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

YES 0.99 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

  ↳ BR

mainModule FiniteMap

((lookupFM :: FiniteMap Ordering a -> Ordering -> Maybe a) :: FiniteMap Ordering a -> Ordering -> Maybe 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

module Maybe where

import qualified FiniteMap
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule FiniteMap

((lookupFM :: FiniteMap Ordering a -> Ordering -> Maybe a) :: FiniteMap Ordering a -> Ordering -> Maybe 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 a => FiniteMap a b -> a -> Maybe b

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

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

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

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

compare0 x y True = GT

compare2 x y True = EQ

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

compare1 x y True = LT

compare1 x y False = compare0 x y otherwise

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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule FiniteMap

(lookupFM :: FiniteMap Ordering a -> Ordering -> Maybe 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	=	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

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="lookupFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="lookupFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="lookupFM wx3 wx4",fontsize=16,color="burlywood",shape="triangle"];152[label="wx3/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];4 -> 152[label="",style="solid", color="burlywood", weight=9];
152 -> 5[label="",style="solid", color="burlywood", weight=3];
153[label="wx3/Branch wx30 wx31 wx32 wx33 wx34",fontsize=10,color="white",style="solid",shape="box"];4 -> 153[label="",style="solid", color="burlywood", weight=9];
153 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="lookupFM EmptyFM wx4",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="lookupFM (Branch wx30 wx31 wx32 wx33 wx34) wx4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="lookupFM4 EmptyFM wx4",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="lookupFM3 (Branch wx30 wx31 wx32 wx33 wx34) wx4",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="Nothing",fontsize=16,color="green",shape="box"];10[label="lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (wx4 < wx30)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (compare wx4 wx30 == LT)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (compare3 wx4 wx30 == LT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (compare2 wx4 wx30 (wx4 == wx30) == LT)",fontsize=16,color="burlywood",shape="box"];154[label="wx4/LT",fontsize=10,color="white",style="solid",shape="box"];13 -> 154[label="",style="solid", color="burlywood", weight=9];
154 -> 14[label="",style="solid", color="burlywood", weight=3];
155[label="wx4/EQ",fontsize=10,color="white",style="solid",shape="box"];13 -> 155[label="",style="solid", color="burlywood", weight=9];
155 -> 15[label="",style="solid", color="burlywood", weight=3];
156[label="wx4/GT",fontsize=10,color="white",style="solid",shape="box"];13 -> 156[label="",style="solid", color="burlywood", weight=9];
156 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="lookupFM2 wx30 wx31 wx32 wx33 wx34 LT (compare2 LT wx30 (LT == wx30) == LT)",fontsize=16,color="burlywood",shape="box"];157[label="wx30/LT",fontsize=10,color="white",style="solid",shape="box"];14 -> 157[label="",style="solid", color="burlywood", weight=9];
157 -> 17[label="",style="solid", color="burlywood", weight=3];
158[label="wx30/EQ",fontsize=10,color="white",style="solid",shape="box"];14 -> 158[label="",style="solid", color="burlywood", weight=9];
158 -> 18[label="",style="solid", color="burlywood", weight=3];
159[label="wx30/GT",fontsize=10,color="white",style="solid",shape="box"];14 -> 159[label="",style="solid", color="burlywood", weight=9];
159 -> 19[label="",style="solid", color="burlywood", weight=3];
15[label="lookupFM2 wx30 wx31 wx32 wx33 wx34 EQ (compare2 EQ wx30 (EQ == wx30) == LT)",fontsize=16,color="burlywood",shape="box"];160[label="wx30/LT",fontsize=10,color="white",style="solid",shape="box"];15 -> 160[label="",style="solid", color="burlywood", weight=9];
160 -> 20[label="",style="solid", color="burlywood", weight=3];
