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

YES 1.147 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_LE :: (Char -> a -> b -> b) -> b -> Char -> FiniteMap Char a -> b) :: (Char -> a -> b -> b) -> b -> Char -> FiniteMap Char 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_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

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_LE :: (Char -> a -> b -> b) -> b -> Char -> FiniteMap Char a -> b) :: (Char -> a -> b -> b) -> b -> Char -> FiniteMap Char 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_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

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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule FiniteMap

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

module Maybe where

import qualified FiniteMap
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="foldFM_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="foldFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="foldFM_LE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="foldFM_LE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="foldFM_LE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];345[label="wz6/EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 345[label="",style="solid", color="burlywood", weight=9];
345 -> 7[label="",style="solid", color="burlywood", weight=3];
346[label="wz6/Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 346[label="",style="solid", color="burlywood", weight=9];
346 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="foldFM_LE wz3 wz4 wz5 EmptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="foldFM_LE 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_LE3 wz3 wz4 wz5 EmptyFM",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldFM_LE2 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_LE1 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_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (compare wz60 wz5 /= GT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare wz60 wz5 == GT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (primCmpChar wz60 wz5 == GT))",fontsize=16,color="burlywood",shape="box"];347[label="wz60/Char wz600",fontsize=10,color="white",style="solid",shape="box"];15 -> 347[label="",style="solid", color="burlywood", weight=9];
347 -> 16[label="",style="solid", color="burlywood", weight=3];
16[label="foldFM_LE1 wz3 wz4 wz5 (Char wz600) wz61 wz62 wz63 wz64 (not (primCmpChar (Char wz600) wz5 == GT))",fontsize=16,color="burlywood",shape="box"];348[label="wz5/Char wz50",fontsize=10,color="white",style="solid",shape="box"];16 -> 348[label="",style="solid", color="burlywood", weight=9];
348 -> 17[label="",style="solid", color="burlywood", weight=3];
17[label="foldFM_LE1 wz3 wz4 (Char wz50) (Char wz600) wz61 wz62 wz63 wz64 (not (primCmpChar (Char wz600) (Char wz50) == GT))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="foldFM_LE1 wz3 wz4 (Char wz50) (Char wz600) wz61 wz62 wz63 wz64 (not (primCmpNat wz600 wz50 == GT))",fontsize=16,color="burlywood",shape="box"];349[label="wz600/Succ wz6000",fontsize=10,color="white",style="solid",shape="box"];18 -> 349[label="",style="solid", color="burlywood", weight=9];
349 -> 19[label="",style="solid", color="burlywood", weight=3];
350[label="wz600/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 350[label="",style="solid", color="burlywood", weight=9];
350 -> 20[label="",style="solid", color="burlywood", weight=3];
19[label="foldFM_LE1 wz3 wz4 (Char wz50) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (primCmpNat (Succ wz6000) wz50 == GT))",fontsize=16,color="burlywood",shape="box"];351[label="wz50/Succ wz500",fontsize=10,color="white",style="solid",shape="box"];19 -> 351[label="",style="solid", color="burlywood", weight=9];
351 -> 21[label="",style="solid", color="burlywood", weight=3];
352[label="wz50/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 352[label="",style="solid", color="burlywood", weight=9];
352 -> 22[label="",style="solid", color="burlywood", weight=3];
20[label="foldFM_LE1 wz3 wz4 (Char wz50) (Char Zero) wz61 wz62 wz63 wz64 (not (primCmpNat Zero wz50 == GT))",fontsize=16,color="burlywood",shape="box"];353[label="wz50/Succ wz500",fontsize=10,color="white",style="solid",shape="box"];20 -> 353[label="",style="solid", color="burlywood", weight=9];
353 -> 23[label="",style="solid", color="burlywood", weight=3];
354[label="wz50/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 354[label="",style="solid", color="burlywood", weight=9];
354 -> 24[label="",style="solid", color="burlywood", weight=3];
21[label="foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (primCmpNat (Succ wz6000) (Succ wz500) == GT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="foldFM_LE1 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (primCmpNat (Succ wz6000) Zero == GT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 (not (primCmpNat Zero (Succ wz500) == GT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="foldFM_LE1 wz3 wz4 (Char Zero) (Char Zero) wz61 wz62 wz63 wz64 (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25 -> 238[label="",style="dashed", color="red", weight=0];
