328 lines
13 KiB
Agda
328 lines
13 KiB
Agda
{-# OPTIONS --prop --rewriting #-}
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open import PropUtil
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module FFOLCompleteness where
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open import Agda.Primitive
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open import FinitaryFirstOrderLogic
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open import ListUtil
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record Family : Set (lsuc (ℓ¹)) where
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field
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World : Set ℓ¹
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_≤_ : World → World → Prop
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≤refl : {w : World} → w ≤ w
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≤tran : {w w' w'' : World} → w ≤ w' → w' ≤ w'' → w ≤ w'
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TM : World → Set ℓ¹
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TM≤ : {w w' : World} → w ≤ w' → TM w → TM w'
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REL : (w : World) → TM w → TM w → Prop ℓ¹
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REL≤ : {w w' : World} → {t u : TM w} → (eq : w ≤ w') → REL w t u → REL w' (TM≤ eq t) (TM≤ eq u)
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infixr 10 _∘_
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Con = World → Set ℓ¹
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Sub : Con → Con → Set ℓ¹
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Sub Δ Γ = (w : World) → Δ w → Γ w
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_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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α ∘ β = λ w γ → α w (β w γ)
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id : {Γ : Con} → Sub Γ Γ
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id = λ w γ → γ
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◇ : Con -- The initial object of the category
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◇ = λ w → ⊤ₛ
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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ε w Γ = ttₛ
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-- Functor Con → Set called Tm
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Tm : Con → Set ℓ¹
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Tm Γ = (w : World) → (Γ w) → TM w
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_[_]t : {Γ Δ : Con} → Tm Γ → Sub Δ Γ → Tm Δ -- The functor's action on morphisms
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t [ σ ]t = λ w → λ γ → t w (σ w γ)
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-- Tm⁺
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_▹ₜ : Con → Con
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Γ ▹ₜ = λ w → (Γ w) × (TM w)
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πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ
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πₜ¹ σ = λ w → λ x → proj×₁ (σ w x)
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πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ
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πₜ² σ = λ w → λ x → proj×₂ (σ w x)
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_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ)
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σ ,ₜ t = λ w → λ x → (σ w x) ,× (t w x)
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-- Functor Con → Set called For
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For : Con → Set (lsuc ℓ¹)
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For Γ = (w : World) → (Γ w) → Prop ℓ¹
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_[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms
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F [ σ ]f = λ w → λ x → F w (σ w x)
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-- Formulas with relation on terms
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R : {Γ : Con} → Tm Γ → Tm Γ → For Γ
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R t u = λ w → λ γ → REL w (t w γ) (u w γ)
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-- Proofs
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_⊢_ : (Γ : Con) → For Γ → Prop ℓ¹
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Γ ⊢ F = ∀ w (γ : Γ w) → F w γ
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_[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms
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prf [ σ ]p = λ w → λ γ → prf w (σ w γ)
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-- Equalities below are useless because Γ ⊢ F is in prop
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-- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf
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-- []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
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-- → Prop⁺
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_▹ₚ_ : (Γ : Con) → For Γ → Con
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Γ ▹ₚ F = λ w → (Γ w) ×'' (F w)
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πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
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πₚ¹ σ w δ = proj×''₁ (σ w δ)
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πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f)
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πₚ² σ w δ = proj×''₂ (σ w δ)
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_,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F)
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_,ₚ_ {F = F} σ pf w δ = (σ w δ) ,×'' pf w δ
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-- Implication
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_⇒_ : {Γ : Con} → For Γ → For Γ → For Γ
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F ⇒ G = λ w → λ γ → (∀ w' → w ≤ w' → (F w γ) → (G w γ))
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-- Forall
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∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ
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∀∀ F = λ w → λ γ → ∀ t → F w (γ ,× t)
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-- Lam & App
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lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G)
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lam prf = λ w γ w' s h → prf w (γ ,×'' h)
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app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G
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app prf prf' = λ w γ → prf w γ w ≤refl (prf' w γ)
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-- Again, we don't write the _[_]p equalities as everything is in Prop
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-- ∀i and ∀e
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∀i : {Γ : Con} → {F : For (Γ ▹ₜ)} → (Γ ▹ₜ) ⊢ F → Γ ⊢ (∀∀ F)
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∀i p w γ = λ t → p w (γ ,× t)
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∀e : {Γ : Con} → {F : For (Γ ▹ₜ)} → Γ ⊢ (∀∀ F) → {t : Tm Γ} → Γ ⊢ ( F [(id {Γ}) ,ₜ t ]f)
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∀e p {t} w γ = p w γ (t w γ)
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tod : FFOL
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tod = record
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{ Con = Con
