Firsts step to an initial model
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@ -84,7 +84,100 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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∀i : {Γ : Con} → {F : For (Γ ▹ₜ)} → (Γ ▹ₜ) ⊢ F → Γ ⊢ (∀∀ F)
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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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module Tarski (TM : Set) (REL : (n : Nat) → R n → (Array TM n → Prop)) (FUN : (n : Nat) → F n → (Array TM n → TM)) where
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module Initial where
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data TmCon : Set₁ where -- isom integer ≡ number of terms in the context
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◇t : TmCon
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_▹t : TmCon → TmCon
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variable
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Γ : TmCon
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n : Nat
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data TmVar : TmCon → Set₁ where -- if Γ ≡ k, then TmVar Γ ≡ ⟦ 0 , k-1 ⟧
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tvzero : TmVar (Γ ▹t)
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tvnext : TmVar Γ → TmVar (Γ ▹t)
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data Tm : TmCon → Set₁ where
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tvar : TmVar Γ → Tm Γ
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tfun : F n → Array (Tm Γ) n → Tm Γ
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data For : TmCon → Set₁ where
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rel : R n → Array (Tm Γ) n → For Γ
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_⇒_ : For Γ → For Γ → For Γ
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∀∀ : For (Γ ▹t) → For Γ
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data PfCon : TmCon → Set₁ where
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◇p : PfCon Γ
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_▹p_ : PfCon Γ → For Γ → PfCon Γ
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variable
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Ψ : PfCon Γ
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data PfVar : PfCon Γ → For Γ → Set₁ where
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pfzero : {A : For Γ} → PfVar (Ψ ▹p A) A
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pfnext : {A B : For Γ} → PfVar Ψ A → PfVar (Ψ ▹p B) A
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data Pf : PfCon Γ → For Γ → Set₁ where
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pvar : {A : For Γ} → PfVar Ψ A → Pf Ψ A
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papp : {A B : For Γ} → Pf Ψ (A ⇒ B) → Pf Ψ A → Pf Ψ B
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plam : {A B : For Γ} → Pf (Ψ ▹p A) B → Pf Ψ (A ⇒ B)
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--p∀∀e : {A : For Γ} → Pf Ψ (∀∀ A) → Pf Ψ (A [ t , id ])
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--p∀∀i : {A : For (Γ ▹t)} → Pf (Ψ [?]) A → Pf Ψ (∀∀ A)
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record Con : Set₁ where
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constructor _,_
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field
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tc : TmCon
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pc : PfCon tc
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imod : FFOL {lsuc lzero} {lzero} {lzero} F R
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imod = record
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{ Con = Con
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; Sub = {!!}
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; _∘_ = {!!}
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; id = {!!}
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; ◇ = {!!}
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; ε = {!!}
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; Tm = {!!}
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; _[_]t = {!!}
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; []t-id = {!!}
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; []t-∘ = {!!}
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; fun = {!!}
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; fun[] = {!!}
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; _▹ₜ = {!!}
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; πₜ¹ = {!!}
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; πₜ² = {!!}
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; _,ₜ_ = {!!}
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; πₜ²∘,ₜ = {!!}
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; πₜ¹∘,ₜ = {!!}
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; ,ₜ∘πₜ = {!!}
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; For = {!!}
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; _[_]f = {!!}
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; []f-id = {!!}
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; []f-∘ = {!!}
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; rel = {!!}
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; rel[] = {!!}
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; _⊢_ = {!!}
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; _▹ₚ_ = {!!}
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; πₚ¹ = {!!}
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; πₚ² = {!!}
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; _,ₚ_ = {!!}
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; ,ₚ∘πₚ = {!!}
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; πₚ¹∘,ₚ = {!!}
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; _⇒_ = {!!}
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; []f-⇒ = {!!}
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; ∀∀ = {!!}
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; []f-∀∀ = {!!}
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; lam = {!!}
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; app = {!!}
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; ∀i = {!!}
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; ∀e = {!!}
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}
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record Tarski : Set₁ where
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field
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TM : Set
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REL : (n : Nat) → R n → (Array TM n → Prop)
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FUN : (n : Nat) → F n → (Array TM n → TM)
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infixr 10 _∘_
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Con = Set
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Sub : Con → Con → Set
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@ -247,16 +340,16 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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; rel[] = rel[]
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}
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module Kripke
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(World : Set)
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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 : Set)
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(REL : (n : Nat) → R n → Array TM n → World → Prop)
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(RELmon : {n : Nat} → {r : R n} → {x : Array TM n} → {w w' : World} → REL n r x w → REL n r x w')