161[label="wx30/EQ",fontsize=10,color="white",style="solid",shape="box"];15 -> 161[label="",style="solid", color="burlywood", weight=9];
161 -> 21[label="",style="solid", color="burlywood", weight=3];
162[label="wx30/GT",fontsize=10,color="white",style="solid",shape="box"];15 -> 162[label="",style="solid", color="burlywood", weight=9];
162 -> 22[label="",style="solid", color="burlywood", weight=3];
16[label="lookupFM2 wx30 wx31 wx32 wx33 wx34 GT (compare2 GT wx30 (GT == wx30) == LT)",fontsize=16,color="burlywood",shape="box"];163[label="wx30/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 163[label="",style="solid", color="burlywood", weight=9];
163 -> 23[label="",style="solid", color="burlywood", weight=3];
164[label="wx30/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 164[label="",style="solid", color="burlywood", weight=9];
164 -> 24[label="",style="solid", color="burlywood", weight=3];
165[label="wx30/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 165[label="",style="solid", color="burlywood", weight=9];
165 -> 25[label="",style="solid", color="burlywood", weight=3];
17[label="lookupFM2 LT wx31 wx32 wx33 wx34 LT (compare2 LT LT (LT == LT) == LT)",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
18[label="lookupFM2 EQ wx31 wx32 wx33 wx34 LT (compare2 LT EQ (LT == EQ) == LT)",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
19[label="lookupFM2 GT wx31 wx32 wx33 wx34 LT (compare2 LT GT (LT == GT) == LT)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
20[label="lookupFM2 LT wx31 wx32 wx33 wx34 EQ (compare2 EQ LT (EQ == LT) == LT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="lookupFM2 EQ wx31 wx32 wx33 wx34 EQ (compare2 EQ EQ (EQ == EQ) == LT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="lookupFM2 GT wx31 wx32 wx33 wx34 EQ (compare2 EQ GT (EQ == GT) == LT)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="lookupFM2 LT wx31 wx32 wx33 wx34 GT (compare2 GT LT (GT == LT) == LT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="lookupFM2 EQ wx31 wx32 wx33 wx34 GT (compare2 GT EQ (GT == EQ) == LT)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="lookupFM2 GT wx31 wx32 wx33 wx34 GT (compare2 GT GT (GT == GT) == LT)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="lookupFM2 LT wx31 wx32 wx33 wx34 LT (compare2 LT LT True == LT)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="lookupFM2 EQ wx31 wx32 wx33 wx34 LT (compare2 LT EQ False == LT)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="lookupFM2 GT wx31 wx32 wx33 wx34 LT (compare2 LT GT False == LT)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
29[label="lookupFM2 LT wx31 wx32 wx33 wx34 EQ (compare2 EQ LT False == LT)",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
30[label="lookupFM2 EQ wx31 wx32 wx33 wx34 EQ (compare2 EQ EQ True == LT)",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
31[label="lookupFM2 GT wx31 wx32 wx33 wx34 EQ (compare2 EQ GT False == LT)",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="lookupFM2 LT wx31 wx32 wx33 wx34 GT (compare2 GT LT False == LT)",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="lookupFM2 EQ wx31 wx32 wx33 wx34 GT (compare2 GT EQ False == LT)",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
34[label="lookupFM2 GT wx31 wx32 wx33 wx34 GT (compare2 GT GT True == LT)",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
35[label="lookupFM2 LT wx31 wx32 wx33 wx34 LT (EQ == LT)",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
36[label="lookupFM2 EQ wx31 wx32 wx33 wx34 LT (compare1 LT EQ (LT <= EQ) == LT)",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3];
37[label="lookupFM2 GT wx31 wx32 wx33 wx34 LT (compare1 LT GT (LT <= GT) == LT)",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3];
38[label="lookupFM2 LT wx31 wx32 wx33 wx34 EQ (compare1 EQ LT (EQ <= LT) == LT)",fontsize=16,color="black",shape="box"];38 -> 47[label="",style="solid", color="black", weight=3];
39[label="lookupFM2 EQ wx31 wx32 wx33 wx34 EQ (EQ == LT)",fontsize=16,color="black",shape="box"];39 -> 48[label="",style="solid", color="black", weight=3];
40[label="lookupFM2 GT wx31 wx32 wx33 wx34 EQ (compare1 EQ GT (EQ <= GT) == LT)",fontsize=16,color="black",shape="box"];40 -> 49[label="",style="solid", color="black", weight=3];
