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

YES 1.24 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

  ↳ LR

mainModule List

((elemIndices :: Char -> [Char] -> [Int]) :: Char -> [Char] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

elemIndices :: Eq a => a -> [a] -> [Int]

elemIndices

findIndices (== x)

findIndices :: (a -> Bool) -> [a] -> [Int]

findIndices

p xs

concatMap (\vv1 ->

case	vv1 of
	(x,i)	->	if p x then i : [] else []
	_	->	[]

) (zip xs (enumFrom 0))

module Maybe where

import qualified List
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\vv1→

case vv1 of
(x,i) → if p x then i : [] else []

_ → []

is transformed to

findIndices0 p vv1 =

case vv1 of
(x,i) → if p x then i : [] else []

_ → []

The following Lambda expression

\ab→(a,b)

is transformed to

zip0 a b = (a,b)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

mainModule List

((elemIndices :: Char -> [Char] -> [Int]) :: Char -> [Char] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

elemIndices :: Eq a => a -> [a] -> [Int]

elemIndices

findIndices (== x)

findIndices :: (a -> Bool) -> [a] -> [Int]

findIndices

p xs

concatMap (findIndices0 p) (zip xs (enumFrom 0))

findIndices0

p vv1

case	vv1 of
	(x,i)	->	if p x then i : [] else []
	_	->	[]

module Maybe where

import qualified List
import qualified Prelude

Case Reductions:
The following Case expression

case vv1 of
(x,i) → if p x then i : [] else []

_ → []

is transformed to

findIndices00 p (x,i) = if p x then i : [] else []

findIndices00 p _ = []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

mainModule List

((elemIndices :: Char -> [Char] -> [Int]) :: Char -> [Char] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

elemIndices :: Eq a => a -> [a] -> [Int]

elemIndices

findIndices (== x)

findIndices :: (a -> Bool) -> [a] -> [Int]

findIndices

p xs

concatMap (findIndices0 p) (zip xs (enumFrom 0))

findIndices0

p vv1

findIndices00 p vv1

findIndices00	p (x,i)	=	if p x then i : [] else []
findIndices00	p _	=	[]

module Maybe where

import qualified List
import qualified Prelude

If Reductions:
The following If expression

if p x then i : [] else []

is transformed to

findIndices000 i True = i : []

findIndices000 i False = []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

mainModule List

((elemIndices :: Char -> [Char] -> [Int]) :: Char -> [Char] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

elemIndices :: Eq a => a -> [a] -> [Int]

elemIndices

findIndices (== x)

findIndices :: (a -> Bool) -> [a] -> [Int]

findIndices

p xs

concatMap (findIndices0 p) (zip xs (enumFrom 0))

findIndices0

p vv1

findIndices00 p vv1

findIndices00	p (x,i)	=	findIndices000 i (p x)
findIndices00	p _	=	[]

findIndices000	i True	=	i : []
findIndices000	i False	=	[]

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

mainModule List

((elemIndices :: Char -> [Char] -> [Int]) :: Char -> [Char] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

elemIndices :: Eq a => a -> [a] -> [Int]

elemIndices

findIndices (== x)

findIndices :: (a -> Bool) -> [a] -> [Int]

findIndices

p xs

concatMap (findIndices0 p) (zip xs (enumFrom 0))

findIndices0

p vv1

findIndices00 p vv1

findIndices00	p (x,i)	=	findIndices000 i (p x)
findIndices00	p vw	=	[]

findIndices000	i True	=	i : []
findIndices000	i False	=	[]

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

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ NumRed

mainModule List

((elemIndices :: Char -> [Char] -> [Int]) :: Char -> [Char] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

elemIndices :: Eq a => a -> [a] -> [Int]

elemIndices

findIndices (== x)

findIndices :: (a -> Bool) -> [a] -> [Int]

findIndices

p xs

concatMap (findIndices0 p) (zip xs (enumFrom 0))

findIndices0

p vv1

findIndices00 p vv1

findIndices00	p (x,i)	=	findIndices000 i (p x)
findIndices00	p vw	=	[]

findIndices000	i True	=	i : []
findIndices000	i False	=	[]

module Maybe where

import qualified List
import qualified Prelude

Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

mainModule List

(elemIndices :: Char -> [Char] -> [Int])

module List where

import qualified Maybe
import qualified Prelude

elemIndices :: Eq a => a -> [a] -> [Int]

elemIndices

findIndices (== x)

findIndices :: (a -> Bool) -> [a] -> [Int]

findIndices

p xs

concatMap (findIndices0 p) (zip xs (enumFrom (Pos Zero)))

findIndices0

p vv1

findIndices00 p vv1

findIndices00	p (x,i)	=	findIndices000 i (p x)
findIndices00	p vw	=	[]

findIndices000	i True	=	i : []
findIndices000	i False	=	[]

