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

YES 0.62 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

((scanl1 :: (a -> a -> a) -> [a] -> [a]) :: (a -> a -> a) -> [a] -> [a])

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

((scanl1 :: (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

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((scanl1 :: (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

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(scanl1 :: (a -> a -> a) -> [a] -> [a])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="scanl1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="scanl1 vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="scanl1 vy3 vy4",fontsize=16,color="burlywood",shape="triangle"];19[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 5[label="",style="solid", color="burlywood", weight=3];
20[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="scanl1 vy3 (vy40 : vy41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="scanl1 vy3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="scanl vy3 vy40 vy41",fontsize=16,color="black",shape="triangle"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="[]",fontsize=16,color="green",shape="box"];9[label="vy40 : scanl0 vy3 vy40 vy41",fontsize=16,color="green",shape="box"];9 -> 10[label="",style="dashed", color="green", weight=3];
10[label="scanl0 vy3 vy40 vy41",fontsize=16,color="burlywood",shape="box"];21[label="vy41/vy410 : vy411",fontsize=10,color="white",style="solid",shape="box"];10 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 11[label="",style="solid", color="burlywood", weight=3];
22[label="vy41/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="scanl0 vy3 vy40 (vy410 : vy411)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="scanl0 vy3 vy40 []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13 -> 7[label="",style="dashed", color="red", weight=0];
13[label="scanl vy3 (vy3 vy40 vy410) vy411",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
13 -> 16[label="",style="dashed", color="magenta", weight=3];
14[label="[]",fontsize=16,color="green",shape="box"];15[label="vy411",fontsize=16,color="green",shape="box"];16[label="vy3 vy40 vy410",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="green", weight=3];
16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="vy40",fontsize=16,color="green",shape="box"];18[label="vy410",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(vy3, :(vy410, vy411), ba) → new_scanl(vy3, vy411, 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_scanl(vy3, :(vy410, vy411), ba) → new_scanl(vy3, vy411, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3