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

YES 0.657 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

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

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

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

foldFM	k z EmptyFM	=	z
foldFM	k z (Branch key elt _ fm_l fm_r)	=	foldFM k (k key elt (foldFM k z fm_r)) fm_l

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

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

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

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

foldFM	k z EmptyFM	=	z
foldFM	k z (Branch key elt vw fm_l fm_r)	=	foldFM k (k key elt (foldFM k z fm_r)) fm_l

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

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

module FiniteMap where

import qualified Maybe
import qualified Prelude

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

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

foldFM	k z EmptyFM	=	z
foldFM	k z (Branch key elt vw fm_l fm_r)	=	foldFM k (k key elt (foldFM k z fm_r)) fm_l

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="foldFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="foldFM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="foldFM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="foldFM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];16[label="vz5/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 6[label="",style="solid", color="burlywood", weight=3];
17[label="vz5/Branch vz50 vz51 vz52 vz53 vz54",fontsize=10,color="white",style="solid",shape="box"];5 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldFM vz3 vz4 EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldFM vz3 vz4 (Branch vz50 vz51 vz52 vz53 vz54)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="vz4",fontsize=16,color="green",shape="box"];9 -> 5[label="",style="dashed", color="red", weight=0];
9[label="foldFM vz3 (vz3 vz50 vz51 (foldFM vz3 vz4 vz54)) vz53",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
9 -> 11[label="",style="dashed", color="magenta", weight=3];
10[label="vz3 vz50 vz51 (foldFM vz3 vz4 vz54)",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
10 -> 13[label="",style="dashed", color="green", weight=3];
10 -> 14[label="",style="dashed", color="green", weight=3];
11[label="vz53",fontsize=16,color="green",shape="box"];12[label="vz50",fontsize=16,color="green",shape="box"];13[label="vz51",fontsize=16,color="green",shape="box"];14 -> 5[label="",style="dashed", color="red", weight=0];
14[label="foldFM vz3 vz4 vz54",fontsize=16,color="magenta"];14 -> 15[label="",style="dashed", color="magenta", weight=3];
15[label="vz54",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_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) → new_foldFM(vz3, vz54, h, ba, bb)
new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) → new_foldFM(vz3, vz53, 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_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) → new_foldFM(vz3, vz54, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5
new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) → new_foldFM(vz3, vz53, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5