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
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3[label="elemIndices ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="elemIndices ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="findIndices (ww3 ==) ww4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concatMap (findIndices0 (ww3 ==)) (zip ww4 (enumFrom (Pos Zero)))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="concat . map (findIndices0 (ww3 ==))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="concat (map (findIndices0 (ww3 ==)) (zip ww4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (++) [] (map (findIndices0 (ww3 ==)) (zip ww4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 ww4 (enumFrom (Pos Zero))))",fontsize=16,color="burlywood",shape="box"];266[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];10 -> 266[label="",style="solid", color="burlywood", weight=9];
266 -> 11[label="",style="solid", color="burlywood", weight=3];
267[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 267[label="",style="solid", color="burlywood", weight=9];
267 -> 12[label="",style="solid", color="burlywood", weight=3];
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12[label="foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 [] (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 (ww40 : ww41) (numericEnumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="foldr (++) [] (map (findIndices0 (ww3 ==)) [])",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 (ww40 : ww41) (Pos Zero : (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="foldr (++) [] []",fontsize=16,color="black",shape="triangle"];16 -> 18[label="",style="solid", color="black", weight=3];
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20[label="(++) findIndices0 (ww3 ==) (zip0 ww40 (Pos Zero)) foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="(++) findIndices00 (ww3 ==) (zip0 ww40 (Pos Zero)) foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="(++) findIndices00 (ww3 ==) (ww40,Pos Zero) foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="(++) findIndices000 (Pos Zero) (ww3 == ww40) foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
24[label="(++) findIndices000 (Pos Zero) (primEqChar ww3 ww40) foldr (++) [] (map (findIndices0 (primEqChar ww3)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];268[label="ww3/Char ww30",fontsize=10,color="white",style="solid",shape="box"];24 -> 268[label="",style="solid", color="burlywood", weight=9];
268 -> 25[label="",style="solid", color="burlywood", weight=3];
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269 -> 26[label="",style="solid", color="burlywood", weight=3];
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27 -> 59[label="",style="dashed", color="red", weight=0];
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271 -> 67[label="",style="solid", color="burlywood", weight=3];
272[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];62 -> 272[label="",style="solid", color="burlywood", weight=9];
272 -> 68[label="",style="solid", color="burlywood", weight=3];
59[label="(++) findIndices000 (Pos Zero) (primEqNat ww3000 ww40000) ww5",fontsize=16,color="burlywood",shape="triangle"];273[label="ww3000/Succ ww30000",fontsize=10,color="white",style="solid",shape="box"];59 -> 273[label="",style="solid", color="burlywood", weight=9];
273 -> 69[label="",style="solid", color="burlywood", weight=3];
274[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 274[label="",style="solid", color="burlywood", weight=9];
274 -> 70[label="",style="solid", color="burlywood", weight=3];
67[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
68[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 [] (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="(++) findIndices000 (Pos Zero) (primEqNat (Succ ww30000) ww40000) ww5",fontsize=16,color="burlywood",shape="box"];275[label="ww40000/Succ ww400000",fontsize=10,color="white",style="solid",shape="box"];69 -> 275[label="",style="solid", color="burlywood", weight=9];
275 -> 73[label="",style="solid", color="burlywood", weight=3];
276[label="ww40000/Zero",fontsize=10,color="white",style="solid",shape="box"];69 -> 276[label="",style="solid", color="burlywood", weight=9];
276 -> 74[label="",style="solid", color="burlywood", weight=3];
70[label="(++) findIndices000 (Pos Zero) (primEqNat Zero ww40000) ww5",fontsize=16,color="burlywood",shape="box"];277[label="ww40000/Succ ww400000",fontsize=10,color="white",style="solid",shape="box"];70 -> 277[label="",style="solid", color="burlywood", weight=9];
277 -> 75[label="",style="solid", color="burlywood", weight=3];
278[label="ww40000/Zero",fontsize=10,color="white",style="solid",shape="box"];70 -> 278[label="",style="solid", color="burlywood", weight=9];
278 -> 76[label="",style="solid", color="burlywood", weight=3];
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73[label="(++) findIndices000 (Pos Zero) (primEqNat (Succ ww30000) (Succ ww400000)) ww5",fontsize=16,color="black",shape="box"];73 -> 79[label="",style="solid", color="black", weight=3];
74[label="(++) findIndices000 (Pos Zero) (primEqNat (Succ ww30000) Zero) ww5",fontsize=16,color="black",shape="box"];74 -> 80[label="",style="solid", color="black", weight=3];