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; Sub = Sub
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; _∘_ = _∘_
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; ∘-ass = refl
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; id = id
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; idl = refl
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; idr = refl
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; ◇ = ◇
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; ε = ε
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; ε-u = refl
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; Tm = Tm
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; _[_]t = _[_]t
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; []t-id = refl
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; []t-∘ = refl
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; _▹ₜ = _▹ₜ
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; πₜ¹ = πₜ¹
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; πₜ² = πₜ²
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; _,ₜ_ = _,ₜ_
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; πₜ²∘,ₜ = refl
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; πₜ¹∘,ₜ = refl
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; ,ₜ∘πₜ = refl
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; ,ₜ∘ = refl
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; For = For
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; _[_]f = _[_]f
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; []f-id = refl
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; []f-∘ = refl
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; _⊢_ = _⊢_
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; _[_]p = _[_]p
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; _▹ₚ_ = _▹ₚ_
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; πₚ¹ = πₚ¹
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; πₚ² = πₚ²
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; _,ₚ_ = _,ₚ_
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; ,ₚ∘πₚ = refl
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; πₚ¹∘,ₚ = refl
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; ,ₚ∘ = refl
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; _⇒_ = _⇒_
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; []f-⇒ = refl
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; ∀∀ = ∀∀
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; []f-∀∀ = refl
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; lam = lam
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; app = app
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; ∀i = ∀i
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; ∀e = ∀e
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; R = R
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; R[] = refl
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}
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record Presheaf : Set (lsuc (ℓ¹)) where
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field
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World : Set ℓ¹
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_≤_ : World → World → Set ℓ¹ -- arrows
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≤refl : {w : World} → w ≤ w -- id arrow
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≤tran : {w w' w'' : World} → w ≤ w' → w' ≤ w'' → w ≤ w'' -- arrow composition
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≤-ass : {w w' w'' w''' : World}{a : w ≤ w'}{b : w' ≤ w''}{c : w'' ≤ w'''}
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→ (≤tran (≤tran a b) c) ≡ (≤tran a (≤tran b c))
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≤-idl : {w w' : World} → {a : w ≤ w'} → ≤tran (≤refl {w}) a ≡ a
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≤-idr : {w w' : World} → {a : w ≤ w'} → ≤tran a (≤refl {w'}) ≡ a
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TM : World → Set ℓ¹
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TM≤ : {w w' : World} → w ≤ w' → TM w → TM w'
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REL : (w : World) → TM w → TM w → Prop ℓ¹
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REL≤ : {w w' : World} → {t u : TM w} → (eq : w ≤ w') → REL w t u → REL w' (TM≤ eq t) (TM≤ eq u)
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infixr 10 _∘_
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Con = World → Set ℓ¹
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Sub : Con → Con → Set ℓ¹
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Sub Δ Γ = (w : World) → Δ w → Γ w
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_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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α ∘ β = λ w γ → α w (β w γ)
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id : {Γ : Con} → Sub Γ Γ
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id = λ w γ → γ
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◇ : Con -- The initial object of the category
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◇ = λ w → ⊤ₛ
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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ε w Γ = ttₛ
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-- Functor Con → Set called Tm
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Tm : Con → Set ℓ¹
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Tm Γ = (w : World) → (Γ w) → TM w
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_[_]t : {Γ Δ : Con} → Tm Γ → Sub Δ Γ → Tm Δ -- The functor's action on morphisms
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t [ σ ]t = λ w → λ γ → t w (σ w γ)
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-- Tm⁺
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_▹ₜ : Con → Con
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Γ ▹ₜ = λ w → (Γ w) × (TM w)
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πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ
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πₜ¹ σ = λ w → λ x → proj×₁ (σ w x)
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πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ
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πₜ² σ = λ w → λ x → proj×₂ (σ w x)
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_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ)
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σ ,ₜ t = λ w → λ x → (σ w x) ,× (t w x)
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-- Functor Con → Set called For
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For : Con → Set (lsuc ℓ¹)
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For Γ = (w : World) → (Γ w) → Prop ℓ¹
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_[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms
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F [ σ ]f = λ w → λ x → F w (σ w x)
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-- Formulas with relation on terms
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R : {Γ : Con} → Tm Γ → Tm Γ → For Γ
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R t u = λ w → λ γ → REL w (t w γ) (u w γ)
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-- Proofs
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_⊢_ : (Γ : Con) → For Γ → Prop ℓ¹
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Γ ⊢ F = ∀ w (γ : Γ w) → F w γ
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_[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms
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prf [ σ ]p = λ w → λ γ → prf w (σ w γ)
