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---

Equation preservation

This is the Base.Varieties.Preservation module of the Agda Universal Algebra Library. In this module we show that identities are preserved by closure operators H, S, and P. This will establish the easy direction of Birkhoff’s HSP Theorem.

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

open import Overture using ( 𝓞 ; 𝓥 ; Signature )

module Base.Varieties.Preservation {𝑆 : Signature 𝓞 𝓥} where

-- Imports from Agda and the Agda Standard Library ----------------------
open  import Agda.Primitive
      using () renaming  ( Set to Type )
open  import Data.Product
      using ( _,_ ; Σ-syntax ; _×_ ) renaming  ( proj₁ to fst ; proj₂ to snd )
open  import Data.Sum
      using ( _⊎_ ) renaming  ( inj₁  to inl ; inj₂  to inr )
open  import Function
      using ( _∘_ )
open  import Level
      using ( Level ; _⊔_ ; suc )
open  import Relation.Unary
      using ( Pred ; _⊆_ ; _∈_ ; ｛_｝ ; _∪_ )
open  import Axiom.Extensionality.Propositional
      using () renaming (Extensionality to funext)
open  import Relation.Binary.PropositionalEquality as ≡
      using ( _≡_ ; module ≡-Reasoning )

-- Imports from the Agda Universal Algebra Library ---------------------------------------------
open import Overture        using ( ∣_∣ ; ∥_∥ ; _⁻¹ )
open import Base.Functions  using ( Inv ; InvIsInverseʳ ; IsInjective )
open import Base.Equality   using ( SwellDef ; hfunext ; DFunExt )

open  import Base.Algebras {𝑆 = 𝑆}
      using ( Algebra ; Lift-Alg ; ov ; ⨅ ; 𝔄 ; class-product )
open  import Base.Homomorphisms {𝑆 = 𝑆}
      using ( is-homomorphism ; _≅_ ; ≅-sym ; Lift-≅ ; ≅-trans ; ⨅≅ ; ≅-refl )
      using ( Lift-Alg-iso ; Lift-Alg-assoc )
open  import Base.Terms {𝑆 = 𝑆}
      using ( Term ; 𝑻 ; _⟦_⟧; comm-hom-term )
open  import Base.Subalgebras {𝑆 = 𝑆}
      using ( _IsSubalgebraOfClass_ ; ≤-Lift ; _IsSubalgebraOf_ ; _≤_ )
      using ( Lift-≤-Lift ; SubalgebraOfClass )
open  import Base.Varieties.EquationalLogic {𝑆 = 𝑆}
      using ( _⊫_≈_ ; _⊧_≈_ ; Th )
open  import Base.Varieties.Properties {𝑆 = 𝑆}
      using ( ⊧-Lift-invar ; ⊧-lower-invar ; ⊧-I-invar ; ⊧-S-invar ; ⊧-P-invar )
      using ( ⊧-S-class-invar ; ⊧-P-lift-invar )
open  import Base.Varieties.Closure {𝑆 = 𝑆}
      using ( H ; S ; P ; V ; P-expa ; S-mono ; S→subalgebra ; Lift-Alg-subP' )
      using ( subalgebra→S ; P-idemp ; module Vlift )

open H ; open S ; open P ; open V
private variable α β : Level

Closure properties

The types defined above represent operators with useful closure properties. We now prove a handful of such properties that we need later.

The next lemma would be too obvious to care about were it not for the fact that we’ll need it later, so it too must be formalized.

S⊆SP :  (𝒦 : Pred (Algebra α 𝑆)(ov α))
 →      S{α}{β} 𝒦 ⊆ S{α ⊔ β}{α ⊔ β} (P{α}{β} 𝒦)

S⊆SP {α} {β} 𝒦 {.(Lift-Alg 𝑨 β)}(sbase{𝑨} x) = siso spllA(≅-sym Lift-≅)
 where
 llA : Algebra (α ⊔ β) 𝑆
 llA = Lift-Alg (Lift-Alg 𝑨 β) (α ⊔ β)

 spllA : llA ∈ S (P 𝒦)
 spllA = sbase{α ⊔ β}{α ⊔ β} (pbase x)

S⊆SP {α} {β} 𝒦 {.(Lift-Alg 𝑨 β)}(slift{𝑨} x) = subalgebra→S lAsc
 where
 splAu : 𝑨 ∈ S(P 𝒦)
 splAu = S⊆SP{α}{α} 𝒦 x

