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

YES 0.748 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

((minFM :: FiniteMap Int a -> Maybe Int) :: FiniteMap Int a -> Maybe Int)

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

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

minFM

EmptyFM

Nothing

minFM

(Branch key _ _ fm_l _)

case	minFM fm_l of
	Nothing	->	Just key
	Just key1	->	Just key1

module Maybe where

import qualified FiniteMap
import qualified Prelude

Case Reductions:
The following Case expression

case minFM fm_l of
Nothing → Just key

Just key1 → Just key1

is transformed to

minFM0 key Nothing = Just key

minFM0 key (Just key1) = Just key1

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

mainModule FiniteMap

((minFM :: FiniteMap Int a -> Maybe Int) :: FiniteMap Int a -> Maybe Int)

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

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

minFM	EmptyFM	=	Nothing
minFM	(Branch key _ _ fm_l _)	=	minFM0 key (minFM fm_l)

minFM0	key Nothing	=	Just key
minFM0	key (Just key1)	=	Just key1

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

((minFM :: FiniteMap Int a -> Maybe Int) :: FiniteMap Int a -> Maybe Int)

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

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

minFM	EmptyFM	=	Nothing
minFM	(Branch key vw vx fm_l vy)	=	minFM0 key (minFM fm_l)

minFM0	key Nothing	=	Just key
minFM0	key (Just key1)	=	Just key1

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

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule FiniteMap

(minFM :: FiniteMap Int a -> Maybe Int)

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

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

minFM	EmptyFM	=	Nothing
minFM	(Branch key vw vx fm_l vy)	=	minFM0 key (minFM fm_l)

minFM0	key Nothing	=	Just key
minFM0	key (Just key1)	=	Just key1

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="minFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="minFM wv3",fontsize=16,color="burlywood",shape="triangle"];15[label="wv3/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];3 -> 15[label="",style="solid", color="burlywood", weight=9];
15 -> 4[label="",style="solid", color="burlywood", weight=3];
16[label="wv3/Branch wv30 wv31 wv32 wv33 wv34",fontsize=10,color="white",style="solid",shape="box"];3 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="minFM EmptyFM",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="minFM (Branch wv30 wv31 wv32 wv33 wv34)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="Nothing",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="red", weight=0];
7[label="minFM0 wv30 (minFM wv33)",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
9 -> 3[label="",style="dashed", color="red", weight=0];
9[label="minFM wv33",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="minFM0 wv30 wv4",fontsize=16,color="burlywood",shape="triangle"];19[label="wv4/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 11[label="",style="solid", color="burlywood", weight=3];
20[label="wv4/Just wv40",fontsize=10,color="white",style="solid",shape="box"];8 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 12[label="",style="solid", color="burlywood", weight=3];
10[label="wv33",fontsize=16,color="green",shape="box"];11[label="minFM0 wv30 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="minFM0 wv30 (Just wv40)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="Just wv30",fontsize=16,color="green",shape="box"];14[label="Just wv40",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h) → new_minFM(wv33, 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_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h) → new_minFM(wv33, h)
The graph contains the following edges 1 > 1, 2 >= 2