="box"];75 -> 92[label="",style="solid", color="black", weight=3];
76[label="deleteBy primEqInt (Pos Zero) ww41",fontsize=16,color="burlywood",shape="triangle"];654[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];76 -> 654[label="",style="solid", color="burlywood", weight=9];
654 -> 93[label="",style="solid", color="burlywood", weight=3];
655[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];76 -> 655[label="",style="solid", color="burlywood", weight=9];
655 -> 94[label="",style="solid", color="burlywood", weight=3];
77 -> 76[label="",style="dashed", color="red", weight=0];
77[label="deleteBy primEqInt (Pos Zero) ww41",fontsize=16,color="magenta"];78[label="deleteBy primEqInt (Neg (Succ ww300)) (ww410 : ww411)",fontsize=16,color="black",shape="box"];78 -> 95[label="",style="solid", color="black", weight=3];
79[label="deleteBy primEqInt (Neg (Succ ww300)) []",fontsize=16,color="black",shape="box"];79 -> 96[label="",style="solid", color="black", weight=3];
579[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) ww55)",fontsize=16,color="burlywood",shape="box"];657[label="ww55/Succ ww550",fontsize=10,color="white",style="solid",shape="box"];579 -> 657[label="",style="solid", color="burlywood", weight=9];
657 -> 585[label="",style="solid", color="burlywood", weight=3];
658[label="ww55/Zero",fontsize=10,color="white",style="solid",shape="box"];579 -> 658[label="",style="solid", color="burlywood", weight=9];
658 -> 586[label="",style="solid", color="burlywood", weight=3];
580[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero ww55)",fontsize=16,color="burlywood",shape="box"];659[label="ww55/Succ ww550",fontsize=10,color="white",style="solid",shape="box"];580 -> 659[label="",style="solid", color="burlywood", weight=9];
659 -> 587[label="",style="solid", color="burlywood", weight=3];
660[label="ww55/Zero",fontsize=10,color="white",style="solid",shape="box"];580 -> 660[label="",style="solid", color="burlywood", weight=9];
660 -> 588[label="",style="solid", color="burlywood", weight=3];
84 -> 61[label="",style="dashed", color="red", weight=0];
84[label="deleteBy primEqInt (Neg (Succ ww300)) ww41",fontsize=16,color="magenta"];85[label="deleteBy primEqInt (Neg Zero) ww41",fontsize=16,color="burlywood",shape="triangle"];662[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];85 -> 662[label="",style="solid", color="burlywood", weight=9];
662 -> 101[label="",style="solid", color="burlywood", weight=3];
663[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];85 -> 663[label="",style="solid", color="burlywood", weight=9];
663 -> 102[label="",style="solid", color="burlywood", weight=3];
86 -> 85[label="",style="dashed", color="red", weight=0];
86[label="deleteBy primEqInt (Neg Zero) ww41",fontsize=16,color="magenta"];581[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) (Succ ww490))",fontsize=16,color="black",shape="box"];581 -> 589[label="",style="solid", color="black", weight=3];
582[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) Zero)",fontsize=16,color="black",shape="box"];582 -> 590[label="",style="solid", color="black", weight=3];
583[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero (Succ ww490))",fontsize=16,color="black",shape="box"];583 -> 591[label="",style="solid", color="black", weight=3];
584[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];584 -> 592[label="",style="solid", color="black", weight=3];
91 -> 10[label="",style="dashed", color="red", weight=0];
91[label="deleteBy0 ww411 ww410 primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) ww410)",fontsize=16,color="magenta"];91 -> 108[label="",style="dashed", color="magenta", weight=3];
91 -> 109[label="",style="dashed", color="magenta", weight=3];
91 -> 110[label="",style="dashed", color="magenta", weight=3];
92[label="[]",fontsize=16,color="green",shape="box"];93[label="deleteBy primEqInt (Pos Zero) (ww410 : ww411)",fontsize=16,color="black",shape="box"];93 -> 111[label="",style="solid", color="black", weight=3];
94[label="deleteBy primEqInt (Pos Zero) []",fontsize=16,color="black",shape="box"];94 -> 112[label="",style="solid", color="black", weight=3];
95 -> 10[label="",style="dashed", color="red", weight=0];
95[label="deleteBy0 ww411 ww410 primEqInt (Neg (Succ ww300)) (primEqInt (Neg (Succ ww300)) ww410)",fontsize=16,color="magenta"];95 -> 113[label="",style="dashed", color="magenta", weight=3];
95 -> 114[label="",style="dashed", color="magenta", weight=3];
95 -> 115[label="",style="dashed", color="magenta", weight=3];
96[label="[]",fontsize=16,color="green",shape="box"];585[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) (Succ ww550))",fontsize=16,color="black",shape="box"];585 -> 593[label="",style="solid", color="black", weight=3];
586[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) Zero)",fontsize=16,color="black",shape="box"];586 -> 594[label="",style="solid", color="black", weight=3];
587[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero (Succ ww550))",fontsize=16,color="black",shape="box"];587 -> 595[label="",style="solid", color="black", weight=3];