 Asc : 𝑨 IsSubalgebraOfClass (P 𝒦)
 Asc = S→subalgebra{α}{P{α}{α} 𝒦}{𝑨} splAu

 lAsc : (Lift-Alg 𝑨 β) IsSubalgebraOfClass (P 𝒦)
 lAsc = Lift-Alg-subP' Asc

S⊆SP {α} {β} 𝒦 {𝑩}(ssub{𝑨} sA B≤A) = ssub (subalgebra→S lAsc) (≤-Lift 𝑨 B≤A )
 where
  lA : Algebra (α ⊔ β) 𝑆
  lA = Lift-Alg 𝑨 β

  splAu : 𝑨 ∈ S (P 𝒦)
  splAu = S⊆SP{α}{α} 𝒦 sA

  Asc : 𝑨 IsSubalgebraOfClass (P 𝒦)
  Asc = S→subalgebra{α}{P{α}{α} 𝒦}{𝑨} splAu

  lAsc : lA IsSubalgebraOfClass (P 𝒦)
  lAsc = Lift-Alg-subP' Asc

S⊆SP {α = α}{β} 𝒦 {𝑩}(siso{𝑨} sA A≅B) = siso{α ⊔ β}{α ⊔ β} lAsp lA≅B
 where
 lA : Algebra (α ⊔ β) 𝑆
 lA = Lift-Alg 𝑨 β

 lAsc : lA IsSubalgebraOfClass (P 𝒦)
 lAsc = Lift-Alg-subP' (S→subalgebra{α}{P{α}{α} 𝒦}{𝑨} (S⊆SP 𝒦 sA))

 lAsp : lA ∈ S(P 𝒦)
 lAsp = subalgebra→S{α ⊔ β}{α ⊔ β}{P{α}{β} 𝒦}{lA} lAsc

 lA≅B : lA ≅ 𝑩
 lA≅B = ≅-trans (≅-sym Lift-≅) A≅B

We need to formalize one more lemma before arriving the main objective of this section, which is the proof of the inclusion PS⊆SP.

module _ {α β : Level} {𝒦 : Pred(Algebra α 𝑆)(ov α)} where

 lemPS⊆SP :  hfunext β α → funext β α → {I : Type β}{ℬ : I → Algebra α 𝑆}
  →          (∀ i → (ℬ i) IsSubalgebraOfClass 𝒦)
  →          ⨅ ℬ IsSubalgebraOfClass (P{α}{β} 𝒦)

 lemPS⊆SP hwu fwu {I}{ℬ} B≤K =  ⨅ 𝒜 , (⨅ SA , ⨅SA≤⨅𝒜) ,
                                 ξ , (⨅≅ {fiu = fwu}{fiw = fwu} B≅SA)
  where
  𝒜 : I → Algebra α 𝑆
  𝒜 = λ i → ∣ B≤K i ∣

  SA : I → Algebra α 𝑆
  SA = λ i → ∣ fst ∥ B≤K i ∥ ∣

  B≅SA : ∀ i → ℬ i ≅ SA i
  B≅SA = λ i → ∥ snd ∥ B≤K i ∥ ∥

  SA≤𝒜 : ∀ i → (SA i) IsSubalgebraOf (𝒜 i)
  SA≤𝒜 = λ i → snd ∣ ∥ B≤K i ∥ ∣

  h : ∀ i → ∣ SA i ∣ → ∣ 𝒜 i ∣
  h = λ i → fst ∣ SA≤𝒜 i ∣

  hinj : ∀ i → IsInjective (h i)
  hinj = λ i → snd (snd ∣ ∥ B≤K i ∥ ∣)

  σ : ∣ ⨅ SA ∣ → ∣ ⨅ 𝒜 ∣
  σ = λ x i → (h i) (x i)
  ν : is-homomorphism (⨅ SA) (⨅ 𝒜) σ
  ν = λ 𝑓 𝒂 → fwu λ i → (snd ∣ SA≤𝒜 i ∣) 𝑓 (λ x → 𝒂 x i)

  σinj : IsInjective σ
  σinj σxσy = fwu λ i → (hinj i)(≡.cong-app σxσy i)

