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

YES 1.415 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

  ↳ CR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

insert :: Ord a => a -> [a] -> [a]

insert

e ls

insertBy compare e ls

insertBy :: (a -> a -> Ordering) -> a -> [a] -> [a]

insertBy

_ x []

x : []

insertBy

cmp x ys@(y : ys')

case	cmp x y of
	GT	->	y : insertBy cmp x ys'
	_	->	x : ys

module Maybe where

import qualified List
import qualified Prelude

Case Reductions:
The following Case expression

case cmp x y of
GT → y : insertBy cmp x ys'

_ → x : ys

is transformed to

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys'

insertBy0 y cmp x ys' ys _ = x : ys

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

insert :: Ord a => a -> [a] -> [a]

insert

e ls

insertBy compare e ls

insertBy :: (a -> a -> Ordering) -> a -> [a] -> [a]

insertBy	_ x []	=	x : []
insertBy	cmp x ys@(y : ys')	=	insertBy0 y cmp x ys' ys (cmp x y)

insertBy0	y cmp x ys' ys GT	=	y : insertBy cmp x ys'
insertBy0	y cmp x ys' ys _	=	x : ys

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.
Binding Reductions:
The bind variable of the following binding Pattern

ys@(vy : vz)

is replaced by the following term

vy : vz

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

insert :: Ord a => a -> [a] -> [a]

insert

e ls

insertBy compare e ls

insertBy :: (a -> a -> Ordering) -> a -> [a] -> [a]

insertBy	vx x []	=	x : []
insertBy	cmp x (vy : vz)	=	insertBy0 vy cmp x vz (vy : vz) (cmp x vy)

insertBy0	y cmp x ys' ys GT	=	y : insertBy cmp x ys'
insertBy0	y cmp x ys' ys vw	=	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

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

insert :: Ord a => a -> [a] -> [a]

insert

e ls

insertBy compare e ls

insertBy :: (a -> a -> Ordering) -> a -> [a] -> [a]

insertBy	vx x []	=	x : []
insertBy	cmp x (vy : vz)	=	insertBy0 vy cmp x vz (vy : vz) (cmp x vy)

insertBy0	y cmp x ys' ys GT	=	y : insertBy cmp x ys'
insertBy0	y cmp x ys' ys vw	=	x : ys

