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

YES 1.059 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

  ↳ BR

mainModule Main

((signum :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((signum :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int)

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

signumReal x

| x == 0

= 0

| x > 0

= 1

| otherwise

= -1

is transformed to

signumReal x = signumReal3 x

signumReal1 x True = 1

signumReal1 x False = signumReal0 x otherwise

signumReal0 x True = -1

signumReal2 x True = 0

signumReal2 x False = signumReal1 x (x > 0)

signumReal3 x = signumReal2 x (x == 0)

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

((signum :: Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int)

module Main where

import qualified Prelude

Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(signum :: Ratio Int -> Ratio Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="signum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="signum vz3",fontsize=16,color="burlywood",shape="triangle"];363[label="vz3/vz30 :% vz31",fontsize=10,color="white",style="solid",shape="box"];3 -> 363[label="",style="solid", color="burlywood", weight=9];
363 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="signum (vz30 :% vz31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="signum vz30 :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
5 -> 7[label="",style="dashed", color="green", weight=3];
6[label="signum vz30",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="signumReal vz30",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];10[label="signumReal3 vz30",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="signumReal2 vz30 (vz30 == fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="signumReal2 vz30 (primEqInt vz30 (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];364[label="vz30/Pos vz300",fontsize=10,color="white",style="solid",shape="box"];12 -> 364[label="",style="solid", color="burlywood", weight=9];
364 -> 13[label="",style="solid", color="burlywood", weight=3];
365[label="vz30/Neg vz300",fontsize=10,color="white",style="solid",shape="box"];12 -> 365[label="",style="solid", color="burlywood", weight=9];
365 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="signumReal2 (Pos vz300) (primEqInt (Pos vz300) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];366[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];13 -> 366[label="",style="solid", color="burlywood", weight=9];
366 -> 15[label="",style="solid", color="burlywood", weight=3];
367[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 367[label="",style="solid", color="burlywood", weight=9];
367 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="signumReal2 (Neg vz300) (primEqInt (Neg vz300) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];368[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];14 -> 368[label="",style="solid", color="burlywood", weight=9];
368 -> 17[label="",style="solid", color="burlywood", weight=3];
369[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 369[label="",style="solid", color="burlywood", weight=9];
369 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="signumReal2 (Pos (Succ vz3000)) (primEqInt (Pos (Succ vz3000)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="signumReal2 (Pos Zero) (primEqInt (Pos Zero) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="signumReal2 (Neg (Succ vz3000)) (primEqInt (Neg (Succ vz3000)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="signumReal2 (Neg Zero) (primEqInt (Neg Zero) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="signumReal2 (Pos (Succ vz3000)) (primEqInt (Pos (Succ vz3000)) (Pos Zero))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="signumReal2 (Pos Zero) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="signumReal2 (Neg (Succ vz3000)) (primEqInt (Neg (Succ vz3000)) (Pos Zero))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="signumReal2 (Neg Zero) (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="signumReal2 (Pos (Succ vz3000)) False",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="signumReal2 (Pos Zero) True",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="signumReal2 (Neg (Succ vz3000)) False",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="signumReal2 (Neg Zero) True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="signumReal1 (Pos (Succ vz3000)) (Pos (Succ vz3000) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="fromInt (Pos Zero)",fontsize=16,color="black",shape="triangle"];28 -> 32[label="",style="solid", color="black", weight=3];