75[label="(++) findIndices000 (Pos Zero) (primEqNat Zero (Succ ww400000)) ww5",fontsize=16,color="black",shape="box"];75 -> 81[label="",style="solid", color="black", weight=3];
76[label="(++) findIndices000 (Pos Zero) (primEqNat Zero Zero) ww5",fontsize=16,color="black",shape="box"];76 -> 82[label="",style="solid", color="black", weight=3];
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78 -> 16[label="",style="dashed", color="red", weight=0];
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79[label="(++) findIndices000 (Pos Zero) (primEqNat ww30000 ww400000) ww5",fontsize=16,color="magenta"];79 -> 84[label="",style="dashed", color="magenta", weight=3];
79 -> 85[label="",style="dashed", color="magenta", weight=3];
80[label="(++) findIndices000 (Pos Zero) False ww5",fontsize=16,color="black",shape="triangle"];80 -> 86[label="",style="solid", color="black", weight=3];
81 -> 80[label="",style="dashed", color="red", weight=0];
81[label="(++) findIndices000 (Pos Zero) False ww5",fontsize=16,color="magenta"];82[label="(++) findIndices000 (Pos Zero) True ww5",fontsize=16,color="black",shape="box"];82 -> 87[label="",style="solid", color="black", weight=3];
83[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (enforceWHNF (WHNF (primPlusInt (Pos Zero) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos Zero) (fromInt (Pos (Succ Zero))))))))",fontsize=16,color="black",shape="box"];83 -> 88[label="",style="solid", color="black", weight=3];
84[label="ww400000",fontsize=16,color="green",shape="box"];85[label="ww30000",fontsize=16,color="green",shape="box"];86[label="(++) [] ww5",fontsize=16,color="black",shape="triangle"];86 -> 89[label="",style="solid", color="black", weight=3];
87[label="(++) (Pos Zero : []) ww5",fontsize=16,color="black",shape="box"];87 -> 90[label="",style="solid", color="black", weight=3];
88[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (enforceWHNF (WHNF (primPlusInt (Pos Zero) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos Zero) (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];88 -> 91[label="",style="solid", color="black", weight=3];
89[label="ww5",fontsize=16,color="green",shape="box"];90[label="Pos Zero : [] ++ ww5",fontsize=16,color="green",shape="box"];90 -> 92[label="",style="dashed", color="green", weight=3];
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92 -> 86[label="",style="dashed", color="red", weight=0];
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93[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (numericEnumFrom (Pos (primPlusNat Zero (Succ Zero))))))",fontsize=16,color="magenta"];93 -> 173[label="",style="dashed", color="magenta", weight=3];
93 -> 174[label="",style="dashed", color="magenta", weight=3];
93 -> 175[label="",style="dashed", color="magenta", weight=3];
173[label="ww411",fontsize=16,color="green",shape="box"];174[label="Zero",fontsize=16,color="green",shape="box"];175[label="ww410",fontsize=16,color="green",shape="box"];172[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww4110 : ww4111) (numericEnumFrom (Pos (primPlusNat ww7 (Succ Zero))))))",fontsize=16,color="black",shape="triangle"];172 -> 177[label="",style="solid", color="black", weight=3];
177[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww4110 : ww4111) (Pos (primPlusNat ww7 (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];177 -> 178[label="",style="solid", color="black", weight=3];
178[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero))) : zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];178 -> 179[label="",style="solid", color="black", weight=3];
179[label="foldr (++) [] (findIndices0 (primEqChar (Char ww30)) (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero)))) : map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];179 -> 180[label="",style="solid", color="black", weight=3];
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286 -> 226[label="",style="solid", color="burlywood", weight=3];
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288 -> 228[label="",style="solid", color="burlywood", weight=3];
289[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];218 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 229[label="",style="solid", color="burlywood", weight=3];
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228[label="(++) findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqNat (Succ ww30000) ww4110000) ww8",fontsize=16,color="burlywood",shape="box"];290[label="ww4110000/Succ ww41100000",fontsize=10,color="white",style="solid",shape="box"];228 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 232[label="",style="solid", color="burlywood", weight=3];
291[label="ww4110000/Zero",fontsize=10,color="white",style="solid",shape="box"];228 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 233[label="",style="solid", color="burlywood", weight=3];
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237 -> 243[label="",style="dashed", color="magenta", weight=3];
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239 -> 238[label="",style="dashed", color="red", weight=0];
