Completed Tarski model for finitary first order logic.
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@ -2,7 +2,7 @@
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open import PropUtil
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module FinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
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module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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open import Agda.Primitive
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open import ListUtil
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@ -10,7 +10,7 @@ module FinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
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variable
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ℓ¹ ℓ² ℓ³ : Level
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record FFOL : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³)) where
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record FFOL (F : Nat → Set) (R : Nat → Set) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³)) where
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infixr 10 _∘_
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field
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Con : Set ℓ¹
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@ -26,6 +26,10 @@ module FinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
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[]t-id : {Γ : Con} → {x : Tm Γ} → x [ id {Γ} ]t ≡ x
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[]t-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {t : Tm Γ} → t [ β ∘ α ]t ≡ (t [ β ]t) [ α ]t
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-- Term extension with functions
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fun : {Γ : Con} → {n : Nat} → F n → Array (Tm Γ) n → Tm Γ
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fun[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {n : Nat} → {f : F n} → {tz : Array (Tm Γ) n} → (fun f tz) [ σ ]t ≡ fun f (map (λ t → t [ σ ]t) tz)
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-- Tm⁺
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_▹ₜ : Con → Con
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πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ
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@ -41,6 +45,10 @@ module FinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
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[]f-id : {Γ : Con} → {F : For Γ} → F [ id {Γ} ]f ≡ F
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[]f-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → F [ β ∘ α ]f ≡ (F [ β ]f) [ α ]f
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-- Formulas with relation on terms
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rel : {Γ : Con} → {n : Nat} → R n → Array (Tm Γ) n → For Γ
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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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--_[_]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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@ -76,7 +84,7 @@ module FinitaryFirstOrderLogic (Term : 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) where
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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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infixr 10 _∘_
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Con = Set
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Sub : Con → Con → Set
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@ -93,12 +101,30 @@ module FinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
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Tm : Con → Set
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Tm Γ = Γ → TM
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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 x → t (σ x)
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t [ σ ]t = λ γ → t (σ γ)
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[]t-id : {Γ : Con} → {x : Tm Γ} → x [ id {Γ} ]t ≡ x
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[]t-id = refl
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[]t-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {t : Tm Γ} → t [ β ∘ α ]t ≡ (t [ β ]t) [ α ]t
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[]t-∘ {α = α} {β} {t} = refl {_} {_} {λ z → t (β (α z))}
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_[_]tz : {Γ Δ : Con} → {n : Nat} → Array (Tm Γ) n → Sub Δ Γ → Array (Tm Δ) n
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tz [ σ ]tz = map (λ s → s [ σ ]t) tz
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[]tz-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {n : Nat} → {tz : Array (Tm Γ) n} → tz [ β ∘ α ]tz ≡ tz [ β ]tz [ α ]tz
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[]tz-∘ {tz = zero} = refl
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[]tz-∘ {α = α} {β = β} {tz = next x tz} = substP (λ tz' → (next ((x [ β ]t) [ α ]t) tz') ≡ (((next x tz) [ β ]tz) [ α ]tz)) (≡sym ([]tz-∘ {α = α} {β = β} {tz = tz})) refl
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[]tz-id : {Γ : Con} → {n : Nat} → {tz : Array (Tm Γ) n} → tz [ id ]tz ≡ tz
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[]tz-id {tz = zero} = refl
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[]tz-id {tz = next x tz} = substP (λ tz' → next x tz' ≡ next x tz) (≡sym ([]tz-id {tz = tz})) refl
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thm : {Γ Δ : Con} → {n : Nat} → {tz : Array (Tm Γ) n} → {σ : Sub Δ Γ} → {δ : Δ} → map (λ t → t δ) (tz [ σ ]tz) ≡ map (λ t → t (σ δ)) tz
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thm {tz = zero} = refl