41[label="lookupFM2 LT wx31 wx32 wx33 wx34 GT (compare1 GT LT (GT <= LT) == LT)",fontsize=16,color="black",shape="box"];41 -> 50[label="",style="solid", color="black", weight=3];
42[label="lookupFM2 EQ wx31 wx32 wx33 wx34 GT (compare1 GT EQ (GT <= EQ) == LT)",fontsize=16,color="black",shape="box"];42 -> 51[label="",style="solid", color="black", weight=3];
43[label="lookupFM2 GT wx31 wx32 wx33 wx34 GT (EQ == LT)",fontsize=16,color="black",shape="box"];43 -> 52[label="",style="solid", color="black", weight=3];
44[label="lookupFM2 LT wx31 wx32 wx33 wx34 LT False",fontsize=16,color="black",shape="box"];44 -> 53[label="",style="solid", color="black", weight=3];
45[label="lookupFM2 EQ wx31 wx32 wx33 wx34 LT (compare1 LT EQ True == LT)",fontsize=16,color="black",shape="box"];45 -> 54[label="",style="solid", color="black", weight=3];
46[label="lookupFM2 GT wx31 wx32 wx33 wx34 LT (compare1 LT GT True == LT)",fontsize=16,color="black",shape="box"];46 -> 55[label="",style="solid", color="black", weight=3];
47[label="lookupFM2 LT wx31 wx32 wx33 wx34 EQ (compare1 EQ LT False == LT)",fontsize=16,color="black",shape="box"];47 -> 56[label="",style="solid", color="black", weight=3];
48[label="lookupFM2 EQ wx31 wx32 wx33 wx34 EQ False",fontsize=16,color="black",shape="box"];48 -> 57[label="",style="solid", color="black", weight=3];
49[label="lookupFM2 GT wx31 wx32 wx33 wx34 EQ (compare1 EQ GT True == LT)",fontsize=16,color="black",shape="box"];49 -> 58[label="",style="solid", color="black", weight=3];
50[label="lookupFM2 LT wx31 wx32 wx33 wx34 GT (compare1 GT LT False == LT)",fontsize=16,color="black",shape="box"];50 -> 59[label="",style="solid", color="black", weight=3];
51[label="lookupFM2 EQ wx31 wx32 wx33 wx34 GT (compare1 GT EQ False == LT)",fontsize=16,color="black",shape="box"];51 -> 60[label="",style="solid", color="black", weight=3];
52[label="lookupFM2 GT wx31 wx32 wx33 wx34 GT False",fontsize=16,color="black",shape="box"];52 -> 61[label="",style="solid", color="black", weight=3];
53[label="lookupFM1 LT wx31 wx32 wx33 wx34 LT (LT > LT)",fontsize=16,color="black",shape="box"];53 -> 62[label="",style="solid", color="black", weight=3];
54[label="lookupFM2 EQ wx31 wx32 wx33 wx34 LT (LT == LT)",fontsize=16,color="black",shape="box"];54 -> 63[label="",style="solid", color="black", weight=3];
55[label="lookupFM2 GT wx31 wx32 wx33 wx34 LT (LT == LT)",fontsize=16,color="black",shape="box"];55 -> 64[label="",style="solid", color="black", weight=3];
56[label="lookupFM2 LT wx31 wx32 wx33 wx34 EQ (compare0 EQ LT otherwise == LT)",fontsize=16,color="black",shape="box"];56 -> 65[label="",style="solid", color="black", weight=3];
57[label="lookupFM1 EQ wx31 wx32 wx33 wx34 EQ (EQ > EQ)",fontsize=16,color="black",shape="box"];57 -> 66[label="",style="solid", color="black", weight=3];
58[label="lookupFM2 GT wx31 wx32 wx33 wx34 EQ (LT == LT)",fontsize=16,color="black",shape="box"];58 -> 67[label="",style="solid", color="black", weight=3];
59[label="lookupFM2 LT wx31 wx32 wx33 wx34 GT (compare0 GT LT otherwise == LT)",fontsize=16,color="black",shape="box"];59 -> 68[label="",style="solid", color="black", weight=3];
60[label="lookupFM2 EQ wx31 wx32 wx33 wx34 GT (compare0 GT EQ otherwise == LT)",fontsize=16,color="black",shape="box"];60 -> 69[label="",style="solid", color="black", weight=3];
61[label="lookupFM1 GT wx31 wx32 wx33 wx34 GT (GT > GT)",fontsize=16,color="black",shape="box"];61 -> 70[label="",style="solid", color="black", weight=3];
62[label="lookupFM1 LT wx31 wx32 wx33 wx34 LT (compare LT LT == GT)",fontsize=16,color="black",shape="box"];62 -> 71[label="",style="solid", color="black", weight=3];
63[label="lookupFM2 EQ wx31 wx32 wx33 wx34 LT True",fontsize=16,color="black",shape="box"];63 -> 72[label="",style="solid", color="black", weight=3];
64[label="lookupFM2 GT wx31 wx32 wx33 wx34 LT True",fontsize=16,color="black",shape="box"];64 -> 73[label="",style="solid", color="black", weight=3];
65[label="lookupFM2 LT wx31 wx32 wx33 wx34 EQ (compare0 EQ LT True == LT)",fontsize=16,color="black",shape="box"];65 -> 74[label="",style="solid", color="black", weight=3];
66[label="lookupFM1 EQ wx31 wx32 wx33 wx34 EQ (compare EQ EQ == GT)",fontsize=16,color="black",shape="box"];66 -> 75[label="",style="solid", color="black", weight=3];
67[label="lookupFM2 GT wx31 wx32 wx33 wx34 EQ True",fontsize=16,color="black",shape="box"];67 -> 76[label="",style="solid", color="black", weight=3];
