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@@ -427,308 +427,3 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
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; rel = rel
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; rel = rel
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; rel[] = rel[]
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; rel[] = rel[]
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}
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}
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{-
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module M where
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data Con : Set
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data For : Con → Set
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data _⊢_ : (Γ : Con) → For Γ → Prop
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data Con where
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◇ : Con
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_▹ₜ : Con → Con
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_▹ₚ_ : (Γ : Con) → (A : For Γ) → Con
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data Sub : Con → Con → Set where
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id : {Γ : Con} → Sub Γ Γ
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next▹ₜ : {Γ Δ : Con} → Sub Δ Γ → Sub Δ (Γ ▹ₜ)
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next▹ₚ : {Γ Δ : Con} → {A : For Γ} → Sub Δ Γ → Sub Δ (Γ ▹ₚ A)
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_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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ε : {Γ : Con} → Sub ◇ Γ
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ε {◇} = id
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ε {Γ ▹ₜ} = next▹ₜ ε
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ε {Γ ▹ₚ A} = next▹ₚ ε
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data For where
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_⇒_ : {Γ : Con} → For Γ → For Γ → For Γ
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∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ
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infixr 10 _∘_
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-- Functor Con → Set called Tm
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data Tm : Con → Set where
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zero : {Γ : Con} → Tm (Γ ▹ₜ)
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next : {Γ : Con} → Tm Γ → Tm (Γ ▹ₜ)
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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-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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-- Tm⁺
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πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ
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πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ
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_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ)
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πₜ²∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ² (σ ,ₜ t) ≡ t
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πₜ¹∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ¹ (σ ,ₜ t) ≡ σ
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,ₜ∘πₜ : {Γ Δ : Con} → {σ : Sub Δ (Γ ▹ₜ)} → (πₜ¹ σ) ,ₜ (πₜ² σ) ≡ σ
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-- Functor Con → Set called For
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_[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms
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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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-- Proofs
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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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-- → Prop⁺
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πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
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πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f)
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_,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F)
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,ₚ∘πₚ : {Γ Δ : Con} → {F : For Γ} → {σ : Sub Δ (Γ ▹ₚ F)} → (πₚ¹ σ) ,ₚ (πₚ² σ) ≡ σ
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πₚ¹∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For Γ} → {prf : Δ ⊢ (F [ σ ]f)} → πₚ¹ (σ ,ₚ prf) ≡ σ
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-- Implication
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[]f-⇒ : {Γ Δ : Con} → {F G : For Γ} → {σ : Sub Δ Γ} → (F ⇒ G) [ σ ]f ≡ (F [ σ ]f) ⇒ (G [ σ ]f)
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-- Forall
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[]f-∀∀ : {Γ Δ : Con} → {F : For (Γ ▹ₜ)} → {σ : Sub Δ Γ} → {t : Tm Γ} → (∀∀ F) [ σ ]f ≡ (∀∀ (F [ ((id {Γ}) ,ₜ t) ∘ σ ∘(πₜ¹ (id {Δ ▹ₜ}))]f))
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-- Lam & App
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lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G)
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app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G
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-- Again, we don't write the _[_]p equalities as everything is in Prop
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-- ∀i and ∀e
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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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mod : FFOL
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mod = record {M}
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-}
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-- tod : FFOL
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-- tod = record {Tarski Term}
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{-
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module FOL (x : Abs) where
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open Abs x
