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

YES 0.892 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

((min :: Char -> Char -> Char) :: Char -> Char -> Char)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((min :: Char -> Char -> Char) :: Char -> Char -> Char)

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

min x y

| x <= y

= x

| otherwise

= y

is transformed to

min x y = min2 x y

min0 x y True = y

min1 x y True = x

min1 x y False = min0 x y otherwise

min2 x y = min1 x y (x <= y)

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 Main

(min :: Char -> Char -> Char)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="min",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="min vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="min vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="min2 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="min1 vx3 vx4 (vx3 <= vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="min1 vx3 vx4 (compare vx3 vx4 /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="min1 vx3 vx4 (not (compare vx3 vx4 == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="min1 vx3 vx4 (not (primCmpChar vx3 vx4 == GT))",fontsize=16,color="burlywood",shape="box"];302[label="vx3/Char vx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 302[label="",style="solid", color="burlywood", weight=9];
302 -> 10[label="",style="solid", color="burlywood", weight=3];
10[label="min1 (Char vx30) vx4 (not (primCmpChar (Char vx30) vx4 == GT))",fontsize=16,color="burlywood",shape="box"];303[label="vx4/Char vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 303[label="",style="solid", color="burlywood", weight=9];
303 -> 11[label="",style="solid", color="burlywood", weight=3];
11[label="min1 (Char vx30) (Char vx40) (not (primCmpChar (Char vx30) (Char vx40) == GT))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="min1 (Char vx30) (Char vx40) (not (primCmpNat vx30 vx40 == GT))",fontsize=16,color="burlywood",shape="box"];304[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];12 -> 304[label="",style="solid", color="burlywood", weight=9];
304 -> 13[label="",style="solid", color="burlywood", weight=3];
305[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 305[label="",style="solid", color="burlywood", weight=9];
305 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="min1 (Char (Succ vx300)) (Char vx40) (not (primCmpNat (Succ vx300) vx40 == GT))",fontsize=16,color="burlywood",shape="box"];306[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 306[label="",style="solid", color="burlywood", weight=9];
306 -> 15[label="",style="solid", color="burlywood", weight=3];
307[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 307[label="",style="solid", color="burlywood", weight=9];
307 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="min1 (Char Zero) (Char vx40) (not (primCmpNat Zero vx40 == GT))",fontsize=16,color="burlywood",shape="box"];308[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 308[label="",style="solid", color="burlywood", weight=9];
308 -> 17[label="",style="solid", color="burlywood", weight=3];
309[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 309[label="",style="solid", color="burlywood", weight=9];
309 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="min1 (Char (Succ vx300)) (Char (Succ vx400)) (not (primCmpNat (Succ vx300) (Succ vx400) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="min1 (Char (Succ vx300)) (Char Zero) (not (primCmpNat (Succ vx300) Zero == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="min1 (Char Zero) (Char (Succ vx400)) (not (primCmpNat Zero (Succ vx400) == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="min1 (Char Zero) (Char Zero) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19 -> 240[label="",style="dashed", color="red", weight=0];
19[label="min1 (Char (Succ vx300)) (Char (Succ vx400)) (not (primCmpNat vx300 vx400 == GT))",fontsize=16,color="magenta"];19 -> 241[label="",style="dashed", color="magenta", weight=3];
19 -> 242[label="",style="dashed", color="magenta", weight=3];
19 -> 243[label="",style="dashed", color="magenta", weight=3];
19 -> 244[label="",style="dashed", color="magenta", weight=3];
20[label="min1 (Char (Succ vx300)) (Char Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 25[label="",style="solid", color="black", weight=3];
21[label="min1 (Char Zero) (Char (Succ vx400)) (not (LT == GT))",fontsize=16,color="black",shape="box"];21 -> 26[label="",style="solid", color="black", weight=3];
22[label="min1 (Char Zero) (Char Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3];
241[label="vx300",fontsize=16,color="green",shape="box"];242[label="vx400",fontsize=16,color="green",shape="box"];243[label="vx300",fontsize=16,color="green",shape="box"];244[label="vx400",fontsize=16,color="green",shape="box"];240[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat vx29 vx30 == GT))",fontsize=16,color="burlywood",shape="triangle"];311[label="vx29/Succ vx290",fontsize=10,color="white",style="solid",shape="box"];240 -> 311[label="",style="solid", color="burlywood", weight=9];
311 -> 281[label="",style="solid", color="burlywood", weight=3];
312[label="vx29/Zero",fontsize=10,color="white",style="solid",shape="box"];240 -> 312[label="",style="solid", color="burlywood", weight=9];
312 -> 282[label="",style="solid", color="burlywood", weight=3];
