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

YES 1.181 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

((minimum :: [Int] -> Int) :: [Int] -> Int)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((minimum :: [Int] -> Int) :: [Int] -> Int)

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

min1 x y True = x

min1 x y False = min0 x y otherwise

min0 x y True = y

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

(minimum :: [Int] -> Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="minimum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="minimum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl1 min vx3",fontsize=16,color="burlywood",shape="box"];601[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 601[label="",style="solid", color="burlywood", weight=9];
601 -> 5[label="",style="solid", color="burlywood", weight=3];
602[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 602[label="",style="solid", color="burlywood", weight=9];
602 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl1 min (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl1 min []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl min vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];603[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 603[label="",style="solid", color="burlywood", weight=9];
603 -> 9[label="",style="solid", color="burlywood", weight=3];
604[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 604[label="",style="solid", color="burlywood", weight=9];
604 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl min vx30 (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl min vx30 []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 7[label="",style="dashed", color="red", weight=0];
11[label="foldl min (min vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="vx30",fontsize=16,color="green",shape="box"];13[label="min vx30 vx310",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="vx311",fontsize=16,color="green",shape="box"];15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="min1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="min1 vx30 vx310 (compare vx30 vx310 /= GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="min1 vx30 vx310 (not (compare vx30 vx310 == GT))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="min1 vx30 vx310 (not (primCmpInt vx30 vx310 == GT))",fontsize=16,color="burlywood",shape="box"];606[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];19 -> 606[label="",style="solid", color="burlywood", weight=9];
606 -> 20[label="",style="solid", color="burlywood", weight=3];
607[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];19 -> 607[label="",style="solid", color="burlywood", weight=9];
607 -> 21[label="",style="solid", color="burlywood", weight=3];
20[label="min1 (Pos vx300) vx310 (not (primCmpInt (Pos vx300) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];608[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];20 -> 608[label="",style="solid", color="burlywood", weight=9];
608 -> 22[label="",style="solid", color="burlywood", weight=3];
609[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 609[label="",style="solid", color="burlywood", weight=9];
609 -> 23[label="",style="solid", color="burlywood", weight=3];
21[label="min1 (Neg vx300) vx310 (not (primCmpInt (Neg vx300) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];610[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];21 -> 610[label="",style="solid", color="burlywood", weight=9];
610 -> 24[label="",style="solid", color="burlywood", weight=3];
611[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 611[label="",style="solid", color="burlywood", weight=9];
611 -> 25[label="",style="solid", color="burlywood", weight=3];
22[label="min1 (Pos (Succ vx3000)) vx310 (not (primCmpInt (Pos (Succ vx3000)) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];612[label="vx310/Pos vx3100",fontsize=10,color="white",style="solid",shape="box"];22 -> 612[label="",style="solid", color="burlywood", weight=9];
612 -> 26[label="",style="solid", color="burlywood", weight=3];
613[label="vx310/Neg vx3100",fontsize=10,color="white",style="solid",shape="box"];22 -> 613[label="",style="solid", color="burlywood", weight=9];
