Simplified the notation, working this time
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@ -2,7 +2,7 @@
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open import PropUtil
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module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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module FinitaryFirstOrderLogic 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 (F : Nat → Set) (R : Nat → Set) where
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variable
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ℓ¹ ℓ² ℓ³ ℓ⁴ ℓ⁵ : Level
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record FFOL (F : Nat → Set) (R : Nat → Set) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴ ⊔ ℓ⁵)) where
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record FFOL : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴ ⊔ ℓ⁵)) where
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infixr 10 _∘_
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infixr 5 _⊢_
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field
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@ -27,10 +27,6 @@ module FinitaryFirstOrderLogic (F : Nat → 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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@ -48,8 +44,8 @@ module FinitaryFirstOrderLogic (F : Nat → 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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-- 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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R : {Γ : Con} → (t u : Tm Γ) → For Γ
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R[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t u : Tm Γ} → (R t u) [ σ ]f ≡ R (t [ σ ]t) (u [ σ ]t)
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-- Proofs
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_⊢_ : (Γ : Con) → For Γ → Prop ℓ⁴
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@ -132,8 +128,7 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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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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REL : TM → TM → Prop
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infixr 10 _∘_
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Con = Set
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Sub : Con → Con → Set
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@ -169,12 +164,6 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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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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@ -203,11 +192,10 @@ module FinitaryFirstOrderLogic (F : Nat → 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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R : {Γ : Con} → Tm Γ → Tm Γ → For Γ
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R t u = λ γ → REL (t γ) (u γ)
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R[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t u : Tm Γ} → (R t u) [ σ ]f ≡ R (t [ σ ]t) (u [ σ ]t)
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R[] {σ = σ} = cong₂ R refl refl
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-- Proofs
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_⊢_ : (Γ : Con) → For Γ → Prop
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@ -258,7 +246,7 @@ module FinitaryFirstOrderLogic (F : Nat → 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 F R
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tod : FFOL
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tod = record
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{ Con = Con
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; Sub = Sub
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@ -299,10 +287,8 @@ module FinitaryFirstOrderLogic (F : Nat → 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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; R = R
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; R[] = λ {Γ} {Δ} {σ} {t} {u} → R[] {Γ} {Δ} {σ} {t} {u}
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}
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@ -331,9 +317,8 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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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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REL : TM → TM → World → Prop
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RELmon : {t u : TM} → {w w' : World} → REL t u w → REL t u w'
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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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@ -372,13 +357,6 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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thm {tz = zero} = refl
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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
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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 w γ → FUN n f (map (λ t → t w 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 (map (λ t → t [ σ ]t) tz)
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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
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-- Tm⁺
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_▹ₜ : Con → Con
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Γ ▹ₜ = λ w → (Γ w) × TM
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@ -408,10 +386,10 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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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 w → λ γ → (REL n r) (map (λ t → t w γ) tz) w
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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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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)))
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R : {Γ : Con} → Tm Γ → Tm Γ → For Γ
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R t u = λ w → λ γ → REL (t w γ) (u w γ) w
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R[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t u : Tm Γ} → (R t u) [ σ ]f ≡ R (t [ σ ]t) (u [ σ ]t)
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R[] {σ = σ} = cong₂ R refl refl
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-- Proofs
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@ -468,7 +446,7 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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∀e p {t} w γ = p w γ (t w γ)
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tod : FFOL F R
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tod : FFOL
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tod = record
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{ Con = Con
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; Sub = Sub
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@ -509,10 +487,8 @@ module FinitaryFirstOrderLogic (F : Nat → 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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; R = R
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; R[] = λ {Γ} {Δ} {σ} {t} {u} → R[] {Γ} {Δ} {σ} {t} {u}
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}
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