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(FUN : (n : Nat) → F n → Array TM n → TM)
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where
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record Kripke : Set₁ where
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field
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World : Set
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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 : Set
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REL : (n : Nat) → R n → Array TM n → World → Prop
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RELmon : {n : Nat} → {r : R n} → {x : Array TM n} → {w w' : World} → REL n r x w → REL n r x w'
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FUN : (n : Nat) → F n → Array TM n → TM
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infixr 10 _∘_
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Con = World → Set
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Sub : Con → Con → Set
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@ -386,44 +479,51 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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tod : FFOL F R
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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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; id = id
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; ◇ = ◇
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; ε = ε
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; Tm = Tm
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; _[_]t = _[_]t
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; []t-id = []t-id
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; []t-∘ = λ {Γ} {Δ} {Ξ} {α} {β} {t} → []t-∘ {Γ} {Δ} {Ξ} {α} {β} {t}
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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 = λ {Γ} {Δ} {σ} → πₜ²∘,ₜ {Γ} {Δ} {σ}
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; πₜ¹∘,ₜ = λ {Γ} {Δ} {σ} {t} → πₜ¹∘,ₜ {Γ} {Δ} {σ} {t}
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; ,ₜ∘πₜ = ,ₜ∘πₜ
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; For = For
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; _[_]f = _[_]f
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; []f-id = []f-id
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; []f-∘ = λ {Γ} {Δ} {Ξ} {α} {β} {F} → []f-∘ {Γ} {Δ} {Ξ} {α} {β} {F}
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; _⊢_ = _⊢_
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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 = ,ₚ∘πₚ
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; πₚ¹∘,ₚ = λ {Γ} {Δ} {F} {σ} {p} → πₚ¹∘,ₚ {Γ} {Δ} {F} {σ} {p}
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; _⇒_ = _⇒_
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; []f-⇒ = λ {Γ} {F} {G} {σ} → []f-⇒ {Γ} {F} {G} {σ}
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; ∀∀ = ∀∀
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; []f-∀∀ = λ {Γ} {Δ} {F} {σ} {t} → []f-∀∀ {Γ} {Δ} {F} {σ} {t}
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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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; fun = fun
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; fun[] = fun[]
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; rel = rel
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; rel[] = rel[]
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}
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{ Con = Con
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; Sub = Sub
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; _∘_ = _∘_
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; id = id
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; ◇ = ◇
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; ε = ε
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; Tm = Tm
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; _[_]t = _[_]t
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; []t-id = []t-id
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; []t-∘ = λ {Γ} {Δ} {Ξ} {α} {β} {t} → []t-∘ {Γ} {Δ} {Ξ} {α} {β} {t}
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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 = λ {Γ} {Δ} {σ} → πₜ²∘,ₜ {Γ} {Δ} {σ}
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; πₜ¹∘,ₜ = λ {Γ} {Δ} {σ} {t} → πₜ¹∘,ₜ {Γ} {Δ} {σ} {t}
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; ,ₜ∘πₜ = ,ₜ∘πₜ
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; For = For
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; _[_]f = _[_]f
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; []f-id = []f-id
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; []f-∘ = λ {Γ} {Δ} {Ξ} {α} {β} {F} → []f-∘ {Γ} {Δ} {Ξ} {α} {β} {F}
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; _⊢_ = _⊢_
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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 = ,ₚ∘πₚ
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; πₚ¹∘,ₚ = λ {Γ} {Δ} {F} {σ} {p} → πₚ¹∘,ₚ {Γ} {Δ} {F} {σ} {p}
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; _⇒_ = _⇒_
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; []f-⇒ = λ {Γ} {F} {G} {σ} → []f-⇒ {Γ} {F} {G} {σ}
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; ∀∀ = ∀∀
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; []f-∀∀ = λ {Γ} {Δ} {F} {σ} {t} → []f-∀∀ {Γ} {Δ} {F} {σ} {t}
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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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; fun = fun
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; fun[] = fun[]
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; rel = rel
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; rel[] = rel[]
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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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@ -72,21 +72,22 @@ module PropUtil where
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succ : Nat → Nat
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{-# BUILTIN NATURAL Nat #-}
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variable
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ℓ ℓ' : Level
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record _×_ (A : Set) (B : Set) : Set where
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record _×_ (A : Set ℓ) (B : Set ℓ) : Set ℓ where
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constructor _,×_
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field
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a : A
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b : B
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record _×'_ (A : Set) (B : Prop) : Set where
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record _×'_ (A : Set ℓ) (B : Prop ℓ) : Set ℓ where
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constructor _,×'_
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field
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a : A
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b : B
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record _×''_ (A : Set) (B : A → Prop) : Set where
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record _×''_ (A : Set ℓ) (B : A → Prop ℓ') : Set (ℓ ⊔ ℓ') where
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constructor _,×''_
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field
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a : A
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