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

YES 1.464 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

((readInt :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])]) :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])])

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\nd→n * radix + d

is transformed to

readInt0 radix n d = n * radix + d

The following Lambda expression

\vu77→

case vu77 of
(ds,r) → (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : []

_ → []

is transformed to

readInt1 radix digToInt vu77 =

case vu77 of
(ds,r) → (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : []

_ → []

The following Lambda expression

\vu68→

case vu68 of
(cs@(_ : _),t) → (cs,t) : []

_ → []

is transformed to

nonnull0 vu68 =

case vu68 of
(cs@(_ : _),t) → (cs,t) : []

_ → []

The following Lambda expression

\(_,zs)→zs

is transformed to

zs0 (_,zs) = zs

The following Lambda expression

\(ys,_)→ys

is transformed to

ys0 (ys,_) = ys

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

mainModule Main

((readInt :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])]) :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])])

module Main where

import qualified Prelude

Case Reductions:
The following Case expression

case vu77 of
(ds,r) → (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : []

_ → []

is transformed to

readInt10 radix digToInt (ds,r) = (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : []

readInt10 radix digToInt _ = []

The following Case expression

case vu68 of
(cs@(_ : _),t) → (cs,t) : []

_ → []

is transformed to

nonnull00 (cs@(_ : _),t) = (cs,t) : []

nonnull00 _ = []

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

mainModule Main

((readInt :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])]) :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.
Binding Reductions:
The bind variable of the following binding Pattern

cs@(vy : vz)

is replaced by the following term

vy : vz

The bind variable of the following binding Pattern

xs@(ww : wx)

is replaced by the following term

ww : wx

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule Main

((readInt :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])]) :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])])

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

The following Function with conditions

span p [] = ([],[])

span p (ww : wx)

| p ww

= (ww : ys,zs)

| otherwise

= ([],ww : wx)

where

vu43 = span p wx

ys = ys0 vu43

ys0 (ys,wy) = ys

zs = zs0 vu43

zs0 (wz,zs) = zs

is transformed to

span p [] = span3 p []

span p (ww : wx) = span2 p (ww : wx)

span2 p (ww : wx) =

span1 p ww wx (p ww)

where

span0 p ww wx True = ([],ww : wx)

span1 p ww wx True = (ww : ys,zs)

span1 p ww wx False = span0 p ww wx otherwise

vu43 = span p wx

ys = ys0 vu43

ys0 (ys,wy) = ys

zs = zs0 vu43

zs0 (wz,zs) = zs

span3 p [] = ([],[])

span3 xx xy = span2 xx xy

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

mainModule Main

((readInt :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])]) :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])])

module Main where

import qualified Prelude

Let/Where Reductions:
The bindings of the following Let/Where expression

span1 p ww wx (p ww)

where

span0 p ww wx True = ([],ww : wx)

span1 p ww wx True = (ww : ys,zs)

span1 p ww wx False = span0 p ww wx otherwise

vu43 = span p wx

ys = ys0 vu43

ys0 (ys,wy) = ys

zs = zs0 vu43

zs0 (wz,zs) = zs

are unpacked to the following functions on top level

span2Ys xz yu = span2Ys0 xz yu (span2Vu43 xz yu)

span2Span0 xz yu p ww wx True = ([],ww : wx)

span2Zs xz yu = span2Zs0 xz yu (span2Vu43 xz yu)

span2Vu43 xz yu = span xz yu

span2Ys0 xz yu (ys,wy) = ys

span2Zs0 xz yu (wz,zs) = zs

span2Span1 xz yu p ww wx True = (ww : span2Ys xz yu,span2Zs xz yu)

