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Equalities

module Types.equality where

open import Lang.dataStructures using (
  Bool; true; false;
;;; List;
  one; two; three; four; five; six; seven; eight; nine; ten; zero; succ; _+_;
  _::_; [])

open import Agda.Primitive using (Level; __; lsuc; lzero)

open import Types.functions using (_on_; flip)

Equality is perhaps one of the most richest but most naively understood concepts. Here we try to provide some structural analysis as to what equality really means in various contexts of mathematics. Equality is treated as a relation in type theory and can be classified broadly as of three kinds:

Definitional Equality

Definitional equality is the most basic notion of equality which appeals to our notion of equality being the sameness of meaning (by definition). For example, 9 and 3 + 3 represent the same thing and hence are definitionally equal 9 ≡ 3². Similarly two ≡ succ (succ zero).

defEqual₁ :
defEqual₁ = seven

defEqual₂ :
defEqual₂ = succ (succ five)

Here, defEqual₁ and defEqual₂ both are equal, with equality of the kind “definitional equality”.

Computational Equality

This kind of equality describes the sameness of types that are not directly equal but can be reduced to be equal. “Reduction” here implies mathematical reduction, referring to rewriting expressions into simpler forms. An example of such an equality is applying a function \[(λ x.x+x)(2) ≡ 2 + 2\] Expansions of recursors also falls under this kind of equality: \[2 + 2 ≡ succ (succ zero) + succ (succ zero) ≡ succ (succ (succ (succ zero)))\] Practically, computational equality is included into definitional equality and is also known as “Judgemental equality”.

Propositional Equality

Definitional and computational equalities describe something intrinsic - a property that does not depend upon a proof. For example, a + b ≡ b + a cannot be proven to be definitionally equal unless the concrete values of a and b are known. However, if they’re natural numbers a, b ∈ ℕ, then the statement a + b ≡ b + a requires a proof to claim its truthfulness. Given a, b ∈ ℕ, we can prove that a + b ≡ b + a or in other words that there exists an identity of type Id : a + b ≡ b + a where Id is a proposition − exhibiting a term belonging to such a type is exhibiting (i.e. proving) such a propositional equality.

However, other notions of equalities can be defined that do require proofs. Consider for example natural language - when we say “all flowers are beautiful” the “all flowers” part of the sentence implies all flowers are equal in some way. Or, consider natural numbers a + b = b + a ∀ a, b ∈ ℕ. Here we would need to prove the symmetry of the + operator in order to prove the equality. Such equalities that require to be specified come under the umbrella of propositional equality. Propositional equality is a kind of equality which requires a proof, and hence the equality itself is also a type :

infix 4 _∼_

data _∼_ {A : Set}(a : A) : {B : Set}  B  Set where
  same : a ∼ a

Reflexivity is defined with the definition of by the keyword same, the others being:

Symmetry

Symmetry is the property where binary a relation’s behavior does not depend upon its argument’s position (left or right):

symmetry :  {A B}{a : A}{b : B}
   a ∼ b
   b ∼ a
symmetry same = same

Transitivity

Transitivity is when a binary relation _∼_ and \(x ∼ y and y ∼ z ⟹ x ∼ z\)

transitivity :  {A B C}{a : A}{b : B}{c : C}
   a ∼ b
   b ∼ c
   a ∼ c
transitivity same p = p

Congruence: functions that preserve equality

Functions that when applied to objects of a type, do not alter the operation of equality can be defined as:

congruence :  {A B : Set} (f : A  B) {x y : A}
   x ∼ y
   f x ∼ f y
congruence f same = same

Substitution

If a = b and if predicate a = truepredicate b = true

substitution :  {A : Set} {x y : A} (Predicate : A  Set)
   x ∼ y
   Predicate x
   Predicate y
substitution Predicate same p = p

Any relation which satisfies the above properties of reflexivity, transitivity and symmetry can be considered an equivalence relation and hence can judge a propositional equality.

Relations, with universe polymorphism

We now present a more formal machinery for relations. We use universe polymorphism throughout to develop this machinery.

