------------------------------------------------------------------------
-- The Agda standard library
--
-- Propositional (intensional) equality
------------------------------------------------------------------------

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

module Relation.Binary.PropositionalEquality where

import Axiom.Extensionality.Propositional as Ext
open import Axiom.UniquenessOfIdentityProofs
open import Function.Base using (id; _∘_)
open import Function.Equality using (Π; _⟶_; ≡-setoid)
open import Level using (Level; _⊔_)
open import Data.Product using ()

open import Relation.Nullary using (yes ; no)
open import Relation.Nullary.Decidable.Core
open import Relation.Binary
open import Relation.Binary.Indexed.Heterogeneous
  using (IndexedSetoid)
import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
  as Trivial

private
  variable
    a b c  p : Level
    A : Set a
    B : Set b
    C : Set c

------------------------------------------------------------------------
-- Re-export contents modules that make up the parts

open import Relation.Binary.PropositionalEquality.Core public
open import Relation.Binary.PropositionalEquality.Properties public
open import Relation.Binary.PropositionalEquality.Algebra public

------------------------------------------------------------------------
-- Pointwise equality

infix 4 _≗_

_→-setoid_ :  (A : Set a) (B : Set b)  Setoid _ _
A →-setoid B = ≡-setoid A (Trivial.indexedSetoid (setoid B))

_≗_ : (f g : A  B)  Set _
_≗_ {A = A} {B = B} = Setoid._≈_ (A →-setoid B)

:→-to-Π :  {A : Set a} {B : IndexedSetoid A b } 
          ((x : A)  IndexedSetoid.Carrier B x)  Π (setoid A) B
:→-to-Π {B = B} f = record
  { _⟨$⟩_ = f
  ; cong  = λ { refl  IndexedSetoid.refl B }
  }
  where open IndexedSetoid B using (_≈_)

→-to-⟶ :  {A : Set a} {B : Setoid b } 
         (A  Setoid.Carrier B)  setoid A  B
→-to-⟶ = :→-to-Π

------------------------------------------------------------------------
-- Inspect

-- Inspect can be used when you want to pattern match on the result r
-- of some expression e, and you also need to "remember" that r ≡ e.

-- See README.Inspect for an explanation of how/why to use this.

record Reveal_·_is_ {A : Set a} {B : A  Set b}
                    (f : (x : A)  B x) (x : A) (y : B x) :
                    Set (a  b) where
  constructor [_]
  field eq : f x  y

inspect :  {A : Set a} {B : A  Set b}
          (f : (x : A)  B x) (x : A)  Reveal f · x is f x
inspect f x = [ refl ]

------------------------------------------------------------------------
-- Propositionality

isPropositional : Set a  Set a
isPropositional A = (a b : A)  a  b

------------------------------------------------------------------------
-- More complex rearrangement lemmas

-- A lemma that is very similar to Lemma 2.4.3 from the HoTT book.

naturality :  {x y} {x≡y : x  y} {f g : A  B}
             (f≡g :  x  f x  g x) 
             trans (cong f x≡y) (f≡g y)  trans (f≡g x) (cong g x≡y)
naturality {x = x} {x≡y = refl} f≡g =
  f≡g x               ≡⟨ sym (trans-reflʳ _) 
  trans (f≡g x) refl  
  where open ≡-Reasoning

-- A lemma that is very similar to Corollary 2.4.4 from the HoTT book.

cong-≡id :  {f : A  A} {x : A} (f≡id :  x  f x  x) 
           cong f (f≡id x)  f≡id (f x)
cong-≡id {f = f} {x} f≡id =
  cong f fx≡x                                    ≡⟨ sym (trans-reflʳ _) 
  trans (cong f fx≡x) refl                       ≡⟨ cong (trans _) (sym (trans-symʳ fx≡x)) 
  trans (cong f fx≡x) (trans fx≡x (sym fx≡x))    ≡⟨ sym (trans-assoc (cong f fx≡x)) 
  trans (trans (cong f fx≡x) fx≡x) (sym fx≡x)    ≡⟨ cong  p  trans p (sym _)) (naturality f≡id) 
  trans (trans f²x≡x (cong id fx≡x)) (sym fx≡x)  ≡⟨ cong  p  trans (trans f²x≡x p) (sym fx≡x)) (cong-id _) 
  trans (trans f²x≡x fx≡x) (sym fx≡x)            ≡⟨ trans-assoc f²x≡x 
  trans f²x≡x (trans fx≡x (sym fx≡x))            ≡⟨ cong (trans _) (trans-symʳ fx≡x) 
  trans f²x≡x refl                               ≡⟨ trans-reflʳ _ 
  f≡id (f x)                                     
  where open ≡-Reasoning; fx≡x = f≡id x; f²x≡x = f≡id (f x)

module _ (_≟_ : DecidableEquality A) {x y : A} where

  ≡-≟-identity : (eq : x  y)  x  y  yes eq
  ≡-≟-identity eq = dec-yes-irr (x  y) (Decidable⇒UIP.≡-irrelevant _≟_) eq

  ≢-≟-identity : (x≢y : x  y)  x  y  no x≢y
  ≢-≟-identity = dec-no (x  y)