613 -> 27[label="",style="solid", color="burlywood", weight=3];
23[label="min1 (Pos Zero) vx310 (not (primCmpInt (Pos Zero) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];614[label="vx310/Pos vx3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 614[label="",style="solid", color="burlywood", weight=9];
614 -> 28[label="",style="solid", color="burlywood", weight=3];
615[label="vx310/Neg vx3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 615[label="",style="solid", color="burlywood", weight=9];
615 -> 29[label="",style="solid", color="burlywood", weight=3];
24[label="min1 (Neg (Succ vx3000)) vx310 (not (primCmpInt (Neg (Succ vx3000)) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];616[label="vx310/Pos vx3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 616[label="",style="solid", color="burlywood", weight=9];
616 -> 30[label="",style="solid", color="burlywood", weight=3];
617[label="vx310/Neg vx3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 617[label="",style="solid", color="burlywood", weight=9];
617 -> 31[label="",style="solid", color="burlywood", weight=3];
25[label="min1 (Neg Zero) vx310 (not (primCmpInt (Neg Zero) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];618[label="vx310/Pos vx3100",fontsize=10,color="white",style="solid",shape="box"];25 -> 618[label="",style="solid", color="burlywood", weight=9];
618 -> 32[label="",style="solid", color="burlywood", weight=3];
619[label="vx310/Neg vx3100",fontsize=10,color="white",style="solid",shape="box"];25 -> 619[label="",style="solid", color="burlywood", weight=9];
619 -> 33[label="",style="solid", color="burlywood", weight=3];
26[label="min1 (Pos (Succ vx3000)) (Pos vx3100) (not (primCmpInt (Pos (Succ vx3000)) (Pos vx3100) == GT))",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
27[label="min1 (Pos (Succ vx3000)) (Neg vx3100) (not (primCmpInt (Pos (Succ vx3000)) (Neg vx3100) == GT))",fontsize=16,color="black",shape="box"];27 -> 35[label="",style="solid", color="black", weight=3];
28[label="min1 (Pos Zero) (Pos vx3100) (not (primCmpInt (Pos Zero) (Pos vx3100) == GT))",fontsize=16,color="burlywood",shape="box"];620[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];28 -> 620[label="",style="solid", color="burlywood", weight=9];
620 -> 36[label="",style="solid", color="burlywood", weight=3];
621[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 621[label="",style="solid", color="burlywood", weight=9];
621 -> 37[label="",style="solid", color="burlywood", weight=3];
29[label="min1 (Pos Zero) (Neg vx3100) (not (primCmpInt (Pos Zero) (Neg vx3100) == GT))",fontsize=16,color="burlywood",shape="box"];622[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];29 -> 622[label="",style="solid", color="burlywood", weight=9];
622 -> 38[label="",style="solid", color="burlywood", weight=3];
623[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 623[label="",style="solid", color="burlywood", weight=9];
623 -> 39[label="",style="solid", color="burlywood", weight=3];
30[label="min1 (Neg (Succ vx3000)) (Pos vx3100) (not (primCmpInt (Neg (Succ vx3000)) (Pos vx3100) == GT))",fontsize=16,color="black",shape="box"];30 -> 40[label="",style="solid", color="black", weight=3];
31[label="min1 (Neg (Succ vx3000)) (Neg vx3100) (not (primCmpInt (Neg (Succ vx3000)) (Neg vx3100) == GT))",fontsize=16,color="black",shape="box"];31 -> 41[label="",style="solid", color="black", weight=3];
32[label="min1 (Neg Zero) (Pos vx3100) (not (primCmpInt (Neg Zero) (Pos vx3100) == GT))",fontsize=16,color="burlywood",shape="box"];624[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];32 -> 624[label="",style="solid", color="burlywood", weight=9];
624 -> 42[label="",style="solid", color="burlywood", weight=3];
625[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 625[label="",style="solid", color="burlywood", weight=9];
625 -> 43[label="",style="solid", color="burlywood", weight=3];
33[label="min1 (Neg Zero) (Neg vx3100) (not (primCmpInt (Neg Zero) (Neg vx3100) == GT))",fontsize=16,color="burlywood",shape="box"];626[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];33 -> 626[label="",style="solid", color="burlywood", weight=9];
626 -> 44[label="",style="solid", color="burlywood", weight=3];
