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

YES 0.882 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/FiniteMap.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ CR

mainModule FiniteMap

((lookupWithDefaultFM :: FiniteMap Bool a -> a -> Bool -> a) :: FiniteMap Bool a -> a -> Bool -> a)

module FiniteMap where

import qualified Maybe
import qualified Prelude

data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b)

instance (Eq a, Eq b) => Eq (FiniteMap b a) where

lookupFM :: Ord b => FiniteMap b a -> b -> Maybe a

lookupFM

EmptyFM key

Nothing

lookupFM

(Branch key elt _ fm_l fm_r) key_to_find

key_to_find < key

lookupFM fm_l key_to_find

key_to_find > key

lookupFM fm_r key_to_find

otherwise

Just elt

lookupWithDefaultFM :: Ord b => FiniteMap b a -> a -> b -> a

lookupWithDefaultFM

fm deflt key

case	lookupFM fm key of
	Nothing	->	deflt
	Just elt	->	elt

module Maybe where

import qualified FiniteMap
import qualified Prelude

Case Reductions:
The following Case expression

case lookupFM fm key of
Nothing → deflt

Just elt → elt

is transformed to

lookupWithDefaultFM0 deflt Nothing = deflt

lookupWithDefaultFM0 deflt (Just elt) = elt

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

mainModule FiniteMap

((lookupWithDefaultFM :: FiniteMap Bool a -> a -> Bool -> a) :: FiniteMap Bool a -> a -> Bool -> a)

module FiniteMap where

import qualified Maybe
import qualified Prelude

data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b)

instance (Eq a, Eq b) => Eq (FiniteMap b a) where

lookupFM :: Ord a => FiniteMap a b -> a -> Maybe b

lookupFM

EmptyFM key

Nothing

lookupFM

(Branch key elt _ fm_l fm_r) key_to_find

key_to_find < key

lookupFM fm_l key_to_find

key_to_find > key

lookupFM fm_r key_to_find

otherwise

Just elt

lookupWithDefaultFM :: Ord a => FiniteMap a b -> b -> a -> b

lookupWithDefaultFM

fm deflt key

lookupWithDefaultFM0 deflt (lookupFM fm key)

lookupWithDefaultFM0	deflt Nothing	=	deflt
lookupWithDefaultFM0	deflt (Just elt)	=	elt

module Maybe where

import qualified FiniteMap
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule FiniteMap

((lookupWithDefaultFM :: FiniteMap Bool a -> a -> Bool -> a) :: FiniteMap Bool a -> a -> Bool -> a)

module FiniteMap where

import qualified Maybe
import qualified Prelude

data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b)

instance (Eq a, Eq b) => Eq (FiniteMap a b) where

lookupFM :: Ord b => FiniteMap b a -> b -> Maybe a

lookupFM

EmptyFM key

Nothing

lookupFM

(Branch key elt vw fm_l fm_r) key_to_find

key_to_find < key

lookupFM fm_l key_to_find

key_to_find > key

lookupFM fm_r key_to_find

otherwise

Just elt

lookupWithDefaultFM :: Ord b => FiniteMap b a -> a -> b -> a

lookupWithDefaultFM

fm deflt key

lookupWithDefaultFM0 deflt (lookupFM fm key)

lookupWithDefaultFM0	deflt Nothing	=	deflt
lookupWithDefaultFM0	deflt (Just elt)	=	elt

module Maybe where

import qualified FiniteMap
import qualified Prelude

Cond Reductions:
The following Function with conditions

lookupFM EmptyFM key = Nothing

lookupFM (Branch key elt vw fm_l fm_r) key_to_find

| key_to_find < key

= lookupFM fm_l key_to_find

| key_to_find > key

= lookupFM fm_r key_to_find

| otherwise

= Just elt

is transformed to

lookupFM EmptyFM key = lookupFM4 EmptyFM key

lookupFM (Branch key elt vw fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vw fm_l fm_r) key_to_find

lookupFM1 key elt vw fm_l fm_r key_to_find True = lookupFM fm_r key_to_find

lookupFM1 key elt vw fm_l fm_r key_to_find False = lookupFM0 key elt vw fm_l fm_r key_to_find otherwise

lookupFM2 key elt vw fm_l fm_r key_to_find True = lookupFM fm_l key_to_find

lookupFM2 key elt vw fm_l fm_r key_to_find False = lookupFM1 key elt vw fm_l fm_r key_to_find (key_to_find > key)

lookupFM0 key elt vw fm_l fm_r key_to_find True = Just elt

lookupFM3 (Branch key elt vw fm_l fm_r) key_to_find = lookupFM2 key elt vw fm_l fm_r key_to_find (key_to_find < key)

lookupFM4 EmptyFM key = Nothing

lookupFM4 wv ww = lookupFM3 wv ww

The following Function with conditions

compare x y

| x == y

= EQ

| x <= y

= LT

| otherwise

= GT

is transformed to

compare x y = compare3 x y

compare1 x y True = LT

compare1 x y False = compare0 x y otherwise

compare2 x y True = EQ

compare2 x y False = compare1 x y (x <= y)

compare0 x y True = GT

compare3 x y = compare2 x y (x == y)