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-- Equalities below are useless because Γ ⊢ F is in prop
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-- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf
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-- []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
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-- → Prop⁺
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_▹ₚ_ : (Γ : Con) → For Γ → Con
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Γ ▹ₚ F = λ w → (Γ w) ×'' (F w)
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πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
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πₚ¹ σ w δ = proj×''₁ (σ w δ)
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πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f)
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πₚ² σ w δ = proj×''₂ (σ w δ)
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_,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F)
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_,ₚ_ {F = F} σ pf w δ = (σ w δ) ,×'' pf w δ
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-- Implication
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_⇒_ : {Γ : Con} → For Γ → For Γ → For Γ
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F ⇒ G = λ w → λ γ → (∀ w' → w ≤ w' → (F w γ) → (G w γ))
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-- Forall
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∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ
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∀∀ F = λ w → λ γ → ∀ t → F w (γ ,× t)
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-- Lam & App
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lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G)
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lam prf = λ w γ w' s h → prf w (γ ,×'' h)
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app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G
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app prf prf' = λ w γ → prf w γ w ≤refl (prf' w γ)
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-- Again, we don't write the _[_]p equalities as everything is in Prop
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-- ∀i and ∀e
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∀i : {Γ : Con} → {F : For (Γ ▹ₜ)} → (Γ ▹ₜ) ⊢ F → Γ ⊢ (∀∀ F)
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∀i p w γ = λ t → p w (γ ,× t)
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∀e : {Γ : Con} → {F : For (Γ ▹ₜ)} → Γ ⊢ (∀∀ F) → {t : Tm Γ} → Γ ⊢ ( F [(id {Γ}) ,ₜ t ]f)
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∀e p {t} w γ = p w γ (t w γ)
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tod : FFOL
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tod = record
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{ Con = Con
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; Sub = Sub
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; _∘_ = _∘_
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; ∘-ass = refl
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; id = id
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; idl = refl
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; idr = refl
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; ◇ = ◇
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; ε = ε
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; ε-u = refl
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; Tm = Tm
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; _[_]t = _[_]t
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; []t-id = refl
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; []t-∘ = refl
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; _▹ₜ = _▹ₜ
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; πₜ¹ = πₜ¹
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; πₜ² = πₜ²
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; _,ₜ_ = _,ₜ_
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; πₜ²∘,ₜ = refl
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; πₜ¹∘,ₜ = refl
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; ,ₜ∘πₜ = refl
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; ,ₜ∘ = refl
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; For = For
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; _[_]f = _[_]f
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; []f-id = refl
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; []f-∘ = refl
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; _⊢_ = _⊢_
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; _[_]p = _[_]p
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; _▹ₚ_ = _▹ₚ_
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; πₚ¹ = πₚ¹
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; πₚ² = πₚ²
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; _,ₚ_ = _,ₚ_
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; ,ₚ∘πₚ = refl
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; πₚ¹∘,ₚ = refl
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; ,ₚ∘ = refl
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; _⇒_ = _⇒_
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; []f-⇒ = refl
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; ∀∀ = ∀∀
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; []f-∀∀ = refl
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; lam = lam
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; app = app
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; ∀i = ∀i
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; ∀e = ∀e
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; R = R
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; R[] = refl
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}
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-- Completeness proof
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-- We first build our universal Kripke model
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module ComplenessProof where
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-- We have a model, we construct the Universal Presheaf model of this model
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open import FFOLInitial as I
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World : Set₁
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World = I.Con
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_≤_ : World → World → Set₁
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Γ ≤ Δ = I.Sub Γ Δ
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UP : Presheaf
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UP = record
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{ World = I.Con
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; _≤_ = I.Sub
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; ≤refl = I.id
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; ≤tran = λ σ σ' → σ' I.∘ σ
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; ≤-ass = λ {w}{w'}{w''}{w'''}{a}{b}{c} → ≡sym I.∘-ass
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; ≤-idl = I.idr
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; ≤-idr = I.idl
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; TM = λ Γ → I.Tm (Con.t Γ)
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; TM≤ = {!!}
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; REL = λ Γ t u → {!I.r t u!}
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; REL≤ = {!!}
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}
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