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

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

module Relation.Nullary.Negation where

open import Effect.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 (λ (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 (λ 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 (λ 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 (λ 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 : ∃ (λ b → P → if b then Q else R) → (P → Q) ⊎ (P → R)
    helper (true  , f) = inj₁ f
    helper (false , f) = inj₂ f