Second steps, but i guess there is a problem
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+116
-58
@@ -8,17 +8,17 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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open import ListUtil
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variable
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ℓ¹ ℓ² ℓ³ : Level
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ℓ¹ ℓ² ℓ³ ℓ⁴̂ ℓ⁵ : Level
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record FFOL (F : Nat → Set) (R : Nat → Set) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³)) where
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record FFOL (F : Nat → Set) (R : Nat → Set) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴̂ ⊔ ℓ⁵)) where
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infixr 10 _∘_
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field
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Con : Set ℓ¹
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Sub : Con → Con → Set -- It makes a posetal category
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Sub : Con → Con → Set ℓ⁵ -- It makes a posetal category
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_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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id : {Γ : Con} → Sub Γ Γ
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◇ : Con -- The initial object of the category
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ε : {Γ : Con} → Sub ◇ Γ -- The morphism from the initial to any object
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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-- Functor Con → Set called Tm
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Tm : Con → Set ℓ²
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@@ -50,7 +50,7 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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rel[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {n : Nat} → {r : R n} → {tz : Array (Tm Γ) n} → (rel r tz) [ σ ]f ≡ rel r (map (λ t → t [ σ ]t) tz)
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-- Proofs
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_⊢_ : (Γ : Con) → For Γ → Prop
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_⊢_ : (Γ : Con) → For Γ → Prop ℓ⁴̂
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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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-- 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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@@ -86,90 +86,146 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) 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 Con : Set₁
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data For : Con → Set₁
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data Con where -- isom integer ≡ number of terms in the context
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◇ : Con
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_▹t : Con → Con
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_▹p_ : (Γ : Con) → For Γ → Con
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data TmVar : TmCon → Set₁ where -- if Γ ≡ k, then TmVar Γ ≡ ⟦ 0 , k-1 ⟧
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variable
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Γ Δ Ξ : Con
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n : Nat
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A : For Γ
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data TmVar : Con → Set₁ where
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tvzero : TmVar (Γ ▹t)
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tvnext : TmVar Γ → TmVar (Γ ▹t)
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tvdisc : TmVar Γ → TmVar (Γ ▹p A)
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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 Tm : Con → Set₁ where
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var : TmVar Γ → Tm Γ
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fun : F n → Array (Tm Γ) n → Tm Γ
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data For : TmCon → Set₁ where
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data For 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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data PfVar : Con → For Γ → Set₁ where
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pvzero : {A : For Γ} → PfVar (Γ ▹p A) A
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pvnext : {A : For Δ} → {B : For Γ} → PfVar Γ A → PfVar (Γ ▹p B) A
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pvdisc : {A : For Δ} → PfVar Γ A → PfVar (Γ ▹t) A
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variable
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Ψ : PfCon Γ
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data Pf : Con → For Γ → Prop₁ where
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var : {A : For Δ} → PfVar Γ A → Pf Γ A
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app : {A B : For Δ} → Pf Γ (A ⇒ B) → Pf Γ A → Pf Γ B
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lam : {A B : For Γ} → Pf (Γ ▹p A) B → Pf Γ (A ⇒ B)
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--p∀∀e : {A : For Γ} → Pf Γ (∀∀ A) → Pf Γ (A [ t , id ])
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--p∀∀i : {A : For (Γ ▹t)} → Pf (Γ [?]) A → Pf Γ (∀∀ A)
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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 Sub : Con → Con → Set₁ where -- TODO replace with prop
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ε : Sub Γ ◇
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wk▹t : Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹t)
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wk▹p : Sub Δ Γ → Pf Δ A → Sub Δ (Γ ▹p 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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-- We subst on terms
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_[_]t : Tm Γ → Sub Δ Γ → Tm Δ
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_[_]tz : Array (Tm Γ) n → Sub Δ Γ → Array (Tm Δ) n
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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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var tvzero [ wk▹t σ t ]t = t
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var (tvnext tv) [ wk▹t σ x ]t = var tv [ σ ]t
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var (tvdisc tv) [ wk▹p σ x ]t = var tv [ σ ]t
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fun f tz [ σ ]t = fun f (tz [ σ ]tz)
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zero [ σ ]tz = zero
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next t tz [ σ ]tz = next (t [ σ ]t) (tz [ σ ]tz)
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imod : FFOL {lsuc lzero} {lzero} {lzero} F R
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-- We subst on proofs