  ⨅SA≤⨅𝒜 : ⨅ SA ≤ ⨅ 𝒜
  ⨅SA≤⨅𝒜 = (σ , ν) , σinj

  ξ : ⨅ 𝒜 ∈ P 𝒦
  ξ = produ (λ i → P-expa (∣ snd ∥ B≤K i ∥ ∣))

PS(𝒦) ⊆ SP(𝒦)

Finally, we are in a position to prove that a product of subalgebras of algebras in a class 𝒦 is a subalgebra of a product of algebras in 𝒦.

module _  {α : Level} {fovu : funext (ov α) (ov α)}
          {𝒦 : Pred (Algebra α 𝑆)(ov α)} where

 PS⊆SP :  -- extensionality assumptions:
          hfunext (ov α)(ov α)

  →       P{ov α}{ov α} (S{α}{ov α} 𝒦) ⊆ S{ov α}{ov α} (P{α}{ov α} 𝒦)

 PS⊆SP _ (pbase (sbase x)) = sbase (pbase x)
 PS⊆SP _ (pbase (slift{𝑨} x)) = slift (S⊆SP{α}{ov α} 𝒦 (slift x))
 PS⊆SP _ (pbase{𝑩}(ssub{𝑨} sA B≤A)) = siso(ssub(S⊆SP 𝒦 (slift sA))(Lift-≤-Lift (ov(α)){𝑨}(ov(α))B≤A)) ≅-refl
 PS⊆SP _ (pbase (siso{𝑨}{𝑩} x A≅B)) = siso (S⊆SP 𝒦 (slift x)) ( Lift-Alg-iso A≅B )
 PS⊆SP hfe (pliftu x) = slift (PS⊆SP hfe x)
 PS⊆SP hfe (pliftw x) = slift (PS⊆SP hfe x)

 PS⊆SP hfe (produ{I}{𝒜} x) = (S-mono (P-idemp)) (subalgebra→S η)
  where
   ξ : (i : I) → (𝒜 i) IsSubalgebraOfClass (P{α}{ov α} 𝒦)
   ξ i = S→subalgebra (PS⊆SP hfe (x i))

   η : ⨅ 𝒜 IsSubalgebraOfClass (P{ov α}{ov α} (P{α}{ov α} 𝒦))
   η = lemPS⊆SP hfe fovu {I} {𝒜} ξ

 PS⊆SP hfe (prodw{I}{𝒜} x) = (S-mono (P-idemp)) (subalgebra→S η)
  where
   ξ : (i : I) → (𝒜 i) IsSubalgebraOfClass (P{α}{ov α} 𝒦)
   ξ i = S→subalgebra (PS⊆SP hfe (x i))

   η : ⨅ 𝒜 IsSubalgebraOfClass (P{ov α}{ov α} (P{α}{ov α} 𝒦))
   η = lemPS⊆SP hfe fovu  {I} {𝒜} ξ

 PS⊆SP hfe (pisow{𝑨}{𝑩} pA A≅B) = siso (PS⊆SP hfe pA) A≅B

More class inclusions

We conclude this subsection with three more inclusion relations that will have bit parts to play later (e.g., in the formal proof of Birkhoff’s Theorem).

P⊆V : {α β : Level}{𝒦 : Pred (Algebra α 𝑆)(ov α)} → P{α}{β} 𝒦 ⊆ V{α}{β} 𝒦

P⊆V (pbase x) = vbase x
P⊆V{α} (pliftu x) = vlift (P⊆V{α}{α} x)
P⊆V{α}{β} (pliftw x) = vliftw (P⊆V{α}{β} x)
P⊆V (produ x) = vprodu (λ i → P⊆V (x i))
P⊆V (prodw x) = vprodw (λ i → P⊆V (x i))
P⊆V (pisow x x₁) = visow (P⊆V x) x₁

SP⊆V :  {α β : Level}{𝒦 : Pred (Algebra α 𝑆)(ov α)}
 →      S{α ⊔ β}{α ⊔ β} (P{α}{β} 𝒦) ⊆ V 𝒦

SP⊆V (sbase{𝑨} PCloA) = P⊆V (pisow PCloA Lift-≅)
SP⊆V (slift{𝑨} x) = vliftw (SP⊆V x)
SP⊆V (ssub{𝑨}{𝑩} spA B≤A) = vssubw (SP⊆V spA) B≤A
SP⊆V (siso x x₁) = visow (SP⊆V x) x₁

V is closed under lift

As mentioned earlier, a technical hurdle that must be overcome when formalizing proofs in Agda is the proper handling of universe levels. In particular, in the proof of the Birkhoff’s theorem, for example, we will need to know that if an algebra 𝑨 belongs to the variety V 𝒦, then so does the lift of 𝑨. Let us get the tedious proof of this technical lemma out of the way.