span2Span1 xz yu p ww wx False = span2Span0 xz yu p ww wx otherwise

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

mainModule Main

(readInt :: Int -> (Char -> Bool) -> (Char -> Int) -> [Char] -> [(Int,[Char])])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="readInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
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4[label="readInt yv3 yv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="readInt yv3 yv4 yv5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="readInt yv3 yv4 yv5 yv6",fontsize=16,color="black",shape="triangle"];6 -> 7[label="",style="solid", color="black", weight=3];
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8[label="concat . map (readInt1 yv3 yv5)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="concat (map (readInt1 yv3 yv5) (nonnull yv4 yv6))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="foldr (++) [] (map (readInt1 yv3 yv5) (nonnull yv4 yv6))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="foldr (++) [] (map (readInt1 yv3 yv5) (concatMap nonnull0 (span yv4 yv6 : [])))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="foldr (++) [] (map (readInt1 yv3 yv5) (concat . map nonnull0))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
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14[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (map nonnull0 (span yv4 yv6 : []))))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (nonnull0 (span yv4 yv6) : map nonnull0 [])))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull0 (span yv4 yv6) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
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278 -> 18[label="",style="solid", color="burlywood", weight=3];
279[label="yv6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 279[label="",style="solid", color="burlywood", weight=9];
279 -> 19[label="",style="solid", color="burlywood", weight=3];
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19[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span yv4 []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
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21[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span3 yv4 []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22 -> 24[label="",style="dashed", color="red", weight=0];
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23[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 ([],[]) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
25[label="yv4 yv60",fontsize=16,color="green",shape="box"];25 -> 30[label="",style="dashed", color="green", weight=3];
24[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 yv7) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="burlywood",shape="triangle"];281[label="yv7/False",fontsize=10,color="white",style="solid",shape="box"];24 -> 281[label="",style="solid", color="burlywood", weight=9];
281 -> 28[label="",style="solid", color="burlywood", weight=3];
282[label="yv7/True",fontsize=10,color="white",style="solid",shape="box"];24 -> 282[label="",style="solid", color="burlywood", weight=9];
282 -> 29[label="",style="solid", color="burlywood", weight=3];
26[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) [] foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="triangle"];26 -> 31[label="",style="solid", color="black", weight=3];
30[label="yv60",fontsize=16,color="green",shape="box"];28[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 False) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 True) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
31[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3];
32[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span0 yv4 yv61 yv4 yv60 yv61 otherwise) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
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34[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] []))",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3];
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36[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) ((yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) : []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
37[label="foldr (++) [] (map (readInt1 yv3 yv5) [])",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
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40[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];40 -> 43[label="",style="solid", color="black", weight=3];
41 -> 26[label="",style="dashed", color="red", weight=0];
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43[label="[]",fontsize=16,color="green",shape="box"];44 -> 45[label="",style="dashed", color="red", weight=0];
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49 -> 51[label="",style="dashed", color="green", weight=3];
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51[label="span2Zs yv4 yv61",fontsize=16,color="black",shape="triangle"];51 -> 54[label="",style="solid", color="black", weight=3];
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54[label="span2Zs0 yv4 yv61 (span2Vu43 yv4 yv61)",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="yv8",fontsize=16,color="green",shape="box"];56[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys yv4 yv61))",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
57[label="span2Zs0 yv4 yv61 (span yv4 yv61)",fontsize=16,color="burlywood",shape="box"];286[label="yv61/yv610 : yv611",fontsize=10,color="white",style="solid",shape="box"];57 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 59[label="",style="solid", color="burlywood", weight=3];
287[label="yv61/[]",fontsize=10,color="white",style="solid",shape="box"];57 -> 287[label="",style="solid", color="burlywood", weight=9];