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
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656 -> 6[label="",style="solid", color="burlywood", weight=3];
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669[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];15 -> 669[label="",style="solid", color="burlywood", weight=9];
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677[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 677[label="",style="solid", color="burlywood", weight=9];
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679[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 679[label="",style="solid", color="burlywood", weight=9];
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683 -> 46[label="",style="solid", color="burlywood", weight=3];
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51 -> 513[label="",style="dashed", color="magenta", weight=3];
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685 -> 67[label="",style="solid", color="burlywood", weight=3];
686[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];53 -> 686[label="",style="solid", color="burlywood", weight=9];
686 -> 68[label="",style="solid", color="burlywood", weight=3];
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57[label="Pos Zero : Neg Zero : ww41",fontsize=16,color="green",shape="box"];58 -> 566[label="",style="dashed", color="red", weight=0];
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58 -> 568[label="",style="dashed", color="magenta", weight=3];
58 -> 569[label="",style="dashed", color="magenta", weight=3];
58 -> 570[label="",style="dashed", color="magenta", weight=3];
58 -> 571[label="",style="dashed", color="magenta", weight=3];
59[label="insertBy0 (Neg Zero) primCmpInt (Neg (Succ ww300)) ww41 (Neg Zero : ww41) LT",fontsize=16,color="black",shape="box"];59 -> 73[label="",style="solid", color="black", weight=3];
60[label="Neg Zero : Pos (Succ ww4000) : ww41",fontsize=16,color="green",shape="box"];61[label="Neg Zero : Pos Zero : ww41",fontsize=16,color="green",shape="box"];62[label="insertBy0 (Neg (Succ ww4000)) primCmpInt (Neg Zero) ww41 (Neg (Succ ww4000) : ww41) GT",fontsize=16,color="black",shape="box"];62 -> 74[label="",style="solid", color="black", weight=3];
63[label="Neg Zero : Neg Zero : ww41",fontsize=16,color="green",shape="box"];512[label="ww300",fontsize=16,color="green",shape="box"];513[label="ww4000",fontsize=16,color="green",shape="box"];514[label="ww41",fontsize=16,color="green",shape="box"];515[label="ww4000",fontsize=16,color="green",shape="box"];516[label="ww300",fontsize=16,color="green",shape="box"];511[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat ww62 ww63)",fontsize=16,color="burlywood",shape="triangle"];688[label="ww62/Succ ww620",fontsize=10,color="white",style="solid",shape="box"];511 -> 688[label="",style="solid", color="burlywood", weight=9];
688 -> 562[label="",style="solid", color="burlywood", weight=3];
689[label="ww62/Zero",fontsize=10,color="white",style="solid",shape="box"];511 -> 689[label="",style="solid", color="burlywood", weight=9];
689 -> 563[label="",style="solid", color="burlywood", weight=3];
66[label="Pos Zero : insertBy primCmpInt (Pos (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];66 -> 79[label="",style="dashed", color="green", weight=3];
67[label="insertBy primCmpInt (Pos (Succ ww300)) (ww410 : ww411)",fontsize=16,color="black",shape="box"];67 -> 80[label="",style="solid", color="black", weight=3];
68[label="insertBy primCmpInt (Pos (Succ ww300)) []",fontsize=16,color="black",shape="box"];68 -> 81[label="",style="solid", color="black", weight=3];
69[label="Pos Zero : Pos (Succ ww4000) : ww41",fontsize=16,color="green",shape="box"];70[label="insertBy primCmpInt (Pos Zero) ww41",fontsize=16,color="burlywood",shape="box"];690[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];70 -> 690[label="",style="solid", color="burlywood", weight=9];
690 -> 82[label="",style="solid", color="burlywood", weight=3];
691[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];70 -> 691[label="",style="solid", color="burlywood", weight=9];
691 -> 83[label="",style="solid", color="burlywood", weight=3];
567[label="ww4000",fontsize=16,color="green",shape="box"];568[label="ww4000",fontsize=16,color="green",shape="box"];569[label="ww300",fontsize=16,color="green",shape="box"];570[label="ww300",fontsize=16,color="green",shape="box"];571[label="ww41",fontsize=16,color="green",shape="box"];566[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat ww68 ww69)",fontsize=16,color="burlywood",shape="triangle"];692[label="ww68/Succ ww680",fontsize=10,color="white",style="solid",shape="box"];566 -> 692[label="",style="solid", color="burlywood", weight=9];
692 -> 617[label="",style="solid", color="burlywood", weight=3];
693[label="ww68/Zero",fontsize=10,color="white",style="solid",shape="box"];566 -> 693[label="",style="solid", color="burlywood", weight=9];
693 -> 618[label="",style="solid", color="burlywood", weight=3];
73[label="Neg (Succ ww300) : Neg Zero : ww41",fontsize=16,color="green",shape="box"];74[label="Neg (Succ ww4000) : insertBy primCmpInt (Neg Zero) ww41",fontsize=16,color="green",shape="box"];74 -> 88[label="",style="dashed", color="green", weight=3];
562[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat (Succ ww620) ww63)",fontsize=16,color="burlywood",shape="box"];694[label="ww63/Succ ww630",fontsize=10,color="white",style="solid",shape="box"];562 -> 694[label="",style="solid", color="burlywood", weight=9];
