160 lines
6.3 KiB
Plaintext
160 lines
6.3 KiB
Plaintext
\begin{code}
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{-# OPTIONS --prop --rewriting #-}
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open import PropUtil
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module ZOLCompleteness where
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open import Agda.Primitive
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open import ZOL2
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open import ListUtil
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record Kripke : Set (lsuc (ℓ¹)) where
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field
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World : Set ℓ¹
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_-w->_ : World → World → Prop ℓ¹ -- arrows
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-w->id : {w : World} → w -w-> w -- id arrow
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_∘-w->_ : {w w' w'' : World} → w -w-> w' → w' -w-> w'' → w -w-> w'' -- arrow composition
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Ι : World → Prop ℓ¹
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Ι≤ : {w w' : World} → w -w-> w' → Ι w → Ι w'
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infixr 10 _∘_
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Con : Set (lsuc ℓ¹)
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Con = (World → Prop ℓ¹) ×'' (λ Γ → {w w' : World} → (w -w-> w')→ Γ w → Γ w')
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Sub : Con → Con → Prop ℓ¹
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Sub Δ Γ = (w : World) → (proj×''₁ Δ) w → (proj×''₁ 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 → ⊤) ,×'' (λ _ _ → tt)
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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ε w Γ = tt
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-- Functor Con → Set called For
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For : Set (lsuc ℓ¹)
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For = (World → Prop ℓ¹) ×'' (λ F → {w w' : World} → (w -w-> w')→ F w → F w')
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-- Proofs
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Pf : (Γ : Con) → For → Prop ℓ¹
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Pf Γ F = ∀ w (γ : (proj×''₁ Γ) w) → (proj×''₁ F) w
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_[_]p : {Γ Δ : Con} → {F : For} → Pf Γ F → (σ : Sub Δ Γ) → Pf Δ 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 → Con
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Γ ▹ₚ F = (λ w → (proj×''₁ Γ) w ∧ (proj×''₁ F) w) ,×'' λ s γ → ⟨ proj×''₂ Γ s (proj₁ γ) , proj×''₂ F s (proj₂ γ) ⟩
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πₚ¹ : {Γ Δ : Con} → {F : For} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
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πₚ¹ σ w δ = proj₁ (σ w δ)
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πₚ² : {Γ Δ : Con} → {F : For} → (σ : Sub Δ (Γ ▹ₚ F)) → Pf Δ F
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πₚ² σ w δ = proj₂ (σ w δ)
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_,ₚ_ : {Γ Δ : Con} → {F : For} → (σ : Sub Δ Γ) → Pf Δ F → Sub Δ (Γ ▹ₚ F)
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(σ ,ₚ pf) w δ = ⟨ (σ w δ) , pf w δ ⟩
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-- Base formula
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ι : For
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ι = (λ w → Ι w) ,×'' λ s f → Ι≤ s f
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-- Implication
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_⇒_ : For → For → For
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(F ⇒ G) = (λ w → {w' : World} → (s : w -w-> w') → ((proj×''₁ F) w') → ((proj×''₁ G) w')) ,×'' λ s f s' f' → f (s ∘-w-> s') f'
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-- Lam & App
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lam : {Γ : Con} → {F : For} → {G : For} → Pf (Γ ▹ₚ F) G → Pf Γ (F ⇒ G)
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lam {Γ} pf w γ {w'} s x = pf w' ⟨ proj×''₂ Γ s γ , x ⟩
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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} → Pf Γ (F ⇒ G) → Pf Γ F → Pf Γ G
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app pf pf' w γ = pf w γ -w->id (pf' w γ)
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-- Again, we don't write the _[_]p equalities as everything is in Prop
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zol : ZOL
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zol = record
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{ Con = Con
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; Sub = Sub
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; _∘_ = λ {Γ}{Δ}{Ξ} σ δ → _∘_ {Γ}{Δ}{Ξ} σ δ
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; id = λ {Γ} → id {Γ}