68[label="lookupFM2 LT wx31 wx32 wx33 wx34 GT (compare0 GT LT True == LT)",fontsize=16,color="black",shape="box"];68 -> 77[label="",style="solid", color="black", weight=3];
69[label="lookupFM2 EQ wx31 wx32 wx33 wx34 GT (compare0 GT EQ True == LT)",fontsize=16,color="black",shape="box"];69 -> 78[label="",style="solid", color="black", weight=3];
70[label="lookupFM1 GT wx31 wx32 wx33 wx34 GT (compare GT GT == GT)",fontsize=16,color="black",shape="box"];70 -> 79[label="",style="solid", color="black", weight=3];
71[label="lookupFM1 LT wx31 wx32 wx33 wx34 LT (compare3 LT LT == GT)",fontsize=16,color="black",shape="box"];71 -> 80[label="",style="solid", color="black", weight=3];
72 -> 4[label="",style="dashed", color="red", weight=0];
72[label="lookupFM wx33 LT",fontsize=16,color="magenta"];72 -> 81[label="",style="dashed", color="magenta", weight=3];
72 -> 82[label="",style="dashed", color="magenta", weight=3];
73 -> 4[label="",style="dashed", color="red", weight=0];
73[label="lookupFM wx33 LT",fontsize=16,color="magenta"];73 -> 83[label="",style="dashed", color="magenta", weight=3];
73 -> 84[label="",style="dashed", color="magenta", weight=3];
74[label="lookupFM2 LT wx31 wx32 wx33 wx34 EQ (GT == LT)",fontsize=16,color="black",shape="box"];74 -> 85[label="",style="solid", color="black", weight=3];
75[label="lookupFM1 EQ wx31 wx32 wx33 wx34 EQ (compare3 EQ EQ == GT)",fontsize=16,color="black",shape="box"];75 -> 86[label="",style="solid", color="black", weight=3];
76 -> 4[label="",style="dashed", color="red", weight=0];
76[label="lookupFM wx33 EQ",fontsize=16,color="magenta"];76 -> 87[label="",style="dashed", color="magenta", weight=3];
76 -> 88[label="",style="dashed", color="magenta", weight=3];
77[label="lookupFM2 LT wx31 wx32 wx33 wx34 GT (GT == LT)",fontsize=16,color="black",shape="box"];77 -> 89[label="",style="solid", color="black", weight=3];
78[label="lookupFM2 EQ wx31 wx32 wx33 wx34 GT (GT == LT)",fontsize=16,color="black",shape="box"];78 -> 90[label="",style="solid", color="black", weight=3];
79[label="lookupFM1 GT wx31 wx32 wx33 wx34 GT (compare3 GT GT == GT)",fontsize=16,color="black",shape="box"];79 -> 91[label="",style="solid", color="black", weight=3];
80[label="lookupFM1 LT wx31 wx32 wx33 wx34 LT (compare2 LT LT (LT == LT) == GT)",fontsize=16,color="black",shape="box"];80 -> 92[label="",style="solid", color="black", weight=3];
81[label="LT",fontsize=16,color="green",shape="box"];82[label="wx33",fontsize=16,color="green",shape="box"];83[label="LT",fontsize=16,color="green",shape="box"];84[label="wx33",fontsize=16,color="green",shape="box"];85[label="lookupFM2 LT wx31 wx32 wx33 wx34 EQ False",fontsize=16,color="black",shape="box"];85 -> 93[label="",style="solid", color="black", weight=3];
86[label="lookupFM1 EQ wx31 wx32 wx33 wx34 EQ (compare2 EQ EQ (EQ == EQ) == GT)",fontsize=16,color="black",shape="box"];86 -> 94[label="",style="solid", color="black", weight=3];
87[label="EQ",fontsize=16,color="green",shape="box"];88[label="wx33",fontsize=16,color="green",shape="box"];89[label="lookupFM2 LT wx31 wx32 wx33 wx34 GT False",fontsize=16,color="black",shape="box"];89 -> 95[label="",style="solid", color="black", weight=3];
90[label="lookupFM2 EQ wx31 wx32 wx33 wx34 GT False",fontsize=16,color="black",shape="box"];90 -> 96[label="",style="solid", color="black", weight=3];
91[label="lookupFM1 GT wx31 wx32 wx33 wx34 GT (compare2 GT GT (GT == GT) == GT)",fontsize=16,color="black",shape="box"];91 -> 97[label="",style="solid", color="black", weight=3];
92[label="lookupFM1 LT wx31 wx32 wx33 wx34 LT (compare2 LT LT True == GT)",fontsize=16,color="black",shape="box"];92 -> 98[label="",style="solid", color="black", weight=3];
93[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (EQ > LT)",fontsize=16,color="black",shape="box"];93 -> 99[label="",style="solid", color="black", weight=3];
94[label="lookupFM1 EQ wx31 wx32 wx33 wx34 EQ (compare2 EQ EQ True == GT)",fontsize=16,color="black",shape="box"];94 -> 100[label="",style="solid", color="black", weight=3];
95[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (GT > LT)",fontsize=16,color="black",shape="box"];95 -> 101[label="",style="solid", color="black", weight=3];
96[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (GT > EQ)",fontsize=16,color="black",shape="box"];96 -> 102[label="",style="solid", color="black", weight=3];