694 -> 619[label="",style="solid", color="burlywood", weight=3];
695[label="ww63/Zero",fontsize=10,color="white",style="solid",shape="box"];562 -> 695[label="",style="solid", color="burlywood", weight=9];
695 -> 620[label="",style="solid", color="burlywood", weight=3];
563[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat Zero ww63)",fontsize=16,color="burlywood",shape="box"];696[label="ww63/Succ ww630",fontsize=10,color="white",style="solid",shape="box"];563 -> 696[label="",style="solid", color="burlywood", weight=9];
696 -> 621[label="",style="solid", color="burlywood", weight=3];
697[label="ww63/Zero",fontsize=10,color="white",style="solid",shape="box"];563 -> 697[label="",style="solid", color="burlywood", weight=9];
697 -> 622[label="",style="solid", color="burlywood", weight=3];
79 -> 53[label="",style="dashed", color="red", weight=0];
79[label="insertBy primCmpInt (Pos (Succ ww300)) ww41",fontsize=16,color="magenta"];80 -> 10[label="",style="dashed", color="red", weight=0];
80[label="insertBy0 ww410 primCmpInt (Pos (Succ ww300)) ww411 (ww410 : ww411) (primCmpInt (Pos (Succ ww300)) ww410)",fontsize=16,color="magenta"];80 -> 93[label="",style="dashed", color="magenta", weight=3];
80 -> 94[label="",style="dashed", color="magenta", weight=3];
80 -> 95[label="",style="dashed", color="magenta", weight=3];
81[label="Pos (Succ ww300) : []",fontsize=16,color="green",shape="box"];82[label="insertBy primCmpInt (Pos Zero) (ww410 : ww411)",fontsize=16,color="black",shape="box"];82 -> 96[label="",style="solid", color="black", weight=3];
83[label="insertBy primCmpInt (Pos Zero) []",fontsize=16,color="black",shape="box"];83 -> 97[label="",style="solid", color="black", weight=3];
617[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat (Succ ww680) ww69)",fontsize=16,color="burlywood",shape="box"];700[label="ww69/Succ ww690",fontsize=10,color="white",style="solid",shape="box"];617 -> 700[label="",style="solid", color="burlywood", weight=9];
700 -> 623[label="",style="solid", color="burlywood", weight=3];
701[label="ww69/Zero",fontsize=10,color="white",style="solid",shape="box"];617 -> 701[label="",style="solid", color="burlywood", weight=9];
701 -> 624[label="",style="solid", color="burlywood", weight=3];
618[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat Zero ww69)",fontsize=16,color="burlywood",shape="box"];702[label="ww69/Succ ww690",fontsize=10,color="white",style="solid",shape="box"];618 -> 702[label="",style="solid", color="burlywood", weight=9];
702 -> 625[label="",style="solid", color="burlywood", weight=3];
703[label="ww69/Zero",fontsize=10,color="white",style="solid",shape="box"];618 -> 703[label="",style="solid", color="burlywood", weight=9];
703 -> 626[label="",style="solid", color="burlywood", weight=3];
88[label="insertBy primCmpInt (Neg Zero) ww41",fontsize=16,color="burlywood",shape="box"];704[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];88 -> 704[label="",style="solid", color="burlywood", weight=9];
704 -> 102[label="",style="solid", color="burlywood", weight=3];
705[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];88 -> 705[label="",style="solid", color="burlywood", weight=9];
705 -> 103[label="",style="solid", color="burlywood", weight=3];
619[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat (Succ ww620) (Succ ww630))",fontsize=16,color="black",shape="box"];619 -> 627[label="",style="solid", color="black", weight=3];
620[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat (Succ ww620) Zero)",fontsize=16,color="black",shape="box"];620 -> 628[label="",style="solid", color="black", weight=3];
621[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat Zero (Succ ww630))",fontsize=16,color="black",shape="box"];621 -> 629[label="",style="solid", color="black", weight=3];
622[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat Zero Zero)",fontsize=16,color="black",shape="box"];622 -> 630[label="",style="solid", color="black", weight=3];
93[label="ww411",fontsize=16,color="green",shape="box"];94[label="ww410",fontsize=16,color="green",shape="box"];95[label="Pos (Succ ww300)",fontsize=16,color="green",shape="box"];96 -> 10[label="",style="dashed", color="red", weight=0];
96[label="insertBy0 ww410 primCmpInt (Pos Zero) ww411 (ww410 : ww411) (primCmpInt (Pos Zero) ww410)",fontsize=16,color="magenta"];96 -> 109[label="",style="dashed", color="magenta", weight=3];
96 -> 110[label="",style="dashed", color="magenta", weight=3];
96 -> 111[label="",style="dashed", color="magenta", weight=3];
97[label="Pos Zero : []",fontsize=16,color="green",shape="box"];623[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat (Succ ww680) (Succ ww690))",fontsize=16,color="black",shape="box"];623 -> 631[label="",style="solid", color="black", weight=3];
624[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat (Succ ww680) Zero)",fontsize=16,color="black",shape="box"];624 -> 632[label="",style="solid", color="black", weight=3];
625[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat Zero (Succ ww690))",fontsize=16,color="black",shape="box"];625 -> 633[label="",style="solid", color="black", weight=3];
626[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat Zero Zero)",fontsize=16,color="black",shape="box"];626 -> 634[label="",style="solid", color="black", weight=3];
