Tried to add First order logic and First order Kripke. Some misconceptions ...
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FirstOrderKripke.agda
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91
FirstOrderKripke.agda
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{-# OPTIONS --prop --no-termination-check #-}
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open import Agda.Builtin.Nat
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open import Agda.Builtin.Bool
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module FirstOrderKripke (PV : Set) (TV : Set) (TV= : TV → TV → Bool) (Fu : Nat → Set) (R : Nat → Set) where
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open import ListUtil
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open import PropUtil
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open import FirstOrderLogic TV TV= Fu R
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private
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variable
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n : Nat
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t : Term
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x : TV
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y : TV
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F : Form
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G : Form
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Γ : Con
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Γ' : Con
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Η : Con
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Η' : Con
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record Kripke : Set₁ where
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field
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Worlds : Set₀
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_≤_ : Worlds → Worlds → Prop
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refl≤ : {w : Worlds} → w ≤ w
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tran≤ : {a b c : Worlds} → a ≤ b → b ≤ c → a ≤ c
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_⊩_[_] : Worlds → R n → Args n → Prop
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mon⊩ : {a b : Worlds} → a ≤ b → {r : R n} {A : Args n} → a ⊩ r [ A ] → b ⊩ r [ A ]
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private
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variable
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w : Worlds
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w' : Worlds
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w₁ : Worlds
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w₂ : Worlds
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w₃ : Worlds
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{- Extending ⊩ to Formulas and Contexts -}
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-- It is in fact
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_⊩ᶠ_ : Worlds → Form → Prop
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w ⊩ᶠ (Rel {n = n} r A) = {B : Args n} → A ⊂sub B → w ⊩ r [ B ]
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w ⊩ᶠ (fp ⇒ fq) = {w' : Worlds} → w ≤ w' → w' ⊩ᶠ fp → w' ⊩ᶠ fq
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w ⊩ᶠ (fp ∧∧ fq) = w ⊩ᶠ fp ∧ w ⊩ᶠ fq
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w ⊩ᶠ ⊤⊤ = ⊤
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w ⊩ᶠ (∀∀ x F) = (t : Term) → w ⊩ᶠ ([ t / x ] F)
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_⊩ᶜ_ : Worlds → Con → Prop
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w ⊩ᶜ [] = ⊤
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w ⊩ᶜ (p ∷ c) = (w ⊩ᶠ p) ∧ (w ⊩ᶜ c)
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-- The extensions are monotonous
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mon⊩ᶠ : w ≤ w' → w ⊩ᶠ F → w' ⊩ᶠ F
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mon⊩ᶠ {F = Rel r A} ww' wF s = mon⊩ ww' (wF s)
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mon⊩ᶠ {F = F ⇒ G} ww' wF w'w'' w''F = wF (tran≤ ww' w'w'') w''F
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mon⊩ᶠ {F = F ∧∧ G} ww' ⟨ wF , wG ⟩ = ⟨ mon⊩ᶠ {F = F} ww' wF , mon⊩ᶠ {F = G} ww' wG ⟩
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mon⊩ᶠ {F = ⊤⊤} ww' wF = tt
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mon⊩ᶠ {F = ∀∀ x F} ww' wF t = mon⊩ᶠ {F = [ t / x ] F} ww' (wF t)
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mon⊩ᶜ : w ≤ w' → w ⊩ᶜ Γ → w' ⊩ᶜ Γ
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mon⊩ᶜ {Γ = []} ww' wΓ = wΓ
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mon⊩ᶜ {Γ = F ∷ Γ} ww' wΓ = ⟨ mon⊩ᶠ {F = F} ww' (proj₁ wΓ) , mon⊩ᶜ ww' (proj₂ wΓ) ⟩
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{- General operator matching the shape of ⊢ -}
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_⊫_ : Con → Form → Prop
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Γ ⊫ F = {w : Worlds} → w ⊩ᶜ Γ → w ⊩ᶠ F
