{-# OPTIONS --prop --rewriting #-} open import PropUtil module FFOLCompleteness where open import Agda.Primitive open import FFOL open import ListUtil record Family : Set (lsuc (ℓ¹)) where field World : Set ℓ¹ _≤_ : World → World → Prop ≤refl : {w : World} → w ≤ w ≤tran : {w w' w'' : World} → w ≤ w' → w' ≤ w'' → w ≤ w' TM : World → Set ℓ¹ TM≤ : {w w' : World} → w ≤ w' → TM w → TM w' REL : (w : World) → TM w → TM w → Prop ℓ¹ REL≤ : {w w' : World} → {t u : TM w} → (eq : w ≤ w') → REL w t u → REL w' (TM≤ eq t) (TM≤ eq u) infixr 10 _∘_ Con = World → Set ℓ¹ Sub : Con → Con → Set ℓ¹ Sub Δ Γ = (w : World) → Δ w → Γ w _∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ α ∘ β = λ w γ → α w (β w γ) id : {Γ : Con} → Sub Γ Γ id = λ w γ → γ ◇ : Con -- The initial object of the category ◇ = λ w → ⊤ₛ ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object ε w Γ = ttₛ -- Functor Con → Set called Tm Tm : Con → Set ℓ¹ Tm Γ = (w : World) → (Γ w) → TM w _[_]t : {Γ Δ : Con} → Tm Γ → Sub Δ Γ → Tm Δ -- The functor's action on morphisms t [ σ ]t = λ w → λ γ → t w (σ w γ) -- Tm⁺ _▹ₜ : Con → Con Γ ▹ₜ = λ w → (Γ w) × (TM w) πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ πₜ¹ σ = λ w → λ x → proj×₁ (σ w x) πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ πₜ² σ = λ w → λ x → proj×₂ (σ w x) _,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ) σ ,ₜ t = λ w → λ x → (σ w x) ,× (t w x) -- Functor Con → Set called For For : Con → Set (lsuc ℓ¹) For Γ = (w : World) → (Γ w) → Prop ℓ¹ _[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms F [ σ ]f = λ w → λ x → F w (σ w x) -- Formulas with relation on terms R : {Γ : Con} → Tm Γ → Tm Γ → For Γ R t u = λ w → λ γ → REL w (t w γ) (u w γ) -- Proofs _⊢_ : (Γ : Con) → For Γ → Prop ℓ¹ Γ ⊢ F = ∀ w (γ : Γ w) → F w γ _[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms prf [ σ ]p = λ w → λ γ → prf w (σ w γ) -- Equalities below are useless because Γ ⊢ F is in prop -- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf -- []p-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ α ∘ β ]p ≡ (prf [ β ]p) [ α ]p -- → Prop⁺ _▹ₚ_ : (Γ : Con) → For Γ → Con Γ ▹ₚ F = λ w → (Γ w) ×'' (F w) πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ πₚ¹ σ w δ = proj×''₁ (σ w δ) πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f) πₚ² σ w δ = proj×''₂ (σ w δ) _,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F) _,ₚ_ {F = F} σ pf w δ = (σ w δ) ,×'' pf w δ -- Implication _⇒_ : {Γ : Con} → For Γ → For Γ → For Γ F ⇒ G = λ w → λ γ → (∀ w' → w ≤ w' → (F w γ) → (G w γ)) -- Forall ∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ ∀∀ F = λ w → λ γ → ∀ t → F w (γ ,× t) -- Lam & App lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G) lam prf = λ w γ w' s h → prf w (γ ,×'' h) app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G