Above we proved that SP(𝒦) ⊆ V(𝒦), and we did so under fairly general assumptions about the universe level parameters. Unfortunately, this is sometimes not quite general enough, so we now prove the inclusion again for the specific universe parameters that align with subsequent applications of this result.

module _  {α : Level}  {fe₀ : funext (ov α) α}
          {fe₁ : funext ((ov α) ⊔ (suc (ov α))) (suc (ov α))}
          {fe₂ : funext (ov α) (ov α)}
          {𝒦 : Pred (Algebra α 𝑆)(ov α)} where
 open Vlift {α}{fe₀}{fe₁}{fe₂}{𝒦}

 SP⊆V' : S{ov α}{suc (ov α)} (P{α}{ov α} 𝒦) ⊆ V 𝒦
 SP⊆V' (sbase{𝑨} x) = visow (VlA (SP⊆V (sbase x))) (≅-sym (Lift-Alg-assoc _ _{𝑨}))
 SP⊆V' (slift x) = VlA (SP⊆V x)

 SP⊆V' (ssub{𝑨}{𝑩} spA B≤A) = vssubw (VlA (SP⊆V spA)) B≤lA
  where
   B≤lA : 𝑩 ≤ Lift-Alg 𝑨 (suc (ov α))
   B≤lA = ≤-Lift 𝑨 B≤A

 SP⊆V' (siso{𝑨}{𝑩} x A≅B) = visow (VlA (SP⊆V x)) Goal
  where
   Goal : Lift-Alg 𝑨 (suc (ov α)) ≅ 𝑩
   Goal = ≅-trans (≅-sym Lift-≅) A≅B

⨅ S(𝒦) ∈ SP(𝒦)

Finally, we prove a result that plays an important role, e.g., in the formal proof of Birkhoff’s Theorem. As we saw in Base.Algebras.Products, the (informal) product ⨅ S(𝒦) of all subalgebras of algebras in 𝒦 is implemented (formally) in the agda-algebras library as ⨅ 𝔄 S(𝒦). Our goal is to prove that this product belongs to SP(𝒦). We do so by first proving that the product belongs to PS(𝒦) and then applying the PS⊆SP lemma.

Before doing so, we need to redefine the class product so that each factor comes with a map from the type X of variable symbols into that factor. We will explain the reason for this below.

module class-products-with-maps {α : Level}
 {X : Type α}
 {fe𝓕α : funext (ov α) α}
 {fe₁ : funext ((ov α) ⊔ (suc (ov α))) (suc (ov α))}
 {fe₂ : funext (ov α) (ov α)}
 (𝒦 : Pred (Algebra α 𝑆)(ov α))
 where

 ℑ' : Type (ov α)
 ℑ' = Σ[ 𝑨 ∈ (Algebra α 𝑆) ] ((𝑨 ∈ S{α}{α} 𝒦) × (X → ∣ 𝑨 ∣))

Notice that the second component of this dependent pair type is (𝑨 ∈ 𝒦) × (X → ∣ 𝑨 ∣). In previous versions of the [UALib][] this second component was simply 𝑨 ∈ 𝒦, until we realized that adding the type X → ∣ 𝑨 ∣ is quite useful. Later we will see exactly why, but for now suffice it to say that a map of type X → ∣ 𝑨 ∣ may be viewed abstractly as an ambient context, or more concretely, as an assignment of values in ∣ 𝑨 ∣ to variable symbols in X. When computing with or reasoning about products, while we don’t want to rigidly impose a context in advance, want do want to lay our hands on whatever context is ultimately assumed. Including the “context map” inside the index type ℑ of the product turns out to be a convenient way to achieve this flexibility.

Taking the product over the index type ℑ requires a function that maps an index i : ℑ to the corresponding algebra. Each i : ℑ is a triple, say, (𝑨 , p , h), where 𝑨 : Algebra α 𝑆, p : 𝑨 ∈ 𝒦, and h : X → ∣ 𝑨 ∣, so the function mapping an index to the corresponding algebra is simply the first projection.