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{- Soundness -}
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th' : w ⊩ᶠ F → w ⊩ᶠ [ t / x ] F
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th' {F = Rel r A} h {B} s = h {B} (tran⊂sub (next zero) s)
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th' {F = F ⇒ F₁} h o hF = {!h o ?!}
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th' {F = F ∧∧ F₁} h = {!!}
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th' {F = ⊤⊤} h = {!!}
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th' {F = ∀∀ x F} h = {!!}
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th : Γ ⊫ F → Γ ⊫ [ t / x ] F
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th {[]} h _ = {!!}
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th {x ∷ Γ} h = {!!}
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⟦_⟧ : Γ ⊢ F → Γ ⊫ F
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⟦ zero zero∈ ⟧ wΓ = proj₁ wΓ
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⟦ zero (next∈ h) ⟧ wΓ = ⟦ zero h ⟧ (proj₂ wΓ)
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⟦ lam p ⟧ = λ wΓ w≤ w'A → ⟦ p ⟧ ⟨ w'A , mon⊩ᶜ w≤ wΓ ⟩
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⟦ app p p₁ ⟧ wΓ = ⟦ p ⟧ wΓ refl≤ (⟦ p₁ ⟧ wΓ)
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⟦ andi p₁ p₂ ⟧ wΓ = ⟨ (⟦ p₁ ⟧ wΓ) , (⟦ p₂ ⟧ wΓ) ⟩
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⟦ ande₁ p ⟧ wΓ = proj₁ $ ⟦ p ⟧ wΓ
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⟦ ande₂ p ⟧ wΓ = proj₂ $ ⟦ p ⟧ wΓ
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⟦ true ⟧ wΓ = tt
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⟦ ∀-i i p ⟧ wΓ t = {!⟦ p ⟧!}
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⟦ ∀-e {t = t} p ⟧ wΓ = ⟦ p ⟧ wΓ t
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@ -1,30 +1,108 @@
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{-# OPTIONS --prop #-}
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open import Agda.Builtin.Nat
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open import Agda.Builtin.Bool
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open import Agda.Primitive using (Level)
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module FirstOrderLogic (TV : Set) (F : Nat → Set) (R : Nat → Set) where
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module FirstOrderLogic (TV : Set) (TV= : TV → TV → Bool) (F : Nat → Set) (R : Nat → Set) where
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open import PropUtil
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open import ListUtil
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mutual
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data FArgs : Nat → Set where
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zero : FArgs 0
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next : {n : Nat} → FArgs n → Term → FArgs (suc n)
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data Args : Nat → Set where
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zero : Args 0
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next : {n : Nat} → Args n → Term → Args (suc n)
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data Term : Set where
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Var : TV → Term
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Fun : {n : Nat} → F n → FArgs n → Term
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Fun : {n : Nat} → F n → Args n → Term
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data RArgs : Nat → Set where
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zero : RArgs 0
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next : {n : Nat} → RArgs n → Term → RArgs (suc n)
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private
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variable
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n : Nat
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if : {ℓ : Level} → {T : Set ℓ} → Bool → T → T → T
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if true a b = a
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if false a b = b
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mutual
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[_/_]ᵗ : Term → TV → Term → Term
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[_/_]ᵃ : Term → TV → Args n → Args n
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[ t / x ]ᵗ (Var x') = if (TV= x x') t (Var x')
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[ t / x ]ᵗ (Fun f A) = Fun f ([ t / x ]ᵃ A)
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[ t / x ]ᵃ zero = zero
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[ t / x ]ᵃ (next A t₁) = next ([ t / x ]ᵃ A) ([ t / x ]ᵗ t₁)
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-- A ⊂sub B if B can be obtained with finite substitution from A
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data _⊂sub_ : Args n → Args n → Prop where