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thm {tz = next x tz} {σ} {δ} = substP (λ tz' → (next (x (σ δ)) (map (λ t → t δ) (map (λ s γ → s (σ γ)) tz))) ≡ (next (x (σ δ)) tz')) (thm {tz = tz}) refl
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-- Term extension with functions
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fun : {Γ : Con} → {n : Nat} → F n → Array (Tm Γ) n → Tm Γ
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fun {n = n} f tz = λ γ → FUN n f (map (λ t → t γ) tz)
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fun[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {n : Nat} → {f : F n} → {tz : Array (Tm Γ) n} → (fun f tz) [ σ ]t ≡ fun f (tz [ σ ]tz)
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fun[] {σ = σ} {n = n} {f = f} {tz = tz} = ≡fun (λ γ → (substP (λ x → (FUN n f) x ≡ (FUN n f) (map (λ t → t γ) (tz [ σ ]tz))) thm refl))
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-- Tm⁺
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_▹ₜ : Con → Con
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Γ ▹ₜ = Γ × TM
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@ -125,6 +151,12 @@ module FinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
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[]f-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → F [ β ∘ α ]f ≡ (F [ β ]f) [ α ]f
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[]f-∘ = refl
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-- Formulas with relation on terms
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rel : {Γ : Con} → {n : Nat} → R n → Array (Tm Γ) n → For Γ
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rel {n = n} r tz = λ γ → REL n r (map (λ t → t γ) tz)
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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 (tz [ σ ]tz)
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rel[] {σ = σ} {n = n} {r = r} {tz = tz} = ≡fun (λ γ → (substP (λ x → (REL n r) x ≡ (REL n r) (map (λ t → t γ) (tz [ σ ]tz))) thm refl))
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-- Proofs
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_⊢_ : (Γ : Con) → For Γ → Prop
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Γ ⊢ F = ∀ (γ : Γ) → F γ
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@ -171,7 +203,7 @@ module FinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
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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} γ = p γ (t γ)
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tod : FFOL
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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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@ -209,6 +241,10 @@ module FinitaryFirstOrderLogic (Term : Set) (R : Nat → Set) where
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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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{-
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module M where
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@ -145,3 +145,16 @@ module ListUtil where
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⊆→∈* : L ⊆ L' → L ∈* L'
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⊆→∈* h = ⊂⁺→∈* (⊂→⊂⁺ (⊆→⊂ h))
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open import PropUtil using (Nat; zero; succ)
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open import Agda.Primitive
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variable
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ℓ : Level
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data Array (T : Set ℓ) : Nat → Set ℓ where
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zero : Array T zero
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next : {n : Nat} → T → Array T n → Array T (succ n)
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map : {T U : Set ℓ} → (T → U) → {n : Nat} → Array T n → Array U n
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map f zero = zero
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map f (next t a) = next (f t) (map f a)
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@ -1,4 +1,4 @@
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{-# OPTIONS --prop #-}
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{-# OPTIONS --prop --rewriting #-}
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module PropUtil where
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@ -51,11 +51,19 @@ module PropUtil where
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_$_ : {T U : Prop} → (T → U) → T → U
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h $ t = h t
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open import Agda.Primitive
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infix 3 _≡_
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data _≡_ {ℓ}{A : Set ℓ}(a : A) : A → Prop ℓ where
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refl : a ≡ a
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≡sym : {ℓ : Level} → {A : Set ℓ}→ {a a' : A} → a ≡ a' → a' ≡ a
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≡sym refl = refl
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postulate ≡fun : {ℓ ℓ' : Level} → {A : Set ℓ} → {B : Set ℓ'} → {f g : A → B} → ((x : A) → (f x ≡ g x)) → f ≡ g
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postulate subst : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Set ℓ'){a a' : A} → a ≡ a' → P a → P a'
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postulate substP : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Prop ℓ'){a a' : A} → a ≡ a' → P a → P a'
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{-# BUILTIN EQUALITY _≡_ #-}
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data Nat : Set where
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