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

YES 0.723 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_GE :: (() -> a -> b -> b) -> b -> () -> FiniteMap () a -> b) :: (() -> a -> b -> b) -> b -> () -> FiniteMap () a -> 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)

instance (Eq a, Eq b) => Eq (FiniteMap b a) where

foldFM_GE :: Ord a => (a -> b -> c -> c) -> c -> a -> FiniteMap a b -> c

foldFM_GE

k z fr EmptyFM

foldFM_GE

k z fr (Branch key elt _ fm_l fm_r)

key >= fr

foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l

otherwise

foldFM_GE k z fr 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

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

instance (Eq a, Eq b) => Eq (FiniteMap b a) where

foldFM_GE :: Ord a => (a -> c -> b -> b) -> b -> a -> FiniteMap a c -> b

foldFM_GE

k z fr EmptyFM

foldFM_GE

k z fr (Branch key elt vw fm_l fm_r)

key >= fr

foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l

otherwise

foldFM_GE k z fr fm_r

module Maybe where

import qualified FiniteMap
import qualified Prelude

Cond Reductions:
The following Function with conditions

foldFM_GE k z fr EmptyFM = z

foldFM_GE k z fr (Branch key elt vw fm_l fm_r)

| key >= fr

= foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l

| otherwise

= foldFM_GE k z fr fm_r

is transformed to

foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM

foldFM_GE k z fr (Branch key elt vw fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vw fm_l fm_r)

foldFM_GE0 k z fr key elt vw fm_l fm_r True = foldFM_GE k z fr fm_r

foldFM_GE1 k z fr key elt vw fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l

foldFM_GE1 k z fr key elt vw fm_l fm_r False = foldFM_GE0 k z fr key elt vw fm_l fm_r otherwise

foldFM_GE2 k z fr (Branch key elt vw fm_l fm_r) = foldFM_GE1 k z fr key elt vw fm_l fm_r (key >= fr)

foldFM_GE3 k z fr EmptyFM = z

foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy

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

instance (Eq a, Eq b) => Eq (FiniteMap a b) where

foldFM_GE :: Ord a => (a -> c -> b -> b) -> b -> a -> FiniteMap a c -> b

foldFM_GE	k z fr EmptyFM	=	foldFM_GE3 k z fr EmptyFM
foldFM_GE	k z fr (Branch key elt vw fm_l fm_r)	=	foldFM_GE2 k z fr (Branch key elt vw fm_l fm_r)

foldFM_GE0

k z fr key elt vw fm_l fm_r True

foldFM_GE k z fr fm_r

foldFM_GE1	k z fr key elt vw fm_l fm_r True	=	foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
foldFM_GE1	k z fr key elt vw fm_l fm_r False	=	foldFM_GE0 k z fr key elt vw fm_l fm_r otherwise

foldFM_GE2

k z fr (Branch key elt vw fm_l fm_r)

foldFM_GE1 k z fr key elt vw fm_l fm_r (key >= fr)

foldFM_GE3	k z fr EmptyFM	=	z
foldFM_GE3	wv ww wx wy	=	foldFM_GE2 wv ww wx wy

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="foldFM_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="foldFM_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="foldFM_GE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="foldFM_GE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="foldFM_GE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];29[label="wz6/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 7[label="",style="solid", color="burlywood", weight=3];
30[label="wz6/Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="foldFM_GE wz3 wz4 wz5 EmptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="foldFM_GE wz3 wz4 wz5 (Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldFM_GE3 wz3 wz4 wz5 EmptyFM",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldFM_GE2 wz3 wz4 wz5 (Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="wz4",fontsize=16,color="green",shape="box"];12[label="foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (wz60 >= wz5)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (compare wz60 wz5 /= LT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare wz60 wz5 == LT))",fontsize=16,color="burlywood",shape="box"];31[label="wz60/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 15[label="",style="solid", color="burlywood", weight=3];
15[label="foldFM_GE1 wz3 wz4 wz5 () wz61 wz62 wz63 wz64 (not (compare () wz5 == LT))",fontsize=16,color="burlywood",shape="box"];32[label="wz5/()",fontsize=10,color="white",style="solid",shape="box"];15 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 16[label="",style="solid", color="burlywood", weight=3];
16[label="foldFM_GE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="foldFM_GE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not (EQ == LT))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="foldFM_GE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="foldFM_GE1 wz3 wz4 () () wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20 -> 6[label="",style="dashed", color="red", weight=0];
20[label="foldFM_GE wz3 (wz3 () wz61 (foldFM_GE wz3 wz4 () wz64)) () wz63",fontsize=16,color="magenta"];20 -> 21[label="",style="dashed", color="magenta", weight=3];
20 -> 22[label="",style="dashed", color="magenta", weight=3];
20 -> 23[label="",style="dashed", color="magenta", weight=3];
21[label="()",fontsize=16,color="green",shape="box"];22[label="wz3 () wz61 (foldFM_GE wz3 wz4 () wz64)",fontsize=16,color="green",shape="box"];22 -> 24[label="",style="dashed", color="green", weight=3];
22 -> 25[label="",style="dashed", color="green", weight=3];
22 -> 26[label="",style="dashed", color="green", weight=3];
23[label="wz63",fontsize=16,color="green",shape="box"];24[label="()",fontsize=16,color="green",shape="box"];25[label="wz61",fontsize=16,color="green",shape="box"];26 -> 6[label="",style="dashed", color="red", weight=0];
26[label="foldFM_GE wz3 wz4 () wz64",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
26 -> 28[label="",style="dashed", color="magenta", weight=3];
27[label="()",fontsize=16,color="green",shape="box"];28[label="wz64",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_GE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) → new_foldFM_GE(wz3, @0, wz64, h, ba)
new_foldFM_GE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) → new_foldFM_GE(wz3, @0, wz63, h, ba)

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_GE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) → new_foldFM_GE(wz3, @0, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
new_foldFM_GE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) → new_foldFM_GE(wz3, @0, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5

foldFM_GE	k z fr EmptyFM	= foldFM_GE3 k z fr EmptyFM
foldFM_GE	k z fr (Branch key elt vw fm_l fm_r)	= foldFM_GE2 k z fr (Branch key elt vw fm_l fm_r)

foldFM_GE1	k z fr key elt vw fm_l fm_r True	= foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
foldFM_GE1	k z fr key elt vw fm_l fm_r False	= foldFM_GE0 k z fr key elt vw fm_l fm_r otherwise