 𝔄' : ℑ' → Algebra α 𝑆
 𝔄' = λ (i : ℑ') → ∣ i ∣

Finally, we define class-product which represents the product of all members of 𝒦.

 class-product' : Algebra (ov α) 𝑆
 class-product' = ⨅ 𝔄'

If p : 𝑨 ∈ 𝒦 and h : X → ∣ 𝑨 ∣, we view the triple (𝑨 , p , h) ∈ ℑ as an index over the class, and so we can think of 𝔄 (𝑨 , p , h) (which is simply 𝑨) as the projection of the product ⨅ 𝔄 onto the (𝑨 , p, h)-th component.

 class-prod-s-∈-ps : class-product' ∈ P{ov α}{ov α}(S 𝒦)
 class-prod-s-∈-ps = pisow psPllA (⨅≅ {fiu = fe₂}{fiw = fe𝓕α} llA≅A)
  where
  lA llA : ℑ' → Algebra (ov α) 𝑆
  lA i =  Lift-Alg (𝔄 i) (ov α)
  llA i = Lift-Alg (lA i) (ov α)

  slA : ∀ i → (lA i) ∈ S 𝒦
  slA i = siso (fst ∥ i ∥) Lift-≅

  psllA : ∀ i → (llA i) ∈ P (S 𝒦)
  psllA i = pbase (slA i)

  psPllA : ⨅ llA ∈ P (S 𝒦)
  psPllA = produ psllA

  llA≅A : ∀ i → (llA i) ≅ (𝔄' i)
  llA≅A i = ≅-trans (≅-sym Lift-≅)(≅-sym Lift-≅)

So, since PS⊆SP, we see that that the product of all subalgebras of a class 𝒦 belongs to SP(𝒦).

 class-prod-s-∈-sp : hfunext (ov α) (ov α) → class-product ∈ S(P 𝒦)
 class-prod-s-∈-sp hfe = PS⊆SP {fovu = fe₂} hfe class-prod-s-∈-ps

H preserves identities

First we prove that the closure operator H is compatible with identities that hold in the given class.

open ≡-Reasoning

private variable 𝓧 : Level
open Term

module _ (wd : SwellDef){X : Type 𝓧} {𝒦 : Pred (Algebra α 𝑆)(ov α)} where

 H-id1 : (p q : Term X) → 𝒦 ⊫ p ≈ q → H{β = α} 𝒦 ⊫ p ≈ q
 H-id1 p q σ (hbase x) = ⊧-Lift-invar wd p q (σ x)
 H-id1 p q σ (hhimg{𝑨}{𝑪} HA (𝑩 , ((φ , φh) , φE))) b = goal
  where
  IH : 𝑨 ⊧ p ≈ q
  IH = (H-id1 p q σ) HA

  preim : X → ∣ 𝑨 ∣
  preim x = Inv φ (φE (b x))

  ζ : ∀ x → φ (preim x) ≡ b x
  ζ x = InvIsInverseʳ (φE (b x))

  goal : (𝑩 ⟦ p ⟧) b ≡ (𝑩 ⟦ q ⟧) b
  goal =  (𝑩 ⟦ p ⟧) b           ≡⟨ wd 𝓧 α (𝑩 ⟦ p ⟧) b (φ ∘ preim )(λ i → (ζ i)⁻¹)⟩
          (𝑩 ⟦ p ⟧)(φ ∘ preim)  ≡⟨(comm-hom-term (wd 𝓥 α) 𝑩 (φ , φh) p preim)⁻¹ ⟩
          φ((𝑨 ⟦ p ⟧) preim)    ≡⟨ ≡.cong φ (IH preim) ⟩
          φ((𝑨 ⟦ q ⟧) preim)    ≡⟨ comm-hom-term (wd 𝓥 α) 𝑩 (φ , φh) q preim ⟩
          (𝑩 ⟦ q ⟧)(φ ∘ preim)  ≡⟨ wd 𝓧 α (𝑩 ⟦ q ⟧)(φ ∘ preim) b ζ ⟩
          (𝑩 ⟦ q ⟧) b           ∎

The converse of the foregoing result is almost too obvious to bother with. Nonetheless, we formalize it for completeness.