29 -> 33[label="",style="dashed", color="red", weight=0];
29[label="signumReal1 (Neg (Succ vz3000)) (Neg (Succ vz3000) > fromInt (Pos Zero))",fontsize=16,color="magenta"];29 -> 34[label="",style="dashed", color="magenta", weight=3];
30 -> 28[label="",style="dashed", color="red", weight=0];
30[label="fromInt (Pos Zero)",fontsize=16,color="magenta"];31 -> 35[label="",style="dashed", color="red", weight=0];
31[label="signumReal1 (Pos (Succ vz3000)) (compare (Pos (Succ vz3000)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3];
32[label="Pos Zero",fontsize=16,color="green",shape="box"];34 -> 28[label="",style="dashed", color="red", weight=0];
34[label="fromInt (Pos Zero)",fontsize=16,color="magenta"];33[label="signumReal1 (Neg (Succ vz3000)) (Neg (Succ vz3000) > vz4)",fontsize=16,color="black",shape="triangle"];33 -> 37[label="",style="solid", color="black", weight=3];
36 -> 28[label="",style="dashed", color="red", weight=0];
36[label="fromInt (Pos Zero)",fontsize=16,color="magenta"];35[label="signumReal1 (Pos (Succ vz3000)) (compare (Pos (Succ vz3000)) vz5 == GT)",fontsize=16,color="black",shape="triangle"];35 -> 38[label="",style="solid", color="black", weight=3];
37[label="signumReal1 (Neg (Succ vz3000)) (compare (Neg (Succ vz3000)) vz4 == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="signumReal1 (Pos (Succ vz3000)) (primCmpInt (Pos (Succ vz3000)) vz5 == GT)",fontsize=16,color="burlywood",shape="box"];375[label="vz5/Pos vz50",fontsize=10,color="white",style="solid",shape="box"];38 -> 375[label="",style="solid", color="burlywood", weight=9];
375 -> 40[label="",style="solid", color="burlywood", weight=3];
376[label="vz5/Neg vz50",fontsize=10,color="white",style="solid",shape="box"];38 -> 376[label="",style="solid", color="burlywood", weight=9];
376 -> 41[label="",style="solid", color="burlywood", weight=3];
39[label="signumReal1 (Neg (Succ vz3000)) (primCmpInt (Neg (Succ vz3000)) vz4 == GT)",fontsize=16,color="burlywood",shape="box"];377[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];39 -> 377[label="",style="solid", color="burlywood", weight=9];
377 -> 42[label="",style="solid", color="burlywood", weight=3];
378[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];39 -> 378[label="",style="solid", color="burlywood", weight=9];
378 -> 43[label="",style="solid", color="burlywood", weight=3];
40[label="signumReal1 (Pos (Succ vz3000)) (primCmpInt (Pos (Succ vz3000)) (Pos vz50) == GT)",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="signumReal1 (Pos (Succ vz3000)) (primCmpInt (Pos (Succ vz3000)) (Neg vz50) == GT)",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="signumReal1 (Neg (Succ vz3000)) (primCmpInt (Neg (Succ vz3000)) (Pos vz40) == GT)",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
43[label="signumReal1 (Neg (Succ vz3000)) (primCmpInt (Neg (Succ vz3000)) (Neg vz40) == GT)",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
44 -> 206[label="",style="dashed", color="red", weight=0];
44[label="signumReal1 (Pos (Succ vz3000)) (primCmpNat (Succ vz3000) vz50 == GT)",fontsize=16,color="magenta"];44 -> 207[label="",style="dashed", color="magenta", weight=3];
44 -> 208[label="",style="dashed", color="magenta", weight=3];
44 -> 209[label="",style="dashed", color="magenta", weight=3];
45[label="signumReal1 (Pos (Succ vz3000)) (GT == GT)",fontsize=16,color="black",shape="triangle"];45 -> 50[label="",style="solid", color="black", weight=3];
