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

YES 0.686 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

((reverse :: [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

((reverse :: [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

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="reverse",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="reverse vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl (flip (:)) [] vx3",fontsize=16,color="burlywood",shape="box"];35[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 5[label="",style="solid", color="burlywood", weight=3];
36[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl (flip (:)) [] (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl (flip (:)) [] []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl (flip (:)) (flip (:) [] vx30) vx31",fontsize=16,color="burlywood",shape="box"];37[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 9[label="",style="solid", color="burlywood", weight=3];
38[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="[]",fontsize=16,color="green",shape="box"];9[label="foldl (flip (:)) (flip (:) [] vx30) (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl (flip (:)) (flip (:) [] vx30) []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 18[label="",style="dashed", color="red", weight=0];
11[label="foldl (flip (:)) (flip (:) (flip (:) [] vx30) vx310) vx311",fontsize=16,color="magenta"];11 -> 19[label="",style="dashed", color="magenta", weight=3];
11 -> 20[label="",style="dashed", color="magenta", weight=3];
11 -> 21[label="",style="dashed", color="magenta", weight=3];
11 -> 22[label="",style="dashed", color="magenta", weight=3];
12[label="flip (:) [] vx30",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
19[label="vx311",fontsize=16,color="green",shape="box"];20[label="[]",fontsize=16,color="green",shape="box"];21[label="vx30",fontsize=16,color="green",shape="box"];22[label="vx310",fontsize=16,color="green",shape="box"];18[label="foldl (flip (:)) (flip (:) (flip (:) vx4 vx310) vx3110) vx3111",fontsize=16,color="burlywood",shape="triangle"];40[label="vx3111/vx31110 : vx31111",fontsize=10,color="white",style="solid",shape="box"];18 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 24[label="",style="solid", color="burlywood", weight=3];
41[label="vx3111/[]",fontsize=10,color="white",style="solid",shape="box"];18 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 25[label="",style="solid", color="burlywood", weight=3];
15[label="(:) vx30 []",fontsize=16,color="green",shape="box"];24[label="foldl (flip (:)) (flip (:) (flip (:) vx4 vx310) vx3110) (vx31110 : vx31111)",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="foldl (flip (:)) (flip (:) (flip (:) vx4 vx310) vx3110) []",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26 -> 18[label="",style="dashed", color="red", weight=0];
26[label="foldl (flip (:)) (flip (:) (flip (:) (flip (:) vx4 vx310) vx3110) vx31110) vx31111",fontsize=16,color="magenta"];26 -> 28[label="",style="dashed", color="magenta", weight=3];
26 -> 29[label="",style="dashed", color="magenta", weight=3];
26 -> 30[label="",style="dashed", color="magenta", weight=3];
26 -> 31[label="",style="dashed", color="magenta", weight=3];
27[label="flip (:) (flip (:) vx4 vx310) vx3110",fontsize=16,color="black",shape="box"];27 -> 32[label="",style="solid", color="black", weight=3];
28[label="vx31111",fontsize=16,color="green",shape="box"];29[label="flip (:) vx4 vx310",fontsize=16,color="black",shape="triangle"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="vx3110",fontsize=16,color="green",shape="box"];31[label="vx31110",fontsize=16,color="green",shape="box"];32[label="(:) vx3110 flip (:) vx4 vx310",fontsize=16,color="green",shape="box"];32 -> 34[label="",style="dashed", color="green", weight=3];
33[label="(:) vx310 vx4",fontsize=16,color="green",shape="box"];34 -> 29[label="",style="dashed", color="red", weight=0];
34[label="flip (:) vx4 vx310",fontsize=16,color="magenta"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

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

new_foldl(vx4, vx310, vx3110, :(vx31110, vx31111), ba) → new_foldl(new_flip(vx4, vx310, ba), vx3110, vx31110, vx31111, ba)

The TRS R consists of the following rules:

new_flip(vx4, vx310, ba) → :(vx310, vx4)

The set Q consists of the following terms:

new_flip(x0, x1, x2)

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(vx4, vx310, vx3110, :(vx31110, vx31111), ba) → new_foldl(new_flip(vx4, vx310, ba), vx3110, vx31110, vx31111, ba)
The graph contains the following edges 3 >= 2, 4 > 3, 4 > 4, 5 >= 5