module Data.List.Any.BagAndSetEquality where
open import Algebra
open import Algebra.FunctionProperties
open import Category.Monad
open import Data.List as List
import Data.List.Properties as LP
open import Data.List.Any as Any using (Any)
open import Data.List.Any.Properties
open import Data.Product
open import Data.Sum
open import Function
open import Function.Equality using (_⟨$⟩_)
import Function.Equivalence as FE
open import Function.Inverse as Inv using (_⇿_; module Inverse)
open import Function.Inverse.TypeIsomorphisms
open import Relation.Binary
import Relation.Binary.EqReasoning as EqR
open import Relation.Binary.PropositionalEquality as P
using (_≡_; _≗_)
open Any.Membership-≡
open RawMonad List.monad
private
module ListMonoid {A : Set} = Monoid (List.monoid A)
module Eq {k} {A : Set} = Setoid ([ k ]-Equality A)
module ×⊎ {k ℓ} = CommutativeSemiring (×⊎-CommutativeSemiring k ℓ)
commutativeMonoid : Kind → Set → CommutativeMonoid _ _
commutativeMonoid k A = record
{ Carrier = List A
; _≈_ = λ xs ys → xs ≈[ k ] ys
; _∙_ = _++_
; ε = []
; isCommutativeMonoid = record
{ isSemigroup = record
{ isEquivalence = Eq.isEquivalence
; assoc = λ xs ys zs →
Eq.reflexive $ ListMonoid.assoc xs ys zs
; ∙-cong = λ {xs₁ xs₂ xs₃ xs₄} xs₁≈xs₂ xs₃≈xs₄ {x} →
x ∈ xs₁ ++ xs₃ ⇿⟨ sym ++⇿ ⟩
(x ∈ xs₁ ⊎ x ∈ xs₃) ≈⟨ xs₁≈xs₂ ⟨ ×⊎.+-cong ⟩ xs₃≈xs₄ ⟩
(x ∈ xs₂ ⊎ x ∈ xs₄) ⇿⟨ ++⇿ ⟩
x ∈ xs₂ ++ xs₄ ∎
}
; identityˡ = λ xs {x} → x ∈ xs ∎
; comm = λ xs ys {x} →
x ∈ xs ++ ys ⇿⟨ ++⇿++ xs ys ⟩
x ∈ ys ++ xs ∎
}
}
where open Inv.EquationalReasoning
++-idempotent : {A : Set} →
Idempotent (λ (xs ys : List A) → xs ≈[ set ] ys) _++_
++-idempotent xs {x} =
x ∈ xs ++ xs ≈⟨ FE.equivalent ([ id , id ]′ ∘ _⟨$⟩_ (Inverse.from ++⇿))
(_⟨$⟩_ (Inverse.to ++⇿) ∘ inj₁) ⟩
x ∈ xs ∎
where open Inv.EquationalReasoning
map-cong : ∀ {k} {A B : Set} {f₁ f₂ : A → B} {xs₁ xs₂} →
f₁ ≗ f₂ → xs₁ ≈[ k ] xs₂ →
List.map f₁ xs₁ ≈[ k ] List.map f₂ xs₂
map-cong {f₁ = f₁} {f₂} {xs₁} {xs₂} f₁≗f₂ xs₁≈xs₂ {x} =
x ∈ List.map f₁ xs₁ ⇿⟨ sym map⇿ ⟩
Any (λ y → x ≡ f₁ y) xs₁ ≈⟨ Any-cong (Inv.⇿⇒ ∘ helper) xs₁≈xs₂ ⟩
Any (λ y → x ≡ f₂ y) xs₂ ⇿⟨ map⇿ ⟩
x ∈ List.map f₂ xs₂ ∎
where
open Inv.EquationalReasoning
helper : ∀ y → x ≡ f₁ y ⇿ x ≡ f₂ y
helper y = record
{ to = P.→-to-⟶ (λ x≡f₁y → P.trans x≡f₁y ( f₁≗f₂ y))
; from = P.→-to-⟶ (λ x≡f₂y → P.trans x≡f₂y (P.sym $ f₁≗f₂ y))