46[label="signumReal1 (Neg (Succ vz3000)) (LT == GT)",fontsize=16,color="black",shape="triangle"];46 -> 51[label="",style="solid", color="black", weight=3];
47 -> 269[label="",style="dashed", color="red", weight=0];
47[label="signumReal1 (Neg (Succ vz3000)) (primCmpNat vz40 (Succ vz3000) == GT)",fontsize=16,color="magenta"];47 -> 270[label="",style="dashed", color="magenta", weight=3];
47 -> 271[label="",style="dashed", color="magenta", weight=3];
47 -> 272[label="",style="dashed", color="magenta", weight=3];
207[label="vz50",fontsize=16,color="green",shape="box"];208[label="Succ vz3000",fontsize=16,color="green",shape="box"];209[label="vz3000",fontsize=16,color="green",shape="box"];206[label="signumReal1 (Pos (Succ vz7)) (primCmpNat vz8 vz9 == GT)\n  Num a",fontsize=16,color="burlywood",shape="triangle"];381[label="vz8/Succ vz80",fontsize=10,color="white",style="solid",shape="box"];206 -> 381[label="",style="solid", color="burlywood", weight=9];
381 -> 228[label="",style="solid", color="burlywood", weight=3];
382[label="vz8/Zero",fontsize=10,color="white",style="solid",shape="box"];206 -> 382[label="",style="solid", color="burlywood", weight=9];
382 -> 229[label="",style="solid", color="burlywood", weight=3];
50[label="signumReal1 (Pos (Succ vz3000)) True",fontsize=16,color="black",shape="box"];50 -> 56[label="",style="solid", color="black", weight=3];
51[label="signumReal1 (Neg (Succ vz3000)) False",fontsize=16,color="black",shape="triangle"];51 -> 57[label="",style="solid", color="black", weight=3];
270[label="vz40",fontsize=16,color="green",shape="box"];271[label="vz3000",fontsize=16,color="green",shape="box"];272[label="Succ vz3000",fontsize=16,color="green",shape="box"];269[label="signumReal1 (Neg (Succ vz13)) (primCmpNat vz14 vz15 == GT)\n  Num a",fontsize=16,color="burlywood",shape="triangle"];383[label="vz14/Succ vz140",fontsize=10,color="white",style="solid",shape="box"];269 -> 383[label="",style="solid", color="burlywood", weight=9];
383 -> 294[label="",style="solid", color="burlywood", weight=3];
384[label="vz14/Zero",fontsize=10,color="white",style="solid",shape="box"];269 -> 384[label="",style="solid", color="burlywood", weight=9];
384 -> 295[label="",style="solid", color="burlywood", weight=3];
228[label="signumReal1 (Pos (Succ vz7)) (primCmpNat (Succ vz80) vz9 == GT)\n  Num a",fontsize=16,color="burlywood",shape="box"];385[label="vz9/Succ vz90",fontsize=10,color="white",style="solid",shape="box"];228 -> 385[label="",style="solid", color="burlywood", weight=9];
385 -> 238[label="",style="solid", color="burlywood", weight=3];
386[label="vz9/Zero",fontsize=10,color="white",style="solid",shape="box"];228 -> 386[label="",style="solid", color="burlywood", weight=9];
386 -> 239[label="",style="solid", color="burlywood", weight=3];
229[label="signumReal1 (Pos (Succ vz7)) (primCmpNat Zero vz9 == GT)\n  Num a",fontsize=16,color="burlywood",shape="box"];387[label="vz9/Succ vz90",fontsize=10,color="white",style="solid",shape="box"];229 -> 387[label="",style="solid", color="burlywood", weight=9];
387 -> 240[label="",style="solid", color="burlywood", weight=3];
388[label="vz9/Zero",fontsize=10,color="white",style="solid",shape="box"];229 -> 388[label="",style="solid", color="burlywood", weight=9];
388 -> 241[label="",style="solid", color="burlywood", weight=3];
56 -> 7[label="",style="dashed", color="red", weight=0];
56[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];57[label="signumReal0 (Neg (Succ vz3000)) otherwise",fontsize=16,color="black",shape="box"];57 -> 62[label="",style="solid", color="black", weight=3];
294[label="signumReal1 (Neg (Succ vz13)) (primCmpNat (Succ vz140) vz15 == GT)\n  Num a",fontsize=16,color="burlywood",shape="box"];390[label="vz15/Succ vz150",fontsize=10,color="white",style="solid",shape="box"];294 -> 390[label="",style="solid", color="burlywood", weight=9];
390 -> 298[label="",style="solid", color="burlywood", weight=3];
391[label="vz15/Zero",fontsize=10,color="white",style="solid",shape="box"];294 -> 391[label="",style="solid", color="burlywood", weight=9];