102[label="insertBy primCmpInt (Neg Zero) (ww410 : ww411)",fontsize=16,color="black",shape="box"];102 -> 117[label="",style="solid", color="black", weight=3];
103[label="insertBy primCmpInt (Neg Zero) []",fontsize=16,color="black",shape="box"];103 -> 118[label="",style="solid", color="black", weight=3];
627 -> 511[label="",style="dashed", color="red", weight=0];
627[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat ww620 ww630)",fontsize=16,color="magenta"];627 -> 635[label="",style="dashed", color="magenta", weight=3];
627 -> 636[label="",style="dashed", color="magenta", weight=3];
628[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) GT",fontsize=16,color="black",shape="box"];628 -> 637[label="",style="solid", color="black", weight=3];
629[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) LT",fontsize=16,color="black",shape="box"];629 -> 638[label="",style="solid", color="black", weight=3];
630[label="insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) EQ",fontsize=16,color="black",shape="box"];630 -> 639[label="",style="solid", color="black", weight=3];
109[label="ww411",fontsize=16,color="green",shape="box"];110[label="ww410",fontsize=16,color="green",shape="box"];111[label="Pos Zero",fontsize=16,color="green",shape="box"];631 -> 566[label="",style="dashed", color="red", weight=0];
631[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat ww680 ww690)",fontsize=16,color="magenta"];631 -> 640[label="",style="dashed", color="magenta", weight=3];
631 -> 641[label="",style="dashed", color="magenta", weight=3];
632[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) GT",fontsize=16,color="black",shape="box"];632 -> 642[label="",style="solid", color="black", weight=3];
633[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) LT",fontsize=16,color="black",shape="box"];633 -> 643[label="",style="solid", color="black", weight=3];
634[label="insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) EQ",fontsize=16,color="black",shape="box"];634 -> 644[label="",style="solid", color="black", weight=3];
117 -> 10[label="",style="dashed", color="red", weight=0];
117[label="insertBy0 ww410 primCmpInt (Neg Zero) ww411 (ww410 : ww411) (primCmpInt (Neg Zero) ww410)",fontsize=16,color="magenta"];117 -> 129[label="",style="dashed", color="magenta", weight=3];
117 -> 130[label="",style="dashed", color="magenta", weight=3];
117 -> 131[label="",style="dashed", color="magenta", weight=3];
118[label="Neg Zero : []",fontsize=16,color="green",shape="box"];635[label="ww630",fontsize=16,color="green",shape="box"];636[label="ww620",fontsize=16,color="green",shape="box"];637[label="Pos (Succ ww59) : insertBy primCmpInt (Pos (Succ ww60)) ww61",fontsize=16,color="green",shape="box"];637 -> 645[label="",style="dashed", color="green", weight=3];
638[label="Pos (Succ ww60) : Pos (Succ ww59) : ww61",fontsize=16,color="green",shape="box"];639[label="Pos (Succ ww60) : Pos (Succ ww59) : ww61",fontsize=16,color="green",shape="box"];640[label="ww680",fontsize=16,color="green",shape="box"];641[label="ww690",fontsize=16,color="green",shape="box"];642[label="Neg (Succ ww65) : insertBy primCmpInt (Neg (Succ ww66)) ww67",fontsize=16,color="green",shape="box"];642 -> 646[label="",style="dashed", color="green", weight=3];
643[label="Neg (Succ ww66) : Neg (Succ ww65) : ww67",fontsize=16,color="green",shape="box"];644[label="Neg (Succ ww66) : Neg (Succ ww65) : ww67",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="Neg Zero",fontsize=16,color="green",shape="box"];645 -> 53[label="",style="dashed", color="red", weight=0];
645[label="insertBy primCmpInt (Pos (Succ ww60)) ww61",fontsize=16,color="magenta"];645 -> 647[label="",style="dashed", color="magenta", weight=3];
645 -> 648[label="",style="dashed", color="magenta", weight=3];
646[label="insertBy primCmpInt (Neg (Succ ww66)) ww67",fontsize=16,color="burlywood",shape="box"];711[label="ww67/ww670 : ww671",fontsize=10,color="white",style="solid",shape="box"];646 -> 711[label="",style="solid", color="burlywood", weight=9];
711 -> 649[label="",style="solid", color="burlywood", weight=3];
712[label="ww67/[]",fontsize=10,color="white",style="solid",shape="box"];646 -> 712[label="",style="solid", color="burlywood", weight=9];
712 -> 650[label="",style="solid", color="burlywood", weight=3];
647[label="ww61",fontsize=16,color="green",shape="box"];648[label="ww60",fontsize=16,color="green",shape="box"];649[label="insertBy primCmpInt (Neg (Succ ww66)) (ww670 : ww671)",fontsize=16,color="black",shape="box"];649 -> 651[label="",style="solid", color="black", weight=3];
650[label="insertBy primCmpInt (Neg (Succ ww66)) []",fontsize=16,color="black",shape="box"];650 -> 652[label="",style="solid", color="black", weight=3];
651 -> 10[label="",style="dashed", color="red", weight=0];
651[label="insertBy0 ww670 primCmpInt (Neg (Succ ww66)) ww671 (ww670 : ww671) (primCmpInt (Neg (Succ ww66)) ww670)",fontsize=16,color="magenta"];651 -> 653[label="",style="dashed", color="magenta", weight=3];
651 -> 654[label="",style="dashed", color="magenta", weight=3];
651 -> 655[label="",style="dashed", color="magenta", weight=3];
652[label="Neg (Succ ww66) : []",fontsize=16,color="green",shape="box"];653[label="ww671",fontsize=16,color="green",shape="box"];654[label="ww670",fontsize=16,color="green",shape="box"];655[label="Neg (Succ ww66)",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