627[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 627[label="",style="solid", color="burlywood", weight=9];
627 -> 45[label="",style="solid", color="burlywood", weight=3];
34[label="min1 (Pos (Succ vx3000)) (Pos vx3100) (not (primCmpNat (Succ vx3000) vx3100 == GT))",fontsize=16,color="burlywood",shape="box"];628[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];34 -> 628[label="",style="solid", color="burlywood", weight=9];
628 -> 46[label="",style="solid", color="burlywood", weight=3];
629[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 629[label="",style="solid", color="burlywood", weight=9];
629 -> 47[label="",style="solid", color="burlywood", weight=3];
35[label="min1 (Pos (Succ vx3000)) (Neg vx3100) (not (GT == GT))",fontsize=16,color="black",shape="box"];35 -> 48[label="",style="solid", color="black", weight=3];
36[label="min1 (Pos Zero) (Pos (Succ vx31000)) (not (primCmpInt (Pos Zero) (Pos (Succ vx31000)) == GT))",fontsize=16,color="black",shape="box"];36 -> 49[label="",style="solid", color="black", weight=3];
37[label="min1 (Pos Zero) (Pos Zero) (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];37 -> 50[label="",style="solid", color="black", weight=3];
38[label="min1 (Pos Zero) (Neg (Succ vx31000)) (not (primCmpInt (Pos Zero) (Neg (Succ vx31000)) == GT))",fontsize=16,color="black",shape="box"];38 -> 51[label="",style="solid", color="black", weight=3];
39[label="min1 (Pos Zero) (Neg Zero) (not (primCmpInt (Pos Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];39 -> 52[label="",style="solid", color="black", weight=3];
40[label="min1 (Neg (Succ vx3000)) (Pos vx3100) (not (LT == GT))",fontsize=16,color="black",shape="box"];40 -> 53[label="",style="solid", color="black", weight=3];
41[label="min1 (Neg (Succ vx3000)) (Neg vx3100) (not (primCmpNat vx3100 (Succ vx3000) == GT))",fontsize=16,color="burlywood",shape="box"];630[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];41 -> 630[label="",style="solid", color="burlywood", weight=9];
630 -> 54[label="",style="solid", color="burlywood", weight=3];
631[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 631[label="",style="solid", color="burlywood", weight=9];
631 -> 55[label="",style="solid", color="burlywood", weight=3];
42[label="min1 (Neg Zero) (Pos (Succ vx31000)) (not (primCmpInt (Neg Zero) (Pos (Succ vx31000)) == GT))",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3];
43[label="min1 (Neg Zero) (Pos Zero) (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
44[label="min1 (Neg Zero) (Neg (Succ vx31000)) (not (primCmpInt (Neg Zero) (Neg (Succ vx31000)) == GT))",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3];
45[label="min1 (Neg Zero) (Neg Zero) (not (primCmpInt (Neg Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3];
46[label="min1 (Pos (Succ vx3000)) (Pos (Succ vx31000)) (not (primCmpNat (Succ vx3000) (Succ vx31000) == GT))",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3];
47[label="min1 (Pos (Succ vx3000)) (Pos Zero) (not (primCmpNat (Succ vx3000) Zero == GT))",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3];
48[label="min1 (Pos (Succ vx3000)) (Neg vx3100) (not True)",fontsize=16,color="black",shape="box"];48 -> 62[label="",style="solid", color="black", weight=3];
49[label="min1 (Pos Zero) (Pos (Succ vx31000)) (not (primCmpNat Zero (Succ vx31000) == GT))",fontsize=16,color="black",shape="box"];49 -> 63[label="",style="solid", color="black", weight=3];
50[label="min1 (Pos Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];50 -> 64[label="",style="solid", color="black", weight=3];
51[label="min1 (Pos Zero) (Neg (Succ vx31000)) (not (GT == GT))",fontsize=16,color="black",shape="box"];51 -> 65[label="",style="solid", color="black", weight=3];
52[label="min1 (Pos Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];52 -> 66[label="",style="solid", color="black", weight=3];
53[label="min1 (Neg (Succ vx3000)) (Pos vx3100) (not False)",fontsize=16,color="black",shape="box"];53 -> 67[label="",style="solid", color="black", weight=3];
54[label="min1 (Neg (Succ vx3000)) (Neg (Succ vx31000)) (not (primCmpNat (Succ vx31000) (Succ vx3000) == GT))",fontsize=16,color="black",shape="box"];54 -> 68[label="",style="solid", color="black", weight=3];
