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

MAYBE 1.395 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

  ↳ IFR

mainModule Main

((until :: (a -> Bool) -> (a -> a) -> a -> a) :: (a -> Bool) -> (a -> a) -> a -> a)

module Main where

import qualified Prelude

If Reductions:
The following If expression

if p x then x else until p f (f x)

is transformed to

until0 x p f True = x

until0 x p f False = until p f (f x)

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

mainModule Main

((until :: (a -> Bool) -> (a -> a) -> a -> a) :: (a -> Bool) -> (a -> a) -> a -> a)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((until :: (a -> Bool) -> (a -> a) -> a -> a) :: (a -> Bool) -> (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

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

              ↳ Narrow

mainModule Main

(until :: (a -> Bool) -> (a -> a) -> a -> a)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="until",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="until vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="until vx3 vx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="until vx3 vx4 vx5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6 -> 7[label="",style="dashed", color="red", weight=0];
6[label="until0 vx5 vx3 vx4 (vx3 vx5)",fontsize=16,color="magenta"];6 -> 8[label="",style="dashed", color="magenta", weight=3];
8[label="vx3 vx5",fontsize=16,color="green",shape="box"];8 -> 12[label="",style="dashed", color="green", weight=3];
7[label="until0 vx5 vx3 vx4 vx6",fontsize=16,color="burlywood",shape="triangle"];18[label="vx6/False",fontsize=10,color="white",style="solid",shape="box"];7 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 10[label="",style="solid", color="burlywood", weight=3];
19[label="vx6/True",fontsize=10,color="white",style="solid",shape="box"];7 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 11[label="",style="solid", color="burlywood", weight=3];
12[label="vx5",fontsize=16,color="green",shape="box"];10[label="until0 vx5 vx3 vx4 False",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="until0 vx5 vx3 vx4 True",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
13 -> 5[label="",style="dashed", color="red", weight=0];
13[label="until vx3 vx4 (vx4 vx5)",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
14[label="vx5",fontsize=16,color="green",shape="box"];15[label="vx4 vx5",fontsize=16,color="green",shape="box"];15 -> 16[label="",style="dashed", color="green", weight=3];
16[label="vx5",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ NonTerminationProof

              ↳ Narrow

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

new_until(vx3, vx4, ba) → new_until0(vx3, vx4, ba)
new_until0(vx3, vx4, ba) → new_until(vx3, vx4, 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 narrowing to the left:

The TRS P consists of the following rules:

new_until(vx3, vx4, ba) → new_until0(vx3, vx4, ba)
new_until0(vx3, vx4, ba) → new_until(vx3, vx4, ba)

The TRS R consists of the following rules:none

s = new_until0(vx3', vx4', ba') evaluates to t =new_until0(vx3', vx4', ba')

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

new_until0(vx3', vx4', ba') → new_until(vx3', vx4', ba')
with rule new_until0(vx3'', vx4'', ba'') → new_until(vx3'', vx4'', ba'') at position [] and matcher [ba'' / ba', vx4'' / vx4', vx3'' / vx3']

new_until(vx3', vx4', ba') → new_until0(vx3', vx4', ba')
with rule new_until(vx3, vx4, ba) → new_until0(vx3, vx4, ba)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

Haskell To QDPs

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

              ↳ Narrow

                ↳ QDP

                  ↳ PisEmptyProof

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.