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

YES 1.169 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 :: () -> [()] -> [Int]) :: () -> [()] -> [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 :: () -> [()] -> [Int]) :: () -> [()] -> [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 :: () -> [()] -> [Int]) :: () -> [()] -> [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 :: () -> [()] -> [Int]) :: () -> [()] -> [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 :: () -> [()] -> [Int]) :: () -> [()] -> [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 :: () -> [()] -> [Int]) :: () -> [()] -> [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 :: () -> [()] -> [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 {
node [outthreshold=100, inthreshold=100];1[label="elemIndices",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
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"];98[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];10 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 11[label="",style="solid", color="burlywood", weight=3];
99[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="foldr (++) [] (map (findIndices0 (ww3 ==)) (zipWith zip0 (ww40 : ww41) (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
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="triangle"];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="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="foldr (++) [] (map (findIndices0 (ww3 ==)) (zip0 ww40 (Pos Zero) : zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="[]",fontsize=16,color="green",shape="box"];19[label="foldr (++) [] (findIndices0 (ww3 ==) (zip0 ww40 (Pos Zero)) : map (findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
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="burlywood",shape="box"];100[label="ww3/()",fontsize=10,color="white",style="solid",shape="box"];23 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 24[label="",style="solid", color="burlywood", weight=3];
24[label="(++) findIndices000 (Pos Zero) (() == ww40) foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];101[label="ww40/()",fontsize=10,color="white",style="solid",shape="box"];24 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 25[label="",style="solid", color="burlywood", weight=3];
25[label="(++) findIndices000 (Pos Zero) (() == ()) foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3];
26[label="(++) findIndices000 (Pos Zero) True foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
27[label="(++) (Pos Zero : []) foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];27 -> 28[label="",style="solid", color="black", weight=3];
28[label="Pos Zero : [] ++ foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];28 -> 29[label="",style="dashed", color="green", weight=3];
29 -> 55[label="",style="dashed", color="red", weight=0];
29[label="[] ++ foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];29 -> 56[label="",style="dashed", color="magenta", weight=3];
29 -> 57[label="",style="dashed", color="magenta", weight=3];
56[label="Zero",fontsize=16,color="green",shape="box"];57[label="ww41",fontsize=16,color="green",shape="box"];55[label="[] ++ foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww411 (numericEnumFrom $! Pos ww5 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="triangle"];55 -> 59[label="",style="solid", color="black", weight=3];
59[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww411 (numericEnumFrom $! Pos ww5 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="triangle"];103[label="ww411/ww4110 : ww4111",fontsize=10,color="white",style="solid",shape="box"];59 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 60[label="",style="solid", color="burlywood", weight=3];
104[label="ww411/[]",fontsize=10,color="white",style="solid",shape="box"];59 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 61[label="",style="solid", color="burlywood", weight=3];
60[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (numericEnumFrom $! Pos ww5 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
61[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 [] (numericEnumFrom $! Pos ww5 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (Pos ww5 + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos ww5 + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];62 -> 64[label="",style="solid", color="black", weight=3];
63 -> 14[label="",style="dashed", color="red", weight=0];
63[label="foldr (++) [] (map (findIndices0 (() ==)) [])",fontsize=16,color="magenta"];63 -> 65[label="",style="dashed", color="magenta", weight=3];
64[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (Pos ww5 + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos ww5 + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];64 -> 66[label="",style="solid", color="black", weight=3];
65[label="()",fontsize=16,color="green",shape="box"];66[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (primPlusInt (Pos ww5) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos ww5) (fromInt (Pos (Succ Zero))))))))",fontsize=16,color="black",shape="box"];66 -> 67[label="",style="solid", color="black", weight=3];
67[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (primPlusInt (Pos ww5) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos ww5) (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];67 -> 68[label="",style="solid", color="black", weight=3];
68[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (Pos (primPlusNat ww5 (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat ww5 (Succ Zero)))))))",fontsize=16,color="black",shape="box"];68 -> 69[label="",style="solid", color="black", weight=3];
69[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (numericEnumFrom (Pos (primPlusNat ww5 (Succ Zero))))))",fontsize=16,color="black",shape="box"];69 -> 70[label="",style="solid", color="black", weight=3];