55[label="min1 (Neg (Succ vx3000)) (Neg Zero) (not (primCmpNat Zero (Succ vx3000) == GT))",fontsize=16,color="black",shape="box"];55 -> 69[label="",style="solid", color="black", weight=3];
56[label="min1 (Neg Zero) (Pos (Succ vx31000)) (not (LT == GT))",fontsize=16,color="black",shape="box"];56 -> 70[label="",style="solid", color="black", weight=3];
57[label="min1 (Neg Zero) (Pos Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];57 -> 71[label="",style="solid", color="black", weight=3];
58[label="min1 (Neg Zero) (Neg (Succ vx31000)) (not (primCmpNat (Succ vx31000) Zero == GT))",fontsize=16,color="black",shape="box"];58 -> 72[label="",style="solid", color="black", weight=3];
59[label="min1 (Neg Zero) (Neg Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];59 -> 73[label="",style="solid", color="black", weight=3];
60 -> 463[label="",style="dashed", color="red", weight=0];
60[label="min1 (Pos (Succ vx3000)) (Pos (Succ vx31000)) (not (primCmpNat vx3000 vx31000 == GT))",fontsize=16,color="magenta"];60 -> 464[label="",style="dashed", color="magenta", weight=3];
60 -> 465[label="",style="dashed", color="magenta", weight=3];
60 -> 466[label="",style="dashed", color="magenta", weight=3];
60 -> 467[label="",style="dashed", color="magenta", weight=3];
61[label="min1 (Pos (Succ vx3000)) (Pos Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];61 -> 76[label="",style="solid", color="black", weight=3];
62[label="min1 (Pos (Succ vx3000)) (Neg vx3100) False",fontsize=16,color="black",shape="box"];62 -> 77[label="",style="solid", color="black", weight=3];
63[label="min1 (Pos Zero) (Pos (Succ vx31000)) (not (LT == GT))",fontsize=16,color="black",shape="box"];63 -> 78[label="",style="solid", color="black", weight=3];
64[label="min1 (Pos Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];64 -> 79[label="",style="solid", color="black", weight=3];
65[label="min1 (Pos Zero) (Neg (Succ vx31000)) (not True)",fontsize=16,color="black",shape="box"];65 -> 80[label="",style="solid", color="black", weight=3];
66[label="min1 (Pos Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];66 -> 81[label="",style="solid", color="black", weight=3];
67[label="min1 (Neg (Succ vx3000)) (Pos vx3100) True",fontsize=16,color="black",shape="box"];67 -> 82[label="",style="solid", color="black", weight=3];
68 -> 533[label="",style="dashed", color="red", weight=0];
68[label="min1 (Neg (Succ vx3000)) (Neg (Succ vx31000)) (not (primCmpNat vx31000 vx3000 == GT))",fontsize=16,color="magenta"];68 -> 534[label="",style="dashed", color="magenta", weight=3];
68 -> 535[label="",style="dashed", color="magenta", weight=3];
68 -> 536[label="",style="dashed", color="magenta", weight=3];
68 -> 537[label="",style="dashed", color="magenta", weight=3];
69[label="min1 (Neg (Succ vx3000)) (Neg Zero) (not (LT == GT))",fontsize=16,color="black",shape="box"];69 -> 85[label="",style="solid", color="black", weight=3];
70[label="min1 (Neg Zero) (Pos (Succ vx31000)) (not False)",fontsize=16,color="black",shape="box"];70 -> 86[label="",style="solid", color="black", weight=3];
71[label="min1 (Neg Zero) (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];71 -> 87[label="",style="solid", color="black", weight=3];
72[label="min1 (Neg Zero) (Neg (Succ vx31000)) (not (GT == GT))",fontsize=16,color="black",shape="box"];72 -> 88[label="",style="solid", color="black", weight=3];
73[label="min1 (Neg Zero) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];73 -> 89[label="",style="solid", color="black", weight=3];
464[label="vx3000",fontsize=16,color="green",shape="box"];465[label="vx3000",fontsize=16,color="green",shape="box"];466[label="vx31000",fontsize=16,color="green",shape="box"];467[label="vx31000",fontsize=16,color="green",shape="box"];463[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat vx42 vx43 == GT))",fontsize=16,color="burlywood",shape="triangle"];634[label="vx42/Succ vx420",fontsize=10,color="white",style="solid",shape="box"];463 -> 634[label="",style="solid", color="burlywood", weight=9];
634 -> 500[label="",style="solid", color="burlywood", weight=3];
635[label="vx42/Zero",fontsize=10,color="white",style="solid",shape="box"];463 -> 635[label="",style="solid", color="burlywood", weight=9];
635 -> 501[label="",style="solid", color="burlywood", weight=3];
