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

YES 0.53 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 () a -> () -> Maybe a) :: FiniteMap () a -> () -> 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 a b) 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

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 () a -> () -> Maybe a) :: FiniteMap () a -> () -> 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 a b) 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

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

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

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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule FiniteMap

(lookupFM :: FiniteMap () a -> () -> 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	=	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"];23[label="wx3/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 5[label="",style="solid", color="burlywood", weight=3];
24[label="wx3/Branch wx30 wx31 wx32 wx33 wx34",fontsize=10,color="white",style="solid",shape="box"];4 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 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="burlywood",shape="box"];25[label="wx4/()",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 12[label="",style="solid", color="burlywood", weight=3];
12[label="lookupFM2 wx30 wx31 wx32 wx33 wx34 () (compare () wx30 == LT)",fontsize=16,color="burlywood",shape="box"];26[label="wx30/()",fontsize=10,color="white",style="solid",shape="box"];12 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 13[label="",style="solid", color="burlywood", weight=3];
13[label="lookupFM2 () wx31 wx32 wx33 wx34 () (compare () () == LT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="lookupFM2 () wx31 wx32 wx33 wx34 () (EQ == LT)",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="lookupFM2 () wx31 wx32 wx33 wx34 () False",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="lookupFM1 () wx31 wx32 wx33 wx34 () (() > ())",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="lookupFM1 () wx31 wx32 wx33 wx34 () (compare () () == GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="lookupFM1 () wx31 wx32 wx33 wx34 () (EQ == GT)",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="lookupFM1 () wx31 wx32 wx33 wx34 () False",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="lookupFM0 () wx31 wx32 wx33 wx34 () otherwise",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="lookupFM0 () wx31 wx32 wx33 wx34 () True",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="Just wx31",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

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)

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

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