------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties related to negation
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Relation.Nullary.Negation where

open import Category.Monad
open import Data.Bool.Base using (Bool; false; true; if_then_else_; not)
open import Data.Empty
open import Data.Product as Prod
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
open import Function
open import Level
open import Relation.Nullary
open import Relation.Nullary.Decidable
open import Relation.Unary

private
  variable
    a p q r w : Level
    A : Set a
    P : Set p
    Q : Set q
    R : Set r
    Whatever : Set w

------------------------------------------------------------------------
-- Re-export public definitions

open import Relation.Nullary.Negation.Core public

------------------------------------------------------------------------
-- Other properties

-- Decidable predicates are stable.

decidable-stable : Dec P  Stable P
decidable-stable (yes p) ¬¬p = p
decidable-stable (no ¬p) ¬¬p = ⊥-elim (¬¬p ¬p)

¬-drop-Dec : Dec (¬ ¬ P)  Dec (¬ P)
¬-drop-Dec ¬¬p? = map′ negated-stable contradiction (¬? ¬¬p?)

-- Double-negation is a monad (if we assume that all elements of ¬ ¬ P
-- are equal).

¬¬-Monad : RawMonad  (P : Set p)  ¬ ¬ P)
¬¬-Monad = record
  { return = contradiction
  ; _>>=_  = λ x f  negated-stable (¬¬-map f x)
  }

¬¬-push :  {P : Set p} {Q : P  Set q} 
          ¬ ¬ ((x : P)  Q x)  (x : P)  ¬ ¬ Q x
¬¬-push ¬¬P⟶Q P ¬Q = ¬¬P⟶Q  P⟶Q  ¬Q (P⟶Q P))

-- A double-negation-translated variant of excluded middle (or: every
-- nullary relation is decidable in the double-negation monad).

excluded-middle : ¬ ¬ Dec P
excluded-middle ¬h = ¬h (no  p  ¬h (yes p)))

-- If Whatever is instantiated with ¬ ¬ something, then this function
-- is call with current continuation in the double-negation monad, or,
-- if you will, a double-negation translation of Peirce's law.
--
-- In order to prove ¬ ¬ P one can assume ¬ P and prove ⊥. However,
-- sometimes it is nice to avoid leaving the double-negation monad; in
-- that case this function can be used (with Whatever instantiated to
-- ⊥).

call/cc : ((P  Whatever)  ¬ ¬ P)  ¬ ¬ P
call/cc hyp ¬p = hyp  p  ⊥-elim (¬p p)) ¬p

-- The "independence of premise" rule, in the double-negation monad.
-- It is assumed that the index set (Q) is inhabited.

independence-of-premise :  {P : Set p} {Q : Set q} {R : Q  Set r} 
                          Q  (P  Σ Q R)  ¬ ¬ (Σ[ x  Q ] (P  R x))
independence-of-premise {P = P} q f = ¬¬-map helper excluded-middle
  where
  helper : Dec P  _
  helper (yes p) = Prod.map id const (f p)
  helper (no ¬p) = (q , ⊥-elim ∘′ ¬p)

-- The independence of premise rule for binary sums.

independence-of-premise-⊎ : (P  Q  R)  ¬ ¬ ((P  Q)  (P  R))
independence-of-premise-⊎ {P = P} f = ¬¬-map helper excluded-middle
  where
  helper : Dec P  _
  helper (yes p) = Sum.map const const (f p)
  helper (no ¬p) = inj₁ (⊥-elim ∘′ ¬p)

private

  -- Note that independence-of-premise-⊎ is a consequence of
  -- independence-of-premise (for simplicity it is assumed that Q and
  -- R have the same type here):

  corollary : {P : Set p} {Q R : Set q} 
              (P  Q  R)  ¬ ¬ ((P  Q)  (P  R))
  corollary {P = P} {Q} {R} f =
    ¬¬-map helper (independence-of-premise
                     true ([ _,_ true , _,_ false ] ∘′ f))
    where
    helper :   b  P  if b then Q else R)  (P  Q)  (P  R)
    helper (true  , f) = inj₁ f
    helper (false , f) = inj₂ f


------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 1.0

Excluded-Middle : ( : Level)  Set (suc )
Excluded-Middle p = {P : Set p}  Dec P
{-# WARNING_ON_USAGE Excluded-Middle
"Warning: Excluded-Middle was deprecated in v1.0.
Please use ExcludedMiddle from `Axiom.ExcludedMiddle` instead."
#-}

Double-Negation-Elimination : ( : Level)  Set (suc )
Double-Negation-Elimination p = {P : Set p}  Stable P
{-# WARNING_ON_USAGE Double-Negation-Elimination
"Warning: Double-Negation-Elimination was deprecated in v1.0.
Please use DoubleNegationElimination from `Axiom.DoubleNegationElimination` instead."
#-}

Generated from commit 2fd14c996b195ef101dff8919e837907ca0a08aa.