------------------------------------------------------------------------
-- 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
open import Function.Equality using (Π; _⟶_; ≡-setoid)
open import Level as L
open import Data.Empty
open import Data.Nat.Base using (; zero; suc)
open import Data.Product

open import Relation.Nullary using (yes ; no)
open import Relation.Nullary.Decidable.Core
open import Relation.Unary using (Pred)
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 of core module

open import Relation.Binary.PropositionalEquality.Core public

------------------------------------------------------------------------
-- Some properties

subst₂ :  (_∼_ : REL A B ) {x y u v}  x  y  u  v  x  u  y  v
subst₂ _ refl refl p = p

cong-app :  {A : Set a} {B : A  Set b} {f g : (x : A)  B x} 
           f  g  (x : A)  f x  g x
cong-app refl x = refl

cong₂ :  (f : A  B  C) {x y u v}  x  y  u  v  f x u  f y v
cong₂ f refl refl = refl

------------------------------------------------------------------------
-- Structure of equality as a binary relation

setoid : Set a  Setoid _ _
setoid A = record
  { Carrier       = A
  ; _≈_           = _≡_
  ; isEquivalence = isEquivalence
  }

decSetoid : Decidable {A = A} _≡_  DecSetoid _ _
decSetoid dec = record
  { _≈_              = _≡_
  ; isDecEquivalence = record
      { isEquivalence = isEquivalence
      ; _≟_           = dec
      }
  }

isPreorder : IsPreorder {A = A} _≡_ _≡_
isPreorder = record
  { isEquivalence = isEquivalence
  ; reflexive     = id
  ; trans         = trans
  }

preorder : Set a  Preorder _ _ _
preorder A = record
  { Carrier    = A
  ; _≈_        = _≡_
  ; _∼_        = _≡_
  ; isPreorder = isPreorder
  }

------------------------------------------------------------------------
-- 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

------------------------------------------------------------------------
-- Various equality rearrangement lemmas

trans-injectiveˡ :  {x y z : A} {p₁ p₂ : x  y} (q : y  z) 
                   trans p₁ q  trans p₂ q  p₁  p₂
trans-injectiveˡ refl = subst₂ _≡_ (trans-reflʳ _) (trans-reflʳ _)

trans-injectiveʳ :  {x y z : A} (p : x  y) {q₁ q₂ : y  z} 
                   trans p q₁  trans p q₂  q₁  q₂
trans-injectiveʳ refl eq = eq

cong-id :  {x y : A} (p : x  y)  cong id p  p
cong-id refl = refl

cong-∘ :  {x y : A} {f : B  C} {g : A  B} (p : x  y) 
         cong (f  g) p  cong f (cong g p)
cong-∘ refl = refl

module _ {P : Pred A p} {x y : A} where

  subst-injective :  (x≡y : x  y) {p q : P x} 
                    subst P x≡y p  subst P x≡y q  p  q
  subst-injective refl p≡q = p≡q

  subst-subst :  {z} (x≡y : x  y) {y≡z : y  z} {p : P x} 
                subst P y≡z (subst P x≡y p)  subst P (trans x≡y y≡z) p
  subst-subst refl = refl

  subst-subst-sym : (x≡y : x  y) {p : P y} 
                    subst P x≡y (subst P (sym x≡y) p)  p
  subst-subst-sym refl = refl

  subst-sym-subst : (x≡y : x  y) {p : P x} 
                    subst P (sym x≡y) (subst P x≡y p)  p
  subst-sym-subst refl = refl

subst-∘ :  {x y : A} {P : Pred B p} {f : A  B}
          (x≡y : x  y) {p : P (f x)} 
          subst (P  f) x≡y p  subst P (cong f x≡y) p
subst-∘ refl = refl

subst-application :  {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
                    (B₁ : A₁  Set b₁) {B₂ : A₂  Set b₂}
                    {f : A₂  A₁} {x₁ x₂ : A₂} {y : B₁ (f x₁)}
                    (g :  x  B₁ (f x)  B₂ x) (eq : x₁  x₂) 
                    subst B₂ eq (g x₁ y)  g x₂ (subst B₁ (cong f eq) y)
subst-application _ _ refl = refl

-- 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 _ (_≟_ : Decidable {A = A} _≡_) where

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

  ≢-≟-identity :  {x y : A}  x  y   λ ¬eq  x  y  no ¬eq
  ≢-≟-identity {x} {y} ¬eq = dec-no (x  y) ¬eq



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

-- Version 1.0

Extensionality = Ext.Extensionality
{-# WARNING_ON_USAGE Extensionality
"Warning: Extensionality was deprecated in v1.0.
Please use Extensionality from `Axiom.Extensionality.Propositional` instead."
#-}
extensionality-for-lower-levels = Ext.lower-extensionality
{-# WARNING_ON_USAGE extensionality-for-lower-levels
"Warning: extensionality-for-lower-levels was deprecated in v1.0.
Please use lower-extensionality from `Axiom.Extensionality.Propositional` instead."
#-}
∀-extensionality = Ext.∀-extensionality
{-# WARNING_ON_USAGE ∀-extensionality
"Warning: ∀-extensionality was deprecated in v1.0.
Please use ∀-extensionality from `Axiom.Extensionality.Propositional` instead."
#-}