app prf prf' = λ w γ → prf w γ w ≤refl (prf' w γ) -- Again, we don't write the _[_]p equalities as everything is in Prop -- ∀i and ∀e ∀i : {Γ : Con} → {F : For (Γ ▹ₜ)} → (Γ ▹ₜ) ⊢ F → Γ ⊢ (∀∀ F) ∀i p w γ = λ t → p w (γ ,× t) ∀e : {Γ : Con} → {F : For (Γ ▹ₜ)} → Γ ⊢ (∀∀ F) → {t : Tm Γ} → Γ ⊢ ( F [(id {Γ}) ,ₜ t ]f) ∀e p {t} w γ = p w γ (t w γ) tod : FFOL tod = record { Con = Con ; Sub = Sub ; _∘_ = _∘_ ; ∘-ass = refl ; id = id ; idl = refl ; idr = refl ; ◇ = ◇ ; ε = ε ; ε-u = refl ; Tm = Tm ; _[_]t = _[_]t ; []t-id = refl ; []t-∘ = refl ; _▹ₜ = _▹ₜ ; πₜ¹ = πₜ¹ ; πₜ² = πₜ² ; _,ₜ_ = _,ₜ_ ; πₜ²∘,ₜ = refl ; πₜ¹∘,ₜ = refl ; ,ₜ∘πₜ = refl ; ,ₜ∘ = refl ; For = For ; _[_]f = _[_]f ; []f-id = refl ; []f-∘ = refl ; _⊢_ = _⊢_ ; _[_]p = _[_]p ; _▹ₚ_ = _▹ₚ_ ; πₚ¹ = πₚ¹ ; πₚ² = πₚ² ; _,ₚ_ = _,ₚ_ ; ,ₚ∘πₚ = refl ; πₚ¹∘,ₚ = refl ; ,ₚ∘ = refl ; _⇒_ = _⇒_ ; []f-⇒ = refl ; ∀∀ = ∀∀ ; []f-∀∀ = refl ; lam = lam ; app = app ; ∀i = ∀i ; ∀e = ∀e ; R = R ; R[] = refl } record Presheaf : Set (lsuc (ℓ¹)) where field World : Set ℓ¹ _-w->_ : World → World → Set ℓ¹ -- arrows -w->id : {w : World} → w -w-> w -- id arrow _∘-w->_ : {w w' w'' : World} → w -w-> w' → w' -w-> w'' → w -w-> w'' -- arrow composition -w->ass : {w w' w'' w''' : World}{a : w -w-> w'}{b : w' -w-> w''}{c : w'' -w-> w'''} → ((a ∘-w-> b) ∘-w-> c) ≡ (a ∘-w-> (b ∘-w-> c)) -w->idl : {w w' : World} → {a : w -w-> w'} → (-w->id {w}) ∘-w-> a ≡ a -w->idr : {w w' : World} → {a : w -w-> w'} → a ∘-w-> (-w->id {w'}) ≡ a TM : World → Set ℓ¹ TM≤ : {w w' : World} → w -w-> w' → TM w' → TM w REL : (w : World) → TM w → TM w → Prop ℓ¹ REL≤ : {w w' : World} → {t u : TM w'} → (eq : w -w-> w') → REL w' t u → REL w (TM≤ eq t) (TM≤ eq u) infixr 10 _∘_ Con = World → Set ℓ¹ Sub : Con → Con → Set ℓ¹ Sub Δ Γ = (w : World) → Δ w → Γ w _∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ α ∘ β = λ w γ → α w (β w γ) id : {Γ : Con} → Sub Γ Γ id = λ w γ → γ ◇ : Con -- The initial object of the category ◇ = λ w → ⊤ₛ ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object ε w Γ = ttₛ -- Functor Con → Set called Tm Tm : Con → Set ℓ¹ Tm Γ = (w : World) → (Γ w) → TM w _[_]t : {Γ Δ : Con} → Tm Γ → Sub Δ Γ → Tm Δ -- The functor's action on morphisms t [ σ ]t = λ w → λ γ → t w (σ w γ) -- Tm⁺ _▹ₜ : Con → Con Γ ▹ₜ = λ w → (Γ w) × (TM w) πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ πₜ¹ σ = λ w → λ x → proj×₁ (σ w x) πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ πₜ² σ = λ w → λ x → proj×₂ (σ w x) _,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ) σ ,ₜ t = λ w → λ x → (σ w x) ,× (t w x) -- Functor Con → Set called For For : Con → Set (lsuc ℓ¹) For Γ = (w : World) → (Γ w) → Prop ℓ¹ _[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms F [ σ ]f = λ w → λ x → F w (σ w x) -- Formulas