97[label="lookupFM1 GT wx31 wx32 wx33 wx34 GT (compare2 GT GT True == GT)",fontsize=16,color="black",shape="box"];97 -> 103[label="",style="solid", color="black", weight=3];
98[label="lookupFM1 LT wx31 wx32 wx33 wx34 LT (EQ == GT)",fontsize=16,color="black",shape="box"];98 -> 104[label="",style="solid", color="black", weight=3];
99[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (compare EQ LT == GT)",fontsize=16,color="black",shape="box"];99 -> 105[label="",style="solid", color="black", weight=3];
100[label="lookupFM1 EQ wx31 wx32 wx33 wx34 EQ (EQ == GT)",fontsize=16,color="black",shape="box"];100 -> 106[label="",style="solid", color="black", weight=3];
101[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (compare GT LT == GT)",fontsize=16,color="black",shape="box"];101 -> 107[label="",style="solid", color="black", weight=3];
102[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (compare GT EQ == GT)",fontsize=16,color="black",shape="box"];102 -> 108[label="",style="solid", color="black", weight=3];
103[label="lookupFM1 GT wx31 wx32 wx33 wx34 GT (EQ == GT)",fontsize=16,color="black",shape="box"];103 -> 109[label="",style="solid", color="black", weight=3];
104[label="lookupFM1 LT wx31 wx32 wx33 wx34 LT False",fontsize=16,color="black",shape="box"];104 -> 110[label="",style="solid", color="black", weight=3];
105[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (compare3 EQ LT == GT)",fontsize=16,color="black",shape="box"];105 -> 111[label="",style="solid", color="black", weight=3];
106[label="lookupFM1 EQ wx31 wx32 wx33 wx34 EQ False",fontsize=16,color="black",shape="box"];106 -> 112[label="",style="solid", color="black", weight=3];
107[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (compare3 GT LT == GT)",fontsize=16,color="black",shape="box"];107 -> 113[label="",style="solid", color="black", weight=3];
108[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (compare3 GT EQ == GT)",fontsize=16,color="black",shape="box"];108 -> 114[label="",style="solid", color="black", weight=3];
109[label="lookupFM1 GT wx31 wx32 wx33 wx34 GT False",fontsize=16,color="black",shape="box"];109 -> 115[label="",style="solid", color="black", weight=3];
110[label="lookupFM0 LT wx31 wx32 wx33 wx34 LT otherwise",fontsize=16,color="black",shape="box"];110 -> 116[label="",style="solid", color="black", weight=3];
111[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (compare2 EQ LT (EQ == LT) == GT)",fontsize=16,color="black",shape="box"];111 -> 117[label="",style="solid", color="black", weight=3];
112[label="lookupFM0 EQ wx31 wx32 wx33 wx34 EQ otherwise",fontsize=16,color="black",shape="box"];112 -> 118[label="",style="solid", color="black", weight=3];
113[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (compare2 GT LT (GT == LT) == GT)",fontsize=16,color="black",shape="box"];113 -> 119[label="",style="solid", color="black", weight=3];
114[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (compare2 GT EQ (GT == EQ) == GT)",fontsize=16,color="black",shape="box"];114 -> 120[label="",style="solid", color="black", weight=3];
115[label="lookupFM0 GT wx31 wx32 wx33 wx34 GT otherwise",fontsize=16,color="black",shape="box"];115 -> 121[label="",style="solid", color="black", weight=3];
116[label="lookupFM0 LT wx31 wx32 wx33 wx34 LT True",fontsize=16,color="black",shape="box"];116 -> 122[label="",style="solid", color="black", weight=3];
117[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (compare2 EQ LT False == GT)",fontsize=16,color="black",shape="box"];117 -> 123[label="",style="solid", color="black", weight=3];
118[label="lookupFM0 EQ wx31 wx32 wx33 wx34 EQ True",fontsize=16,color="black",shape="box"];118 -> 124[label="",style="solid", color="black", weight=3];
119[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (compare2 GT LT False == GT)",fontsize=16,color="black",shape="box"];119 -> 125[label="",style="solid", color="black", weight=3];
120[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (compare2 GT EQ False == GT)",fontsize=16,color="black",shape="box"];120 -> 126[label="",style="solid", color="black", weight=3];
121[label="lookupFM0 GT wx31 wx32 wx33 wx34 GT True",fontsize=16,color="black",shape="box"];121 -> 127[label="",style="solid", color="black", weight=3];
122[label="Just wx31",fontsize=16,color="green",shape="box"];123[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (compare1 EQ LT (EQ <= LT) == GT)",fontsize=16,color="black",shape="box"];123 -> 128[label="",style="solid", color="black", weight=3];
