H-Termination proof of ../tpdb/FP/full_haskell/Prelude_minimum

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_minimum_8.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

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

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)

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

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

module Main where

import qualified Prelude

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(minimum :: [Bool] -> Bool)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="minimum\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="minimum vx3\n",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl1 min vx3\n",fontsize=16,color="burlywood",shape="box"];33[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 5[label="",style="solid", color="burlywood", weight=3];
34[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl1 min (vx30 : vx31)\n",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl1 min []\n",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl min vx30 vx31\n",fontsize=16,color="burlywood",shape="triangle"];35[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 9[label="",style="solid", color="burlywood", weight=3];
36[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="error []\n",fontsize=16,color="red",shape="box"];9[label="foldl min vx30 (vx310 : vx311)\n",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl min vx30 []\n",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\n",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\n",fontsize=16,color="green",shape="box"];13[label="vx311\n",fontsize=16,color="green",shape="box"];14[label="min vx30 vx310\n",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="min2 vx30 vx310\n",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="min1 vx30 vx310 (vx30 <= vx310)\n",fontsize=16,color="burlywood",shape="box"];38[label="vx30/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 17[label="",style="solid", color="burlywood", weight=3];
39[label="vx30/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 18[label="",style="solid", color="burlywood", weight=3];
17[label="min1 False vx310 (False <= vx310)\n",fontsize=16,color="burlywood",shape="box"];40[label="vx310/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 19[label="",style="solid", color="burlywood", weight=3];
41[label="vx310/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 20[label="",style="solid", color="burlywood", weight=3];
18[label="min1 True vx310 (True <= vx310)\n",fontsize=16,color="burlywood",shape="box"];42[label="vx310/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 21[label="",style="solid", color="burlywood", weight=3];
43[label="vx310/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 22[label="",style="solid", color="burlywood", weight=3];
19[label="min1 False False (False <= False)\n",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="min1 False True (False <= True)\n",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="min1 True False (True <= False)\n",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="min1 True True (True <= True)\n",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="min1 False False True\n",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="min1 False True True\n",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="min1 True False False\n",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="min1 True True True\n",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="False\n",fontsize=16,color="green",shape="box"];28[label="False\n",fontsize=16,color="green",shape="box"];29[label="min0 True False otherwise\n",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="True\n",fontsize=16,color="green",shape="box"];31[label="min0 True False True\n",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32[label="False\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_foldl(vx30, :(vx310, vx311)) → new_foldl(new_min1(vx30, vx310), vx311)

The TRS R consists of the following rules:

new_min1(False, False) → False
new_min1(True, True) → True
new_min1(False, True) → False
new_min1(True, False) → False

The set Q consists of the following terms:

new_min1(False, True)
new_min1(True, False)
new_min1(True, True)
new_min1(False, False)

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_min1(vx30, vx310), vx311)
The graph contains the following edges 2 > 2