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

YES 0.678 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

  ↳ LR

mainModule Main

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

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\_→q

is transformed to

gtGt0 q _ = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

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

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(>>)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(>>) vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(>>) vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="vy3 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];5 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 6[label="",style="solid", color="burlywood", weight=3];
21[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="vy30 : vy31 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="[] >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 10[label="",style="dashed", color="red", weight=0];
8[label="gtGt0 vy4 vy30 ++ (vy31 >>= gtGt0 vy4)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="[]",fontsize=16,color="green",shape="box"];11 -> 5[label="",style="dashed", color="red", weight=0];
11[label="vy31 >>= gtGt0 vy4",fontsize=16,color="magenta"];11 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="gtGt0 vy4 vy30 ++ vy5",fontsize=16,color="black",shape="triangle"];10 -> 13[label="",style="solid", color="black", weight=3];
12[label="vy31",fontsize=16,color="green",shape="box"];13[label="vy4 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];24[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];13 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 14[label="",style="solid", color="burlywood", weight=3];
25[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="(vy40 : vy41) ++ vy5",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="[] ++ vy5",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="vy40 : vy41 ++ vy5",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="vy5",fontsize=16,color="green",shape="box"];18 -> 13[label="",style="dashed", color="red", weight=0];
18[label="vy41 ++ vy5",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3];
19[label="vy41",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_psPs(:(vy40, vy41), vy5, ba) → new_psPs(vy41, vy5, 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_psPs(:(vy40, vy41), vy5, ba) → new_psPs(vy41, vy5, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_gtGtEs(:(vy30, vy31), vy4, ba, bb) → new_gtGtEs(vy31, vy4, ba, bb)

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

new_gtGtEs(:(vy30, vy31), vy4, ba, bb) → new_gtGtEs(vy31, vy4, ba, bb)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4