25[label="foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (primCmpNat wz6000 wz500 == GT))",fontsize=16,color="magenta"];25 -> 239[label="",style="dashed", color="magenta", weight=3];
25 -> 240[label="",style="dashed", color="magenta", weight=3];
25 -> 241[label="",style="dashed", color="magenta", weight=3];
25 -> 242[label="",style="dashed", color="magenta", weight=3];
25 -> 243[label="",style="dashed", color="magenta", weight=3];
25 -> 244[label="",style="dashed", color="magenta", weight=3];
25 -> 245[label="",style="dashed", color="magenta", weight=3];
25 -> 246[label="",style="dashed", color="magenta", weight=3];
25 -> 247[label="",style="dashed", color="magenta", weight=3];
25 -> 248[label="",style="dashed", color="magenta", weight=3];
26[label="foldFM_LE1 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (GT == GT))",fontsize=16,color="black",shape="box"];26 -> 31[label="",style="solid", color="black", weight=3];
27[label="foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 (not (LT == GT))",fontsize=16,color="black",shape="box"];27 -> 32[label="",style="solid", color="black", weight=3];
28[label="foldFM_LE1 wz3 wz4 (Char Zero) (Char Zero) wz61 wz62 wz63 wz64 (not (EQ == GT))",fontsize=16,color="black",shape="box"];28 -> 33[label="",style="solid", color="black", weight=3];
239[label="wz6000",fontsize=16,color="green",shape="box"];240[label="wz62",fontsize=16,color="green",shape="box"];241[label="wz500",fontsize=16,color="green",shape="box"];242[label="wz61",fontsize=16,color="green",shape="box"];243[label="wz6000",fontsize=16,color="green",shape="box"];244[label="wz4",fontsize=16,color="green",shape="box"];245[label="wz63",fontsize=16,color="green",shape="box"];246[label="wz500",fontsize=16,color="green",shape="box"];247[label="wz3",fontsize=16,color="green",shape="box"];248[label="wz64",fontsize=16,color="green",shape="box"];238[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat wz42 wz43 == GT))",fontsize=16,color="burlywood",shape="triangle"];356[label="wz42/Succ wz420",fontsize=10,color="white",style="solid",shape="box"];238 -> 356[label="",style="solid", color="burlywood", weight=9];
356 -> 309[label="",style="solid", color="burlywood", weight=3];
357[label="wz42/Zero",fontsize=10,color="white",style="solid",shape="box"];238 -> 357[label="",style="solid", color="burlywood", weight=9];
357 -> 310[label="",style="solid", color="burlywood", weight=3];
31[label="foldFM_LE1 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not True)",fontsize=16,color="black",shape="box"];31 -> 38[label="",style="solid", color="black", weight=3];
32[label="foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];32 -> 39[label="",style="solid", color="black", weight=3];
33[label="foldFM_LE1 wz3 wz4 (Char Zero) (Char Zero) wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3];
309[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat (Succ wz420) wz43 == GT))",fontsize=16,color="burlywood",shape="box"];358[label="wz43/Succ wz430",fontsize=10,color="white",style="solid",shape="box"];309 -> 358[label="",style="solid", color="burlywood", weight=9];
358 -> 311[label="",style="solid", color="burlywood", weight=3];
359[label="wz43/Zero",fontsize=10,color="white",style="solid",shape="box"];309 -> 359[label="",style="solid", color="burlywood", weight=9];
359 -> 312[label="",style="solid", color="burlywood", weight=3];
310[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat Zero wz43 == GT))",fontsize=16,color="burlywood",shape="box"];360[label="wz43/Succ wz430",fontsize=10,color="white",style="solid",shape="box"];310 -> 360[label="",style="solid", color="burlywood", weight=9];
360 -> 313[label="",style="solid", color="burlywood", weight=3];
361[label="wz43/Zero",fontsize=10,color="white",style="solid",shape="box"];310 -> 361[label="",style="solid", color="burlywood", weight=9];
361 -> 314[label="",style="solid", color="burlywood", weight=3];
38[label="foldFM_LE1 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39[label="foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
40[label="foldFM_LE1 wz3 wz4 (Char Zero) (Char Zero) wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
311[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat (Succ wz420) (Succ wz430) == GT))",fontsize=16,color="black",shape="box"];311 -> 315[label="",style="solid", color="black", weight=3];
312[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat (Succ wz420) Zero == GT))",fontsize=16,color="black",shape="box"];312 -> 316[label="",style="solid", color="black", weight=3];
313[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat Zero (Succ wz430) == GT))",fontsize=16,color="black",shape="box"];313 -> 317[label="",style="solid", color="black", weight=3];
