{-# 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 Arr : World → World → Set ℓ¹ -- arrows id-a : {w : World} → Arr w w -- id arrow _∘a_ : {w w' w'' : World} → Arr w w' → Arr w' w'' → Arr w w'' -- arrow composition ∘a-ass : {w w' w'' w''' : World}{a : Arr w w'}{b : Arr w' w''}{c : Arr w'' w'''} → ((a ∘a b) ∘a c) ≡ (a ∘a (b ∘a c)) idl-a : {w w' : World} → {a : Arr w w'} → (id-a {w}) ∘a a ≡ a idr-a : {w w' : World} → {a : Arr w w'} → a ∘a (id-a {w'}) ≡ a TM : World → Set ℓ¹ TM≤ : {w w' : World} → Arr w w' → TM w' → TM w REL : (w : World) → TM w → TM w → Prop ℓ¹ REL≤ : {w w' : World} → {t u : TM w'} → (eq : Arr 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' → Arr 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 id-a (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 Arr : World → World → Set ℓ¹ -- arrows id-a : {w : World} → Arr w w -- id arrow _∘a_ : {w w' w'' : World} → Arr w w' → Arr w' w'' → Arr w w'' -- arrow composition ∘a-ass : {w w' w'' w''' : World}{a : Arr w w'}{b : Arr w' w''}{c : Arr w'' w'''} → ((a ∘a b) ∘a c) ≡ (a ∘a (b ∘a c)) idl-a : {w w' : World} → {a : Arr w w'} → (id-a {w}) ∘a a ≡ a idr-a : {w w' : World} → {a : Arr w w'} → a ∘a (id-a {w'}) ≡ a TM : World → Set ℓ¹ TM≤ : {w w' : World} → Arr w w' → TM w' → TM w REL : (w : World) → TM w → TM w → Set ℓ¹ REL≤ : {w w' : World} → {t u : TM w'} → (eq : Arr w w') → REL w' t u → REL w (TM≤ eq t) (TM≤ eq u) {- -- Now we try to interpret formulæ and contexts import FFOLInitial as I _⊩ᶠ_ : World → (I.For I.◇t) → Set ℓ¹ w ⊩ᶠ I.R t u = REL w {!!} {!!} w ⊩ᶠ (A I.⇒ B) = ∀ (w' : World) → Arr w w' → w' ⊩ᶠ A → w' ⊩ᶠ B w ⊩ᶠ I.∀∀ A = ∀ (t : TM w) → {!!} -} record Kripke : Set (lsuc (ℓ¹)) where field World : Set ℓ¹ Arr : World → World → Prop ℓ¹ -- arrows id-a : {w : World} → Arr w w -- id arrow _∘a_ : {w w' w'' : World} → Arr w w' → Arr w' w'' → Arr w w'' -- arrow composition -- associativity and id rules are trivial cause arrows are in prop TM : World → Set ℓ¹ TM≤ : {w w' : World} → Arr w w' → TM w' → TM w REL : (w : World) → TM w → TM w → Prop ℓ¹ REL≤ : {w w' : World} → {t u : TM w'} → (eq : Arr w w') → REL w' t u → REL w (TM≤ eq t) (TM≤ eq u) -- 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 UniversalPresheaf : Kripke UniversalPresheaf = record { World = (Γₜ : I.Cont) → I.Conp Γₜ ; Arr = λ w₁ w₂ → (Γₜ : I.Cont) → (I.Pf* Γₜ (w₂ Γₜ) (w₁ Γₜ)) ; id-a = λ {w} Γₜ → I.Pf*-id ; _∘a_ = λ σ₁ σ₂ Γₜ → I.Pf*-∘ (σ₁ Γₜ) (σ₂ Γₜ) --; ∘a-ass = λ {w} → ≡fun' (λ Γₜ → ≡sym (I.