287 -> 60[label="",style="solid", color="burlywood", weight=3];
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60[label="span2Zs0 yv4 [] (span yv4 [])",fontsize=16,color="black",shape="box"];60 -> 63[label="",style="solid", color="black", weight=3];
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288 -> 64[label="",style="solid", color="burlywood", weight=3];
289[label="yv61/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 65[label="",style="solid", color="burlywood", weight=3];
62[label="span2Zs0 yv4 (yv610 : yv611) (span2 yv4 (yv610 : yv611))",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3];
63[label="span2Zs0 yv4 [] (span3 yv4 [])",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
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66 -> 70[label="",style="dashed", color="red", weight=0];
66[label="span2Zs0 yv4 (yv610 : yv611) (span2Span1 yv4 yv611 yv4 yv610 yv611 (yv4 yv610))",fontsize=16,color="magenta"];66 -> 71[label="",style="dashed", color="magenta", weight=3];
67[label="span2Zs0 yv4 [] ([],[])",fontsize=16,color="black",shape="box"];67 -> 72[label="",style="solid", color="black", weight=3];
68[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys0 yv4 (yv610 : yv611) (span2 yv4 (yv610 : yv611))))",fontsize=16,color="black",shape="box"];68 -> 73[label="",style="solid", color="black", weight=3];
69[label="foldl (readInt0 yv3) (fromIntegral . yv5) (map (fromIntegral . yv5) (span2Ys0 yv4 [] (span3 yv4 [])))",fontsize=16,color="black",shape="box"];69 -> 74[label="",style="solid", color="black", weight=3];
71[label="yv4 yv610",fontsize=16,color="green",shape="box"];71 -> 78[label="",style="dashed", color="green", weight=3];
70[label="span2Zs0 yv4 (yv610 : yv611) (span2Span1 yv4 yv611 yv4 yv610 yv611 yv9)",fontsize=16,color="burlywood",shape="triangle"];291[label="yv9/False",fontsize=10,color="white",style="solid",shape="box"];70 -> 291[label="",style="solid", color="burlywood", weight=9];
291 -> 76[label="",style="solid", color="burlywood", weight=3];
292[label="yv9/True",fontsize=10,color="white",style="solid",shape="box"];70 -> 292[label="",style="solid", color="burlywood", weight=9];
292 -> 77[label="",style="solid", color="burlywood", weight=3];
72[label="[]",fontsize=16,color="green",shape="box"];73 -> 79[label="",style="dashed", color="red", weight=0];
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333[label="yv140/Zero",fontsize=10,color="white",style="solid",shape="box"];222 -> 333[label="",style="solid", color="burlywood", weight=9];
333 -> 230[label="",style="solid", color="burlywood", weight=3];
221[label="primMinusNat yv18 yv150",fontsize=16,color="burlywood",shape="triangle"];334[label="yv18/Succ yv180",fontsize=10,color="white",style="solid",shape="box"];221 -> 334[label="",style="solid", color="burlywood", weight=9];
334 -> 231[label="",style="solid", color="burlywood", weight=3];
335[label="yv18/Zero",fontsize=10,color="white",style="solid",shape="box"];221 -> 335[label="",style="solid", color="burlywood", weight=9];
335 -> 232[label="",style="solid", color="burlywood", weight=3];
223 -> 222[label="",style="dashed", color="red", weight=0];
223[label="primMulNat yv140 yv30",fontsize=16,color="magenta"];223 -> 233[label="",style="dashed", color="magenta", weight=3];
224[label="yv150",fontsize=16,color="green",shape="box"];188 -> 234[label="",style="dashed", color="red", weight=0];
188[label="primPlusNat (primMulNat yv140 yv30) yv150",fontsize=16,color="magenta"];188 -> 236[label="",style="dashed", color="magenta", weight=3];
188 -> 237[label="",style="dashed", color="magenta", weight=3];
235 -> 222[label="",style="dashed", color="red", weight=0];
235[label="primMulNat yv140 yv30",fontsize=16,color="magenta"];234[label="primPlusNat yv19 yv150",fontsize=16,color="burlywood",shape="triangle"];339[label="yv19/Succ yv190",fontsize=10,color="white",style="solid",shape="box"];234 -> 339[label="",style="solid", color="burlywood", weight=9];
339 -> 242[label="",style="solid", color="burlywood", weight=3];
340[label="yv19/Zero",fontsize=10,color="white",style="solid",shape="box"];234 -> 340[label="",style="solid", color="burlywood", weight=9];
340 -> 243[label="",style="solid", color="burlywood", weight=3];
229[label="primMulNat (Succ yv1400) yv30",fontsize=16,color="burlywood",shape="box"];341[label="yv30/Succ yv300",fontsize=10,color="white",style="solid",shape="box"];229 -> 341[label="",style="solid", color="burlywood", weight=9];
341 -> 244[label="",style="solid", color="burlywood", weight=3];
342[label="yv30/Zero",fontsize=10,color="white",style="solid",shape="box"];229 -> 342[label="",style="solid", color="burlywood", weight=9];
342 -> 245[label="",style="solid", color="burlywood", weight=3];
230[label="primMulNat Zero yv30",fontsize=16,color="burlywood",shape="box"];343[label="yv30/Succ yv300",fontsize=10,color="white",style="solid",shape="box"];230 -> 343[label="",style="solid", color="burlywood", weight=9];
343 -> 246[label="",style="solid", color="burlywood", weight=3];
344[label="yv30/Zero",fontsize=10,color="white",style="solid",shape="box"];230 -> 344[label="",style="solid", color="burlywood", weight=9];
344 -> 247[label="",style="solid", color="burlywood", weight=3];
231[label="primMinusNat (Succ yv180) yv150",fontsize=16,color="burlywood",shape="box"];345[label="yv150/Succ yv1500",fontsize=10,color="white",style="solid",shape="box"];231 -> 345[label="",style="solid", color="burlywood", weight=9];
345 -> 248[label="",style="solid", color="burlywood", weight=3];
346[label="yv150/Zero",fontsize=10,color="white",style="solid",shape="box"];231 -> 346[label="",style="solid", color="burlywood", weight=9];
346 -> 249[label="",style="solid", color="burlywood", weight=3];