588[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];588 -> 596[label="",style="solid", color="black", weight=3];
101[label="deleteBy primEqInt (Neg Zero) (ww410 : ww411)",fontsize=16,color="black",shape="box"];101 -> 121[label="",style="solid", color="black", weight=3];
102[label="deleteBy primEqInt (Neg Zero) []",fontsize=16,color="black",shape="box"];102 -> 122[label="",style="solid", color="black", weight=3];
589 -> 475[label="",style="dashed", color="red", weight=0];
589[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat ww480 ww490)",fontsize=16,color="magenta"];589 -> 597[label="",style="dashed", color="magenta", weight=3];
589 -> 598[label="",style="dashed", color="magenta", weight=3];
590[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) False",fontsize=16,color="black",shape="triangle"];590 -> 599[label="",style="solid", color="black", weight=3];
591 -> 590[label="",style="dashed", color="red", weight=0];
591[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) False",fontsize=16,color="magenta"];592[label="deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) True",fontsize=16,color="black",shape="box"];592 -> 600[label="",style="solid", color="black", weight=3];
108[label="ww411",fontsize=16,color="green",shape="box"];109[label="ww410",fontsize=16,color="green",shape="box"];110[label="Pos (Succ ww300)",fontsize=16,color="green",shape="box"];111 -> 10[label="",style="dashed", color="red", weight=0];
111[label="deleteBy0 ww411 ww410 primEqInt (Pos Zero) (primEqInt (Pos Zero) ww410)",fontsize=16,color="magenta"];111 -> 129[label="",style="dashed", color="magenta", weight=3];
111 -> 130[label="",style="dashed", color="magenta", weight=3];
111 -> 131[label="",style="dashed", color="magenta", weight=3];
112[label="[]",fontsize=16,color="green",shape="box"];113[label="ww411",fontsize=16,color="green",shape="box"];114[label="ww410",fontsize=16,color="green",shape="box"];115[label="Neg (Succ ww300)",fontsize=16,color="green",shape="box"];593 -> 528[label="",style="dashed", color="red", weight=0];
593[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat ww540 ww550)",fontsize=16,color="magenta"];593 -> 601[label="",style="dashed", color="magenta", weight=3];
593 -> 602[label="",style="dashed", color="magenta", weight=3];
594[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) False",fontsize=16,color="black",shape="triangle"];594 -> 603[label="",style="solid", color="black", weight=3];
595 -> 594[label="",style="dashed", color="red", weight=0];
595[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) False",fontsize=16,color="magenta"];596[label="deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) True",fontsize=16,color="black",shape="box"];596 -> 604[label="",style="solid", color="black", weight=3];
121 -> 10[label="",style="dashed", color="red", weight=0];
121[label="deleteBy0 ww411 ww410 primEqInt (Neg Zero) (primEqInt (Neg Zero) ww410)",fontsize=16,color="magenta"];121 -> 138[label="",style="dashed", color="magenta", weight=3];
121 -> 139[label="",style="dashed", color="magenta", weight=3];
121 -> 140[label="",style="dashed", color="magenta", weight=3];
122[label="[]",fontsize=16,color="green",shape="box"];597[label="ww490",fontsize=16,color="green",shape="box"];598[label="ww480",fontsize=16,color="green",shape="box"];599[label="Pos (Succ ww46) : deleteBy primEqInt (Pos (Succ ww47)) ww45",fontsize=16,color="green",shape="box"];599 -> 605[label="",style="dashed", color="green", weight=3];
600[label="ww45",fontsize=16,color="green",shape="box"];129[label="ww411",fontsize=16,color="green",shape="box"];130[label="ww410",fontsize=16,color="green",shape="box"];131[label="Pos Zero",fontsize=16,color="green",shape="box"];601[label="ww540",fontsize=16,color="green",shape="box"];602[label="ww550",fontsize=16,color="green",shape="box"];603[label="Neg (Succ ww52) : deleteBy primEqInt (Neg (Succ ww53)) ww51",fontsize=16,color="green",shape="box"];603 -> 606[label="",style="dashed", color="green", weight=3];
604[label="ww51",fontsize=16,color="green",shape="box"];138[label="ww411",fontsize=16,color="green",shape="box"];139[label="ww410",fontsize=16,color="green",shape="box"];140[label="Neg Zero",fontsize=16,color="green",shape="box"];605 -> 56[label="",style="dashed", color="red", weight=0];
605[label="deleteBy primEqInt (Pos (Succ ww47)) ww45",fontsize=16,color="magenta"];605 -> 607[label="",style="dashed", color="magenta", weight=3];
605 -> 608[label="",style="dashed", color="magenta", weight=3];
606 -> 61[label="",style="dashed", color="red", weight=0];
606[label="deleteBy primEqInt (Neg (Succ ww53)) ww51",fontsize=16,color="magenta"];606 -> 609[label="",style="dashed", color="magenta", weight=3];
606 -> 610[label="",style="dashed", color="magenta", weight=3];
607[label="ww45",fontsize=16,color="green",shape="box"];608[label="ww47",fontsize=16,color="green",shape="box"];609[label="ww51",fontsize=16,color="green",shape="box"];610[label="ww53",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