76[label="min1 (Pos (Succ vx3000)) (Pos Zero) (not True)",fontsize=16,color="black",shape="box"];76 -> 94[label="",style="solid", color="black", weight=3];
77[label="min0 (Pos (Succ vx3000)) (Neg vx3100) otherwise",fontsize=16,color="black",shape="box"];77 -> 95[label="",style="solid", color="black", weight=3];
78[label="min1 (Pos Zero) (Pos (Succ vx31000)) (not False)",fontsize=16,color="black",shape="box"];78 -> 96[label="",style="solid", color="black", weight=3];
79[label="min1 (Pos Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];79 -> 97[label="",style="solid", color="black", weight=3];
80[label="min1 (Pos Zero) (Neg (Succ vx31000)) False",fontsize=16,color="black",shape="box"];80 -> 98[label="",style="solid", color="black", weight=3];
81[label="min1 (Pos Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];81 -> 99[label="",style="solid", color="black", weight=3];
82[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];534[label="vx31000",fontsize=16,color="green",shape="box"];535[label="vx3000",fontsize=16,color="green",shape="box"];536[label="vx3000",fontsize=16,color="green",shape="box"];537[label="vx31000",fontsize=16,color="green",shape="box"];533[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat vx51 vx52 == GT))",fontsize=16,color="burlywood",shape="triangle"];636[label="vx51/Succ vx510",fontsize=10,color="white",style="solid",shape="box"];533 -> 636[label="",style="solid", color="burlywood", weight=9];
636 -> 574[label="",style="solid", color="burlywood", weight=3];
637[label="vx51/Zero",fontsize=10,color="white",style="solid",shape="box"];533 -> 637[label="",style="solid", color="burlywood", weight=9];
637 -> 575[label="",style="solid", color="burlywood", weight=3];
85[label="min1 (Neg (Succ vx3000)) (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];85 -> 104[label="",style="solid", color="black", weight=3];
86[label="min1 (Neg Zero) (Pos (Succ vx31000)) True",fontsize=16,color="black",shape="box"];86 -> 105[label="",style="solid", color="black", weight=3];
87[label="min1 (Neg Zero) (Pos Zero) True",fontsize=16,color="black",shape="box"];87 -> 106[label="",style="solid", color="black", weight=3];
88[label="min1 (Neg Zero) (Neg (Succ vx31000)) (not True)",fontsize=16,color="black",shape="box"];88 -> 107[label="",style="solid", color="black", weight=3];
89[label="min1 (Neg Zero) (Neg Zero) True",fontsize=16,color="black",shape="box"];89 -> 108[label="",style="solid", color="black", weight=3];
500[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat (Succ vx420) vx43 == GT))",fontsize=16,color="burlywood",shape="box"];638[label="vx43/Succ vx430",fontsize=10,color="white",style="solid",shape="box"];500 -> 638[label="",style="solid", color="burlywood", weight=9];
638 -> 506[label="",style="solid", color="burlywood", weight=3];
639[label="vx43/Zero",fontsize=10,color="white",style="solid",shape="box"];500 -> 639[label="",style="solid", color="burlywood", weight=9];
639 -> 507[label="",style="solid", color="burlywood", weight=3];
501[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat Zero vx43 == GT))",fontsize=16,color="burlywood",shape="box"];640[label="vx43/Succ vx430",fontsize=10,color="white",style="solid",shape="box"];501 -> 640[label="",style="solid", color="burlywood", weight=9];
640 -> 508[label="",style="solid", color="burlywood", weight=3];
641[label="vx43/Zero",fontsize=10,color="white",style="solid",shape="box"];501 -> 641[label="",style="solid", color="burlywood", weight=9];
641 -> 509[label="",style="solid", color="burlywood", weight=3];
94[label="min1 (Pos (Succ vx3000)) (Pos Zero) False",fontsize=16,color="black",shape="box"];94 -> 113[label="",style="solid", color="black", weight=3];
95[label="min0 (Pos (Succ vx3000)) (Neg vx3100) True",fontsize=16,color="black",shape="box"];95 -> 114[label="",style="solid", color="black", weight=3];
96[label="min1 (Pos Zero) (Pos (Succ vx31000)) True",fontsize=16,color="black",shape="box"];96 -> 115[label="",style="solid", color="black", weight=3];
97[label="Pos Zero",fontsize=16,color="green",shape="box"];98[label="min0 (Pos Zero) (Neg (Succ vx31000)) otherwise",fontsize=16,color="black",shape="box"];98 -> 116[label="",style="solid", color="black", weight=3];