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zero : {A : Args n} → A ⊂sub A
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next : {A B : Args n} → {t : Term} → {x : TV} → A ⊂sub B → A ⊂sub ([ t / x ]ᵃ B)
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tran⊂sub : {A B C : Args n} → A ⊂sub B → B ⊂sub C → A ⊂sub C
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tran⊂sub zero h₂ = h₂
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tran⊂sub h₁ zero = h₁
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tran⊂sub h₁ (next h₂) = next (tran⊂sub h₁ h₂)
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data Form : Set where
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Rel : {n : Nat} → R n → (RArgs n) → Form
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Rel : {n : Nat} → R n → (Args n) → Form
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_⇒_ : Form → Form → Form
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_∧∧_ : Form → Form → Form
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_∀∀_ : TV → Form → Form
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⊤⊤ : Form
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∀∀ : TV → Form → Form
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infixr 10 _∧∧_
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infixr 8 _⇒_
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Con = List Form
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private
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variable
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A : Form
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A' : Form
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B : Form
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B' : Form
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C : Form
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Γ : Con
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Γ' : Con
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Γ'' : Con
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Δ : Con
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Δ' : Con
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x : TV
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t : Term
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[_/_] : Term → TV → Form → Form
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[ t / x ] (Rel r tz) = Rel r ([ t / x ]ᵃ tz)
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[ t / x ] (A ⇒ A₁) = ([ t / x ] A) ⇒ ([ t / x ] A₁)
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[ t / x ] (A ∧∧ A₁) = ([ t / x ] A) ∧∧ ([ t / x ] A₁)
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[ t / x ] ⊤⊤ = ⊤⊤
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[ t / x ] (∀∀ x₁ A) = if (TV= x x₁) A ([ t / x ] A)
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mutual
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_∉FVᵗ_ : TV → Term → Prop
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_∉FVᵃ_ : TV → Args n → Prop
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x ∉FVᵗ Var x₁ = if (TV= x x₁) ⊥ ⊤
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x ∉FVᵗ Fun f A = x ∉FVᵃ A
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x ∉FVᵃ zero = ⊤
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x ∉FVᵃ next A t = (x ∉FVᵃ A) ∧ (x ∉FVᵗ t)
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_∉FVᶠ_ : TV → Form → Prop
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x ∉FVᶠ Rel R A = x ∉FVᵃ A
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x ∉FVᶠ (A ⇒ A₁) = x ∉FVᶠ A ∧ x ∉FVᶠ A₁
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x ∉FVᶠ (A ∧∧ A₁) = x ∉FVᶠ A ∧ x ∉FVᶠ A₁
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x ∉FVᶠ ⊤⊤ = ⊤
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x ∉FVᶠ ∀∀ x₁ A = if (TV= x x₁) ⊤ (x ∉FVᶠ A)
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_∉FVᶜ_ : TV → Con → Prop
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x ∉FVᶜ [] = ⊤
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x ∉FVᶜ (A ∷ Γ) = x ∉FVᶠ A ∧ x ∉FVᶜ Γ
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data _⊢_ : Con → Form → Prop where
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zero : A ∈ Γ → Γ ⊢ A
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lam : (A ∷ Γ) ⊢ B → Γ ⊢ (A ⇒ B)
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app : Γ ⊢ (A ⇒ B) → Γ ⊢ A → Γ ⊢ B
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andi : Γ ⊢ A → Γ ⊢ B → Γ ⊢ A ∧∧ B
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ande₁ : Γ ⊢ A ∧∧ B → Γ ⊢ A
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ande₂ : Γ ⊢ A ∧∧ B → Γ ⊢ B
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true : Γ ⊢ ⊤⊤
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∀-i : x ∉FVᶜ Γ → Γ ⊢ A → Γ ⊢ ∀∀ x A
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∀-e : Γ ⊢ ∀∀ x A → Γ ⊢ [ t / x ] A
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infix 5 _⊢_
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