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

YES 0.884 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

((maximum :: [Ordering] -> Ordering) :: [Ordering] -> Ordering)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((maximum :: [Ordering] -> Ordering) :: [Ordering] -> Ordering)

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

max x y

| x <= y

= y

| otherwise

= x

is transformed to

max x y = max2 x y

max1 x y True = y

max1 x y False = max0 x y otherwise

max0 x y True = x

max2 x y = max1 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

(maximum :: [Ordering] -> Ordering)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="maximum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="maximum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl1 max vx3",fontsize=16,color="burlywood",shape="box"];53[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 5[label="",style="solid", color="burlywood", weight=3];
54[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl1 max (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl1 max []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl max vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];55[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 9[label="",style="solid", color="burlywood", weight=3];
56[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl max vx30 (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl max 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 max (max 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="max 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="max2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="max1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="burlywood",shape="box"];58[label="vx30/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 17[label="",style="solid", color="burlywood", weight=3];
59[label="vx30/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 18[label="",style="solid", color="burlywood", weight=3];
60[label="vx30/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 19[label="",style="solid", color="burlywood", weight=3];
17[label="max1 LT vx310 (LT <= vx310)",fontsize=16,color="burlywood",shape="box"];61[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];17 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 20[label="",style="solid", color="burlywood", weight=3];
62[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];17 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 21[label="",style="solid", color="burlywood", weight=3];
63[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];17 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 22[label="",style="solid", color="burlywood", weight=3];
18[label="max1 EQ vx310 (EQ <= vx310)",fontsize=16,color="burlywood",shape="box"];64[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 23[label="",style="solid", color="burlywood", weight=3];
65[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 24[label="",style="solid", color="burlywood", weight=3];
66[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 25[label="",style="solid", color="burlywood", weight=3];
19[label="max1 GT vx310 (GT <= vx310)",fontsize=16,color="burlywood",shape="box"];67[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 26[label="",style="solid", color="burlywood", weight=3];
68[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 27[label="",style="solid", color="burlywood", weight=3];
69[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 28[label="",style="solid", color="burlywood", weight=3];
20[label="max1 LT LT (LT <= LT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="max1 LT EQ (LT <= EQ)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="max1 LT GT (LT <= GT)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="max1 EQ LT (EQ <= LT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="max1 EQ EQ (EQ <= EQ)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="max1 EQ GT (EQ <= GT)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="max1 GT LT (GT <= LT)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="max1 GT EQ (GT <= EQ)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="max1 GT GT (GT <= GT)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
29[label="max1 LT LT True",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
30[label="max1 LT EQ True",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
31[label="max1 LT GT True",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="max1 EQ LT False",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="max1 EQ EQ True",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
34[label="max1 EQ GT True",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
35[label="max1 GT LT False",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
36[label="max1 GT EQ False",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3];
37[label="max1 GT GT True",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3];
38[label="LT",fontsize=16,color="green",shape="box"];39[label="EQ",fontsize=16,color="green",shape="box"];40[label="GT",fontsize=16,color="green",shape="box"];41[label="max0 EQ LT otherwise",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
42[label="EQ",fontsize=16,color="green",shape="box"];43[label="GT",fontsize=16,color="green",shape="box"];44[label="max0 GT LT otherwise",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
45[label="max0 GT EQ otherwise",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
46[label="GT",fontsize=16,color="green",shape="box"];47[label="max0 EQ LT True",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
48[label="max0 GT LT True",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
49[label="max0 GT EQ True",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
50[label="EQ",fontsize=16,color="green",shape="box"];51[label="GT",fontsize=16,color="green",shape="box"];52[label="GT",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_foldl(vx30, :(vx310, vx311)) → new_foldl(new_max1(vx30, vx310), vx311)

The TRS R consists of the following rules:

new_max1(LT, GT) → GT
new_max1(GT, LT) → GT
new_max1(EQ, GT) → GT
new_max1(GT, EQ) → GT
new_max1(GT, GT) → GT
new_max1(LT, LT) → LT
new_max1(EQ, EQ) → EQ
new_max1(LT, EQ) → EQ
new_max1(EQ, LT) → EQ

The set Q consists of the following terms:

new_max1(LT, GT)
new_max1(GT, LT)
new_max1(EQ, EQ)
new_max1(EQ, GT)
new_max1(GT, EQ)
new_max1(LT, LT)
new_max1(GT, GT)
new_max1(LT, EQ)
new_max1(EQ, LT)

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_max1(vx30, vx310), vx311)
The graph contains the following edges 2 > 2