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

YES 0.517 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

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

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 a b) where

elemFM :: Ord a => a -> FiniteMap a b -> Bool

elemFM

key fm

case	lookupFM fm key of
	Nothing	->	False
	Just elt	->	True

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

Case Reductions:
The following Case expression

case lookupFM fm key of
Nothing → False

Just elt → True

is transformed to

elemFM0 Nothing = False

elemFM0 (Just elt) = True

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

mainModule FiniteMap

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

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 a b) where

elemFM :: Ord b => b -> FiniteMap b a -> Bool

elemFM

key fm

elemFM0 (lookupFM fm key)

elemFM0	Nothing	=	False
elemFM0	(Just elt)	=	True

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

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

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

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

elemFM :: Ord a => a -> FiniteMap a b -> Bool

elemFM

key fm

elemFM0 (lookupFM fm key)

elemFM0	Nothing	=	False
elemFM0	(Just elt)	=	True

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

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

lookupFM0 key elt vw fm_l fm_r key_to_find True = Just elt

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

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule FiniteMap

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

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

elemFM :: Ord b => b -> FiniteMap b a -> Bool

elemFM

key fm

elemFM0 (lookupFM fm key)

elemFM0	Nothing	=	False
elemFM0	(Just elt)	=	True

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

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="elemFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="elemFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="elemFM wx3 wx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="elemFM0 (lookupFM wx4 wx3)",fontsize=16,color="burlywood",shape="box"];26[label="wx4/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 6[label="",style="solid", color="burlywood", weight=3];
27[label="wx4/Branch wx40 wx41 wx42 wx43 wx44",fontsize=10,color="white",style="solid",shape="box"];5 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="elemFM0 (lookupFM EmptyFM wx3)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="elemFM0 (lookupFM (Branch wx40 wx41 wx42 wx43 wx44) wx3)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="elemFM0 (lookupFM4 EmptyFM wx3)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="elemFM0 (lookupFM3 (Branch wx40 wx41 wx42 wx43 wx44) wx3)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="elemFM0 Nothing",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="elemFM0 (lookupFM2 wx40 wx41 wx42 wx43 wx44 wx3 (wx3 < wx40))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="False",fontsize=16,color="green",shape="box"];13[label="elemFM0 (lookupFM2 wx40 wx41 wx42 wx43 wx44 wx3 (compare wx3 wx40 == LT))",fontsize=16,color="burlywood",shape="box"];28[label="wx3/()",fontsize=10,color="white",style="solid",shape="box"];13 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 14[label="",style="solid", color="burlywood", weight=3];
14[label="elemFM0 (lookupFM2 wx40 wx41 wx42 wx43 wx44 () (compare () wx40 == LT))",fontsize=16,color="burlywood",shape="box"];29[label="wx40/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 15[label="",style="solid", color="burlywood", weight=3];
15[label="elemFM0 (lookupFM2 () wx41 wx42 wx43 wx44 () (compare () () == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="elemFM0 (lookupFM2 () wx41 wx42 wx43 wx44 () (EQ == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="elemFM0 (lookupFM2 () wx41 wx42 wx43 wx44 () False)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="elemFM0 (lookupFM1 () wx41 wx42 wx43 wx44 () (() > ()))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="elemFM0 (lookupFM1 () wx41 wx42 wx43 wx44 () (compare () () == GT))",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="elemFM0 (lookupFM1 () wx41 wx42 wx43 wx44 () (EQ == GT))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="elemFM0 (lookupFM1 () wx41 wx42 wx43 wx44 () False)",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="elemFM0 (lookupFM0 () wx41 wx42 wx43 wx44 () otherwise)",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="elemFM0 (lookupFM0 () wx41 wx42 wx43 wx44 () True)",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
24[label="elemFM0 (Just wx41)",fontsize=16,color="black",shape="box"];24 -> 25[label="",style="solid", color="black", weight=3];
25[label="True",fontsize=16,color="green",shape="box"];}

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