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; ◇ = ◇
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; ε = λ {Γ} → ε {Γ}
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; For = λ Γ → For
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; _[_]f = λ A σ → A
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; []f-id = refl
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; []f-∘ = refl
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; Pf = Pf
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; _[_]p = λ {Γ}{Δ}{F} pf σ → _[_]p {Γ}{Δ}{F} pf σ
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; _▹ₚ_ = _▹ₚ_
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; πₚ¹ = λ {Γ}{Δ}{F}σ → πₚ¹ {Γ}{Δ}{F} σ
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; πₚ² = λ {Γ}{Δ}{F}σ → πₚ² {Γ}{Δ}{F} σ
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; _,ₚ_ = λ {Γ}{Δ}{F} σ pf → _,ₚ_ {Γ}{Δ}{F}σ pf
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; ι = ι
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; []f-ι = refl
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; _⇒_ = _⇒_
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; []f-⇒ = refl
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; lam = λ {Γ}{F}{G} pf → lam {Γ}{F}{G} pf
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; app = λ {Γ}{F}{G} pf pf' → app {Γ}{F}{G} pf pf'
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}
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module U where
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import ZOLInitial as I
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U : Kripke
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U = record
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{ World = I.Con
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; _-w->_ = λ Γ Δ → I.Sub Δ Γ
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; -w->id = I.id
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; _∘-w->_ = λ σ σ' → σ I.∘ σ'
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; Ι = λ Γ → I.Pf Γ I.ι
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; Ι≤ = λ s pf → pf I.[ s ]p
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}
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open Kripke U
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y : Mapping I.zol zol
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y = record
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{ mCon = λ Γ → (λ Δ → I.Sub Δ Γ) ,×'' λ σ δ → δ I.∘ σ
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; mSub = λ σ Ξ δ → σ I.∘ δ
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; mFor = λ A → (λ Ξ → I.Pf Ξ A) ,×'' λ σ pf → pf I.[ σ ]p
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; mPf = λ pf Ξ σ → pf I.[ σ ]p
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}
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m : Morphism I.zol zol
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m = I.InitialMorphism.mor zol
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q : (Γ : I.Con) → Sub (Morphism.mCon m Γ) (Mapping.mCon y Γ)
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u : (Γ : I.Con) → Sub (Mapping.mCon y Γ) (Morphism.mCon m Γ)
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⟦_⟧c = Morphism.mCon m
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⟦_,_⟧f = λ A Γ → Morphism.mFor m {Γ} A
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q⁰ : {F : I.For} → {Γ Γ₀ : I.Con} → proj×''₁ ⟦ F , Γ₀ ⟧f Γ → I.Pf Γ F
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u⁰ : {F : I.For} → {Γ Γ₀ : I.Con} → I.Pf Γ F → proj×''₁ ⟦ F , Γ₀ ⟧f Γ
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q⁰ {I.ι} {Γ} h = h
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q⁰ {A I.⇒ B} {Γ} h = I.lam (q⁰ {B} (h (I.πₚ¹ I.id) (u⁰ {A} (I.var I.pvzero))))
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u⁰ {I.ι} {Γ} pf = pf
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u⁰ {A I.⇒ B} {Γ} pf iq hF = u⁰ {B} (I.app (pf I.[ iq ]p) (q⁰ hF) )
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q I.◇ w γ = I.ε
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q (Γ I.▹ₚ A) w γ = (q Γ w (proj₁ γ) I.,ₚ q⁰ (proj₂ γ))
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u I.◇ w σ = tt
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u (Γ I.▹ₚ A) w σ = ⟨ (u Γ w (I.πₚ¹ σ)) , u⁰ (I.πₚ² σ) ⟩
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ηq : TrNat (Morphism.m m) y
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ηq = record { f = q }
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ηu : TrNat y (Morphism.m m)
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ηu = record { f = u }
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eq : ηu ∘TrNat ηq ≡ idTrNat
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eq = refl
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\end{code}
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