391 -> 299[label="",style="solid", color="burlywood", weight=3];
295[label="signumReal1 (Neg (Succ vz13)) (primCmpNat Zero vz15 == GT)\n  Num a",fontsize=16,color="burlywood",shape="box"];392[label="vz15/Succ vz150",fontsize=10,color="white",style="solid",shape="box"];295 -> 392[label="",style="solid", color="burlywood", weight=9];
392 -> 300[label="",style="solid", color="burlywood", weight=3];
393[label="vz15/Zero",fontsize=10,color="white",style="solid",shape="box"];295 -> 393[label="",style="solid", color="burlywood", weight=9];
393 -> 301[label="",style="solid", color="burlywood", weight=3];
238[label="signumReal1 (Pos (Succ vz7)) (primCmpNat (Succ vz80) (Succ vz90) == GT)\n  Num a",fontsize=16,color="black",shape="box"];238 -> 249[label="",style="solid", color="black", weight=3];
239[label="signumReal1 (Pos (Succ vz7)) (primCmpNat (Succ vz80) Zero == GT)\n  Num a",fontsize=16,color="black",shape="box"];239 -> 250[label="",style="solid", color="black", weight=3];
240[label="signumReal1 (Pos (Succ vz7)) (primCmpNat Zero (Succ vz90) == GT)\n  Num a",fontsize=16,color="black",shape="box"];240 -> 251[label="",style="solid", color="black", weight=3];
241[label="signumReal1 (Pos (Succ vz7)) (primCmpNat Zero Zero == GT)\n  Num a",fontsize=16,color="black",shape="box"];241 -> 252[label="",style="solid", color="black", weight=3];
62[label="signumReal0 (Neg (Succ vz3000)) True",fontsize=16,color="black",shape="box"];62 -> 69[label="",style="solid", color="black", weight=3];
298[label="signumReal1 (Neg (Succ vz13)) (primCmpNat (Succ vz140) (Succ vz150) == GT)\n  Num a",fontsize=16,color="black",shape="box"];298 -> 308[label="",style="solid", color="black", weight=3];
299[label="signumReal1 (Neg (Succ vz13)) (primCmpNat (Succ vz140) Zero == GT)\n  Num a",fontsize=16,color="black",shape="box"];299 -> 309[label="",style="solid", color="black", weight=3];
300[label="signumReal1 (Neg (Succ vz13)) (primCmpNat Zero (Succ vz150) == GT)\n  Num a",fontsize=16,color="black",shape="box"];300 -> 310[label="",style="solid", color="black", weight=3];
301[label="signumReal1 (Neg (Succ vz13)) (primCmpNat Zero Zero == GT)\n  Num a",fontsize=16,color="black",shape="box"];301 -> 311[label="",style="solid", color="black", weight=3];
249 -> 206[label="",style="dashed", color="red", weight=0];
249[label="signumReal1 (Pos (Succ vz7)) (primCmpNat vz80 vz90 == GT)\n  Num a",fontsize=16,color="magenta"];249 -> 259[label="",style="dashed", color="magenta", weight=3];
249 -> 260[label="",style="dashed", color="magenta", weight=3];
250[label="signumReal1 (Pos (Succ vz7)) (GT == GT)\n  Num a",fontsize=16,color="black",shape="box"];250 -> 261[label="",style="solid", color="black", weight=3];
251[label="signumReal1 (Pos (Succ vz7)) (LT == GT)\n  Num a",fontsize=16,color="black",shape="box"];251 -> 262[label="",style="solid", color="black", weight=3];
252[label="signumReal1 (Pos (Succ vz7)) (EQ == GT)\n  Num a",fontsize=16,color="black",shape="box"];252 -> 263[label="",style="solid", color="black", weight=3];
69[label="fromInt (Neg (Succ Zero))",fontsize=16,color="black",shape="triangle"];69 -> 78[label="",style="solid", color="black", weight=3];
308 -> 269[label="",style="dashed", color="red", weight=0];
308[label="signumReal1 (Neg (Succ vz13)) (primCmpNat vz140 vz150 == GT)\n  Num a",fontsize=16,color="magenta"];308 -> 317[label="",style="dashed", color="magenta", weight=3];
308 -> 318[label="",style="dashed", color="magenta", weight=3];
309[label="signumReal1 (Neg (Succ vz13)) (GT == GT)\n  Num a",fontsize=16,color="black",shape="box"];309 -> 319[label="",style="solid", color="black", weight=3];
310[label="signumReal1 (Neg (Succ vz13)) (LT == GT)\n  Num a",fontsize=16,color="black",shape="box"];310 -> 320[label="",style="solid", color="black", weight=3];
311[label="signumReal1 (Neg (Succ vz13)) (EQ == GT)\n  Num a",fontsize=16,color="black",shape="box"];311 -> 321[label="",style="solid", color="black", weight=3];