314[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];314 -> 318[label="",style="solid", color="black", weight=3];
45[label="foldFM_LE0 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3];
46 -> 6[label="",style="dashed", color="red", weight=0];
46[label="foldFM_LE wz3 (wz3 (Char Zero) wz61 (foldFM_LE wz3 wz4 (Char (Succ wz500)) wz63)) (Char (Succ wz500)) wz64",fontsize=16,color="magenta"];46 -> 54[label="",style="dashed", color="magenta", weight=3];
46 -> 55[label="",style="dashed", color="magenta", weight=3];
46 -> 56[label="",style="dashed", color="magenta", weight=3];
47 -> 6[label="",style="dashed", color="red", weight=0];
47[label="foldFM_LE wz3 (wz3 (Char Zero) wz61 (foldFM_LE wz3 wz4 (Char Zero) wz63)) (Char Zero) wz64",fontsize=16,color="magenta"];47 -> 57[label="",style="dashed", color="magenta", weight=3];
47 -> 58[label="",style="dashed", color="magenta", weight=3];
47 -> 59[label="",style="dashed", color="magenta", weight=3];
315 -> 238[label="",style="dashed", color="red", weight=0];
315[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat wz420 wz430 == GT))",fontsize=16,color="magenta"];315 -> 319[label="",style="dashed", color="magenta", weight=3];
315 -> 320[label="",style="dashed", color="magenta", weight=3];
316[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (GT == GT))",fontsize=16,color="black",shape="box"];316 -> 321[label="",style="solid", color="black", weight=3];
317[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (LT == GT))",fontsize=16,color="black",shape="box"];317 -> 322[label="",style="solid", color="black", weight=3];
318[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (EQ == GT))",fontsize=16,color="black",shape="box"];318 -> 323[label="",style="solid", color="black", weight=3];
53[label="foldFM_LE0 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];53 -> 67[label="",style="solid", color="black", weight=3];
54[label="wz64",fontsize=16,color="green",shape="box"];55[label="wz3 (Char Zero) wz61 (foldFM_LE wz3 wz4 (Char (Succ wz500)) wz63)",fontsize=16,color="green",shape="box"];55 -> 68[label="",style="dashed", color="green", weight=3];
55 -> 69[label="",style="dashed", color="green", weight=3];
55 -> 70[label="",style="dashed", color="green", weight=3];
56[label="Char (Succ wz500)",fontsize=16,color="green",shape="box"];57[label="wz64",fontsize=16,color="green",shape="box"];58[label="wz3 (Char Zero) wz61 (foldFM_LE wz3 wz4 (Char Zero) wz63)",fontsize=16,color="green",shape="box"];58 -> 71[label="",style="dashed", color="green", weight=3];
58 -> 72[label="",style="dashed", color="green", weight=3];
58 -> 73[label="",style="dashed", color="green", weight=3];
59[label="Char Zero",fontsize=16,color="green",shape="box"];319[label="wz420",fontsize=16,color="green",shape="box"];320[label="wz430",fontsize=16,color="green",shape="box"];321[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not True)",fontsize=16,color="black",shape="box"];321 -> 324[label="",style="solid", color="black", weight=3];
322[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not False)",fontsize=16,color="black",shape="triangle"];322 -> 325[label="",style="solid", color="black", weight=3];
323 -> 322[label="",style="dashed", color="red", weight=0];
323[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not False)",fontsize=16,color="magenta"];67 -> 6[label="",style="dashed", color="red", weight=0];
67[label="foldFM_LE wz3 wz4 (Char Zero) wz63",fontsize=16,color="magenta"];67 -> 81[label="",style="dashed", color="magenta", weight=3];
67 -> 82[label="",style="dashed", color="magenta", weight=3];
68[label="Char Zero",fontsize=16,color="green",shape="box"];69[label="wz61",fontsize=16,color="green",shape="box"];70 -> 6[label="",style="dashed", color="red", weight=0];
70[label="foldFM_LE wz3 wz4 (Char (Succ wz500)) wz63",fontsize=16,color="magenta"];70 -> 83[label="",style="dashed", color="magenta", weight=3];
70 -> 84[label="",style="dashed", color="magenta", weight=3];
71[label="Char Zero",fontsize=16,color="green",shape="box"];72[label="wz61",fontsize=16,color="green",shape="box"];73 -> 6[label="",style="dashed", color="red", weight=0];
73[label="foldFM_LE wz3 wz4 (Char Zero) wz63",fontsize=16,color="magenta"];73 -> 85[label="",style="dashed", color="magenta", weight=3];
73 -> 86[label="",style="dashed", color="magenta", weight=3];
324[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 False",fontsize=16,color="black",shape="box"];324 -> 326[label="",style="solid", color="black", weight=3];
325[label="foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 True",fontsize=16,color="black",shape="box"];325 -> 327[label="",style="solid", color="black", weight=3];