25[label="min1 (Char (Succ vx300)) (Char Zero) (not True)",fontsize=16,color="black",shape="box"];25 -> 32[label="",style="solid", color="black", weight=3];
26[label="min1 (Char Zero) (Char (Succ vx400)) (not False)",fontsize=16,color="black",shape="box"];26 -> 33[label="",style="solid", color="black", weight=3];
27[label="min1 (Char Zero) (Char Zero) (not False)",fontsize=16,color="black",shape="box"];27 -> 34[label="",style="solid", color="black", weight=3];
281[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat (Succ vx290) vx30 == GT))",fontsize=16,color="burlywood",shape="box"];313[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];281 -> 313[label="",style="solid", color="burlywood", weight=9];
313 -> 283[label="",style="solid", color="burlywood", weight=3];
314[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];281 -> 314[label="",style="solid", color="burlywood", weight=9];
314 -> 284[label="",style="solid", color="burlywood", weight=3];
282[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat Zero vx30 == GT))",fontsize=16,color="burlywood",shape="box"];315[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];282 -> 315[label="",style="solid", color="burlywood", weight=9];
315 -> 285[label="",style="solid", color="burlywood", weight=3];
316[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];282 -> 316[label="",style="solid", color="burlywood", weight=9];
316 -> 286[label="",style="solid", color="burlywood", weight=3];
32[label="min1 (Char (Succ vx300)) (Char Zero) False",fontsize=16,color="black",shape="box"];32 -> 39[label="",style="solid", color="black", weight=3];
33[label="min1 (Char Zero) (Char (Succ vx400)) True",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3];
34[label="min1 (Char Zero) (Char Zero) True",fontsize=16,color="black",shape="box"];34 -> 41[label="",style="solid", color="black", weight=3];
283[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat (Succ vx290) (Succ vx300) == GT))",fontsize=16,color="black",shape="box"];283 -> 287[label="",style="solid", color="black", weight=3];
284[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat (Succ vx290) Zero == GT))",fontsize=16,color="black",shape="box"];284 -> 288[label="",style="solid", color="black", weight=3];
285[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat Zero (Succ vx300) == GT))",fontsize=16,color="black",shape="box"];285 -> 289[label="",style="solid", color="black", weight=3];
286[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];286 -> 290[label="",style="solid", color="black", weight=3];
39[label="min0 (Char (Succ vx300)) (Char Zero) otherwise",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3];
40[label="Char Zero",fontsize=16,color="green",shape="box"];41[label="Char Zero",fontsize=16,color="green",shape="box"];287 -> 240[label="",style="dashed", color="red", weight=0];
287[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (primCmpNat vx290 vx300 == GT))",fontsize=16,color="magenta"];287 -> 291[label="",style="dashed", color="magenta", weight=3];
287 -> 292[label="",style="dashed", color="magenta", weight=3];
288[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (GT == GT))",fontsize=16,color="black",shape="box"];288 -> 293[label="",style="solid", color="black", weight=3];
289[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (LT == GT))",fontsize=16,color="black",shape="box"];289 -> 294[label="",style="solid", color="black", weight=3];
290[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];290 -> 295[label="",style="solid", color="black", weight=3];
47[label="min0 (Char (Succ vx300)) (Char Zero) True",fontsize=16,color="black",shape="box"];47 -> 55[label="",style="solid", color="black", weight=3];
291[label="vx300",fontsize=16,color="green",shape="box"];292[label="vx290",fontsize=16,color="green",shape="box"];293[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not True)",fontsize=16,color="black",shape="box"];293 -> 296[label="",style="solid", color="black", weight=3];
294[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not False)",fontsize=16,color="black",shape="triangle"];294 -> 297[label="",style="solid", color="black", weight=3];
295 -> 294[label="",style="dashed", color="red", weight=0];
295[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) (not False)",fontsize=16,color="magenta"];55[label="Char Zero",fontsize=16,color="green",shape="box"];296[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) False",fontsize=16,color="black",shape="box"];296 -> 298[label="",style="solid", color="black", weight=3];
297[label="min1 (Char (Succ vx27)) (Char (Succ vx28)) True",fontsize=16,color="black",shape="box"];297 -> 299[label="",style="solid", color="black", weight=3];
298[label="min0 (Char (Succ vx27)) (Char (Succ vx28)) otherwise",fontsize=16,color="black",shape="box"];298 -> 300[label="",style="solid", color="black", weight=3];
299[label="Char (Succ vx27)",fontsize=16,color="green",shape="box"];300[label="min0 (Char (Succ vx27)) (Char (Succ vx28)) True",fontsize=16,color="black",shape="box"];300 -> 301[label="",style="solid", color="black", weight=3];
301[label="Char (Succ vx28)",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

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

new_min1(vx27, vx28, Succ(vx290), Succ(vx300)) → new_min1(vx27, vx28, vx290, vx300)

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_min1(vx27, vx28, Succ(vx290), Succ(vx300)) → new_min1(vx27, vx28, vx290, vx300)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4