 Extensionality

This is the Equality.Extensionality module of the Agda Universal Algebra Library.

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

module Equality.Extensionality where

-- imports from Agda and the Agda Standard Library ------------------------------------
open import Axiom.Extensionality.Propositional
using () renaming ( Extensionality to funext )
open import Agda.Primitive         using ( _⊔_ ; lsuc ; Level )
renaming ( Set to Type ; Setω to Typeω )
open import Data.Product           using ( _,_ ) renaming ( _×_ to _∧_ )
open import Relation.Binary        using ( IsEquivalence ) renaming ( Rel to BinRel )
open import Relation.Unary         using ( Pred ; _⊆_ )
open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl )

-- -- imports from agda-algebras --------------------------------------------------------------
open import Overture.Preliminaries using ( transport )
open import Relations.Quotients    using ( [_] ; []-⊆ ; []-⊇ ; IsBlock ; ⟪_⟫ )
open import Equality.Truncation    using ( blk-uip ; to-Σ-≡ )

private variable α β γ ρ 𝓥 : Level

Function Extensionality

Previous versions of the agda-algebras library made heavy use of a global function extensionality principle asserting that function extensionality holds at all universe levels. However, we have removed all instances of global function extensionality from the current version of the library and we now limit ourselves to local applications of the principle. This has the advantage of making transparent precisely how and where the library depends on function extensionality. Eventually we hope to be able to remove these postulates altogether in favor of an alternative approach to extensionality (e.g., by working with setoids or by reimplementing the entire library in Cubical Agda).

The following definition is useful for postulating function extensionality when and where needed.

DFunExt : Typeω
DFunExt = (𝓤 𝓥 : Level)  funext 𝓤 𝓥

An alternative way to express function extensionality

A useful alternative for expressing dependent function extensionality, which is essentially equivalent to dfunext, is to assert that the happly function is actually an equivalence.

The principle of proposition extensionality asserts that logically equivalent propositions are equivalent. That is, if P and Q are propositions and if P ⊆ Q and Q ⊆ P, then P ≡ Q. For our purposes, it will suffice to formalize this notion for general predicates, rather than for propositions (i.e., truncated predicates).

_≐_ : {α β : Level}{A : Type α}(P Q : Pred A β )  Type _
P  Q = (P  Q)  (Q  P)

pred-ext : (α β : Level)  Type (lsuc (α  β))
pred-ext α β =  {A : Type α}{P Q : Pred A β }  P  Q  Q  P  P  Q

Note that pred-ext merely defines an extensionality principle. It does not postulate that the principle holds. If we wish to postulate pred-ext, then we do so by assuming that type is inhabited (see block-ext below, for example).

Quotient extensionality

We need an identity type for congruence classes (blocks) over sets so that two different presentations of the same block (e.g., using different representatives) may be identified. This requires two postulates: (1) predicate extensionality, manifested by the pred-ext type; (2) equivalence class truncation or “uniqueness of block identity proofs”, manifested by the blk-uip type defined in the [Relations.Truncation][] module. We now use pred-ext and blk-uip to define a type called block-ext|uip which we require for the proof of the First Homomorphism Theorem presented in Homomorphisms.Noether.

module _ {A : Type α}{R : BinRel A ρ} where

block-ext : pred-ext α ρ  IsEquivalence{a = α}{ = ρ} R
{u v : A}  R u v  [ u ] R  [ v ] R

block-ext pe Req {u}{v} Ruv = pe ([]-⊆ {R = (R , Req)} u v Ruv)
([]-⊇ {R = (R , Req)} u v Ruv)

private
to-subtype|uip : blk-uip A R
{C D : Pred A ρ}{c : IsBlock C {R}}{d : IsBlock D {R}}
C  D  (C , c)  (D , d)

to-subtype|uip buip {C}{D}{c}{d} CD =
to-Σ-≡ (CD , buip D (transport  B  IsBlock B) CD c) d)

block-ext|uip : pred-ext α ρ  blk-uip A R
IsEquivalence R  ∀{u}{v}  R u v   u    v

block-ext|uip pe buip Req Ruv = to-subtype|uip buip (block-ext pe Req Ruv)