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

YES 0.768 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

  ↳ LR

mainModule FiniteMap

((keysFM_LE :: FiniteMap () a -> () -> [()]) :: FiniteMap () a -> () -> [()])

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 a b) where

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

foldFM_LE

k z fr EmptyFM

foldFM_LE

k z fr (Branch key elt _ fm_l fm_r)

key <= fr

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

otherwise

foldFM_LE k z fr fm_l

keysFM_LE :: Ord a => FiniteMap a b -> a -> [a]

keysFM_LE

fm fr

foldFM_LE (\key elt rest ->key : rest) [] fr fm

module Maybe where

import qualified FiniteMap
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\keyeltrest→key : rest

is transformed to

keysFM_LE0 key elt rest = key : rest

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule FiniteMap

((keysFM_LE :: FiniteMap () a -> () -> [()]) :: FiniteMap () a -> () -> [()])

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

foldFM_LE

k z fr EmptyFM

foldFM_LE

k z fr (Branch key elt _ fm_l fm_r)

key <= fr

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

otherwise

foldFM_LE k z fr fm_l

keysFM_LE :: Ord a => FiniteMap a b -> a -> [a]

keysFM_LE

fm fr

foldFM_LE keysFM_LE0 [] fr fm

keysFM_LE0

key elt rest

key : rest

module Maybe where

import qualified FiniteMap
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule FiniteMap

((keysFM_LE :: FiniteMap () a -> () -> [()]) :: FiniteMap () a -> () -> [()])

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

foldFM_LE

k z fr EmptyFM

foldFM_LE

k z fr (Branch key elt vw fm_l fm_r)

key <= fr

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

otherwise

foldFM_LE k z fr fm_l

keysFM_LE :: Ord b => FiniteMap b a -> b -> [b]

keysFM_LE

fm fr

foldFM_LE keysFM_LE0 [] fr fm

keysFM_LE0

key elt rest

key : rest

module Maybe where

import qualified FiniteMap
import qualified Prelude

Cond Reductions:
The following Function with conditions

foldFM_LE k z fr EmptyFM = z

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

| key <= fr

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

| otherwise

= foldFM_LE k z fr fm_l

is transformed to

foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM

foldFM_LE k z fr (Branch key elt vw fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vw fm_l fm_r)

foldFM_LE1 k z fr key elt vw fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r

foldFM_LE1 k z fr key elt vw fm_l fm_r False = foldFM_LE0 k z fr key elt vw fm_l fm_r otherwise

foldFM_LE0 k z fr key elt vw fm_l fm_r True = foldFM_LE k z fr fm_l

foldFM_LE2 k z fr (Branch key elt vw fm_l fm_r) = foldFM_LE1 k z fr key elt vw fm_l fm_r (key <= fr)

foldFM_LE3 k z fr EmptyFM = z

foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy

The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule FiniteMap

(keysFM_LE :: FiniteMap () a -> () -> [()])

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

foldFM_LE	k z fr EmptyFM	=	foldFM_LE3 k z fr EmptyFM
foldFM_LE	k z fr (Branch key elt vw fm_l fm_r)	=	foldFM_LE2 k z fr (Branch key elt vw fm_l fm_r)

foldFM_LE0

k z fr key elt vw fm_l fm_r True

foldFM_LE k z fr fm_l

foldFM_LE1	k z fr key elt vw fm_l fm_r True	=	foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
foldFM_LE1	k z fr key elt vw fm_l fm_r False	=	foldFM_LE0 k z fr key elt vw fm_l fm_r otherwise

foldFM_LE2

k z fr (Branch key elt vw fm_l fm_r)

foldFM_LE1 k z fr key elt vw fm_l fm_r (key <= fr)

foldFM_LE3	k z fr EmptyFM	=	z
foldFM_LE3	wv ww wx wy	=	foldFM_LE2 wv ww wx wy

keysFM_LE :: Ord a => FiniteMap a b -> a -> [a]