99[label="Pos Zero",fontsize=16,color="green",shape="box"];574[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat (Succ vx510) vx52 == GT))",fontsize=16,color="burlywood",shape="box"];642[label="vx52/Succ vx520",fontsize=10,color="white",style="solid",shape="box"];574 -> 642[label="",style="solid", color="burlywood", weight=9];
642 -> 578[label="",style="solid", color="burlywood", weight=3];
643[label="vx52/Zero",fontsize=10,color="white",style="solid",shape="box"];574 -> 643[label="",style="solid", color="burlywood", weight=9];
643 -> 579[label="",style="solid", color="burlywood", weight=3];
575[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat Zero vx52 == GT))",fontsize=16,color="burlywood",shape="box"];644[label="vx52/Succ vx520",fontsize=10,color="white",style="solid",shape="box"];575 -> 644[label="",style="solid", color="burlywood", weight=9];
644 -> 580[label="",style="solid", color="burlywood", weight=3];
645[label="vx52/Zero",fontsize=10,color="white",style="solid",shape="box"];575 -> 645[label="",style="solid", color="burlywood", weight=9];
645 -> 581[label="",style="solid", color="burlywood", weight=3];
104[label="min1 (Neg (Succ vx3000)) (Neg Zero) True",fontsize=16,color="black",shape="box"];104 -> 121[label="",style="solid", color="black", weight=3];
105[label="Neg Zero",fontsize=16,color="green",shape="box"];106[label="Neg Zero",fontsize=16,color="green",shape="box"];107[label="min1 (Neg Zero) (Neg (Succ vx31000)) False",fontsize=16,color="black",shape="box"];107 -> 122[label="",style="solid", color="black", weight=3];
108[label="Neg Zero",fontsize=16,color="green",shape="box"];506[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat (Succ vx420) (Succ vx430) == GT))",fontsize=16,color="black",shape="box"];506 -> 514[label="",style="solid", color="black", weight=3];
507[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat (Succ vx420) Zero == GT))",fontsize=16,color="black",shape="box"];507 -> 515[label="",style="solid", color="black", weight=3];
508[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat Zero (Succ vx430) == GT))",fontsize=16,color="black",shape="box"];508 -> 516[label="",style="solid", color="black", weight=3];
509[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];509 -> 517[label="",style="solid", color="black", weight=3];
113[label="min0 (Pos (Succ vx3000)) (Pos Zero) otherwise",fontsize=16,color="black",shape="box"];113 -> 128[label="",style="solid", color="black", weight=3];
114[label="Neg vx3100",fontsize=16,color="green",shape="box"];115[label="Pos Zero",fontsize=16,color="green",shape="box"];116[label="min0 (Pos Zero) (Neg (Succ vx31000)) True",fontsize=16,color="black",shape="box"];116 -> 129[label="",style="solid", color="black", weight=3];
578[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat (Succ vx510) (Succ vx520) == GT))",fontsize=16,color="black",shape="box"];578 -> 584[label="",style="solid", color="black", weight=3];
579[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat (Succ vx510) Zero == GT))",fontsize=16,color="black",shape="box"];579 -> 585[label="",style="solid", color="black", weight=3];
580[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat Zero (Succ vx520) == GT))",fontsize=16,color="black",shape="box"];580 -> 586[label="",style="solid", color="black", weight=3];
581[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];581 -> 587[label="",style="solid", color="black", weight=3];
121[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];122[label="min0 (Neg Zero) (Neg (Succ vx31000)) otherwise",fontsize=16,color="black",shape="box"];122 -> 135[label="",style="solid", color="black", weight=3];
514 -> 463[label="",style="dashed", color="red", weight=0];
514[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (primCmpNat vx420 vx430 == GT))",fontsize=16,color="magenta"];514 -> 522[label="",style="dashed", color="magenta", weight=3];
514 -> 523[label="",style="dashed", color="magenta", weight=3];
515[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (GT == GT))",fontsize=16,color="black",shape="box"];515 -> 524[label="",style="solid", color="black", weight=3];
516[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (LT == GT))",fontsize=16,color="black",shape="box"];516 -> 525[label="",style="solid", color="black", weight=3];