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

new_insertBy(ww300, :(ww410, ww411)) → new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
new_insertBy0(ww59, ww60, ww61, Succ(ww620), Zero) → new_insertBy(ww60, ww61)
new_insertBy00(Neg(Succ(ww4000)), Neg(Zero), :(ww410, ww411)) → new_insertBy00(ww410, Neg(Zero), ww411)
new_insertBy00(Neg(Succ(ww4000)), Pos(Zero), :(ww410, ww411)) → new_insertBy00(ww410, Pos(Zero), ww411)
new_insertBy00(Pos(Succ(ww4000)), Pos(Succ(ww300)), ww41) → new_insertBy0(ww4000, ww300, ww41, ww300, ww4000)
new_insertBy00(Neg(ww400), Pos(Succ(ww300)), :(ww410, ww411)) → new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
new_insertBy01(ww65, ww66, :(ww670, ww671), Succ(ww680), Zero) → new_insertBy00(ww670, Neg(Succ(ww66)), ww671)
new_insertBy00(Pos(Zero), Pos(Succ(ww300)), ww41) → new_insertBy(ww300, ww41)
new_insertBy00(Neg(Succ(ww4000)), Neg(Succ(ww300)), ww41) → new_insertBy01(ww4000, ww300, ww41, ww4000, ww300)
new_insertBy0(ww59, ww60, ww61, Succ(ww620), Succ(ww630)) → new_insertBy0(ww59, ww60, ww61, ww620, ww630)
new_insertBy01(ww65, ww66, ww67, Succ(ww680), Succ(ww690)) → new_insertBy01(ww65, ww66, ww67, ww680, ww690)