232[label="primMinusNat Zero yv150",fontsize=16,color="burlywood",shape="box"];347[label="yv150/Succ yv1500",fontsize=10,color="white",style="solid",shape="box"];232 -> 347[label="",style="solid", color="burlywood", weight=9];
347 -> 250[label="",style="solid", color="burlywood", weight=3];
348[label="yv150/Zero",fontsize=10,color="white",style="solid",shape="box"];232 -> 348[label="",style="solid", color="burlywood", weight=9];
348 -> 251[label="",style="solid", color="burlywood", weight=3];
233[label="yv30",fontsize=16,color="green",shape="box"];236[label="yv150",fontsize=16,color="green",shape="box"];237 -> 222[label="",style="dashed", color="red", weight=0];
237[label="primMulNat yv140 yv30",fontsize=16,color="magenta"];237 -> 252[label="",style="dashed", color="magenta", weight=3];
242[label="primPlusNat (Succ yv190) yv150",fontsize=16,color="burlywood",shape="box"];350[label="yv150/Succ yv1500",fontsize=10,color="white",style="solid",shape="box"];242 -> 350[label="",style="solid", color="burlywood", weight=9];
350 -> 253[label="",style="solid", color="burlywood", weight=3];
351[label="yv150/Zero",fontsize=10,color="white",style="solid",shape="box"];242 -> 351[label="",style="solid", color="burlywood", weight=9];
351 -> 254[label="",style="solid", color="burlywood", weight=3];
243[label="primPlusNat Zero yv150",fontsize=16,color="burlywood",shape="box"];352[label="yv150/Succ yv1500",fontsize=10,color="white",style="solid",shape="box"];243 -> 352[label="",style="solid", color="burlywood", weight=9];
352 -> 255[label="",style="solid", color="burlywood", weight=3];
353[label="yv150/Zero",fontsize=10,color="white",style="solid",shape="box"];243 -> 353[label="",style="solid", color="burlywood", weight=9];
353 -> 256[label="",style="solid", color="burlywood", weight=3];
244[label="primMulNat (Succ yv1400) (Succ yv300)",fontsize=16,color="black",shape="box"];244 -> 257[label="",style="solid", color="black", weight=3];
245[label="primMulNat (Succ yv1400) Zero",fontsize=16,color="black",shape="box"];245 -> 258[label="",style="solid", color="black", weight=3];
246[label="primMulNat Zero (Succ yv300)",fontsize=16,color="black",shape="box"];246 -> 259[label="",style="solid", color="black", weight=3];
247[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];247 -> 260[label="",style="solid", color="black", weight=3];
248[label="primMinusNat (Succ yv180) (Succ yv1500)",fontsize=16,color="black",shape="box"];248 -> 261[label="",style="solid", color="black", weight=3];
249[label="primMinusNat (Succ yv180) Zero",fontsize=16,color="black",shape="box"];249 -> 262[label="",style="solid", color="black", weight=3];
250[label="primMinusNat Zero (Succ yv1500)",fontsize=16,color="black",shape="box"];250 -> 263[label="",style="solid", color="black", weight=3];
251[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];251 -> 264[label="",style="solid", color="black", weight=3];
252[label="yv30",fontsize=16,color="green",shape="box"];253[label="primPlusNat (Succ yv190) (Succ yv1500)",fontsize=16,color="black",shape="box"];253 -> 265[label="",style="solid", color="black", weight=3];
254[label="primPlusNat (Succ yv190) Zero",fontsize=16,color="black",shape="box"];254 -> 266[label="",style="solid", color="black", weight=3];
255[label="primPlusNat Zero (Succ yv1500)",fontsize=16,color="black",shape="box"];255 -> 267[label="",style="solid", color="black", weight=3];
256[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];256 -> 268[label="",style="solid", color="black", weight=3];
257 -> 234[label="",style="dashed", color="red", weight=0];
257[label="primPlusNat (primMulNat yv1400 (Succ yv300)) (Succ yv300)",fontsize=16,color="magenta"];257 -> 269[label="",style="dashed", color="magenta", weight=3];
257 -> 270[label="",style="dashed", color="magenta", weight=3];
258[label="Zero",fontsize=16,color="green",shape="box"];259[label="Zero",fontsize=16,color="green",shape="box"];260[label="Zero",fontsize=16,color="green",shape="box"];261 -> 221[label="",style="dashed", color="red", weight=0];
261[label="primMinusNat yv180 yv1500",fontsize=16,color="magenta"];261 -> 271[label="",style="dashed", color="magenta", weight=3];
261 -> 272[label="",style="dashed", color="magenta", weight=3];
262[label="Pos (Succ yv180)",fontsize=16,color="green",shape="box"];263[label="Neg (Succ yv1500)",fontsize=16,color="green",shape="box"];264[label="Pos Zero",fontsize=16,color="green",shape="box"];265[label="Succ (Succ (primPlusNat yv190 yv1500))",fontsize=16,color="green",shape="box"];265 -> 273[label="",style="dashed", color="green", weight=3];
266[label="Succ yv190",fontsize=16,color="green",shape="box"];267[label="Succ yv1500",fontsize=16,color="green",shape="box"];268[label="Zero",fontsize=16,color="green",shape="box"];269[label="Succ yv300",fontsize=16,color="green",shape="box"];270 -> 222[label="",style="dashed", color="red", weight=0];
270[label="primMulNat yv1400 (Succ yv300)",fontsize=16,color="magenta"];270 -> 274[label="",style="dashed", color="magenta", weight=3];
270 -> 275[label="",style="dashed", color="magenta", weight=3];
271[label="yv1500",fontsize=16,color="green",shape="box"];272[label="yv180",fontsize=16,color="green",shape="box"];273 -> 234[label="",style="dashed", color="red", weight=0];
273[label="primPlusNat yv190 yv1500",fontsize=16,color="magenta"];273 -> 276[label="",style="dashed", color="magenta", weight=3];
273 -> 277[label="",style="dashed", color="magenta", weight=3];
274[label="yv1400",fontsize=16,color="green",shape="box"];275[label="Succ yv300",fontsize=16,color="green",shape="box"];276[label="yv1500",fontsize=16,color="green",shape="box"];277[label="yv190",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