259[label="vz90",fontsize=16,color="green",shape="box"];260[label="vz80",fontsize=16,color="green",shape="box"];261[label="signumReal1 (Pos (Succ vz7)) True\n  Num a",fontsize=16,color="black",shape="box"];261 -> 296[label="",style="solid", color="black", weight=3];
262[label="signumReal1 (Pos (Succ vz7)) False\n  Num a",fontsize=16,color="black",shape="triangle"];262 -> 297[label="",style="solid", color="black", weight=3];
263 -> 262[label="",style="dashed", color="red", weight=0];
263[label="signumReal1 (Pos (Succ vz7)) False\n  Num a",fontsize=16,color="magenta"];78[label="Neg (Succ Zero)",fontsize=16,color="green",shape="box"];317[label="vz140",fontsize=16,color="green",shape="box"];318[label="vz150",fontsize=16,color="green",shape="box"];319[label="signumReal1 (Neg (Succ vz13)) True\n  Num a",fontsize=16,color="black",shape="box"];319 -> 330[label="",style="solid", color="black", weight=3];
320[label="signumReal1 (Neg (Succ vz13)) False\n  Num a",fontsize=16,color="black",shape="triangle"];320 -> 331[label="",style="solid", color="black", weight=3];
321 -> 320[label="",style="dashed", color="red", weight=0];
321[label="signumReal1 (Neg (Succ vz13)) False\n  Num a",fontsize=16,color="magenta"];296[label="fromInt (Pos (Succ Zero))\n  Num a",fontsize=16,color="blue",shape="box"];398[label="fromInt :: -> Int Int",fontsize=10,color="white",style="solid",shape="box"];296 -> 398[label="",style="solid", color="blue", weight=9];
398 -> 302[label="",style="solid", color="blue", weight=3];
399[label="fromInt :: -> Int Float",fontsize=10,color="white",style="solid",shape="box"];296 -> 399[label="",style="solid", color="blue", weight=9];
399 -> 303[label="",style="solid", color="blue", weight=3];
400[label="fromInt :: -> Int Double",fontsize=10,color="white",style="solid",shape="box"];296 -> 400[label="",style="solid", color="blue", weight=9];
400 -> 304[label="",style="solid", color="blue", weight=3];
401[label="fromInt :: -> Int (Ratio a)",fontsize=10,color="white",style="solid",shape="box"];296 -> 401[label="",style="solid", color="blue", weight=9];
401 -> 305[label="",style="solid", color="blue", weight=3];
402[label="fromInt :: -> Int Integer",fontsize=10,color="white",style="solid",shape="box"];296 -> 402[label="",style="solid", color="blue", weight=9];
402 -> 306[label="",style="solid", color="blue", weight=3];
297[label="signumReal0 (Pos (Succ vz7)) otherwise\n  Num a",fontsize=16,color="black",shape="box"];297 -> 307[label="",style="solid", color="black", weight=3];
330[label="fromInt (Pos (Succ Zero))\n  Num a",fontsize=16,color="blue",shape="box"];403[label="fromInt :: -> Int Int",fontsize=10,color="white",style="solid",shape="box"];330 -> 403[label="",style="solid", color="blue", weight=9];
403 -> 338[label="",style="solid", color="blue", weight=3];
404[label="fromInt :: -> Int Float",fontsize=10,color="white",style="solid",shape="box"];330 -> 404[label="",style="solid", color="blue", weight=9];
404 -> 339[label="",style="solid", color="blue", weight=3];
405[label="fromInt :: -> Int Double",fontsize=10,color="white",style="solid",shape="box"];330 -> 405[label="",style="solid", color="blue", weight=9];
405 -> 340[label="",style="solid", color="blue", weight=3];
406[label="fromInt :: -> Int (Ratio a)",fontsize=10,color="white",style="solid",shape="box"];330 -> 406[label="",style="solid", color="blue", weight=9];
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↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_signumReal1(vz13, Succ(vz140), Succ(vz150), ba) → new_signumReal1(vz13, vz140, vz150, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

new_signumReal1(vz13, Succ(vz140), Succ(vz150), ba) → new_signumReal1(vz13, vz140, vz150, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_signumReal10(vz7, Succ(vz80), Succ(vz90), ba) → new_signumReal10(vz7, vz80, vz90, ba)

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

new_signumReal10(vz7, Succ(vz80), Succ(vz90), ba) → new_signumReal10(vz7, vz80, vz90, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4

signumReal1	x True	= 1
signumReal1	x False	= signumReal0 x otherwise

signumReal2	x True	= 0
signumReal2	x False	= signumReal1 x (x > 0)