∘ₚ-ass {Γₚ = w Γₜ})) --; idl-a = λ {w} {w'} → ≡fun' (λ Γₜ → I.idrₚ {Γₚ = w Γₜ} {Δₚ = w' Γₜ}) --; idr-a = λ {w} {w'} → ≡fun' (λ Γₜ → I.idlₚ {Γₚ = w Γₜ} {Δₚ = w' Γₜ}) ; TM = λ w → (Γₜ : I.Cont) → (I.Tm Γₜ) ; TM≤ = λ σ t → t ; REL = λ w t u → (Γₜ : I.Cont) → I.Pf Γₜ (w Γₜ) (I.R (t Γₜ) (u Γₜ)) ; REL≤ = λ σ pf → λ Γₜ → I.Pf*Pf {!!} (pf Γₜ) } -- I.xx are from initial, xx are from up open Kripke UniversalPresheaf -- We now create the forcing relation for our Universal presheaf -- We need the world to depend of a term context (i guess), so i think i cannot make it so -- the forcing relation works for every Kripke Model. _⊩f_ : (w : World) → {Γₜ : I.Cont} → I.For Γₜ → Prop₁ _⊩f_ w {Γₜ} (I.R t v) = I.Pf Γₜ (w Γₜ) (I.R t v) w ⊩f (A I.⇒ B) = ∀ w' → Arr w w' → w' ⊩f A → w' ⊩f B w ⊩f I.∀∀ A = w ⊩f A ⊩f-mon : {w w' : World} → Arr w w' → {Γₜ : I.Cont} → {A : I.For Γₜ} → w ⊩f A → w' ⊩f A ⊩f-mon s {Γₜ} {I.R t v} wh = I.Pf*Pf (s Γₜ) wh ⊩f-mon s {A = A I.⇒ B} wh w'' s' w''h = wh w'' (s ∘a s' ) w''h ⊩f-mon s {A = I.∀∀ A} wh = ⊩f-mon s {A = A} wh ⊩fPf : {Γₜ : I.Cont}{w : World}{A : I.For Γₜ} → w ⊩f A → I.Pf Γₜ (w Γₜ) A ⊩fPf {A = I.R t v} pf = pf ⊩fPf {A = A I.⇒ A₁} pf = {!I.app!} ⊩fPf {A = I.∀∀ A} pf = I.p∀∀i (substP (λ X → I.Pf _ X A) {!!} (⊩fPf pf)) _⊩c_ : (w : World) → {Γₜ : I.Cont} → I.Conp Γₜ → Prop₁ w ⊩c I.◇p = ⊤ w ⊩c (Γₚ I.▹p⁰ A) = (w ⊩c Γₚ) ∧ (w ⊩f A) ⊩c-mon : {w w' : World} → Arr w w' → {Γₜ : I.Cont} → {Γₚ : I.Conp Γₜ} → w ⊩c Γₚ → w' ⊩c Γₚ ⊩c-mon s {Γₚ = I.◇p} wh = tt ⊩c-mon s {Γₜ} {Γₚ = Γₚ I.▹p⁰ A} wh = ⟨ (⊩c-mon s (proj₁ wh)) , ⊩f-mon s {Γₜ} {A} (proj₂ wh) ⟩ ⊩cPf* : {Γₜ : I.Cont}{w : World}{Γₚ : I.Conp Γₜ} → w ⊩c Γₚ → I.Pf* Γₜ (w Γₜ) Γₚ ⊩cPf* {Γₚ = I.◇p} h = tt ⊩cPf* {Γₚ = Γₚ I.▹p⁰ x} h = ⟨ (⊩cPf* (proj₁ h)) , {!proj₂ h!} ⟩ _⊫_ : {Γₜ : I.Cont} → (I.Conp Γₜ) → I.For Γₜ → Prop₁ Γₚ ⊫ A = ∀ w → w ⊩c Γₚ → w ⊩f A -- Now we want to show universality of this model, that is -- if you have a proof in UP, you have the same in I. u : {Γₜ : I.Cont}{Γₚ : I.Conp Γₜ}{A : I.For Γₜ} → I.Pf Γₜ Γₚ A → Γₚ ⊫ A q : {Γₜ : I.Cont}{Γₚ : I.Conp Γₜ}{A : I.For Γₜ} → Γₚ ⊫ A → I.Pf Γₜ Γₚ A u {A = I.R t s} pf w wh = {!!} u {A = A I.⇒ B} pf w wh w' s w'h = u {A = B} (I.app pf (q λ w'' w''h → {!!})) w' (⊩c-mon s wh) u {A = I.∀∀ A} pf w wh = {!!} q {A = I.R t s} h = {!!} q {A = A I.⇒ B} h = {!!} q {A = I.∀∀ A} h = {!!}