 H-id2 : ∀ {β} → (p q : Term X) → H{β = β} 𝒦 ⊫ p ≈ q → 𝒦 ⊫ p ≈ q
 H-id2 p q Hpq KA = ⊧-lower-invar wd p q (Hpq (hbase KA))

S preserves identities

 S-id1 : (p q : Term X) → 𝒦 ⊫ p ≈ q → S{β = α} 𝒦 ⊫ p ≈ q
 S-id1 p q σ (sbase x) = ⊧-Lift-invar wd p q (σ x)
 S-id1 p q σ (slift x) = ⊧-Lift-invar wd p q ((S-id1 p q σ) x)
 S-id1 p q σ (ssub{𝑨}{𝑩} sA B≤A) = ⊧-S-class-invar wd p q goal ν
  where --Apply S-⊧ to the class 𝒦 ∪ ｛ 𝑨 ｝
  τ : 𝑨 ⊧ p ≈ q
  τ = S-id1 p q σ sA

  Apq : ｛ 𝑨 ｝ ⊫ p ≈ q
  Apq ≡.refl = τ

  goal : (𝒦 ∪ ｛ 𝑨 ｝) ⊫ p ≈ q
  goal {𝑩} (inl x) = σ x
  goal {𝑩} (inr y) = Apq y

  ν : SubalgebraOfClass  (λ z → (𝒦 ∪ ｛ 𝑨 ｝)
                         (Data.Product.proj₁ z , Data.Product.proj₂ z))

  ν = (𝑩 , 𝑨 , (𝑩 , B≤A) , _⊎_.inj₂ ≡.refl , ≅-refl)

 S-id1 p q σ (siso{𝑨}{𝑩} x x₁) = ⊧-I-invar wd 𝑩 p q (S-id1 p q σ x) x₁

Again, the obvious converse is barely worth the bits needed to formalize it.

 S-id2 : ∀{β}(p q : Term X) → S{β = β}𝒦 ⊫ p ≈ q → 𝒦 ⊫ p ≈ q
 S-id2 p q Spq {𝑨} KA = ⊧-lower-invar wd p q (Spq (sbase KA))

P preserves identities

module _  (fe : DFunExt)(wd : SwellDef){X : Type 𝓧}
          {𝒦 : Pred (Algebra α 𝑆)(ov α)} where

 P-id1 : (p q : Term X) → 𝒦 ⊫ p ≈ q → P{β = α} 𝒦 ⊫ p ≈ q

 P-id1 p q σ (pbase x) = ⊧-Lift-invar wd p q (σ x)
 P-id1 p q σ (pliftu x) = ⊧-Lift-invar wd p q ((P-id1 p q σ) x)
 P-id1 p q σ (pliftw x) = ⊧-Lift-invar wd p q ((P-id1 p q σ) x)

 P-id1 p q σ (produ{I}{𝒜} x) = ⊧-P-lift-invar fe wd 𝒜  p q IH
  where
  IH : ∀ i → (Lift-Alg (𝒜 i) α) ⊧ p ≈ q
  IH i = ⊧-Lift-invar wd  p q ((P-id1 p q σ) (x i))

 P-id1 p q σ (prodw{I}{𝒜} x) = ⊧-P-lift-invar fe wd 𝒜  p q IH
  where
  IH : ∀ i → (Lift-Alg (𝒜 i) α) ⊧ p ≈ q
  IH i = ⊧-Lift-invar wd  p q ((P-id1 p q σ) (x i))

 P-id1 p q σ (pisow{𝑨}{𝑩} x y) = ⊧-I-invar wd 𝑩 p q (P-id1 p q σ x) y

and conversely,

module _  (wd : SwellDef){X : Type 𝓧} {𝒦 : Pred (Algebra α 𝑆)(ov α)} where

 P-id2 : ∀ {β}(p q : Term X) → P{β = β} 𝒦 ⊫ p ≈ q → 𝒦 ⊫ p ≈ q
 P-id2 p q PKpq KA = ⊧-lower-invar wd p q (PKpq (pbase KA))

V preserves identities

Finally, we prove the analogous preservation lemmas for the closure operator V.

module Vid  (fe : DFunExt)(wd : SwellDef)
            {𝓧 : Level} {X : Type 𝓧}{𝒦 : Pred (Algebra α 𝑆)(ov α)} where