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_[_]p : Pf Γ A → Sub Δ Γ → Pf Δ A
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_[_]p = {!!}
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-- We subst on formulæ
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_[_]f : For Γ → Sub Δ Γ → For Δ
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_[_]f = {!!}
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_∘_ : Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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ε ∘ β = ε
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wk▹t α t ∘ β = wk▹t (α ∘ β) (t [ β ]t)
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wk▹p α pf ∘ β = wk▹p (α ∘ β) (pf [ β ]p)
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pgcd : Con → Con → Con
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pgcd ◇ Δ = ◇
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pgcd (Γ ▹t) ◇ = ◇
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pgcd (Γ ▹t) (Δ ▹t) = pgcd Γ Δ
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pgcd (Γ ▹t) (Δ ▹p x) = pgcd Γ Δ
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pgcd (Γ ▹p x) ◇ = ◇
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pgcd (Γ ▹p x) (Δ ▹t) = pgcd Γ Δ
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pgcd (Γ ▹p x) (Δ ▹p x₁) = pgcd Γ Δ
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len : Con → Nat
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len ◇ = 0
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len (Γ ▹t) = succ (len Γ)
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len (Γ ▹p A) = succ (len Γ)
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lift▹tPf : Pf Γ A → Pf (Γ ▹t) A
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lift▹tPf (var x) = var (pvdisc x)
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lift▹tPf (app p p₁) = app (lift▹tPf p) (lift▹tPf p₁)
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lift▹tPf (lam p) = {!!}
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lift▹t : Sub Γ Δ → Sub (Γ ▹t) Δ
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lift▹t ε = ε
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lift▹t (wk▹t σ t) = wk▹t (lift▹t σ) (var tvzero)
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lift▹t (wk▹p {A = A} σ x) = wk▹p (lift▹t σ) (var (pvdisc {!x!}))
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id : Sub Γ Γ
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id {◇} = ε
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id {Γ ▹t} = wk▹t {!!} (var tvzero)
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id {◇ ▹p A} = wk▹p ε (var pvzero)
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id {(Γ ▹t) ▹p A} = wk▹p (wk▹t {!!} (var (tvdisc tvzero))) (var pvzero)
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id {(Γ ▹p x) ▹p A} = wk▹p {!!} (var pvzero)
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imod : FFOL {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc 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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; 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 = {!!}
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; []t-∘ = {!!}
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; fun = {!!}
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; fun = fun
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; fun[] = {!!}
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; _▹ₜ = {!!}
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; _▹ₜ = _▹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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; For = 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 = rel
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; rel[] = {!!}
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; _⊢_ = {!!}
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; _▹ₚ_ = {!!}
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; _⊢_ = λ (Γ : Con) (A : For Γ) → Pf Γ A
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; _▹ₚ_ = _▹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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; _⇒_ = _⇒_
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; []f-⇒ = {!!}
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; ∀∀ = {!!}
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; ∀∀ = ∀∀
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; []f-∀∀ = {!!}
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; lam = {!!}
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; app = {!!}
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; app = app
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; ∀i = {!!}
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; ∀e = {!!}
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}
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@@ -186,9 +242,10 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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f ∘ g = λ x → f (g x)
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id : {Γ : Con} → Sub Γ Γ
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id = λ x → x
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data ◇ : Con where
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ε : {Γ : Con} → Sub ◇ Γ -- The morphism from the initial to any object
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ε ()
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record ◇ : Con where
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constructor ◇◇
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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ε Γ = ◇◇
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-- Functor Con → Set called Tm
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Tm : Con → Set
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@@ -358,11 +415,12 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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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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data ◇⁰ : Set where
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record ◇⁰ : Set where
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constructor ◇◇⁰
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◇ : Con -- The initial object of the category
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◇ = λ w → ◇⁰
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ε : {Γ : Con} → Sub ◇ Γ -- The morphism from the initial to any object
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ε w ()
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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ε w Γ = ◇◇⁰
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-- Functor Con → Set called Tm
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Tm : Con → Set
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