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

MAYBE 1.305 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could not be shown:

↳ HASKELL

  ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((cycle :: [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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

mainModule Main

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

module Main where

import qualified Prelude

Let/Where Reductions:
The bindings of the following Let/Where expression

xs'

where

xs' = xs ++ xs'

are unpacked to the following functions on top level

cycleXs' vx = vx ++ cycleXs' vx

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

              ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="cycle",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="cycle 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="cycle (vy30 : vy31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="cycle []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="cycleXs' (vy30 : vy31)",fontsize=16,color="black",shape="triangle"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="error []",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="(vy30 : vy31) ++ cycleXs' (vy30 : vy31)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="error []",fontsize=16,color="red",shape="box"];11 -> 6[label="",style="dashed", color="red", weight=0];
11[label="cycleXs' (vy30 : vy31)",fontsize=16,color="magenta"];10[label="(vy30 : vy31) ++ vy4",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
12[label="vy30 : vy31 ++ vy4",fontsize=16,color="green",shape="box"];12 -> 13[label="",style="dashed", color="green", weight=3];
13[label="vy31 ++ vy4",fontsize=16,color="burlywood",shape="triangle"];24[label="vy31/vy310 : vy311",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="vy31/[]",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="(vy310 : vy311) ++ vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="[] ++ vy4",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="vy310 : vy311 ++ vy4",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="vy4",fontsize=16,color="green",shape="box"];18 -> 13[label="",style="dashed", color="red", weight=0];
18[label="vy311 ++ vy4",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3];
19[label="vy311",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

              ↳ Narrow

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

new_psPs(:(vy310, vy311), vy4, ba) → new_psPs(vy311, vy4, 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(:(vy310, vy311), vy4, ba) → new_psPs(vy311, vy4, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ NonTerminationProof

              ↳ Narrow

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

new_cycleXs'(vy30, vy31, ba) → new_cycleXs'(vy30, vy31, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_cycleXs'(vy30, vy31, ba) → new_cycleXs'(vy30, vy31, ba)

The TRS R consists of the following rules:none

s = new_cycleXs'(vy30, vy31, ba) evaluates to t =new_cycleXs'(vy30, vy31, ba)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_cycleXs'(vy30, vy31, ba) to new_cycleXs'(vy30, vy31, ba).

Haskell To QDPs