with relation on terms R : {Γ : Con} → Tm Γ → Tm Γ → For Γ R t u = λ w → λ γ → REL w (t w γ) (u w γ) -- Proofs _⊢_ : (Γ : Con) → For Γ → Prop ℓ¹ Γ ⊢ F = ∀ w (γ : Γ w) → F w γ _[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms prf [ σ ]p = λ w → λ γ → prf w (σ w γ) -- Equalities below are useless because Γ ⊢ F is in prop -- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf -- []p-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ α ∘ β ]p ≡ (prf [ β ]p) [ α ]p -- → Prop⁺ _▹ₚ_ : (Γ : Con) → For Γ → Con Γ ▹ₚ F = λ w → (Γ w) ×'' (F w) πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ πₚ¹ σ w δ = proj×''₁ (σ w δ) πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f) πₚ² σ w δ = proj×''₂ (σ w δ) _,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F) _,ₚ_ {F = F} σ pf w δ = (σ w δ) ,×'' pf w δ -- Implication _⇒_ : {Γ : Con} → For Γ → For Γ → For Γ F ⇒ G = λ w → λ γ → (∀ w' → w -w-> w' → (F w γ) → (G w γ)) -- Forall ∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ ∀∀ F = λ w → λ γ → ∀ t → F w (γ ,× t) -- Lam & App lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G) lam prf = λ w γ w' s h → prf w (γ ,×'' h) app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G app prf prf' = λ w γ → prf w γ w -w->id (prf' w γ) -- Again, we don't write the _[_]p equalities as everything is in Prop -- ∀i and ∀e ∀i : {Γ : Con} → {F : For (Γ ▹ₜ)} → (Γ ▹ₜ) ⊢ F → Γ ⊢ (∀∀ F) ∀i p w γ = λ t → p w (γ ,× t) ∀e : {Γ : Con} → {F : For (Γ ▹ₜ)} → Γ ⊢ (∀∀ F) → {t : Tm Γ} → Γ ⊢ ( F [(id {Γ}) ,ₜ t ]f) ∀e p {t} w γ = p w γ (t w γ) tod : FFOL tod = record { Con = Con ; Sub = Sub ; _∘_ = _∘_ ; ∘-ass = refl ; id = id ; idl = refl ; idr = refl ; ◇ = ◇ ; ε = ε ; ε-u = refl ; Tm = Tm ; _[_]t = _[_]t ; []t-id = refl ; []t-∘ = refl ; _▹ₜ = _▹ₜ ; πₜ¹ = πₜ¹ ; πₜ² = πₜ² ; _,ₜ_ = _,ₜ_ ; πₜ²∘,ₜ = refl ; πₜ¹∘,ₜ = refl ; ,ₜ∘πₜ = refl ; ,ₜ∘ = refl ; For = For ; _[_]f = _[_]f ; []f-id = refl ; []f-∘ = refl ; _⊢_ = _⊢_ ; _[_]p = _[_]p ; _▹ₚ_ = _▹ₚ_ ; πₚ¹ = πₚ¹ ; πₚ² = πₚ² ; _,ₚ_ = _,ₚ_ ; ,ₚ∘πₚ = refl ; πₚ¹∘,ₚ = refl ; ,ₚ∘ = refl ; _⇒_ = _⇒_ ; []f-⇒ = refl ; ∀∀ = ∀∀ ; []f-∀∀ = refl ; lam = lam ; app = app ; ∀i = ∀i ; ∀e = ∀e ; R = R ; R[] = refl } module U where import FFOLInitial as I psh : Presheaf psh = record { World = I.Con ; _-w->_ = I.Sub ; -w->id = I.id ; _∘-w->_ = λ σ σ' → σ' I.∘ σ ; -w->ass = ≡sym I.∘-ass ; -w->idl = I.idr ; -w->idr = I.idl ; TM = λ Γ → I.Tm (I.Con.t Γ) ; TM≤ = λ σ t → t I.[ I.Sub.t σ ]t ; REL = λ Γ t u → I.Pf (I.Con.t Γ) (I.Con.p Γ) (I.R t u) ; REL≤ = λ s pf → (pf I.[ I.Sub.t s ]pₜ) I.[ I.Sub.p s ]p } open Presheaf psh public -- Completeness proof -- We first build our universal Kripke model module ComplenessProof where -- We have a model, we construct the Universal Presheaf model of this model import FFOLInitial as I