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Properties of the models relation for setoid algebras

We prove some closure and invariance properties of the relation ⊧. In particular, we prove the following facts (which are needed, for example, in the proof the Birkhoff HSP Theorem).

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

open import Overture using (π“ž ; π“₯ ; Signature)

module Setoid.Varieties.Properties {𝑆 : Signature π“ž π“₯} where

-- Imports from Agda and the Agda Standard Library -------------------------------------------
open import Agda.Primitive   using () renaming ( Set to Type )
open import Data.Product     using ( _,_ )
open import Function         using ( _∘_ ; Func )
open import Level            using ( Level )
open import Relation.Binary  using ( Setoid )
open import Relation.Unary   using ( Pred ; _∈_ )

import Relation.Binary.Reasoning.Setoid as SetoidReasoning

-- Imports from the Agda Universal Algebra Library ---------------------------------------------
open  import Overture                       using  ( ∣_∣ ; βˆ₯_βˆ₯ )
open  import Setoid.Functions               using  ( InvIsInverseΚ³ ; SurjInv )
open  import Base.Terms            {𝑆 = 𝑆}  using  ( Term ; β„Š )
open  import Setoid.Algebras       {𝑆 = 𝑆}
      using  ( Algebra ; Lift-AlgΛ‘ ; ov ; π•Œ[_] ; 𝔻[_] ; β¨… )

open  import Setoid.Homomorphisms  {𝑆 = 𝑆}
      using  ( hom ; _β‰…_ ; mkiso ; Lift-β‰…Λ‘ ; β‰…-sym ; _IsHomImageOf_ )

open  import Setoid.Terms          {𝑆 = 𝑆}
      using  ( 𝑻 ; module Environment ; comm-hom-term ; interp-prod ; term-agreement )

open  import Setoid.Subalgebras    {𝑆 = 𝑆}  using  ( _≀_ ; SubalgebrasOfClass )

open  import Setoid.Varieties.SoundAndComplete {𝑆 = 𝑆}
      using ( _⊧_ ; _⊨_ ; _⊫_ ; Eq ; _β‰ˆΜ‡_ ; lhs ; rhs ; _⊒_β–Ή_β‰ˆ_ )

private variable Ξ± ρᡃ Ξ² ρᡇ Ο‡ β„“ : Level

open Func     using ( cong ) renaming (f to _⟨$⟩_ )
open Algebra  using ( Domain )

Algebraic invariance of ⊧

The binary relation ⊧ would be practically useless if it were not an algebraic invariant (i.e., invariant under isomorphism).

module _ {X : Type Ο‡}{𝑨 : Algebra Ξ± ρᡃ}(𝑩 : Algebra Ξ² ρᡇ)(p q : Term X) where
 open Environment 𝑨      using () renaming ( ⟦_⟧   to ⟦_βŸ§β‚ )
 open Environment 𝑩      using () renaming ( ⟦_⟧   to ⟦_βŸ§β‚‚ )
 open Setoid (Domain 𝑨)  using () renaming ( _β‰ˆ_   to _β‰ˆβ‚_ )
 open Setoid (Domain 𝑩)  using ( _β‰ˆ_ ; sym )
 open SetoidReasoning (Domain 𝑩)

 ⊧-I-invar : 𝑨 ⊧ (p β‰ˆΜ‡ q)  β†’  𝑨 β‰… 𝑩  β†’  𝑩 ⊧ (p β‰ˆΜ‡ q)
 ⊧-I-invar Apq (mkiso fh gh f∼g g∼f) ρ =
  begin
   ⟦ p βŸ§β‚‚ ⟨$⟩ ρ              β‰ˆΛ˜βŸ¨ cong ⟦ p βŸ§β‚‚ (Ξ» x β†’ f∼g (ρ x)) ⟩
   ⟦ p βŸ§β‚‚ ⟨$⟩ (f ∘ (g ∘ ρ))  β‰ˆΛ˜βŸ¨ comm-hom-term fh p (g ∘ ρ) ⟩
   f (⟦ p βŸ§β‚ ⟨$⟩ (g ∘ ρ))    β‰ˆβŸ¨ cong ∣ fh ∣ (Apq (g ∘ ρ)) ⟩
   f (⟦ q βŸ§β‚ ⟨$⟩ (g ∘ ρ))    β‰ˆβŸ¨ comm-hom-term fh q (g ∘ ρ) ⟩
   ⟦ q βŸ§β‚‚ ⟨$⟩ (f ∘ (g ∘ ρ))  β‰ˆβŸ¨ cong ⟦ q βŸ§β‚‚ (Ξ» x β†’ f∼g (ρ x)) ⟩
   ⟦ q βŸ§β‚‚ ⟨$⟩ ρ              ∎
  where private f = _⟨$⟩_ ∣ fh ∣ ; g = _⟨$⟩_ ∣ gh ∣