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241[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww41110 : ww41111) (enforceWHNF (WHNF (primPlusInt (Pos (primPlusNat ww7 (Succ Zero))) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos (primPlusNat ww7 (Succ Zero))) (fromInt (Pos (Succ Zero))))))))",fontsize=16,color="black",shape="box"];241 -> 246[label="",style="solid", color="black", weight=3];
242[label="ww30000",fontsize=16,color="green",shape="box"];243[label="ww41100000",fontsize=16,color="green",shape="box"];244 -> 86[label="",style="dashed", color="red", weight=0];
244[label="(++) [] ww8",fontsize=16,color="magenta"];244 -> 247[label="",style="dashed", color="magenta", weight=3];
245[label="(++) (Pos (primPlusNat ww7 (Succ Zero)) : []) ww8",fontsize=16,color="black",shape="box"];245 -> 248[label="",style="solid", color="black", weight=3];
246[label="foldr (++) [] (map (findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww41110 : ww41111) (enforceWHNF (WHNF (primPlusInt (Pos (primPlusNat ww7 (Succ Zero))) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos (primPlusNat ww7 (Succ Zero))) (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];246 -> 249[label="",style="solid", color="black", weight=3];
247[label="ww8",fontsize=16,color="green",shape="box"];248[label="Pos (primPlusNat ww7 (Succ Zero)) : [] ++ ww8",fontsize=16,color="green",shape="box"];248 -> 250[label="",style="dashed", color="green", weight=3];
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250[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];298[label="ww7/Succ ww70",fontsize=10,color="white",style="solid",shape="box"];250 -> 298[label="",style="solid", color="burlywood", weight=9];
298 -> 253[label="",style="solid", color="burlywood", weight=3];
299[label="ww7/Zero",fontsize=10,color="white",style="solid",shape="box"];250 -> 299[label="",style="solid", color="burlywood", weight=9];
299 -> 254[label="",style="solid", color="burlywood", weight=3];
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252 -> 257[label="",style="dashed", color="magenta", weight=3];
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253[label="primPlusNat (Succ ww70) (Succ Zero)",fontsize=16,color="black",shape="box"];253 -> 259[label="",style="solid", color="black", weight=3];
254[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];254 -> 260[label="",style="solid", color="black", weight=3];
255[label="ww8",fontsize=16,color="green",shape="box"];256[label="ww41111",fontsize=16,color="green",shape="box"];257 -> 250[label="",style="dashed", color="red", weight=0];
257[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];258[label="ww41110",fontsize=16,color="green",shape="box"];259[label="Succ (Succ (primPlusNat ww70 Zero))",fontsize=16,color="green",shape="box"];259 -> 261[label="",style="dashed", color="green", weight=3];
260[label="Succ Zero",fontsize=16,color="green",shape="box"];261[label="primPlusNat ww70 Zero",fontsize=16,color="burlywood",shape="box"];303[label="ww70/Succ ww700",fontsize=10,color="white",style="solid",shape="box"];261 -> 303[label="",style="solid", color="burlywood", weight=9];
303 -> 262[label="",style="solid", color="burlywood", weight=3];
304[label="ww70/Zero",fontsize=10,color="white",style="solid",shape="box"];261 -> 304[label="",style="solid", color="burlywood", weight=9];
304 -> 263[label="",style="solid", color="burlywood", weight=3];
262[label="primPlusNat (Succ ww700) Zero",fontsize=16,color="black",shape="box"];262 -> 264[label="",style="solid", color="black", weight=3];
263[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];263 -> 265[label="",style="solid", color="black", weight=3];
264[label="Succ ww700",fontsize=16,color="green",shape="box"];265[label="Zero",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

                            ↳ AND

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

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

new_psPs(ww7, Succ(ww30000), Succ(ww41100000), ww8) → new_psPs(ww7, ww30000, ww41100000, ww8)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

new_psPs(ww7, Succ(ww30000), Succ(ww41100000), ww8) → new_psPs(ww7, ww30000, ww41100000, ww8)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                              ↳ QDP

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

new_foldr(ww30, Char(ww41100), :(ww41110, ww41111), ww7) → new_foldr(ww30, ww41110, ww41111, new_primPlusNat(ww7))

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(ww700)) → Succ(ww700)
new_primPlusNat(Succ(ww70)) → Succ(Succ(new_primPlusNat0(ww70)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat0(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat(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_foldr(ww30, Char(ww41100), :(ww41110, ww41111), ww7) → new_foldr(ww30, ww41110, ww41111, new_primPlusNat(ww7))
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ QDPSizeChangeProof

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

new_psPs0(Succ(ww30000), Succ(ww400000), ww5) → new_psPs0(ww30000, ww400000, ww5)

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

new_psPs0(Succ(ww30000), Succ(ww400000), ww5) → new_psPs0(ww30000, ww400000, ww5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

findIndices00	p (x,i)	= if p x then i : [] else []
findIndices00	p _	= []