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

new_deleteBy3(:(ww410, ww411)) → new_deleteBy01(ww411, ww410, Neg(Zero))
new_deleteBy2(ww300, :(ww410, ww411)) → new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
new_deleteBy0(ww45, ww46, ww47, Zero, Succ(ww490)) → new_deleteBy00(ww45, ww46, ww47)
new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Zero)) → new_deleteBy3(ww41)
new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Succ(ww550)) → new_deleteBy02(ww51, ww52, ww53, ww540, ww550)
new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Zero) → new_deleteBy2(ww53, ww51)
new_deleteBy00(ww45, ww46, ww47) → new_deleteBy(ww47, ww45)
new_deleteBy01(:(ww410, ww411), Neg(ww400), Pos(Succ(ww300))) → new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
new_deleteBy01(ww41, Neg(Zero), Neg(Succ(ww300))) → new_deleteBy2(ww300, ww41)
new_deleteBy01(ww41, Neg(Succ(ww4000)), Pos(Zero)) → new_deleteBy1(ww41)
new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Neg(Zero)) → new_deleteBy01(ww411, ww410, Neg(Zero))
new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Pos(Zero)) → new_deleteBy01(ww411, ww410, Pos(Zero))
new_deleteBy(ww300, :(ww410, ww411)) → new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
new_deleteBy1(:(ww410, ww411)) → new_deleteBy01(ww411, ww410, Pos(Zero))
new_deleteBy02(ww51, ww52, ww53, Zero, Succ(ww550)) → new_deleteBy03(ww51, ww52, ww53)
new_deleteBy01(:(ww410, ww411), Pos(ww400), Neg(Succ(ww300))) → new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Zero) → new_deleteBy(ww47, ww45)
new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Succ(ww490)) → new_deleteBy0(ww45, ww46, ww47, ww480, ww490)
new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Succ(ww300))) → new_deleteBy02(ww41, ww4000, ww300, ww300, ww4000)
new_deleteBy01(ww41, Pos(Succ(ww4000)), Pos(Succ(ww300))) → new_deleteBy0(ww41, ww4000, ww300, ww300, ww4000)
new_deleteBy01(ww41, Pos(Zero), Pos(Succ(ww300))) → new_deleteBy(ww300, ww41)
new_deleteBy03(ww51, ww52, ww53) → new_deleteBy2(ww53, ww51)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 4 SCCs.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

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

new_deleteBy01(ww41, Neg(Succ(ww4000)), Pos(Zero)) → new_deleteBy1(ww41)
new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Pos(Zero)) → new_deleteBy01(ww411, ww410, Pos(Zero))
new_deleteBy1(:(ww410, ww411)) → new_deleteBy01(ww411, ww410, Pos(Zero))

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_deleteBy01(ww41, Neg(Succ(ww4000)), Pos(Zero)) → new_deleteBy1(ww41)
The graph contains the following edges 1 >= 1
new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Pos(Zero)) → new_deleteBy01(ww411, ww410, Pos(Zero))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
new_deleteBy1(:(ww410, ww411)) → new_deleteBy01(ww411, ww410, Pos(Zero))
The graph contains the following edges 1 > 1, 1 > 2