keysFM_LE

fm fr

foldFM_LE keysFM_LE0 [] fr fm

keysFM_LE0

key elt rest

key : rest

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="keysFM_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="keysFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="keysFM_LE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="foldFM_LE keysFM_LE0 [] wz4 wz3",fontsize=16,color="burlywood",shape="triangle"];54[label="wz3/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 6[label="",style="solid", color="burlywood", weight=3];
55[label="wz3/Branch wz30 wz31 wz32 wz33 wz34",fontsize=10,color="white",style="solid",shape="box"];5 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldFM_LE keysFM_LE0 [] wz4 EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldFM_LE keysFM_LE0 [] wz4 (Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="foldFM_LE3 keysFM_LE0 [] wz4 EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldFM_LE2 keysFM_LE0 [] wz4 (Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="[]",fontsize=16,color="green",shape="box"];11[label="foldFM_LE1 keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (wz30 <= wz4)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="foldFM_LE1 keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (compare wz30 wz4 /= GT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="foldFM_LE1 keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == GT))",fontsize=16,color="burlywood",shape="box"];56[label="wz30/()",fontsize=10,color="white",style="solid",shape="box"];13 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 14[label="",style="solid", color="burlywood", weight=3];
14[label="foldFM_LE1 keysFM_LE0 [] wz4 () wz31 wz32 wz33 wz34 (not (compare () wz4 == GT))",fontsize=16,color="burlywood",shape="box"];57[label="wz4/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 15[label="",style="solid", color="burlywood", weight=3];
15[label="foldFM_LE1 keysFM_LE0 [] () () wz31 wz32 wz33 wz34 (not (compare () () == GT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="foldFM_LE1 keysFM_LE0 [] () () wz31 wz32 wz33 wz34 (not (EQ == GT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="foldFM_LE1 keysFM_LE0 [] () () wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="foldFM_LE1 keysFM_LE0 [] () () wz31 wz32 wz33 wz34 True",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19 -> 20[label="",style="dashed", color="red", weight=0];
19[label="foldFM_LE keysFM_LE0 (keysFM_LE0 () wz31 (foldFM_LE keysFM_LE0 [] () wz33)) () wz34",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
21 -> 5[label="",style="dashed", color="red", weight=0];
21[label="foldFM_LE keysFM_LE0 [] () wz33",fontsize=16,color="magenta"];21 -> 22[label="",style="dashed", color="magenta", weight=3];
21 -> 23[label="",style="dashed", color="magenta", weight=3];
20[label="foldFM_LE keysFM_LE0 (keysFM_LE0 () wz31 wz5) () wz34",fontsize=16,color="burlywood",shape="triangle"];60[label="wz34/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];20 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 24[label="",style="solid", color="burlywood", weight=3];
61[label="wz34/Branch wz340 wz341 wz342 wz343 wz344",fontsize=10,color="white",style="solid",shape="box"];20 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 25[label="",style="solid", color="burlywood", weight=3];
22[label="()",fontsize=16,color="green",shape="box"];23[label="wz33",fontsize=16,color="green",shape="box"];24[label="foldFM_LE keysFM_LE0 (keysFM_LE0 () wz31 wz5) () EmptyFM",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="foldFM_LE keysFM_LE0 (keysFM_LE0 () wz31 wz5) () (Branch wz340 wz341 wz342 wz343 wz344)",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="foldFM_LE3 keysFM_LE0 (keysFM_LE0 () wz31 wz5) () EmptyFM",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="foldFM_LE2 keysFM_LE0 (keysFM_LE0 () wz31 wz5) () (Branch wz340 wz341 wz342 wz343 wz344)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="keysFM_LE0 () wz31 wz5",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
29 -> 31[label="",style="dashed", color="red", weight=0];
29[label="foldFM_LE1 keysFM_LE0 (keysFM_LE0 () wz31 wz5) () wz340 wz341 wz342 wz343 wz344 (wz340 <= ())",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3];
30[label="() : wz5",fontsize=16,color="green",shape="box"];32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="keysFM_LE0 () wz31 wz5",fontsize=16,color="magenta"];31[label="foldFM_LE1 keysFM_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (wz340 <= ())",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
33[label="foldFM_LE1 keysFM_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (compare wz340 () /= GT)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
34[label="foldFM_LE1 keysFM_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (not (compare wz340 () == GT))",fontsize=16,color="burlywood",shape="box"];64[label="wz340/()",fontsize=10,color="white",style="solid",shape="box"];34 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 35[label="",style="solid", color="burlywood", weight=3];
35[label="foldFM_LE1 keysFM_LE0 wz6 () () wz341 wz342 wz343 wz344 (not (compare () () == GT))",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
36[label="foldFM_LE1 keysFM_LE0 wz6 () () wz341 wz342 wz343 wz344 (not (EQ == GT))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="foldFM_LE1 keysFM_LE0 wz6 () () wz341 wz342 wz343 wz344 (not False)",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
38[label="foldFM_LE1 keysFM_LE0 wz6 () () wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
39 -> 20[label="",style="dashed", color="red", weight=0];
39[label="foldFM_LE keysFM_LE0 (keysFM_LE0 () wz341 (foldFM_LE keysFM_LE0 wz6 () wz343)) () wz344",fontsize=16,color="magenta"];39 -> 40[label="",style="dashed", color="magenta", weight=3];
39 -> 41[label="",style="dashed", color="magenta", weight=3];
39 -> 42[label="",style="dashed", color="magenta", weight=3];
40[label="foldFM_LE keysFM_LE0 wz6 () wz343",fontsize=16,color="burlywood",shape="box"];66[label="wz343/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];40 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 43[label="",style="solid", color="burlywood", weight=3];
67[label="wz343/Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];40 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 44[label="",style="solid", color="burlywood", weight=3];
41[label="wz341",fontsize=16,color="green",shape="box"];42[label="wz344",fontsize=16,color="green",shape="box"];43[label="foldFM_LE keysFM_LE0 wz6 () EmptyFM",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="foldFM_LE keysFM_LE0 wz6 () (Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="foldFM_LE3 keysFM_LE0 wz6 () EmptyFM",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
46[label="foldFM_LE2 keysFM_LE0 wz6 () (Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="wz6",fontsize=16,color="green",shape="box"];48 -> 31[label="",style="dashed", color="red", weight=0];
48[label="foldFM_LE1 keysFM_LE0 wz6 () wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= ())",fontsize=16,color="magenta"];48 -> 49[label="",style="dashed", color="magenta", weight=3];
48 -> 50[label="",style="dashed", color="magenta", weight=3];
48 -> 51[label="",style="dashed", color="magenta", weight=3];
48 -> 52[label="",style="dashed", color="magenta", weight=3];
48 -> 53[label="",style="dashed", color="magenta", weight=3];
49[label="wz3433",fontsize=16,color="green",shape="box"];50[label="wz3434",fontsize=16,color="green",shape="box"];51[label="wz3432",fontsize=16,color="green",shape="box"];52[label="wz3431",fontsize=16,color="green",shape="box"];53[label="wz3430",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_foldFM_LE1(wz6, @0, wz341, wz342, wz343, wz344, h) → new_foldFM_LE(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
new_foldFM_LE1(wz6, @0, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) → new_foldFM_LE1(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE1(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