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 FiniteMap

(lookupWithDefaultFM :: FiniteMap Bool a -> a -> Bool -> a)

module FiniteMap where

import qualified Maybe
import qualified Prelude

data FiniteMap b a = EmptyFM | Branch b a Int (FiniteMap b a) (FiniteMap b a)

instance (Eq a, Eq b) => Eq (FiniteMap b a) where

lookupFM :: Ord b => FiniteMap b a -> b -> Maybe a

lookupFM	EmptyFM key	=	lookupFM4 EmptyFM key
lookupFM	(Branch key elt vw fm_l fm_r) key_to_find	=	lookupFM3 (Branch key elt vw fm_l fm_r) key_to_find

lookupFM0

key elt vw fm_l fm_r key_to_find True

Just elt

lookupFM1	key elt vw fm_l fm_r key_to_find True	=	lookupFM fm_r key_to_find
lookupFM1	key elt vw fm_l fm_r key_to_find False	=	lookupFM0 key elt vw fm_l fm_r key_to_find otherwise

lookupFM2	key elt vw fm_l fm_r key_to_find True	=	lookupFM fm_l key_to_find
lookupFM2	key elt vw fm_l fm_r key_to_find False	=	lookupFM1 key elt vw fm_l fm_r key_to_find (key_to_find > key)

lookupFM3

(Branch key elt vw fm_l fm_r) key_to_find

lookupFM2 key elt vw fm_l fm_r key_to_find (key_to_find < key)

lookupFM4	EmptyFM key	=	Nothing
lookupFM4	wv ww	=	lookupFM3 wv ww

lookupWithDefaultFM :: Ord a => FiniteMap a b -> b -> a -> b

lookupWithDefaultFM

fm deflt key

lookupWithDefaultFM0 deflt (lookupFM fm key)

lookupWithDefaultFM0	deflt Nothing	=	deflt
lookupWithDefaultFM0	deflt (Just elt)	=	elt

