From 6fcaabc4db9186977272be56bcd77946f3a71a7c Mon Sep 17 00:00:00 2001 From: Mysaa Date: Tue, 27 Jun 2023 16:08:36 +0200 Subject: [PATCH] Simplified the notation, working this time --- FinitaryFirstOrderLogic.agda | 74 ++++++++++++------------------------ 1 file changed, 25 insertions(+), 49 deletions(-) diff --git a/FinitaryFirstOrderLogic.agda b/FinitaryFirstOrderLogic.agda index cc5318d..9e79228 100644 --- a/FinitaryFirstOrderLogic.agda +++ b/FinitaryFirstOrderLogic.agda @@ -2,7 +2,7 @@ open import PropUtil -module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where +module FinitaryFirstOrderLogic where open import Agda.Primitive open import ListUtil @@ -10,7 +10,7 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where variable ℓ¹ ℓ² ℓ³ ℓ⁴ ℓ⁵ : Level - record FFOL (F : Nat → Set) (R : Nat → Set) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴ ⊔ ℓ⁵)) where + record FFOL : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴ ⊔ ℓ⁵)) where infixr 10 _∘_ infixr 5 _⊢_ field @@ -27,10 +27,6 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where []t-id : {Γ : Con} → {x : Tm Γ} → x [ id {Γ} ]t ≡ x []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 - -- Term extension with functions - fun : {Γ : Con} → {n : Nat} → F n → Array (Tm Γ) n → Tm Γ - 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) - -- Tm⁺ _▹ₜ : Con → Con πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ @@ -48,8 +44,8 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where []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 -- Formulas with relation on terms - rel : {Γ : Con} → {n : Nat} → R n → Array (Tm Γ) n → For Γ - 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) + R : {Γ : Con} → (t u : Tm Γ) → For Γ + R[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t u : Tm Γ} → (R t u) [ σ ]f ≡ R (t [ σ ]t) (u [ σ ]t) -- Proofs _⊢_ : (Γ : Con) → For Γ → Prop ℓ⁴ @@ -132,8 +128,7 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where record Tarski : Set₁ where field TM : Set - REL : (n : Nat) → R n → (Array TM n → Prop) - FUN : (n : Nat) → F n → (Array TM n → TM) + REL : TM → TM → Prop infixr 10 _∘_ Con = Set Sub : Con → Con → Set @@ -169,12 +164,6 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where thm {tz = zero} = refl 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 - -- Term extension with functions - fun : {Γ : Con} → {n : Nat} → F n → Array (Tm Γ) n → Tm Γ - fun {n = n} f tz = λ γ → FUN n f (map (λ t → t γ) tz) - 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) - 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)) - -- Tm⁺ _▹ₜ : Con → Con Γ ▹ₜ = Γ × TM @@ -203,12 +192,11 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where []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 []f-∘ = refl - -- Formulas with relation on terms - rel : {Γ : Con} → {n : Nat} → R n → Array (Tm Γ) n → For Γ - rel {n = n} r tz = λ γ → REL n r (map (λ t → t γ) tz) - 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) - 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)) - + R : {Γ : Con} → Tm Γ → Tm Γ → For Γ + R t u = λ γ → REL (t γ) (u γ) + R[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t u : Tm Γ} → (R t u) [ σ ]f ≡ R (t [ σ ]t) (u [ σ ]t) + R[] {σ = σ} = cong₂ R refl refl + -- Proofs _⊢_ : (Γ : Con) → For Γ → Prop Γ ⊢ F = ∀ (γ : Γ) → F γ @@ -258,7 +246,7 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where ∀e : {Γ : Con} → {F : For (Γ ▹ₜ)} → Γ ⊢ (∀∀ F) → {t : Tm Γ} → Γ ⊢ ( F [(id {Γ}) ,ₜ t ]f) ∀e p {t} γ = p γ (t γ) - tod : FFOL F R + tod : FFOL tod = record { Con = Con ; Sub = Sub @@ -299,10 +287,8 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where ; app = app ; ∀i = ∀i ; ∀e = ∀e - ; fun = fun - ; fun[] = fun[] - ; rel = rel - ; rel[] = rel[] + ; R = R + ; R[] = λ {Γ} {Δ} {σ} {t} {u} → R[] {Γ} {Δ} {σ} {t} {u} } @@ -331,9 +317,8 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where ≤refl : {w : World} → w ≤ w ≤tran : {w w' w'' : World} → w ≤ w' → w' ≤ w'' → w ≤ w' TM : Set - REL : (n : Nat) → R n → Array TM n → World → Prop - RELmon : {n : Nat} → {r : R n} → {x : Array TM n} → {w w' : World} → REL n r x w → REL n r x w' - FUN : (n : Nat) → F n → Array TM n → TM + REL : TM → TM → World → Prop + RELmon : {t u : TM} → {w w' : World} → REL t u w → REL t u w' infixr 10 _∘_ Con = World → Set Sub : Con → Con → Set @@ -372,13 +357,6 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where thm {tz = zero} = refl thm {tz = next x tz} {σ} {w} {δ} = substP (λ tz' → (next (x w (σ w δ)) (map (λ t → t w δ) (map (λ s w γ → s w (σ w γ)) tz))) ≡ (next (x w (σ w δ)) tz')) (thm {tz = tz}) refl -- substP (λ tz' → (next (x w (σ w δ)) (map (λ t → t δ) (map (λ s γ → s w (σ w γ)) tz))) ≡ (next (x w (σ w δ)) tz')) (thm {tz = tz}) refl - - -- Term extension with functions - fun : {Γ : Con} → {n : Nat} → F n → Array (Tm Γ) n → Tm Γ - fun {n = n} f tz = λ w γ → FUN n f (map (λ t → t w γ) tz) - 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) - fun[] {Γ = Γ} {Δ = Δ} {σ = σ} {n = n} {f = f} {tz = tz} = ≡fun' λ w → ≡fun λ γ → substP ((λ x → (FUN n f) x ≡ (FUN n f) (map (λ t → t w γ) (tz [ σ ]tz)))) (thm {tz = tz}) refl - -- Tm⁺ _▹ₜ : Con → Con Γ ▹ₜ = λ w → (Γ w) × TM @@ -406,14 +384,14 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where []f-id = refl []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 []f-∘ = refl - + -- Formulas with relation on terms - rel : {Γ : Con} → {n : Nat} → R n → Array (Tm Γ) n → For Γ - rel {n = n} r tz = λ w → λ γ → (REL n r) (map (λ t → t w γ) tz) w - 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) - rel[] {σ = σ} {n = n} {r = r} {tz = tz} = ≡fun' ( λ w → ≡fun (λ γ → (substP (λ x → (REL n r) x w ≡ (REL n r) (map (λ t → t w γ) (tz [ σ ]tz)) w) thm refl))) - - + R : {Γ : Con} → Tm Γ → Tm Γ → For Γ + R t u = λ w → λ γ → REL (t w γ) (u w γ) w + R[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t u : Tm Γ} → (R t u) [ σ ]f ≡ R (t [ σ ]t) (u [ σ ]t) + R[] {σ = σ} = cong₂ R refl refl + + -- Proofs _⊢_ : (Γ : Con) → For Γ → Prop Γ ⊢ F = ∀ w (γ : Γ w) → F w γ @@ -468,7 +446,7 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where ∀e p {t} w γ = p w γ (t w γ) - tod : FFOL F R + tod : FFOL tod = record { Con = Con ; Sub = Sub @@ -509,10 +487,8 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where ; app = app ; ∀i = ∀i ; ∀e = ∀e - ; fun = fun - ; fun[] = fun[] - ; rel = rel - ; rel[] = rel[] + ; R = R + ; R[] = λ {Γ} {Δ} {σ} {t} {u} → R[] {Γ} {Δ} {σ} {t} {u} }