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

new_span2Zs(yv4, :(yv610, yv611)) → new_span2Zs0(yv4, yv610, yv611)
new_span2Zs0(yv4, yv610, yv611) → new_span2Zs(yv4, yv611)

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_span2Zs0(yv4, yv610, yv611) → new_span2Zs(yv4, yv611)
The graph contains the following edges 1 >= 1, 3 >= 2
new_span2Zs(yv4, :(yv610, yv611)) → new_span2Zs0(yv4, yv610, yv611)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

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

new_primMinusNat(Succ(yv180), Succ(yv1500)) → new_primMinusNat(yv180, yv1500)

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

new_primMinusNat(Succ(yv180), Succ(yv1500)) → new_primMinusNat(yv180, yv1500)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                          ↳ QDP

                          ↳ QDP

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

new_primPlusNat(Succ(yv190), Succ(yv1500)) → new_primPlusNat(yv190, yv1500)

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

new_primPlusNat(Succ(yv190), Succ(yv1500)) → new_primPlusNat(yv190, yv1500)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                          ↳ QDP

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

new_primMulNat(Succ(yv1400), Succ(yv300)) → new_primMulNat(yv1400, Succ(yv300))

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

new_primMulNat(Succ(yv1400), Succ(yv300)) → new_primMulNat(yv1400, Succ(yv300))
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ CR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ LetRed

                    ↳ HASKELL

                      ↳ Narrow

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPSizeChangeProof

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

new_foldl(yv3, yv14, yv5, yv4, yv610, yv611, yv12) → new_foldl0(yv3, yv14, new_pt(yv5, yv610), yv5, yv4, yv611)
new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111) → new_foldl(yv3, new_readInt0(yv3, yv14, yv15), yv5, yv4, yv6110, yv6111, new_span2Zs1(yv4, yv6111))
new_foldl0(yv3, yv14, yv15, yv5, yv4, :(yv6110, yv6111)) → new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111)

