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

YES 0.45 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

((isEmptyFM :: FiniteMap b a -> Bool) :: FiniteMap b 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 b a) where

isEmptyFM :: FiniteMap a b -> Bool

isEmptyFM

sizeFM fm == 0

sizeFM :: FiniteMap b a -> Int

sizeFM	EmptyFM	=	0
sizeFM	(Branch _ _ size _ _)	=	size

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

((isEmptyFM :: FiniteMap a b -> Bool) :: FiniteMap a b -> 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 b a) where

isEmptyFM :: FiniteMap b a -> Bool

isEmptyFM

sizeFM fm == 0

sizeFM :: FiniteMap b a -> Int

sizeFM	EmptyFM	=	0
sizeFM	(Branch vw vx size vy vz)	=	size

module Maybe where

import qualified FiniteMap
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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule FiniteMap

((isEmptyFM :: FiniteMap a b -> Bool) :: FiniteMap a b -> 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

isEmptyFM :: FiniteMap a b -> Bool

isEmptyFM

sizeFM fm == 0

sizeFM :: FiniteMap a b -> Int

sizeFM	EmptyFM	=	0
sizeFM	(Branch vw vx size vy vz)	=	size

module Maybe where

import qualified FiniteMap
import qualified Prelude

Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule FiniteMap

(isEmptyFM :: FiniteMap a b -> 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

isEmptyFM :: FiniteMap a b -> Bool

isEmptyFM

sizeFM fm == Pos Zero

sizeFM :: FiniteMap a b -> Int

sizeFM	EmptyFM	=	Pos Zero
sizeFM	(Branch vw vx size vy vz)	=	size

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="isEmptyFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="isEmptyFM wy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="sizeFM wy3 == Pos Zero",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="primEqInt (sizeFM wy3) (Pos Zero)",fontsize=16,color="burlywood",shape="box"];21[label="wy3/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 6[label="",style="solid", color="burlywood", weight=3];
22[label="wy3/Branch wy30 wy31 wy32 wy33 wy34",fontsize=10,color="white",style="solid",shape="box"];5 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="primEqInt (sizeFM EmptyFM) (Pos Zero)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="primEqInt (sizeFM (Branch wy30 wy31 wy32 wy33 wy34)) (Pos Zero)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="primEqInt wy32 (Pos Zero)",fontsize=16,color="burlywood",shape="box"];23[label="wy32/Pos wy320",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 11[label="",style="solid", color="burlywood", weight=3];
24[label="wy32/Neg wy320",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 12[label="",style="solid", color="burlywood", weight=3];
10[label="True",fontsize=16,color="green",shape="box"];11[label="primEqInt (Pos wy320) (Pos Zero)",fontsize=16,color="burlywood",shape="box"];25[label="wy320/Succ wy3200",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 13[label="",style="solid", color="burlywood", weight=3];
26[label="wy320/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="primEqInt (Neg wy320) (Pos Zero)",fontsize=16,color="burlywood",shape="box"];27[label="wy320/Succ wy3200",fontsize=10,color="white",style="solid",shape="box"];12 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 15[label="",style="solid", color="burlywood", weight=3];
28[label="wy320/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 16[label="",style="solid", color="burlywood", weight=3];
13[label="primEqInt (Pos (Succ wy3200)) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="primEqInt (Neg (Succ wy3200)) (Pos Zero)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="primEqInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="False",fontsize=16,color="green",shape="box"];18[label="True",fontsize=16,color="green",shape="box"];19[label="False",fontsize=16,color="green",shape="box"];20[label="True",fontsize=16,color="green",shape="box"];}