As the proof makes clear, we show 𝑩 ⊧ p β‰ˆ q by showing that 𝑩 ⟦ p ⟧ ≑ 𝑩 ⟦ q ⟧ holds *extensionally*, that is, βˆ€ x, 𝑩 ⟦ p ⟧ x ≑ 𝑩 ⟦q ⟧ x`.

Lift-invariance of ⊧

The ⊧ relation is also invariant under the algebraic lift and lower operations.

module _ {X : Type Ο‡}{𝑨 : Algebra Ξ± ρᡃ} where

 ⊧-Lift-invar : (p q : Term X) β†’ 𝑨 ⊧ (p β‰ˆΜ‡ q) β†’ Lift-AlgΛ‘ 𝑨 Ξ² ⊧ (p β‰ˆΜ‡ q)
 ⊧-Lift-invar p q Apq = ⊧-I-invar (Lift-AlgΛ‘ 𝑨 _) p q Apq Lift-β‰…Λ‘

 ⊧-lower-invar : (p q : Term X) β†’ Lift-AlgΛ‘ 𝑨 Ξ² ⊧ (p β‰ˆΜ‡ q)  β†’  𝑨 ⊧ (p β‰ˆΜ‡ q)
 ⊧-lower-invar p q lApq = ⊧-I-invar 𝑨 p q lApq (β‰…-sym Lift-β‰…Λ‘)

Homomorphic invariance of ⊧

Identities modeled by an algebra 𝑨 are also modeled by every homomorphic image of 𝑨, which fact can be formalized as follows.

module _ {X : Type Ο‡}{𝑨 : Algebra Ξ± ρᡃ}{𝑩 : Algebra Ξ² ρᡇ}{p q : Term X} where

 ⊧-H-invar : 𝑨 ⊧ (p β‰ˆΜ‡ q) β†’ 𝑩 IsHomImageOf 𝑨 β†’ 𝑩 ⊧ (p β‰ˆΜ‡ q)
 ⊧-H-invar Apq (Ο†h , Ο†E) ρ =
  begin
       ⟦ p ⟧   ⟨$⟩               ρ    β‰ˆΛ˜βŸ¨  cong ⟦ p ⟧(Ξ» _ β†’ InvIsInverseΚ³ Ο†E)  ⟩
       ⟦ p ⟧   ⟨$⟩ (Ο† ∘  φ⁻¹  ∘  ρ)   β‰ˆΛ˜βŸ¨  comm-hom-term Ο†h p (φ⁻¹ ∘ ρ)        ⟩
   Ο†(  ⟦ p ⟧ᴬ  ⟨$⟩ (     φ⁻¹  ∘  ρ))  β‰ˆβŸ¨   cong ∣ Ο†h ∣ (Apq (φ⁻¹ ∘ ρ))         ⟩
   Ο†(  ⟦ q ⟧ᴬ  ⟨$⟩ (     φ⁻¹  ∘  ρ))  β‰ˆβŸ¨   comm-hom-term Ο†h q (φ⁻¹ ∘ ρ)        ⟩
       ⟦ q ⟧   ⟨$⟩ (Ο† ∘  φ⁻¹  ∘  ρ)   β‰ˆβŸ¨   cong ⟦ q ⟧(Ξ» _ β†’ InvIsInverseΚ³ Ο†E)  ⟩
       ⟦ q ⟧   ⟨$⟩               ρ    ∎
  where
  φ⁻¹ : π•Œ[ 𝑩 ] β†’ π•Œ[ 𝑨 ]
  φ⁻¹ = SurjInv ∣ Ο†h ∣ Ο†E
  private Ο† = (_⟨$⟩_ ∣ Ο†h ∣)
  open Environment 𝑨  using () renaming ( ⟦_⟧ to ⟦_⟧ᴬ)
  open Environment 𝑩  using ( ⟦_⟧ )
  open SetoidReasoning 𝔻[ 𝑩 ]

Subalgebraic invariance of ⊧

Identities modeled by an algebra 𝑨 are also modeled by every subalgebra of 𝑨, which fact can be formalized as follows.