81[label="wz63",fontsize=16,color="green",shape="box"];82[label="Char Zero",fontsize=16,color="green",shape="box"];83[label="wz63",fontsize=16,color="green",shape="box"];84[label="Char (Succ wz500)",fontsize=16,color="green",shape="box"];85[label="wz63",fontsize=16,color="green",shape="box"];86[label="Char Zero",fontsize=16,color="green",shape="box"];326[label="foldFM_LE0 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 otherwise",fontsize=16,color="black",shape="box"];326 -> 328[label="",style="solid", color="black", weight=3];
327 -> 6[label="",style="dashed", color="red", weight=0];
327[label="foldFM_LE wz34 (wz34 (Char (Succ wz37)) wz38 (foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40)) (Char (Succ wz36)) wz41",fontsize=16,color="magenta"];327 -> 329[label="",style="dashed", color="magenta", weight=3];
327 -> 330[label="",style="dashed", color="magenta", weight=3];
327 -> 331[label="",style="dashed", color="magenta", weight=3];
327 -> 332[label="",style="dashed", color="magenta", weight=3];
328[label="foldFM_LE0 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 True",fontsize=16,color="black",shape="box"];328 -> 333[label="",style="solid", color="black", weight=3];
329[label="wz41",fontsize=16,color="green",shape="box"];330[label="wz34 (Char (Succ wz37)) wz38 (foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40)",fontsize=16,color="green",shape="box"];330 -> 334[label="",style="dashed", color="green", weight=3];
330 -> 335[label="",style="dashed", color="green", weight=3];
330 -> 336[label="",style="dashed", color="green", weight=3];
331[label="wz34",fontsize=16,color="green",shape="box"];332[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];333 -> 6[label="",style="dashed", color="red", weight=0];
333[label="foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40",fontsize=16,color="magenta"];333 -> 337[label="",style="dashed", color="magenta", weight=3];
333 -> 338[label="",style="dashed", color="magenta", weight=3];
333 -> 339[label="",style="dashed", color="magenta", weight=3];
333 -> 340[label="",style="dashed", color="magenta", weight=3];
334[label="Char (Succ wz37)",fontsize=16,color="green",shape="box"];335[label="wz38",fontsize=16,color="green",shape="box"];336 -> 6[label="",style="dashed", color="red", weight=0];
336[label="foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40",fontsize=16,color="magenta"];336 -> 341[label="",style="dashed", color="magenta", weight=3];
336 -> 342[label="",style="dashed", color="magenta", weight=3];
336 -> 343[label="",style="dashed", color="magenta", weight=3];
336 -> 344[label="",style="dashed", color="magenta", weight=3];
337[label="wz40",fontsize=16,color="green",shape="box"];338[label="wz35",fontsize=16,color="green",shape="box"];339[label="wz34",fontsize=16,color="green",shape="box"];340[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];341[label="wz40",fontsize=16,color="green",shape="box"];342[label="wz35",fontsize=16,color="green",shape="box"];343[label="wz34",fontsize=16,color="green",shape="box"];344[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

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

new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) → new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba)
new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
new_foldFM_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) → new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba)
new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc)
new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
new_foldFM_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)

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_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) → new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba)
new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) → new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba)
new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc)

From the DPs we obtained the following set of size-change graphs:

new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 3 > 8, 2 > 9, 4 >= 10, 5 >= 11
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) → new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 > 8, 9 > 9, 10 >= 10, 11 >= 11
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) → new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 8, 11 >= 9
new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) → new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
The graph contains the following edges 1 >= 1, 7 >= 3, 10 >= 4, 11 >= 5
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4, 11 >= 5
new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4, 11 >= 5
new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
The graph contains the following edges 1 >= 1, 6 >= 3, 8 >= 4, 9 >= 5
new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) → new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
The graph contains the following edges 1 >= 1, 7 >= 3, 8 >= 4, 9 >= 5

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