124[label="Just wx31",fontsize=16,color="green",shape="box"];125[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (compare1 GT LT (GT <= LT) == GT)",fontsize=16,color="black",shape="box"];125 -> 129[label="",style="solid", color="black", weight=3];
126[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (compare1 GT EQ (GT <= EQ) == GT)",fontsize=16,color="black",shape="box"];126 -> 130[label="",style="solid", color="black", weight=3];
127[label="Just wx31",fontsize=16,color="green",shape="box"];128[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (compare1 EQ LT False == GT)",fontsize=16,color="black",shape="box"];128 -> 131[label="",style="solid", color="black", weight=3];
129[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (compare1 GT LT False == GT)",fontsize=16,color="black",shape="box"];129 -> 132[label="",style="solid", color="black", weight=3];
130[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (compare1 GT EQ False == GT)",fontsize=16,color="black",shape="box"];130 -> 133[label="",style="solid", color="black", weight=3];
131[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (compare0 EQ LT otherwise == GT)",fontsize=16,color="black",shape="box"];131 -> 134[label="",style="solid", color="black", weight=3];
132[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (compare0 GT LT otherwise == GT)",fontsize=16,color="black",shape="box"];132 -> 135[label="",style="solid", color="black", weight=3];
133[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (compare0 GT EQ otherwise == GT)",fontsize=16,color="black",shape="box"];133 -> 136[label="",style="solid", color="black", weight=3];
134[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (compare0 EQ LT True == GT)",fontsize=16,color="black",shape="box"];134 -> 137[label="",style="solid", color="black", weight=3];
135[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (compare0 GT LT True == GT)",fontsize=16,color="black",shape="box"];135 -> 138[label="",style="solid", color="black", weight=3];
136[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (compare0 GT EQ True == GT)",fontsize=16,color="black",shape="box"];136 -> 139[label="",style="solid", color="black", weight=3];
137[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ (GT == GT)",fontsize=16,color="black",shape="box"];137 -> 140[label="",style="solid", color="black", weight=3];
138[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT (GT == GT)",fontsize=16,color="black",shape="box"];138 -> 141[label="",style="solid", color="black", weight=3];
139[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT (GT == GT)",fontsize=16,color="black",shape="box"];139 -> 142[label="",style="solid", color="black", weight=3];
140[label="lookupFM1 LT wx31 wx32 wx33 wx34 EQ True",fontsize=16,color="black",shape="box"];140 -> 143[label="",style="solid", color="black", weight=3];
141[label="lookupFM1 LT wx31 wx32 wx33 wx34 GT True",fontsize=16,color="black",shape="box"];141 -> 144[label="",style="solid", color="black", weight=3];
142[label="lookupFM1 EQ wx31 wx32 wx33 wx34 GT True",fontsize=16,color="black",shape="box"];142 -> 145[label="",style="solid", color="black", weight=3];
143 -> 4[label="",style="dashed", color="red", weight=0];
143[label="lookupFM wx34 EQ",fontsize=16,color="magenta"];143 -> 146[label="",style="dashed", color="magenta", weight=3];
143 -> 147[label="",style="dashed", color="magenta", weight=3];
144 -> 4[label="",style="dashed", color="red", weight=0];
144[label="lookupFM wx34 GT",fontsize=16,color="magenta"];144 -> 148[label="",style="dashed", color="magenta", weight=3];
144 -> 149[label="",style="dashed", color="magenta", weight=3];
145 -> 4[label="",style="dashed", color="red", weight=0];
145[label="lookupFM wx34 GT",fontsize=16,color="magenta"];145 -> 150[label="",style="dashed", color="magenta", weight=3];
145 -> 151[label="",style="dashed", color="magenta", weight=3];
146[label="EQ",fontsize=16,color="green",shape="box"];147[label="wx34",fontsize=16,color="green",shape="box"];148[label="GT",fontsize=16,color="green",shape="box"];149[label="wx34",fontsize=16,color="green",shape="box"];150[label="GT",fontsize=16,color="green",shape="box"];151[label="wx34",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