module _ {X : Type Ο‡}{p q : Term X}{𝑨 : Algebra Ξ± ρᡃ}{𝑩 : Algebra Ξ² ρᡇ} where
 open Environment 𝑨      using () renaming ( ⟦_⟧ to ⟦_βŸ§β‚ )
 open Environment 𝑩      using () renaming ( ⟦_⟧ to ⟦_βŸ§β‚‚ )
 open Setoid (Domain 𝑨)  using ( _β‰ˆ_ )
 open Setoid (Domain 𝑩)  using () renaming ( _β‰ˆ_ to _β‰ˆβ‚‚_ )
 open SetoidReasoning (Domain 𝑨)

 ⊧-S-invar : 𝑨 ⊧ (p β‰ˆΜ‡ q) β†’  𝑩 ≀ 𝑨  β†’  𝑩 ⊧ (p β‰ˆΜ‡ q)
 ⊧-S-invar Apq B≀A b = goal
  where
  hh : hom 𝑩 𝑨
  hh = ∣ B≀A ∣
  h = _⟨$⟩_ ∣ hh ∣
  ΞΎ : βˆ€ b β†’ h (⟦ p βŸ§β‚‚ ⟨$⟩ b) β‰ˆ h (⟦ q βŸ§β‚‚ ⟨$⟩ b)
  ΞΎ b = begin
         h (⟦ p βŸ§β‚‚ ⟨$⟩ b)    β‰ˆβŸ¨ comm-hom-term hh p b ⟩
         ⟦ p βŸ§β‚ ⟨$⟩ (h ∘ b)  β‰ˆβŸ¨ Apq (h ∘ b) ⟩
         ⟦ q βŸ§β‚ ⟨$⟩ (h ∘ b)  β‰ˆΛ˜βŸ¨ comm-hom-term hh q b ⟩
         h (⟦ q βŸ§β‚‚ ⟨$⟩ b)    ∎

  goal : ⟦ p βŸ§β‚‚ ⟨$⟩ b β‰ˆβ‚‚ ⟦ q βŸ§β‚‚ ⟨$⟩ b
  goal = βˆ₯ B≀A βˆ₯ (ΞΎ b)

Next, identities modeled by a class of algebras is also modeled by all subalgebras of the class. In other terms, every term equation (p β‰ˆΜ‡ q) that is satisfied by all 𝑨 ∈ 𝒦 is also satisfied by every subalgebra of a member of 𝒦.

module _ {X : Type Ο‡}{p q : Term X} where

 ⊧-S-class-invar :  {𝒦 : Pred (Algebra Ξ± ρᡃ) β„“}
  β†’                 (𝒦 ⊫ (p β‰ˆΜ‡ q)) β†’ ((𝑩 , _) : SubalgebrasOfClass 𝒦 {Ξ²}{ρᡇ})
  β†’                 𝑩 ⊧ (p β‰ˆΜ‡ q)
 ⊧-S-class-invar Kpq (𝑩 , 𝑨 , kA , B≀A) = ⊧-S-invar{p = p}{q} (Kpq 𝑨 kA) B≀A

Product invariance of ⊧

An identity satisfied by all algebras in an indexed collection is also satisfied by the product of algebras in that collection.

module _ {X : Type Ο‡}{p q : Term X}{I : Type β„“}(π’œ : I β†’ Algebra Ξ± ρᡃ) where

 ⊧-P-invar : (βˆ€ i β†’ π’œ i ⊧ (p β‰ˆΜ‡ q)) β†’ β¨… π’œ ⊧ (p β‰ˆΜ‡ q)
 ⊧-P-invar π’œpq a = goal
  where
  open Algebra (β¨… π’œ)      using () renaming ( Domain to β¨…A )
  open Environment (β¨… π’œ)  using () renaming ( ⟦_⟧ to ⟦_βŸ§β‚ )
  open Environment        using ( ⟦_⟧ )
  open Setoid β¨…A          using ( _β‰ˆ_ )
  open SetoidReasoning β¨…A