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="lookupWithDefaultFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="lookupWithDefaultFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="lookupWithDefaultFM wx3 wx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="lookupWithDefaultFM wx3 wx4 wx5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="lookupWithDefaultFM0 wx4 (lookupFM wx3 wx5)",fontsize=16,color="burlywood",shape="triangle"];79[label="wx3/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 7[label="",style="solid", color="burlywood", weight=3];
80[label="wx3/Branch wx30 wx31 wx32 wx33 wx34",fontsize=10,color="white",style="solid",shape="box"];6 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="lookupWithDefaultFM0 wx4 (lookupFM EmptyFM wx5)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="lookupWithDefaultFM0 wx4 (lookupFM (Branch wx30 wx31 wx32 wx33 wx34) wx5)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="lookupWithDefaultFM0 wx4 (lookupFM4 EmptyFM wx5)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="lookupWithDefaultFM0 wx4 (lookupFM3 (Branch wx30 wx31 wx32 wx33 wx34) wx5)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="lookupWithDefaultFM0 wx4 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="lookupWithDefaultFM0 wx4 (lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (wx5 < wx30))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="wx4",fontsize=16,color="green",shape="box"];14[label="lookupWithDefaultFM0 wx4 (lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (compare wx5 wx30 == LT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="lookupWithDefaultFM0 wx4 (lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (compare3 wx5 wx30 == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="lookupWithDefaultFM0 wx4 (lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (compare2 wx5 wx30 (wx5 == wx30) == LT))",fontsize=16,color="burlywood",shape="box"];81[label="wx5/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 17[label="",style="solid", color="burlywood", weight=3];
82[label="wx5/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 18[label="",style="solid", color="burlywood", weight=3];
17[label="lookupWithDefaultFM0 wx4 (lookupFM2 wx30 wx31 wx32 wx33 wx34 False (compare2 False wx30 (False == wx30) == LT))",fontsize=16,color="burlywood",shape="box"];83[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 19[label="",style="solid", color="burlywood", weight=3];
84[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 20[label="",style="solid", color="burlywood", weight=3];
18[label="lookupWithDefaultFM0 wx4 (lookupFM2 wx30 wx31 wx32 wx33 wx34 True (compare2 True wx30 (True == wx30) == LT))",fontsize=16,color="burlywood",shape="box"];85[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 21[label="",style="solid", color="burlywood", weight=3];
86[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 22[label="",style="solid", color="burlywood", weight=3];
19[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 False (compare2 False False (False == False) == LT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 False (compare2 False True (False == True) == LT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 True (compare2 True False (True == False) == LT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 True (compare2 True True (True == True) == LT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 False (compare2 False False True == LT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 False (compare2 False True False == LT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 True (compare2 True False False == LT))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 True (compare2 True True True == LT))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 False (EQ == LT))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 False (compare1 False True (False <= True) == LT))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 True (compare1 True False (True <= False) == LT))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 True (EQ == LT))",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 False False)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 False (compare1 False True True == LT))",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 True (compare1 True False False == LT))",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 True False)",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 False (False > False))",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 False (LT == LT))",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 True (compare0 True False otherwise == LT))",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="lookupWithDefaultFM0 wx4 (lookupFM1 True wx31 wx32 wx33 wx34 True (True > True))",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 False (compare False False == GT))",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
40[label="lookupWithDefaultFM0 wx4 (lookupFM2 True wx31 wx32 wx33 wx34 False True)",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 True (compare0 True False True == LT))",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="lookupWithDefaultFM0 wx4 (lookupFM1 True wx31 wx32 wx33 wx34 True (compare True True == GT))",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
43[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 False (compare3 False False == GT))",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
44 -> 6[label="",style="dashed", color="red", weight=0];
44[label="lookupWithDefaultFM0 wx4 (lookupFM wx33 False)",fontsize=16,color="magenta"];44 -> 48[label="",style="dashed", color="magenta", weight=3];
44 -> 49[label="",style="dashed", color="magenta", weight=3];
45[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 True (GT == LT))",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
46[label="lookupWithDefaultFM0 wx4 (lookupFM1 True wx31 wx32 wx33 wx34 True (compare3 True True == GT))",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
47[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 False (compare2 False False (False == False) == GT))",fontsize=16,color="black",shape="box"];47 -> 52[label="",style="solid", color="black", weight=3];
48[label="False",fontsize=16,color="green",shape="box"];49[label="wx33",fontsize=16,color="green",shape="box"];50[label="lookupWithDefaultFM0 wx4 (lookupFM2 False wx31 wx32 wx33 wx34 True False)",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3];
51[label="lookupWithDefaultFM0 wx4 (lookupFM1 True wx31 wx32 wx33 wx34 True (compare2 True True (True == True) == GT))",fontsize=16,color="black",shape="box"];51 -> 54[label="",style="solid", color="black", weight=3];
52[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 False (compare2 False False True == GT))",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
53[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (True > False))",fontsize=16,color="black",shape="box"];53 -> 56[label="",style="solid", color="black", weight=3];
54[label="lookupWithDefaultFM0 wx4 (lookupFM1 True wx31 wx32 wx33 wx34 True (compare2 True True True == GT))",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 False (EQ == GT))",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
56[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (compare True False == GT))",fontsize=16,color="black",shape="box"];56 -> 59[label="",style="solid", color="black", weight=3];
57[label="lookupWithDefaultFM0 wx4 (lookupFM1 True wx31 wx32 wx33 wx34 True (EQ == GT))",fontsize=16,color="black",shape="box"];57 -> 60[label="",style="solid", color="black", weight=3];
58[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 False False)",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
59[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (compare3 True False == GT))",fontsize=16,color="black",shape="box"];59 -> 62[label="",style="solid", color="black", weight=3];
60[label="lookupWithDefaultFM0 wx4 (lookupFM1 True wx31 wx32 wx33 wx34 True False)",fontsize=16,color="black",shape="box"];60 -> 63[label="",style="solid", color="black", weight=3];
61[label="lookupWithDefaultFM0 wx4 (lookupFM0 False wx31 wx32 wx33 wx34 False otherwise)",fontsize=16,color="black",shape="box"];61 -> 64[label="",style="solid", color="black", weight=3];
62[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (compare2 True False (True == False) == GT))",fontsize=16,color="black",shape="box"];62 -> 65[label="",style="solid", color="black", weight=3];
63[label="lookupWithDefaultFM0 wx4 (lookupFM0 True wx31 wx32 wx33 wx34 True otherwise)",fontsize=16,color="black",shape="box"];63 -> 66[label="",style="solid", color="black", weight=3];
64[label="lookupWithDefaultFM0 wx4 (lookupFM0 False wx31 wx32 wx33 wx34 False True)",fontsize=16,color="black",shape="box"];64 -> 67[label="",style="solid", color="black", weight=3];
65[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (compare2 True False False == GT))",fontsize=16,color="black",shape="box"];65 -> 68[label="",style="solid", color="black", weight=3];
66[label="lookupWithDefaultFM0 wx4 (lookupFM0 True wx31 wx32 wx33 wx34 True True)",fontsize=16,color="black",shape="box"];66 -> 69[label="",style="solid", color="black", weight=3];
67[label="lookupWithDefaultFM0 wx4 (Just wx31)",fontsize=16,color="black",shape="triangle"];67 -> 70[label="",style="solid", color="black", weight=3];
68[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (compare1 True False (True <= False) == GT))",fontsize=16,color="black",shape="box"];68 -> 71[label="",style="solid", color="black", weight=3];
69 -> 67[label="",style="dashed", color="red", weight=0];
69[label="lookupWithDefaultFM0 wx4 (Just wx31)",fontsize=16,color="magenta"];70[label="wx31",fontsize=16,color="green",shape="box"];71[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (compare1 True False False == GT))",fontsize=16,color="black",shape="box"];71 -> 72[label="",style="solid", color="black", weight=3];
72[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (compare0 True False otherwise == GT))",fontsize=16,color="black",shape="box"];72 -> 73[label="",style="solid", color="black", weight=3];
73[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (compare0 True False True == GT))",fontsize=16,color="black",shape="box"];73 -> 74[label="",style="solid", color="black", weight=3];
74[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True (GT == GT))",fontsize=16,color="black",shape="box"];74 -> 75[label="",style="solid", color="black", weight=3];
75[label="lookupWithDefaultFM0 wx4 (lookupFM1 False wx31 wx32 wx33 wx34 True True)",fontsize=16,color="black",shape="box"];75 -> 76[label="",style="solid", color="black", weight=3];
76 -> 6[label="",style="dashed", color="red", weight=0];
76[label="lookupWithDefaultFM0 wx4 (lookupFM wx34 True)",fontsize=16,color="magenta"];76 -> 77[label="",style="dashed", color="magenta", weight=3];
76 -> 78[label="",style="dashed", color="magenta", weight=3];
77[label="True",fontsize=16,color="green",shape="box"];78[label="wx34",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_lookupWithDefaultFM0(wx4, Branch(False, wx31, wx32, wx33, wx34), True, h) → new_lookupWithDefaultFM0(wx4, wx34, True, h)
new_lookupWithDefaultFM0(wx4, Branch(True, wx31, wx32, wx33, wx34), False, h) → new_lookupWithDefaultFM0(wx4, wx33, False, h)