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

  ↳ CR

    ↳ 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_insertBy01(ww65, ww66, :(ww670, ww671), Succ(ww680), Zero) → new_insertBy00(ww670, Neg(Succ(ww66)), ww671)
new_insertBy00(Neg(Succ(ww4000)), Neg(Succ(ww300)), ww41) → new_insertBy01(ww4000, ww300, ww41, ww4000, ww300)
new_insertBy01(ww65, ww66, ww67, Succ(ww680), Succ(ww690)) → new_insertBy01(ww65, ww66, ww67, ww680, ww690)

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_insertBy01(ww65, ww66, :(ww670, ww671), Succ(ww680), Zero) → new_insertBy00(ww670, Neg(Succ(ww66)), ww671)
The graph contains the following edges 3 > 1, 3 > 3
new_insertBy01(ww65, ww66, ww67, Succ(ww680), Succ(ww690)) → new_insertBy01(ww65, ww66, ww67, ww680, ww690)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
new_insertBy00(Neg(Succ(ww4000)), Neg(Succ(ww300)), ww41) → new_insertBy01(ww4000, ww300, ww41, ww4000, ww300)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 1 > 4, 2 > 5

↳ HASKELL

  ↳ CR

    ↳ 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_insertBy00(Neg(Succ(ww4000)), Pos(Zero), :(ww410, ww411)) → new_insertBy00(ww410, Pos(Zero), ww411)

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

new_insertBy00(Neg(Succ(ww4000)), Pos(Zero), :(ww410, ww411)) → new_insertBy00(ww410, Pos(Zero), ww411)
The graph contains the following edges 3 > 1, 2 >= 2, 3 > 3

↳ HASKELL

  ↳ CR

    ↳ 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_insertBy00(Neg(Succ(ww4000)), Neg(Zero), :(ww410, ww411)) → new_insertBy00(ww410, Neg(Zero), ww411)

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

new_insertBy00(Neg(Succ(ww4000)), Neg(Zero), :(ww410, ww411)) → new_insertBy00(ww410, Neg(Zero), ww411)
The graph contains the following edges 3 > 1, 2 >= 2, 3 > 3

↳ HASKELL

  ↳ CR

    ↳ 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_insertBy(ww300, :(ww410, ww411)) → new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
new_insertBy0(ww59, ww60, ww61, Succ(ww620), Zero) → new_insertBy(ww60, ww61)
new_insertBy00(Pos(Succ(ww4000)), Pos(Succ(ww300)), ww41) → new_insertBy0(ww4000, ww300, ww41, ww300, ww4000)
new_insertBy00(Neg(ww400), Pos(Succ(ww300)), :(ww410, ww411)) → new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
new_insertBy00(Pos(Zero), Pos(Succ(ww300)), ww41) → new_insertBy(ww300, ww41)
new_insertBy0(ww59, ww60, ww61, Succ(ww620), Succ(ww630)) → new_insertBy0(ww59, ww60, ww61, ww620, ww630)

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

new_insertBy00(Neg(ww400), Pos(Succ(ww300)), :(ww410, ww411)) → new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
The graph contains the following edges 3 > 1, 2 >= 2, 3 > 3
new_insertBy(ww300, :(ww410, ww411)) → new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
The graph contains the following edges 2 > 1, 2 > 3
new_insertBy00(Pos(Succ(ww4000)), Pos(Succ(ww300)), ww41) → new_insertBy0(ww4000, ww300, ww41, ww300, ww4000)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 2 > 4, 1 > 5
new_insertBy00(Pos(Zero), Pos(Succ(ww300)), ww41) → new_insertBy(ww300, ww41)
The graph contains the following edges 2 > 1, 3 >= 2
new_insertBy0(ww59, ww60, ww61, Succ(ww620), Succ(ww630)) → new_insertBy0(ww59, ww60, ww61, ww620, ww630)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
new_insertBy0(ww59, ww60, ww61, Succ(ww620), Zero) → new_insertBy(ww60, ww61)
The graph contains the following edges 2 >= 1, 3 >= 2

insertBy0	y cmp x ys' ys GT	= y : insertBy cmp x ys'
insertBy0	y cmp x ys' ys _	= x : ys