Equality

We first re-define propositional equality within the framework of universe polymorphism:

infix 4 __
data __ {a} {A : Set a} (x : A) : A  Set a where
  instance refl : x ≡ x

Types of relations

Nullary relations

Nullary relations are functions that can take any object and return an empty set :

¬  :  {}  Set Set
¬ P = P 

Unary relations

In logic, a predicate can essentially be defined as a function that returns a binary value - whether the proposition that the predicate represents is true or false. In type theory, however, we define predicate in a different way. A predicate for us is a function that exists (and hence, is true):

Pred :  {a}  Set a  (: Level)  Set (a ⊔ lsuc ℓ)
Pred A ℓ = A  Set

The empty (or false) predicate becomes:

:  {a} {A : Set a}  Pred A lzero
= λ _ 

The singleton predicate (constructor):

is_sameAs :  {a} {A : Set a}
         A
         Pred A a
is x sameAs = x ≡_
equal? : is six sameAs (succ five)
equal? = refl

Binary relations

A heterogeneous binary relation is defined as:

REL :  {a b}  Set a  Set b  (: Level)  Set (a ⊔ b ⊔ lsuc ℓ)
REL A B ℓ = A  B  Set

and a homogenous one as:

Rel :  {a}  Set a  (: Level)  Set (a ⊔ lsuc ℓ)
Rel A ℓ = REL A A ℓ

Properties of binary relations

In type theory, an implication $ A ⟹ B $ is just a function type $ f: A → B $, and if f exists, the implication does too. We define implication between two relations in agda as:

__ :  {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}
         REL A B ℓ₁
         REL A B ℓ₂
         Set _
P ⇒ Q =  {i j}  P i j  Q i j

A function f : A → B is invariant to two homogenous relations Rel A ℓ₁ and Rel B ℓ₂ if $ ∀ x, y ∈ A and f(x), f(y) ∈ B, f(Rel x y) ⟹ (Rel f(x) f(y)) $:

_=[_]⇒_ :  {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}
           Rel A ℓ₁
           (A  B)
           Rel B ℓ₂
           Set _
P =[ f ]⇒ Q = P ⇒ (Q on f)

A function f preserves an underlying relation while operating on a datatype if:

_Preserves__ :  {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}
         (A  B)
         Rel A ℓ₁
         Rel B ℓ₂
         Set _
f Preserves P ⟶ Q = P =[ f ]⇒ Q

Similarly, a binary operation _+_ preserves the underlying relation if:

_Preserves₂___ :  {a b c ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : Set b} {C : Set c}
         (A  B  C)
         Rel A ℓ₁
         Rel B ℓ₂
         Rel C ℓ₃
         Set _
_+_ Preserves₂ P ⟶ Q ⟶ R =  {x y u v}  P x y  Q u v  R (x + u) (y + v)

Properties of binary relations:

Reflexive :  {a ℓ} {A : Set a}
         Rel A ℓ
         Set _
Reflexive __ =  {x}  x ∼ x
Sym :  {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}
         REL A B ℓ₁
         REL B A ℓ₂
         Set _
Sym P Q = P ⇒ flip Q

Symmetric :  {a ℓ} {A : Set a}
         Rel A ℓ
         Set _
Symmetric __ = Sym __ __
Trans :  {a b c ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : Set b} {C : Set c}
         REL A B ℓ₁
         REL B C ℓ₂
         REL A C ℓ₃
         Set _
Trans P Q R =  {i j k}  P i j  Q j k  R i k

Transitive :  {a ℓ} {A : Set a}
         Rel A ℓ
         Set _
Transitive __ = Trans __ __ __

Finally, we define an equivalence relation for binary relations:

record IsEquivalence {a ℓ} {A : Set a}
                     (__ : Rel A ℓ) : Set (a ⊔ ℓ) where
  field
    rfl   : Reflexive __
    sym   : Symmetric __
    trans : Transitive __

  reflexive : ____
  reflexive refl = rfl

Properties of equality

We use the new structures to re-define the properties of propositional equality.

module ≡-properties {a} {A : Set a} where
  sym-≡ : Symmetric {A = A} __
  sym-≡ refl = refl

  trans-≡ : Transitive {A = A} __
  trans-≡ refl p = p

  isEquivalence : IsEquivalence {A = A} __
  isEquivalence = record
    { rfl  = refl
    ; sym   = sym-≡
    ; trans = trans-≡
    }

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

subs-≡ :  {a} {A : Set a}{x y : A} (Predicate : A  Set)
   x ≡ y
   Predicate x
   Predicate y
subs-≡ Predicate refl p = p

Setoids

Equality, or specifically, equivalence is at the heart of mathematics. In order to build more complex structures, we introduce a new datatype, which essentially encapsulates any datatype and it’s equivalence operation:

record Setoid c ℓ : Set (lsuc (c ⊔ ℓ)) where
  infix 4 __
  field
    Data          : Set c
    __           : Rel Data ℓ
    isEquivalence : IsEquivalence __

  open IsEquivalence isEquivalence public

Product Types / Σ-types