517[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];517 -> 526[label="",style="solid", color="black", weight=3];
128[label="min0 (Pos (Succ vx3000)) (Pos Zero) True",fontsize=16,color="black",shape="box"];128 -> 143[label="",style="solid", color="black", weight=3];
129[label="Neg (Succ vx31000)",fontsize=16,color="green",shape="box"];584 -> 533[label="",style="dashed", color="red", weight=0];
584[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (primCmpNat vx510 vx520 == GT))",fontsize=16,color="magenta"];584 -> 589[label="",style="dashed", color="magenta", weight=3];
584 -> 590[label="",style="dashed", color="magenta", weight=3];
585[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (GT == GT))",fontsize=16,color="black",shape="box"];585 -> 591[label="",style="solid", color="black", weight=3];
586[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (LT == GT))",fontsize=16,color="black",shape="box"];586 -> 592[label="",style="solid", color="black", weight=3];
587[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];587 -> 593[label="",style="solid", color="black", weight=3];
135[label="min0 (Neg Zero) (Neg (Succ vx31000)) True",fontsize=16,color="black",shape="box"];135 -> 151[label="",style="solid", color="black", weight=3];
522[label="vx420",fontsize=16,color="green",shape="box"];523[label="vx430",fontsize=16,color="green",shape="box"];524[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not True)",fontsize=16,color="black",shape="box"];524 -> 576[label="",style="solid", color="black", weight=3];
525[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not False)",fontsize=16,color="black",shape="triangle"];525 -> 577[label="",style="solid", color="black", weight=3];
526 -> 525[label="",style="dashed", color="red", weight=0];
526[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) (not False)",fontsize=16,color="magenta"];143[label="Pos Zero",fontsize=16,color="green",shape="box"];589[label="vx520",fontsize=16,color="green",shape="box"];590[label="vx510",fontsize=16,color="green",shape="box"];591[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not True)",fontsize=16,color="black",shape="box"];591 -> 595[label="",style="solid", color="black", weight=3];
592[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not False)",fontsize=16,color="black",shape="triangle"];592 -> 596[label="",style="solid", color="black", weight=3];
593 -> 592[label="",style="dashed", color="red", weight=0];
593[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) (not False)",fontsize=16,color="magenta"];151[label="Neg (Succ vx31000)",fontsize=16,color="green",shape="box"];576[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) False",fontsize=16,color="black",shape="box"];576 -> 582[label="",style="solid", color="black", weight=3];
577[label="min1 (Pos (Succ vx40)) (Pos (Succ vx41)) True",fontsize=16,color="black",shape="box"];577 -> 583[label="",style="solid", color="black", weight=3];
595[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) False",fontsize=16,color="black",shape="box"];595 -> 597[label="",style="solid", color="black", weight=3];
596[label="min1 (Neg (Succ vx49)) (Neg (Succ vx50)) True",fontsize=16,color="black",shape="box"];596 -> 598[label="",style="solid", color="black", weight=3];
582[label="min0 (Pos (Succ vx40)) (Pos (Succ vx41)) otherwise",fontsize=16,color="black",shape="box"];582 -> 588[label="",style="solid", color="black", weight=3];
583[label="Pos (Succ vx40)",fontsize=16,color="green",shape="box"];597[label="min0 (Neg (Succ vx49)) (Neg (Succ vx50)) otherwise",fontsize=16,color="black",shape="box"];597 -> 599[label="",style="solid", color="black", weight=3];
598[label="Neg (Succ vx49)",fontsize=16,color="green",shape="box"];588[label="min0 (Pos (Succ vx40)) (Pos (Succ vx41)) True",fontsize=16,color="black",shape="box"];588 -> 594[label="",style="solid", color="black", weight=3];
599[label="min0 (Neg (Succ vx49)) (Neg (Succ vx50)) True",fontsize=16,color="black",shape="box"];599 -> 600[label="",style="solid", color="black", weight=3];
594[label="Pos (Succ vx41)",fontsize=16,color="green",shape="box"];600[label="Neg (Succ vx50)",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

