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

YES 0.625 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

((unwords :: [[Char]] -> [Char]) :: [[Char]] -> [Char])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((unwords :: [[Char]] -> [Char]) :: [[Char]] -> [Char])

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

          ↳ NumRed

mainModule Main

((unwords :: [[Char]] -> [Char]) :: [[Char]] -> [Char])

module Main where

import qualified Prelude

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(unwords :: [[Char]] -> [Char])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="unwords",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="unwords vx3",fontsize=16,color="burlywood",shape="triangle"];28[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 4[label="",style="solid", color="burlywood", weight=3];
29[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="unwords (vx30 : vx31)",fontsize=16,color="burlywood",shape="box"];30[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 6[label="",style="solid", color="burlywood", weight=3];
31[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 7[label="",style="solid", color="burlywood", weight=3];
5[label="unwords []",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
6[label="unwords (vx30 : vx310 : vx311)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="unwords (vx30 : [])",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="[]",fontsize=16,color="green",shape="box"];9 -> 16[label="",style="dashed", color="red", weight=0];
9[label="vx30 ++ Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))) : unwords (vx310 : vx311)",fontsize=16,color="magenta"];9 -> 17[label="",style="dashed", color="magenta", weight=3];
9 -> 18[label="",style="dashed", color="magenta", weight=3];
9 -> 19[label="",style="dashed", color="magenta", weight=3];
10[label="vx30",fontsize=16,color="green",shape="box"];17[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];18[label="vx30",fontsize=16,color="green",shape="box"];19 -> 3[label="",style="dashed", color="red", weight=0];
19[label="unwords (vx310 : vx311)",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
16[label="vx5 ++ Char (Succ vx6) : vx9",fontsize=16,color="burlywood",shape="triangle"];34[label="vx5/vx50 : vx51",fontsize=10,color="white",style="solid",shape="box"];16 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 22[label="",style="solid", color="burlywood", weight=3];
35[label="vx5/[]",fontsize=10,color="white",style="solid",shape="box"];16 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 23[label="",style="solid", color="burlywood", weight=3];
21[label="vx310 : vx311",fontsize=16,color="green",shape="box"];22[label="(vx50 : vx51) ++ Char (Succ vx6) : vx9",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="[] ++ Char (Succ vx6) : vx9",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="vx50 : vx51 ++ Char (Succ vx6) : vx9",fontsize=16,color="green",shape="box"];24 -> 26[label="",style="dashed", color="green", weight=3];
25[label="Char (Succ vx6) : vx9",fontsize=16,color="green",shape="box"];26 -> 16[label="",style="dashed", color="red", weight=0];
26[label="vx51 ++ Char (Succ vx6) : vx9",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
27[label="vx51",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_psPs(:(vx50, vx51), vx6, vx9) → new_psPs(vx51, vx6, vx9)

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_psPs(:(vx50, vx51), vx6, vx9) → new_psPs(vx51, vx6, vx9)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_unwords(:(vx30, :(vx310, vx311))) → new_unwords(:(vx310, vx311))

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

new_unwords(:(vx30, :(vx310, vx311))) → new_unwords(:(vx310, vx311))
The graph contains the following edges 1 > 1