new_keysFM_LE0(wz31, wz5, h) → :(@0, wz5)
new_foldFM_LE10(wz6, @0, wz341, wz342, wz343, wz344, h) → new_foldFM_LE2(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
new_foldFM_LE0(wz6, EmptyFM, h) → wz6
new_foldFM_LE0(wz6, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) → new_foldFM_LE10(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
new_foldFM_LE2(wz31, wz5, EmptyFM, h) → new_keysFM_LE0(wz31, wz5, h)
new_foldFM_LE2(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE10(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)

The set Q consists of the following terms:

new_foldFM_LE2(x0, x1, EmptyFM, x2)
new_foldFM_LE0(x0, EmptyFM, x1)
new_keysFM_LE0(x0, x1, x2)
new_foldFM_LE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
new_foldFM_LE10(x0, @0, x1, x2, x3, x4, x5)

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_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) → new_foldFM_LE1(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
new_foldFM_LE1(wz6, @0, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) → new_foldFM_LE1(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
The graph contains the following edges 1 >= 1, 5 > 2, 5 > 3, 5 > 4, 5 > 5, 5 > 6, 7 >= 7
new_foldFM_LE1(wz6, @0, wz341, wz342, wz343, wz344, h) → new_foldFM_LE(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_foldFM_LE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) → new_foldFM_LE3(@0, wz33, 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_foldFM_LE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) → new_foldFM_LE3(@0, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3

foldFM_LE	k z fr EmptyFM	= foldFM_LE3 k z fr EmptyFM
foldFM_LE	k z fr (Branch key elt vw fm_l fm_r)	= foldFM_LE2 k z fr (Branch key elt vw fm_l fm_r)

foldFM_LE1	k z fr key elt vw fm_l fm_r True	= foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
foldFM_LE1	k z fr key elt vw fm_l fm_r False	= foldFM_LE0 k z fr key elt vw fm_l fm_r otherwise