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

new_lookupFM(Branch(GT, wx31, wx32, wx33, wx34), LT, h) → new_lookupFM(wx33, LT, h)
new_lookupFM(Branch(LT, wx31, wx32, wx33, wx34), GT, h) → new_lookupFM(wx34, GT, h)
new_lookupFM(Branch(EQ, wx31, wx32, wx33, wx34), GT, h) → new_lookupFM(wx34, GT, h)
new_lookupFM(Branch(GT, wx31, wx32, wx33, wx34), EQ, h) → new_lookupFM(wx33, EQ, h)
new_lookupFM(Branch(LT, wx31, wx32, wx33, wx34), EQ, h) → new_lookupFM(wx34, EQ, h)
new_lookupFM(Branch(EQ, wx31, wx32, wx33, wx34), LT, h) → new_lookupFM(wx33, LT, 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 3 SCCs.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_lookupFM(Branch(GT, wx31, wx32, wx33, wx34), EQ, h) → new_lookupFM(wx33, EQ, h)
new_lookupFM(Branch(LT, wx31, wx32, wx33, wx34), EQ, h) → new_lookupFM(wx34, EQ, 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_lookupFM(Branch(LT, wx31, wx32, wx33, wx34), EQ, h) → new_lookupFM(wx34, EQ, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
new_lookupFM(Branch(GT, wx31, wx32, wx33, wx34), EQ, h) → new_lookupFM(wx33, EQ, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_lookupFM(Branch(LT, wx31, wx32, wx33, wx34), GT, h) → new_lookupFM(wx34, GT, h)
new_lookupFM(Branch(EQ, wx31, wx32, wx33, wx34), GT, h) → new_lookupFM(wx34, GT, h)

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

new_lookupFM(Branch(EQ, wx31, wx32, wx33, wx34), GT, h) → new_lookupFM(wx34, GT, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
new_lookupFM(Branch(LT, wx31, wx32, wx33, wx34), GT, h) → new_lookupFM(wx34, GT, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_lookupFM(Branch(GT, wx31, wx32, wx33, wx34), LT, h) → new_lookupFM(wx33, LT, h)
new_lookupFM(Branch(EQ, wx31, wx32, wx33, wx34), LT, h) → new_lookupFM(wx33, LT, h)

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

new_lookupFM(Branch(EQ, wx31, wx32, wx33, wx34), LT, h) → new_lookupFM(wx33, LT, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
new_lookupFM(Branch(GT, wx31, wx32, wx33, wx34), LT, h) → new_lookupFM(wx33, LT, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

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

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

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