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

YES 0.811 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

inits :: [a] -> [[a]]

inits	[]	=	[] : []
inits	(x : xs)	=	([] : []) ++ map (: x) (inits xs)

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

inits :: [a] -> [[a]]

inits	[]	=	[] : []
inits	(x : xs)	=	([] : []) ++ map (: x) (inits xs)

module Maybe where

import qualified List
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 List

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

module List where

import qualified Maybe
import qualified Prelude

inits :: [a] -> [[a]]

inits	[]	=	[] : []
inits	(x : xs)	=	([] : []) ++ map (: x) (inits xs)

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_map(vy30, :(vy40, vy41), ba) → new_map(vy30, vy41, 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_map(vy30, :(vy40, vy41), ba) → new_map(vy30, vy41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_inits(:(vy30, vy31), ba) → new_inits(vy31, ba)

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

new_inits(:(vy30, vy31), ba) → new_inits(vy31, ba)
The graph contains the following edges 1 > 1, 2 >= 2