  ΞΎ : (Ξ» i β†’ (⟦ π’œ i ⟧ p) ⟨$⟩ (Ξ» x β†’ (a x) i)) β‰ˆ (Ξ» i β†’ (⟦ π’œ i ⟧ q) ⟨$⟩ (Ξ» x β†’ (a x) i))
  ΞΎ = Ξ» i β†’ π’œpq i (Ξ» x β†’ (a x) i)
  goal : ⟦ p βŸ§β‚ ⟨$⟩ a β‰ˆ ⟦ q βŸ§β‚ ⟨$⟩ a
  goal = begin
          ⟦ p βŸ§β‚ ⟨$⟩ a                             β‰ˆβŸ¨ interp-prod π’œ p a ⟩
          (Ξ» i β†’ (⟦ π’œ i ⟧ p) ⟨$⟩ (Ξ» x β†’ (a x) i))  β‰ˆβŸ¨ ΞΎ ⟩
          (Ξ» i β†’ (⟦ π’œ i ⟧ q) ⟨$⟩ (Ξ» x β†’ (a x) i))  β‰ˆΛ˜βŸ¨ interp-prod π’œ q a ⟩
          ⟦ q βŸ§β‚ ⟨$⟩ a                             ∎

An identity satisfied by all algebras in a class is also satisfied by the product of algebras in the class.

 ⊧-P-class-invar :  (𝒦 : Pred (Algebra Ξ± ρᡃ)(ov Ξ±))
  β†’                 𝒦 ⊫ (p β‰ˆΜ‡ q) β†’ (βˆ€ i β†’ π’œ i ∈ 𝒦) β†’ β¨… π’œ ⊧ (p β‰ˆΜ‡ q)

 ⊧-P-class-invar 𝒦 Οƒ Kπ’œ = ⊧-P-invar (Ξ» i ρ β†’ Οƒ (π’œ i) (Kπ’œ i) ρ)

Another fact that will turn out to be useful is that a product of a collection of algebras models (p β‰ˆΜ‡ q) if the lift of each algebra in the collection models (p β‰ˆΜ‡ q).

 ⊧-P-lift-invar : (βˆ€ i β†’ Lift-AlgΛ‘ (π’œ i) Ξ² ⊧ (p β‰ˆΜ‡ q))  β†’  β¨… π’œ ⊧ (p β‰ˆΜ‡ q)
 ⊧-P-lift-invar α = ⊧-P-invar Aipq
  where
  Aipq : βˆ€ i β†’ (π’œ i) ⊧ (p β‰ˆΜ‡ q)
  Aipq i = ⊧-lower-invar{𝑨 = (π’œ i)} p q (Ξ± i)

Homomorphic invariance of ⊧

If an algebra 𝑨 models an identity (p β‰ˆΜ‡ q), then the pair (p , q) belongs to the kernel of every homomorphism Ο† : hom (𝑻 X) 𝑨 from the term algebra to 𝑨; that is, every homomorphism from 𝑻 X to 𝑨 maps p and q to the same element of 𝑨.

module _ {X : Type Ο‡}{p q : Term X}{𝑨 : Algebra Ξ± ρᡃ}(Ο†h : hom (𝑻 X) 𝑨) where
 open Setoid (Domain 𝑨) using ( _β‰ˆ_ )
 private Ο† = _⟨$⟩_ ∣ Ο†h ∣

 ⊧-H-ker : 𝑨 ⊧ (p β‰ˆΜ‡ q) β†’ Ο† p β‰ˆ Ο† q
 ⊧-H-ker β =
  begin
   Ο† p                 β‰ˆβŸ¨ cong ∣ Ο†h ∣ (term-agreement p)⟩
   Ο† (⟦ p ⟧ ⟨$⟩ β„Š)     β‰ˆβŸ¨ comm-hom-term Ο†h p β„Š ⟩
   ⟦ p βŸ§β‚‚ ⟨$⟩ (Ο† ∘ β„Š)  β‰ˆβŸ¨ Ξ² (Ο† ∘ β„Š) ⟩
   ⟦ q βŸ§β‚‚ ⟨$⟩ (Ο† ∘ β„Š)  β‰ˆΛ˜βŸ¨ comm-hom-term Ο†h q β„Š ⟩
   Ο† (⟦ q ⟧ ⟨$⟩ β„Š)     β‰ˆΛ˜βŸ¨ cong ∣ Ο†h ∣ (term-agreement q)⟩
   Ο† q                 ∎

  where
  open SetoidReasoning (Domain 𝑨)
  open Environment 𝑨      using () renaming ( ⟦_⟧ to ⟦_βŸ§β‚‚ )
  open Environment (𝑻 X)  using ( ⟦_⟧ )

← Setoid.Varieties.Closure Setoid.Varieties.Preservation β†’