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 2 SCCs.

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                      ↳ QDP

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

new_lookupWithDefaultFM0(wx4, Branch(True, wx31, wx32, wx33, wx34), False, h) → new_lookupWithDefaultFM0(wx4, wx33, False, h)

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_lookupWithDefaultFM0(wx4, Branch(True, wx31, wx32, wx33, wx34), False, h) → new_lookupWithDefaultFM0(wx4, wx33, False, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

new_lookupWithDefaultFM0(wx4, Branch(False, wx31, wx32, wx33, wx34), True, h) → new_lookupWithDefaultFM0(wx4, wx34, True, h)

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

new_lookupWithDefaultFM0(wx4, Branch(False, wx31, wx32, wx33, wx34), True, h) → new_lookupWithDefaultFM0(wx4, wx34, True, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4

lookupWithDefaultFM0	deflt Nothing	= deflt
lookupWithDefaultFM0	deflt (Just elt)	= elt

lookupFM	EmptyFM key	= lookupFM4 EmptyFM key
lookupFM	(Branch key elt vw fm_l fm_r) key_to_find	= lookupFM3 (Branch key elt vw fm_l fm_r) key_to_find

lookupFM1	key elt vw fm_l fm_r key_to_find True	= lookupFM fm_r key_to_find
lookupFM1	key elt vw fm_l fm_r key_to_find False	= lookupFM0 key elt vw fm_l fm_r key_to_find otherwise

lookupFM2	key elt vw fm_l fm_r key_to_find True	= lookupFM fm_l key_to_find
lookupFM2	key elt vw fm_l fm_r key_to_find False	= lookupFM1 key elt vw fm_l fm_r key_to_find (key_to_find > key)

lookupFM4	EmptyFM key	= Nothing
lookupFM4	wv ww	= lookupFM3 wv ww

compare1	x y True	= LT
compare1	x y False	= compare0 x y otherwise

compare2	x y True	= EQ
compare2	x y False	= compare1 x y (x <= y)