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

YES 0.674 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

((mapFM :: (a -> c -> b) -> FiniteMap a c -> FiniteMap a b) :: (a -> c -> b) -> FiniteMap a c -> FiniteMap a b)

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

emptyFM :: FiniteMap b a

emptyFM

EmptyFM

mapFM :: (b -> c -> a) -> FiniteMap b c -> FiniteMap b a

mapFM	f EmptyFM	=	emptyFM
mapFM	f (Branch key elt size fm_l fm_r)	=	Branch key (f key elt) size (mapFM f fm_l) (mapFM f fm_r)

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

((mapFM :: (a -> c -> b) -> FiniteMap a c -> FiniteMap a b) :: (a -> c -> b) -> FiniteMap a c -> FiniteMap a b)

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

emptyFM :: FiniteMap a b

emptyFM

EmptyFM

mapFM :: (b -> a -> c) -> FiniteMap b a -> FiniteMap b c

mapFM	f EmptyFM	=	emptyFM
mapFM	f (Branch key elt size fm_l fm_r)	=	Branch key (f key elt) size (mapFM f fm_l) (mapFM f fm_r)

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

          ↳ Narrow

mainModule FiniteMap

(mapFM :: (a -> b -> c) -> FiniteMap a b -> FiniteMap a c)

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

emptyFM :: FiniteMap a b

emptyFM

EmptyFM

mapFM :: (b -> a -> c) -> FiniteMap b a -> FiniteMap b c

mapFM	f EmptyFM	=	emptyFM
mapFM	f (Branch key elt size fm_l fm_r)	=	Branch key (f key elt) size (mapFM f fm_l) (mapFM f fm_r)

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="mapFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="mapFM vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="mapFM vy3 vy4",fontsize=16,color="burlywood",shape="triangle"];17[label="vy4/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 5[label="",style="solid", color="burlywood", weight=3];
18[label="vy4/Branch vy40 vy41 vy42 vy43 vy44",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="mapFM vy3 EmptyFM",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="mapFM vy3 (Branch vy40 vy41 vy42 vy43 vy44)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="emptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="Branch vy40 (vy3 vy40 vy41) vy42 (mapFM vy3 vy43) (mapFM vy3 vy44)",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
8 -> 11[label="",style="dashed", color="green", weight=3];
8 -> 12[label="",style="dashed", color="green", weight=3];
9[label="EmptyFM",fontsize=16,color="green",shape="box"];10[label="vy3 vy40 vy41",fontsize=16,color="green",shape="box"];10 -> 13[label="",style="dashed", color="green", weight=3];
10 -> 14[label="",style="dashed", color="green", weight=3];
11 -> 4[label="",style="dashed", color="red", weight=0];
11[label="mapFM vy3 vy43",fontsize=16,color="magenta"];11 -> 15[label="",style="dashed", color="magenta", weight=3];
12 -> 4[label="",style="dashed", color="red", weight=0];
12[label="mapFM vy3 vy44",fontsize=16,color="magenta"];12 -> 16[label="",style="dashed", color="magenta", weight=3];
13[label="vy40",fontsize=16,color="green",shape="box"];14[label="vy41",fontsize=16,color="green",shape="box"];15[label="vy43",fontsize=16,color="green",shape="box"];16[label="vy44",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

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

new_mapFM(vy3, Branch(vy40, vy41, vy42, vy43, vy44), h, ba, bb) → new_mapFM(vy3, vy43, h, ba, bb)
new_mapFM(vy3, Branch(vy40, vy41, vy42, vy43, vy44), h, ba, bb) → new_mapFM(vy3, vy44, h, ba, bb)

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_mapFM(vy3, Branch(vy40, vy41, vy42, vy43, vy44), h, ba, bb) → new_mapFM(vy3, vy44, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5
new_mapFM(vy3, Branch(vy40, vy41, vy42, vy43, vy44), h, ba, bb) → new_mapFM(vy3, vy43, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5