new_min1(vx49, vx50, Succ(vx510), Succ(vx520)) → new_min1(vx49, vx50, vx510, vx520)

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(vx49, vx50, Succ(vx510), Succ(vx520)) → new_min1(vx49, vx50, vx510, vx520)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_min10(vx40, vx41, Succ(vx420), Succ(vx430)) → new_min10(vx40, vx41, vx420, vx430)

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

new_min10(vx40, vx41, Succ(vx420), Succ(vx430)) → new_min10(vx40, vx41, vx420, vx430)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_foldl(vx30, :(vx310, vx311)) → new_foldl(new_min11(vx30, vx310), vx311)

The TRS R consists of the following rules:

new_min11(Pos(Zero), Pos(Succ(vx31000))) → Pos(Zero)
new_min11(Pos(Succ(vx3000)), Pos(Zero)) → Pos(Zero)
new_min11(Neg(Succ(vx3000)), Pos(vx3100)) → Neg(Succ(vx3000))
new_min11(Neg(Zero), Pos(Succ(vx31000))) → Neg(Zero)
new_min11(Pos(Zero), Neg(Zero)) → Pos(Zero)
new_min15(vx40, vx41, Zero, Zero) → new_min13(vx40, vx41)
new_min14(vx49, vx50, Succ(vx510), Succ(vx520)) → new_min14(vx49, vx50, vx510, vx520)
new_min11(Neg(Zero), Neg(Zero)) → Neg(Zero)
new_min11(Neg(Zero), Pos(Zero)) → Neg(Zero)
new_min11(Pos(Succ(vx3000)), Neg(vx3100)) → Neg(vx3100)
new_min11(Neg(Succ(vx3000)), Neg(Succ(vx31000))) → new_min14(vx3000, vx31000, vx31000, vx3000)
new_min15(vx40, vx41, Succ(vx420), Succ(vx430)) → new_min15(vx40, vx41, vx420, vx430)
new_min15(vx40, vx41, Succ(vx420), Zero) → Pos(Succ(vx41))
new_min14(vx49, vx50, Zero, Succ(vx520)) → new_min12(vx49, vx50)
new_min13(vx40, vx41) → Pos(Succ(vx40))
new_min11(Pos(Zero), Neg(Succ(vx31000))) → Neg(Succ(vx31000))
new_min14(vx49, vx50, Zero, Zero) → new_min12(vx49, vx50)
new_min11(Pos(Zero), Pos(Zero)) → Pos(Zero)
new_min12(vx49, vx50) → Neg(Succ(vx49))
new_min14(vx49, vx50, Succ(vx510), Zero) → Neg(Succ(vx50))
new_min15(vx40, vx41, Zero, Succ(vx430)) → new_min13(vx40, vx41)
new_min11(Pos(Succ(vx3000)), Pos(Succ(vx31000))) → new_min15(vx3000, vx31000, vx3000, vx31000)
new_min11(Neg(Succ(vx3000)), Neg(Zero)) → Neg(Succ(vx3000))
new_min11(Neg(Zero), Neg(Succ(vx31000))) → Neg(Succ(vx31000))

The set Q consists of the following terms:

new_min11(Neg(Succ(x0)), Neg(Succ(x1)))
new_min14(x0, x1, Succ(x2), Succ(x3))
new_min11(Pos(Zero), Pos(Zero))
new_min11(Pos(Zero), Pos(Succ(x0)))
new_min11(Pos(Succ(x0)), Pos(Succ(x1)))
new_min11(Neg(Zero), Neg(Zero))
new_min11(Neg(Zero), Pos(Zero))
new_min11(Pos(Zero), Neg(Zero))
new_min11(Neg(Succ(x0)), Neg(Zero))
new_min15(x0, x1, Zero, Succ(x2))
new_min15(x0, x1, Succ(x2), Succ(x3))
new_min14(x0, x1, Zero, Succ(x2))
new_min13(x0, x1)
new_min14(x0, x1, Succ(x2), Zero)
new_min15(x0, x1, Zero, Zero)
new_min11(Pos(Succ(x0)), Neg(x1))
new_min11(Neg(Succ(x0)), Pos(x1))
new_min11(Pos(Succ(x0)), Pos(Zero))
new_min11(Neg(Zero), Pos(Succ(x0)))
new_min11(Pos(Zero), Neg(Succ(x0)))
new_min14(x0, x1, Zero, Zero)
new_min15(x0, x1, Succ(x2), Zero)
new_min12(x0, x1)
new_min11(Neg(Zero), Neg(Succ(x0)))

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_foldl(vx30, :(vx310, vx311)) → new_foldl(new_min11(vx30, vx310), vx311)
The graph contains the following edges 2 > 2