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

YES 0.641 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

  ↳ CR

mainModule Main

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

module Main where

import qualified Prelude

Case Reductions:
The following Case expression

case xs of
[] → []

x : xs → scanl f (f q x) xs

is transformed to

scanl0 f q [] = []

scanl0 f q (x : xs) = scanl f (f q x) xs

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

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

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="scanl",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="scanl vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="scanl vx3 vx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="scanl vx3 vx4 vx5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="vx4 : scanl0 vx3 vx4 vx5",fontsize=16,color="green",shape="box"];6 -> 7[label="",style="dashed", color="green", weight=3];
7[label="scanl0 vx3 vx4 vx5",fontsize=16,color="burlywood",shape="box"];16[label="vx5/vx50 : vx51",fontsize=10,color="white",style="solid",shape="box"];7 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 8[label="",style="solid", color="burlywood", weight=3];
17[label="vx5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="scanl0 vx3 vx4 (vx50 : vx51)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="scanl0 vx3 vx4 []",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10 -> 5[label="",style="dashed", color="red", weight=0];
10[label="scanl vx3 (vx3 vx4 vx50) vx51",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
10 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="[]",fontsize=16,color="green",shape="box"];12[label="vx51",fontsize=16,color="green",shape="box"];13[label="vx3 vx4 vx50",fontsize=16,color="green",shape="box"];13 -> 14[label="",style="dashed", color="green", weight=3];
13 -> 15[label="",style="dashed", color="green", weight=3];
14[label="vx4",fontsize=16,color="green",shape="box"];15[label="vx50",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_scanl(vx3, :(vx50, vx51), ba, bb) → new_scanl(vx3, vx51, ba, bb)

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_scanl(vx3, :(vx50, vx51), ba, bb) → new_scanl(vx3, vx51, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4