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

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

new_deleteBy(ww300, :(ww410, ww411)) → new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
new_deleteBy0(ww45, ww46, ww47, Zero, Succ(ww490)) → new_deleteBy00(ww45, ww46, ww47)
new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Zero) → new_deleteBy(ww47, ww45)
new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Succ(ww490)) → new_deleteBy0(ww45, ww46, ww47, ww480, ww490)
new_deleteBy01(:(ww410, ww411), Neg(ww400), Pos(Succ(ww300))) → new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
new_deleteBy00(ww45, ww46, ww47) → new_deleteBy(ww47, ww45)
new_deleteBy01(ww41, Pos(Succ(ww4000)), Pos(Succ(ww300))) → new_deleteBy0(ww41, ww4000, ww300, ww300, ww4000)
new_deleteBy01(ww41, Pos(Zero), Pos(Succ(ww300))) → new_deleteBy(ww300, ww41)

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

new_deleteBy(ww300, :(ww410, ww411)) → new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
The graph contains the following edges 2 > 1, 2 > 2
new_deleteBy0(ww45, ww46, ww47, Zero, Succ(ww490)) → new_deleteBy00(ww45, ww46, ww47)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
new_deleteBy01(ww41, Pos(Zero), Pos(Succ(ww300))) → new_deleteBy(ww300, ww41)
The graph contains the following edges 3 > 1, 1 >= 2
new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Succ(ww490)) → new_deleteBy0(ww45, ww46, ww47, ww480, ww490)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
new_deleteBy01(:(ww410, ww411), Neg(ww400), Pos(Succ(ww300))) → new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
new_deleteBy01(ww41, Pos(Succ(ww4000)), Pos(Succ(ww300))) → new_deleteBy0(ww41, ww4000, ww300, ww300, ww4000)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 2 > 5
new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Zero) → new_deleteBy(ww47, ww45)
The graph contains the following edges 3 >= 1, 1 >= 2
new_deleteBy00(ww45, ww46, ww47) → new_deleteBy(ww47, ww45)
The graph contains the following edges 3 >= 1, 1 >= 2

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

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

new_deleteBy2(ww300, :(ww410, ww411)) → new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
new_deleteBy02(ww51, ww52, ww53, Zero, Succ(ww550)) → new_deleteBy03(ww51, ww52, ww53)
new_deleteBy01(:(ww410, ww411), Pos(ww400), Neg(Succ(ww300))) → new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Succ(ww550)) → new_deleteBy02(ww51, ww52, ww53, ww540, ww550)
new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Zero) → new_deleteBy2(ww53, ww51)
new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Succ(ww300))) → new_deleteBy02(ww41, ww4000, ww300, ww300, ww4000)
new_deleteBy03(ww51, ww52, ww53) → new_deleteBy2(ww53, ww51)
new_deleteBy01(ww41, Neg(Zero), Neg(Succ(ww300))) → new_deleteBy2(ww300, ww41)

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

new_deleteBy2(ww300, :(ww410, ww411)) → new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
The graph contains the following edges 2 > 1, 2 > 2
new_deleteBy01(:(ww410, ww411), Pos(ww400), Neg(Succ(ww300))) → new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
new_deleteBy01(ww41, Neg(Zero), Neg(Succ(ww300))) → new_deleteBy2(ww300, ww41)
The graph contains the following edges 3 > 1, 1 >= 2
new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Succ(ww300))) → new_deleteBy02(ww41, ww4000, ww300, ww300, ww4000)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 2 > 5
new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Zero) → new_deleteBy2(ww53, ww51)
The graph contains the following edges 3 >= 1, 1 >= 2
new_deleteBy03(ww51, ww52, ww53) → new_deleteBy2(ww53, ww51)
The graph contains the following edges 3 >= 1, 1 >= 2
new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Succ(ww550)) → new_deleteBy02(ww51, ww52, ww53, ww540, ww550)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
new_deleteBy02(ww51, ww52, ww53, Zero, Succ(ww550)) → new_deleteBy03(ww51, ww52, ww53)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

new_deleteBy3(:(ww410, ww411)) → new_deleteBy01(ww411, ww410, Neg(Zero))
new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Neg(Zero)) → new_deleteBy01(ww411, ww410, Neg(Zero))
new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Zero)) → new_deleteBy3(ww41)

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

new_deleteBy3(:(ww410, ww411)) → new_deleteBy01(ww411, ww410, Neg(Zero))
The graph contains the following edges 1 > 1, 1 > 2
new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Neg(Zero)) → new_deleteBy01(ww411, ww410, Neg(Zero))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Zero)) → new_deleteBy3(ww41)
The graph contains the following edges 1 >= 1

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