70[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (Pos (primPlusNat ww5 (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];70 -> 71[label="",style="solid", color="black", weight=3];
71[label="foldr (++) [] (map (findIndices0 (() ==)) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero))) : zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];71 -> 72[label="",style="solid", color="black", weight=3];
72[label="foldr (++) [] (findIndices0 (() ==) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero)))) : map (findIndices0 (() ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];72 -> 73[label="",style="solid", color="black", weight=3];
73 -> 74[label="",style="dashed", color="red", weight=0];
73[label="(++) findIndices0 (() ==) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero)))) foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];73 -> 75[label="",style="dashed", color="magenta", weight=3];
75 -> 59[label="",style="dashed", color="red", weight=0];
75[label="foldr (++) [] (map (findIndices0 (() ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];75 -> 76[label="",style="dashed", color="magenta", weight=3];
75 -> 77[label="",style="dashed", color="magenta", weight=3];
74[label="(++) findIndices0 (() ==) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero)))) ww6",fontsize=16,color="black",shape="triangle"];74 -> 78[label="",style="solid", color="black", weight=3];
76[label="primPlusNat ww5 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];108[label="ww5/Succ ww50",fontsize=10,color="white",style="solid",shape="box"];76 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 79[label="",style="solid", color="burlywood", weight=3];
109[label="ww5/Zero",fontsize=10,color="white",style="solid",shape="box"];76 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 80[label="",style="solid", color="burlywood", weight=3];
77[label="ww4111",fontsize=16,color="green",shape="box"];78 -> 81[label="",style="dashed", color="red", weight=0];
78[label="(++) findIndices00 (() ==) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero)))) ww6",fontsize=16,color="magenta"];78 -> 82[label="",style="dashed", color="magenta", weight=3];
79[label="primPlusNat (Succ ww50) (Succ Zero)",fontsize=16,color="black",shape="box"];79 -> 83[label="",style="solid", color="black", weight=3];
80[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];80 -> 84[label="",style="solid", color="black", weight=3];
82 -> 76[label="",style="dashed", color="red", weight=0];
82[label="primPlusNat ww5 (Succ Zero)",fontsize=16,color="magenta"];81[label="(++) findIndices00 (() ==) (zip0 ww4110 (Pos ww7)) ww6",fontsize=16,color="black",shape="triangle"];81 -> 85[label="",style="solid", color="black", weight=3];
83[label="Succ (Succ (primPlusNat ww50 Zero))",fontsize=16,color="green",shape="box"];83 -> 86[label="",style="dashed", color="green", weight=3];
84[label="Succ Zero",fontsize=16,color="green",shape="box"];85[label="(++) findIndices00 (() ==) (ww4110,Pos ww7) ww6",fontsize=16,color="black",shape="box"];85 -> 87[label="",style="solid", color="black", weight=3];
86[label="primPlusNat ww50 Zero",fontsize=16,color="burlywood",shape="box"];112[label="ww50/Succ ww500",fontsize=10,color="white",style="solid",shape="box"];86 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 88[label="",style="solid", color="burlywood", weight=3];
113[label="ww50/Zero",fontsize=10,color="white",style="solid",shape="box"];86 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 89[label="",style="solid", color="burlywood", weight=3];
87[label="(++) findIndices000 (Pos ww7) (() == ww4110) ww6",fontsize=16,color="burlywood",shape="box"];114[label="ww4110/()",fontsize=10,color="white",style="solid",shape="box"];87 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 90[label="",style="solid", color="burlywood", weight=3];
88[label="primPlusNat (Succ ww500) Zero",fontsize=16,color="black",shape="box"];88 -> 91[label="",style="solid", color="black", weight=3];
89[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];89 -> 92[label="",style="solid", color="black", weight=3];
90[label="(++) findIndices000 (Pos ww7) (() == ()) ww6",fontsize=16,color="black",shape="box"];90 -> 93[label="",style="solid", color="black", weight=3];
91[label="Succ ww500",fontsize=16,color="green",shape="box"];92[label="Zero",fontsize=16,color="green",shape="box"];93[label="(++) findIndices000 (Pos ww7) True ww6",fontsize=16,color="black",shape="box"];93 -> 94[label="",style="solid", color="black", weight=3];
94[label="(++) (Pos ww7 : []) ww6",fontsize=16,color="black",shape="box"];94 -> 95[label="",style="solid", color="black", weight=3];
95[label="Pos ww7 : [] ++ ww6",fontsize=16,color="green",shape="box"];95 -> 96[label="",style="dashed", color="green", weight=3];
96[label="[] ++ ww6",fontsize=16,color="black",shape="box"];96 -> 97[label="",style="solid", color="black", weight=3];
97[label="ww6",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ IFR

            ↳ HASKELL

              ↳ BR

                ↳ HASKELL

                  ↳ COR

                    ↳ HASKELL

                      ↳ NumRed

                        ↳ HASKELL

                          ↳ Narrow

                            ↳ QDP

                              ↳ QDPSizeChangeProof

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

new_foldr(:(ww4110, ww4111), ww5) → new_foldr(ww4111, new_primPlusNat(ww5))

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(ww500)) → Succ(ww500)
new_primPlusNat(Succ(ww50)) → Succ(Succ(new_primPlusNat0(ww50)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat0(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(:(ww4110, ww4111), ww5) → new_foldr(ww4111, new_primPlusNat(ww5))
The graph contains the following edges 1 > 1

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