 V-id1 : (p q : Term X) → 𝒦 ⊫ p ≈ q → V{β = α} 𝒦 ⊫ p ≈ q
 V-id1 p q σ (vbase x) = ⊧-Lift-invar wd p q (σ x)
 V-id1 p q σ (vlift{𝑨} x) = ⊧-Lift-invar wd p q ((V-id1 p q σ) x)
 V-id1 p q σ (vliftw{𝑨} x) = ⊧-Lift-invar wd p q ((V-id1 p q σ) x)
 V-id1 p q σ (vhimg{𝑨}{𝑪}VA (𝑩 , ((φ , φh) , φE))) b = goal
  where
  IH : 𝑨 ⊧ p ≈ q
  IH = V-id1 p q σ VA

  preim : X → ∣ 𝑨 ∣
  preim x = Inv φ (φE (b x))

  ζ : ∀ x → φ (preim x) ≡ b x
  ζ x = InvIsInverseʳ (φE (b x))

  goal : (𝑩 ⟦ p ⟧) b ≡ (𝑩 ⟦ q ⟧) b
  goal =  (𝑩 ⟦ p ⟧) b           ≡⟨ wd 𝓧 α (𝑩 ⟦ p ⟧) b (φ ∘ preim )(λ i → (ζ i)⁻¹)⟩
          (𝑩 ⟦ p ⟧)(φ ∘ preim)  ≡⟨(comm-hom-term (wd 𝓥 α) 𝑩 (φ , φh) p preim)⁻¹ ⟩
          φ((𝑨 ⟦ p ⟧) preim)    ≡⟨ ≡.cong φ (IH preim) ⟩
          φ((𝑨 ⟦ q ⟧) preim)    ≡⟨ comm-hom-term (wd 𝓥 α) 𝑩 (φ , φh) q preim ⟩
          (𝑩 ⟦ q ⟧)(φ ∘ preim)  ≡⟨ wd 𝓧 α (𝑩 ⟦ q ⟧)(φ ∘ preim) b ζ ⟩
          (𝑩 ⟦ q ⟧) b           ∎

 V-id1 p q σ ( vssubw {𝑨}{𝑩} VA B≤A ) =
  ⊧-S-class-invar wd p q goal (𝑩 , 𝑨 , (𝑩 , B≤A) , inr ≡.refl , ≅-refl)
   where
   IH : 𝑨 ⊧ p ≈ q
   IH = V-id1 p q σ VA

   Asinglepq : ｛ 𝑨 ｝ ⊫ p ≈ q
   Asinglepq ≡.refl = IH

   goal : (𝒦 ∪ ｛ 𝑨 ｝) ⊫ p ≈ q
   goal {𝑩} (inl x) = σ x
   goal {𝑩} (inr y) = Asinglepq y

 V-id1 p q σ (vprodu{I}{𝒜} V𝒜) = ⊧-P-invar fe wd 𝒜  p q λ i → V-id1 p q σ (V𝒜 i)
 V-id1 p q σ (vprodw{I}{𝒜} V𝒜) = ⊧-P-invar fe wd 𝒜  p q λ i → V-id1 p q σ (V𝒜 i)
 V-id1 p q σ (visou{𝑨}{𝑩} VA A≅B) = ⊧-I-invar wd 𝑩 p q (V-id1 p q σ VA) A≅B
 V-id1 p q σ (visow{𝑨}{𝑩} VA A≅B) = ⊧-I-invar wd 𝑩 p q (V-id1 p q σ VA) A≅B

module Vid'  (fe : DFunExt)(wd : SwellDef)
             {𝓧 : Level}{X : Type 𝓧}{𝒦 : Pred (Algebra α 𝑆)(ov α)} where
 open Vid fe wd {𝓧}{X}{𝒦} public
 V-id1' : (p q : Term X) → 𝒦 ⊫ p ≈ q → V{β = β} 𝒦 ⊫ p ≈ q
 V-id1' p q σ (vbase x) = ⊧-Lift-invar wd p q (σ x)
 V-id1' p q σ (vlift{𝑨} x) = ⊧-Lift-invar wd p q ((V-id1 p q σ) x)
 V-id1' p q σ (vliftw{𝑨} x) = ⊧-Lift-invar wd p q ((V-id1' p q σ) x)
 V-id1' p q σ (vhimg{𝑨}{𝑪} VA (𝑩 , ((φ , φh) , φE))) b = goal
  where
  IH : 𝑨 ⊧ p ≈ q
  IH = V-id1' p q σ VA

  preim : X → ∣ 𝑨 ∣
  preim x = Inv φ (φE (b x))