The TRS R consists of the following rules:

new_primMulNat0(Zero, Zero) → Zero
new_readInt0(Neg(yv30), Pos(yv140), yv15) → new_primPlusInt(yv140, yv30, yv15)
new_readInt0(Pos(yv30), Neg(yv140), yv15) → new_primPlusInt(yv140, yv30, yv15)
new_primPlusInt(yv140, yv30, Neg(yv150)) → Neg(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusInt(yv140, yv30, Pos(yv150)) → new_primMinusNat0(yv150, new_primMulNat0(yv140, yv30))
new_readInt0(Neg(yv30), Neg(yv140), yv15) → new_primPlusInt0(yv140, yv30, yv15)
new_primPlusInt0(yv140, yv30, Pos(yv150)) → Pos(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
new_primMinusNat0(Zero, Zero) → Pos(Zero)
new_primMinusNat0(Succ(yv180), Zero) → Pos(Succ(yv180))
new_primPlusInt0(yv140, yv30, Neg(yv150)) → new_primMinusNat0(new_primMulNat0(yv140, yv30), yv150)
new_readInt0(Pos(yv30), Pos(yv140), yv15) → new_primPlusInt0(yv140, yv30, yv15)
new_primMulNat0(Zero, Succ(yv300)) → Zero
new_primMulNat0(Succ(yv1400), Zero) → Zero
new_primMulNat0(Succ(yv1400), Succ(yv300)) → new_primPlusNat0(new_primMulNat0(yv1400, Succ(yv300)), Succ(yv300))
new_primPlusNat0(Zero, Succ(yv1500)) → Succ(yv1500)
new_primPlusNat0(Succ(yv190), Zero) → Succ(yv190)
new_primMinusNat0(Zero, Succ(yv1500)) → Neg(Succ(yv1500))
new_primPlusNat0(Succ(yv190), Succ(yv1500)) → Succ(Succ(new_primPlusNat0(yv190, yv1500)))
new_primMinusNat0(Succ(yv180), Succ(yv1500)) → new_primMinusNat0(yv180, yv1500)

The set Q consists of the following terms:

new_primPlusNat0(Zero, Succ(x0))
new_primPlusInt0(x0, x1, Neg(x2))
new_primPlusInt(x0, x1, Pos(x2))
new_primPlusNat0(Succ(x0), Zero)
new_primPlusInt(x0, x1, Neg(x2))
new_primMulNat0(Succ(x0), Succ(x1))
new_readInt0(Pos(x0), Neg(x1), x2)
new_readInt0(Neg(x0), Pos(x1), x2)
new_primMulNat0(Zero, Zero)
new_primPlusInt0(x0, x1, Pos(x2))
new_readInt0(Pos(x0), Pos(x1), x2)
new_primMinusNat0(Zero, Succ(x0))
new_primPlusNat0(Zero, Zero)
new_primMinusNat0(Succ(x0), Zero)
new_primMulNat0(Zero, Succ(x0))
new_primMulNat0(Succ(x0), Zero)
new_readInt0(Neg(x0), Neg(x1), x2)
new_primMinusNat0(Zero, Zero)
new_primMinusNat0(Succ(x0), Succ(x1))
new_primPlusNat0(Succ(x0), Succ(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_foldl0(yv3, yv14, yv15, yv5, yv4, :(yv6110, yv6111)) → new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 > 6, 6 > 7
new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111) → new_foldl(yv3, new_readInt0(yv3, yv14, yv15), yv5, yv4, yv6110, yv6111, new_span2Zs1(yv4, yv6111))
The graph contains the following edges 1 >= 1, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6
new_foldl(yv3, yv14, yv5, yv4, yv610, yv611, yv12) → new_foldl0(yv3, yv14, new_pt(yv5, yv610), yv5, yv4, yv611)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4, 4 >= 5, 6 >= 6

case	vu77 of
	(ds,r)	→ (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : []
	_	→ []

readInt10	radix digToInt (ds,r)	= (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : []
readInt10	radix digToInt _	= []

span2Span1	xz yu p ww wx True	= (ww : span2Ys xz yu,span2Zs xz yu)
span2Span1	xz yu p ww wx False	= span2Span0 xz yu p ww wx otherwise