; inverse-of = record
{ left-inverse-of = λ _ → P.proof-irrelevance _ _
; right-inverse-of = λ _ → P.proof-irrelevance _ _
}
}
concat-cong : ∀ {k} {A : Set} {xss₁ xss₂ : List (List A)} →
xss₁ ≈[ k ] xss₂ → concat xss₁ ≈[ k ] concat xss₂
concat-cong {xss₁ = xss₁} {xss₂} xss₁≈xss₂ {x} =
x ∈ concat xss₁ ⇿⟨ sym concat⇿ ⟩
Any (Any (_≡_ x)) xss₁ ≈⟨ Any-cong (λ _ → _ ∎) xss₁≈xss₂ ⟩
Any (Any (_≡_ x)) xss₂ ⇿⟨ concat⇿ ⟩
x ∈ concat xss₂ ∎
where open Inv.EquationalReasoning
>>=-cong : ∀ {k} {A B : Set} {xs₁ xs₂} {f₁ f₂ : A → List B} →
xs₁ ≈[ k ] xs₂ → (∀ x → f₁ x ≈[ k ] f₂ x) →
(xs₁ >>= f₁) ≈[ k ] (xs₂ >>= f₂)
>>=-cong {xs₁ = xs₁} {xs₂} {f₁} {f₂} xs₁≈xs₂ f₁≈f₂ {x} =
x ∈ (xs₁ >>= f₁) ⇿⟨ sym >>=⇿ ⟩
Any (λ y → x ∈ f₁ y) xs₁ ≈⟨ Any-cong (λ x → f₁≈f₂ x) xs₁≈xs₂ ⟩
Any (λ y → x ∈ f₂ y) xs₂ ⇿⟨ >>=⇿ ⟩
x ∈ (xs₂ >>= f₂) ∎
where open Inv.EquationalReasoning
⊛-cong : ∀ {k A B} {fs₁ fs₂ : List (A → B)} {xs₁ xs₂} →
fs₁ ≈[ k ] fs₂ → xs₁ ≈[ k ] xs₂ → fs₁ ⊛ xs₁ ≈[ k ] fs₂ ⊛ xs₂
⊛-cong fs₁≈fs₂ xs₁≈xs₂ =
>>=-cong fs₁≈fs₂ λ f →
>>=-cong xs₁≈xs₂ λ x →
_ ∎
where open Inv.EquationalReasoning
⊗-cong : ∀ {k A B} {xs₁ xs₂ : List A} {ys₁ ys₂ : List B} →
xs₁ ≈[ k ] xs₂ → ys₁ ≈[ k ] ys₂ →
(xs₁ ⊗ ys₁) ≈[ k ] (xs₂ ⊗ ys₂)
⊗-cong xs₁≈xs₂ ys₁≈ys₂ =
⊛-cong (⊛-cong (Eq.refl {x = [ _,_ ]}) xs₁≈xs₂) ys₁≈ys₂
>>=-left-distributive :
∀ {A B : Set} (xs : List A) {f g : A → List B} →
(xs >>= λ x → f x ++ g x) ≈[ bag ] (xs >>= f) ++ (xs >>= g)
>>=-left-distributive xs {f} {g} {y} =
y ∈ (xs >>= λ x → f x ++ g x) ⇿⟨ sym >>=⇿ ⟩
Any (λ x → y ∈ f x ++ g x) xs ⇿⟨ sym (Any-cong (λ _ → ++⇿) (_ ∎)) ⟩
Any (λ x → y ∈ f x ⊎ y ∈ g x) xs ⇿⟨ sym ⊎⇿ ⟩
(Any (λ x → y ∈ f x) xs ⊎ Any (λ x → y ∈ g x) xs) ⇿⟨ >>=⇿ ⟨ ×⊎.+-cong ⟩ >>=⇿ ⟩
(y ∈ (xs >>= f) ⊎ y ∈ (xs >>= g)) ⇿⟨ ++⇿ ⟩
y ∈ (xs >>= f) ++ (xs >>= g) ∎
where open Inv.EquationalReasoning
⊛-left-distributive :
∀ {A B} (fs : List (A → B)) xs₁ xs₂ →
fs ⊛ (xs₁ ++ xs₂) ≈[ bag ] (fs ⊛ xs₁) ++ (fs ⊛ xs₂)
⊛-left-distributive fs xs₁ xs₂ = begin
fs ⊛ (xs₁ ++ xs₂) ≡⟨ P.refl ⟩
(fs >>= λ f → xs₁ ++ xs₂ >>= return ∘ f) ≡⟨ (LP.Monad.cong (P.refl {x = fs}) λ f →
LP.Monad.right-distributive xs₁ xs₂ (return ∘ f)) ⟩
(fs >>= λ f → (xs₁ >>= return ∘ f) ++
(xs₂ >>= return ∘ f)) ≈⟨ >>=-left-distributive fs ⟩
(fs >>= λ f → xs₁ >>= return ∘ f) ++
(fs >>= λ f → xs₂ >>= return ∘ f) ≡⟨ P.refl ⟩
(fs ⊛ xs₁) ++ (fs ⊛ xs₂) ∎
where open EqR ([ bag ]-Equality _)