  ζ : ∀ x → φ (preim x) ≡ b x
  ζ x = InvIsInverseʳ (φE (b x))

  goal : (𝑩 ⟦ p ⟧) b ≡ (𝑩 ⟦ q ⟧) b
  goal =  (𝑩 ⟦ p ⟧) b           ≡⟨ wd 𝓧 _ (𝑩 ⟦ p ⟧) b (φ ∘ preim )(λ i → (ζ i)⁻¹)⟩
          (𝑩 ⟦ p ⟧)(φ ∘ preim)  ≡⟨(comm-hom-term (wd 𝓥 _) 𝑩 (φ , φh) p preim)⁻¹ ⟩
          φ((𝑨 ⟦ p ⟧) preim)    ≡⟨ ≡.cong φ (IH preim) ⟩
          φ((𝑨 ⟦ q ⟧) preim)    ≡⟨ comm-hom-term (wd 𝓥 _) 𝑩 (φ , φh) q preim ⟩
          (𝑩 ⟦ q ⟧)(φ ∘ preim)  ≡⟨ wd 𝓧 _ (𝑩 ⟦ q ⟧)(φ ∘ preim) b ζ ⟩
          (𝑩 ⟦ q ⟧) b           ∎

 V-id1' p q σ (vssubw {𝑨}{𝑩} VA B≤A) = ⊧-S-invar wd 𝑩 {p}{q}(V-id1' p q σ VA) B≤A
 V-id1' p q σ (vprodu{I}{𝒜} V𝒜) = ⊧-P-invar fe wd 𝒜  p q λ i → V-id1 p q σ (V𝒜 i)
 V-id1' p q σ (vprodw{I}{𝒜} V𝒜) = ⊧-P-invar fe wd 𝒜  p q λ i → V-id1' p q σ (V𝒜 i)
 V-id1' p q σ (visou {𝑨}{𝑩} VA A≅B) = ⊧-I-invar wd 𝑩 p q (V-id1 p q σ VA) A≅B
 V-id1' p q σ (visow{𝑨}{𝑩} VA A≅B) = ⊧-I-invar wd 𝑩 p q (V-id1' p q σ VA)A≅B

Class identities

From V-id1 it follows that if 𝒦 is a class of structures, then the set of identities modeled by all structures in 𝒦 is equivalent to the set of identities modeled by all structures in V 𝒦. In other terms, Th (V 𝒦) is precisely the set of identities modeled by 𝒦. We formalize this observation as follows.

module _  (fe : DFunExt)(wd : SwellDef)
          {𝓧 : Level}{X : Type 𝓧} {𝒦 : Pred (Algebra α 𝑆)(ov α)} where
 ovu lovu : Level
 ovu = ov α
 lovu = suc (ov α)
 𝕍 : Pred (Algebra lovu 𝑆) (suc lovu)
 𝕍 = V{α}{lovu} 𝒦
 𝒱 : Pred (Algebra ovu 𝑆) lovu
 𝒱 = V{β = ovu} 𝒦

 open Vid' fe wd {𝓧}{X}{𝒦} public
 class-ids-⇒ : (p q : ∣ 𝑻 X ∣) → 𝒦 ⊫ p ≈ q  →  (p , q) ∈ Th 𝒱
 class-ids-⇒ p q pKq VCloA = V-id1' p q pKq VCloA

 class-ids : (p q : ∣ 𝑻 X ∣) → 𝒦 ⊫ p ≈ q  →  (p , q) ∈ Th 𝕍
 class-ids p q pKq VCloA = V-id1' p q pKq VCloA


 class-ids-⇐ : (p q : ∣ 𝑻 X ∣) → (p , q) ∈ Th 𝒱 →  𝒦 ⊫ p ≈ q
 class-ids-⇐ p q Thpq {𝑨} KA = ⊧-lower-invar wd p q (Thpq (vbase KA))

Once again, and for the last time, completeness dictates that we formalize the coverse of V-id1, however obvious it may be.

module _ (wd : SwellDef){X : Type 𝓧}{𝒦 : Pred (Algebra α 𝑆)(ov α)} where

 V-id2 : (p q : Term X) → (V{β = β} 𝒦 ⊫ p ≈ q) → (𝒦 ⊫ p ≈ q)
 V-id2 p q Vpq {𝑨} KA = ⊧-lower-invar wd p q (Vpq (vbase KA))

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← Base.Varieties.Properties Base.Varieties.FreeAlgebras →