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

YES 0.613 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

((foldr1 :: (a -> a -> a) -> [a] -> a) :: (a -> a -> a) -> [a] -> a)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((foldr1 :: (a -> a -> a) -> [a] -> a) :: (a -> a -> a) -> [a] -> a)

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(foldr1 :: (a -> a -> a) -> [a] -> a)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="foldr1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="foldr1 vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="foldr1 vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];15[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 15[label="",style="solid", color="burlywood", weight=9];
15 -> 5[label="",style="solid", color="burlywood", weight=3];
16[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr1 vx3 (vx40 : vx41)",fontsize=16,color="burlywood",shape="box"];17[label="vx41/vx410 : vx411",fontsize=10,color="white",style="solid",shape="box"];5 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 7[label="",style="solid", color="burlywood", weight=3];
18[label="vx41/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 8[label="",style="solid", color="burlywood", weight=3];
6[label="foldr1 vx3 []",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="foldr1 vx3 (vx40 : vx410 : vx411)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="foldr1 vx3 (vx40 : [])",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="error []",fontsize=16,color="red",shape="box"];10[label="vx3 vx40 (foldr1 vx3 (vx410 : vx411))",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
10 -> 13[label="",style="dashed", color="green", weight=3];
11[label="vx40",fontsize=16,color="green",shape="box"];12[label="vx40",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
13[label="foldr1 vx3 (vx410 : vx411)",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
14[label="vx410 : vx411",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_foldr1(vx3, :(vx40, :(vx410, vx411)), ba) → new_foldr1(vx3, :(vx410, vx411), 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_foldr1(vx3, :(vx40, :(vx410, vx411)), ba) → new_foldr1(vx3, :(vx410, vx411), ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3