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

YES 0.782 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_GE :: 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_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 _ 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

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

keysFM_GE

fm fr

foldFM_GE (\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_GE0 key elt rest = key : rest

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule FiniteMap

((keysFM_GE :: 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_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

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

keysFM_GE

fm fr

foldFM_GE keysFM_GE0 [] fr fm

keysFM_GE0

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

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

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

keysFM_GE

fm fr

foldFM_GE keysFM_GE0 [] fr fm

keysFM_GE0

key elt rest

key : rest

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_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_GE0 k z fr key elt vw fm_l fm_r True = foldFM_GE k z fr 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 (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

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule FiniteMap

(keysFM_GE :: 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 b a) 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

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

keysFM_GE

fm fr

foldFM_GE keysFM_GE0 [] fr fm

keysFM_GE0

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_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="keysFM_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="keysFM_GE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="foldFM_GE keysFM_GE0 [] 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_GE keysFM_GE0 [] wz4 EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldFM_GE keysFM_GE0 [] 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_GE3 keysFM_GE0 [] wz4 EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldFM_GE2 keysFM_GE0 [] 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_GE1 keysFM_GE0 [] 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_GE1 keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (compare wz30 wz4 /= LT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="foldFM_GE1 keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == LT))",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_GE1 keysFM_GE0 [] wz4 () wz31 wz32 wz33 wz34 (not (compare () wz4 == LT))",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_GE1 keysFM_GE0 [] () () wz31 wz32 wz33 wz34 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="foldFM_GE1 keysFM_GE0 [] () () wz31 wz32 wz33 wz34 (not (EQ == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="foldFM_GE1 keysFM_GE0 [] () () wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="foldFM_GE1 keysFM_GE0 [] () () 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_GE keysFM_GE0 (keysFM_GE0 () wz31 (foldFM_GE keysFM_GE0 [] () wz34)) () wz33",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_GE keysFM_GE0 [] () wz34",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_GE keysFM_GE0 (keysFM_GE0 () wz31 wz5) () wz33",fontsize=16,color="burlywood",shape="triangle"];60[label="wz33/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="wz33/Branch wz330 wz331 wz332 wz333 wz334",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="wz34",fontsize=16,color="green",shape="box"];23[label="()",fontsize=16,color="green",shape="box"];24[label="foldFM_GE keysFM_GE0 (keysFM_GE0 () wz31 wz5) () EmptyFM",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="foldFM_GE keysFM_GE0 (keysFM_GE0 () wz31 wz5) () (Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="foldFM_GE3 keysFM_GE0 (keysFM_GE0 () wz31 wz5) () EmptyFM",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="foldFM_GE2 keysFM_GE0 (keysFM_GE0 () wz31 wz5) () (Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="keysFM_GE0 () 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_GE1 keysFM_GE0 (keysFM_GE0 () wz31 wz5) () wz330 wz331 wz332 wz333 wz334 (wz330 >= ())",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_GE0 () wz31 wz5",fontsize=16,color="magenta"];31[label="foldFM_GE1 keysFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (wz330 >= ())",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
33[label="foldFM_GE1 keysFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (compare wz330 () /= LT)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
34[label="foldFM_GE1 keysFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (not (compare wz330 () == LT))",fontsize=16,color="burlywood",shape="box"];64[label="wz330/()",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_GE1 keysFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
36[label="foldFM_GE1 keysFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not (EQ == LT))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="foldFM_GE1 keysFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not False)",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
38[label="foldFM_GE1 keysFM_GE0 wz6 () () wz331 wz332 wz333 wz334 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_GE keysFM_GE0 (keysFM_GE0 () wz331 (foldFM_GE keysFM_GE0 wz6 () wz334)) () wz333",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="wz331",fontsize=16,color="green",shape="box"];41[label="wz333",fontsize=16,color="green",shape="box"];42[label="foldFM_GE keysFM_GE0 wz6 () wz334",fontsize=16,color="burlywood",shape="box"];66[label="wz334/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];42 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 43[label="",style="solid", color="burlywood", weight=3];
67[label="wz334/Branch wz3340 wz3341 wz3342 wz3343 wz3344",fontsize=10,color="white",style="solid",shape="box"];42 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 44[label="",style="solid", color="burlywood", weight=3];
43[label="foldFM_GE keysFM_GE0 wz6 () EmptyFM",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="foldFM_GE keysFM_GE0 wz6 () (Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="foldFM_GE3 keysFM_GE0 wz6 () EmptyFM",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
46[label="foldFM_GE2 keysFM_GE0 wz6 () (Branch wz3340 wz3341 wz3342 wz3343 wz3344)",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_GE1 keysFM_GE0 wz6 () wz3340 wz3341 wz3342 wz3343 wz3344 (wz3340 >= ())",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="wz3341",fontsize=16,color="green",shape="box"];50[label="wz3344",fontsize=16,color="green",shape="box"];51[label="wz3343",fontsize=16,color="green",shape="box"];52[label="wz3340",fontsize=16,color="green",shape="box"];53[label="wz3342",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_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) → new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE1(new_keysFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)

The TRS R consists of the following rules:

new_keysFM_GE0(wz31, wz5, h) → :(@0, wz5)
new_foldFM_GE0(wz6, EmptyFM, h) → wz6
new_foldFM_GE2(wz31, wz5, EmptyFM, h) → new_keysFM_GE0(wz31, wz5, h)
new_foldFM_GE2(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE10(new_keysFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)
new_foldFM_GE10(wz6, @0, wz331, wz332, wz333, wz334, h) → new_foldFM_GE2(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
new_foldFM_GE0(wz6, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE10(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)

The set Q consists of the following terms:

new_foldFM_GE10(x0, @0, x1, x2, x3, x4, x5)
new_foldFM_GE0(x0, Branch(x1, x2, x3, x4, x5), x6)
new_foldFM_GE2(x0, x1, EmptyFM, x2)
new_keysFM_GE0(x0, x1, x2)
new_foldFM_GE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
new_foldFM_GE0(x0, EmptyFM, x1)

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(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) → new_foldFM_GE1(new_keysFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) → new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
The graph contains the following edges 1 >= 1, 6 > 2, 6 > 3, 6 > 4, 6 > 5, 6 > 6, 7 >= 7
new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) → new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
The graph contains the following edges 3 >= 1, 5 >= 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_GE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) → new_foldFM_GE3(@0, wz34, 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_GE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) → new_foldFM_GE3(@0, wz34, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3

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