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variable
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Γ Δ : Con
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data Form : Con → Set where
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_⇒_ : Form Γ → Form Γ → Form Γ
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infixr 8 _⇒_
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vv : Set
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vv = Nat
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record λcalculus : Set₁ where
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field
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Con : Set
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Sub : Con → Con → Set -- Prop makes a posetal category
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_=s_ : {Γ Δ : Con} → Sub Γ Δ → Sub Γ Δ → Prop
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_∘_ : {Γ Δ Ξ : Con} → Sub Γ Δ → Sub Δ Ξ → Sub Γ Ξ
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id : {Γ : Con} → Sub Γ Γ
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◇ : Con
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ε : {Γ : Con} → Sub ◇ Γ
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Tm : Con → Set
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_=t_ : {Γ : Con} → Tm Γ → Tm Γ → Prop
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_[_] : {Γ Δ : Con} → Tm Δ → Sub Γ Δ → Tm Γ
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[∘] : {Γ Δ Ξ : Con} → {σ : Sub Γ Δ} → {δ : Sub Δ Ξ} → {t : Tm Ξ} → (t [ (σ ∘ δ) ]) =t ((t [ δ ]) [ σ ])
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[id] : {Γ : Con} → {t : Tm Γ} → (t [ id {Γ} ]) =t t
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app : {Γ : Con} → Tm Γ → Tm Γ → Tm Γ
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app[] : {Γ Δ : Con} → {σ : Sub Γ Δ} → {x y : Tm Δ} → ((app x y) [ σ ]) =t (app (x [ σ ]) (y [ σ ]))
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_▻_ : (Γ : Con) → Tm Γ → Con
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π₁₁ : {Γ Δ : Con} → {t : Tm Γ} → Sub Δ (Γ ▻ t) → (Sub Δ Γ)
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π₁₂ : {Γ Δ : Con} → {t : Tm Γ} → Sub Δ (Γ ▻ t) → (Tm (Γ ▻ t))
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π₂ : {Γ Δ : Con} → {t : Tm Γ} → Sub Δ Γ → Tm (Γ ▻ t) → Sub Δ (Γ ▻ t)
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inj1 : {Γ Δ : Con} → {t : Tm Γ} → {σ : Sub Δ (Γ ▻ t)} → (π₂ (π₁₁ σ) (π₁₂ σ)) =s σ
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inj2 : {Γ Δ : Con} → {t : Tm Γ} → {σ : Sub Δ Γ} → {x : Tm (Γ ▻ t)} → (π₁₁ (π₂ σ x) =s σ) ∧ (π₁₂ (π₂ σ x) =t x )
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lam : {Γ : Con} → {t : Tm Γ} → Tm (Γ ▻ t) → Tm Γ
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-- lam[] : {Γ Δ : Con} → {t : Tm Γ} → {σ : Sub Δ Γ} → {x : Tm (Γ ▻ t)} → ((lam x) [ σ ]) =t (lam (x [ σ ∘ (π₂ (id {Γ}) x) ]))
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data λterm : Set where
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lam : (λterm → λterm) → λterm
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app : λterm → λterm → λterm
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E : λterm
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E = app (lam (λ x → app x x)) (lam (λ x → app x x))
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data _→β_ : λterm → λterm → Prop where
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βrule : {t : λterm → λterm} → {x : λterm} → (app (lam t) x) →β (t x)
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-- βtran : {x y z : λterm} → x →β y → y →β z → x →β z
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βcong1 : {x y z : λterm} → x →β y → app x z →β app y z
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βcong2 : {x y z : λterm} → x →β y → app z x →β app z y
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βcong3 : {t : λterm → λterm} → ({x y : λterm} → x →β y → t x →β t y) → lam t →β lam t
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thm : E →β E
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thm = βrule
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-- Proofs
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private
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variable
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A B : Form Γ
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data ⊢ : Form Γ → Prop where
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lam : (⊢ A → ⊢ B) → ⊢ (A ⇒ B)
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app : ⊢ (A ⇒ B) → (⊢ A → ⊢ B)
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-- We can add hypotheses to a proof
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addhyp⊢ : Γ ∈* Γ' → Γ ⊢ A → Γ' ⊢ A
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addhyp⊢ s (zero x) = zero (mon∈∈* x s)
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addhyp⊢ s (lam h) = lam (addhyp⊢ (both∈* s) h)
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addhyp⊢ s (app h h₁) = app (addhyp⊢ s h) (addhyp⊢ s h₁)
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addhyp⊢ s (andi h₁ h₂) = andi (addhyp⊢ s h₁) (addhyp⊢ s h₂)
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addhyp⊢ s (ande₁ h) = ande₁ (addhyp⊢ s h)
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addhyp⊢ s (ande₂ h) = ande₂ (addhyp⊢ s h)
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addhyp⊢ s (true) = true
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addhyp⊢ s (∀i h) = ∀i (addhyp⊢ s h)
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addhyp⊢ s (∀e h) = ∀e (addhyp⊢ s h)
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-- Extension of ⊢ to contexts
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_⊢⁺_ : Con → Con → Prop
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Γ ⊢⁺ [] = ⊤
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Γ ⊢⁺ (F ∷ Γ') = (Γ ⊢ F) ∧ (Γ ⊢⁺ Γ')
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infix 5 _⊢⁺_
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-- We show that the relation respects ∈*
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mon∈*⊢⁺ : Γ' ∈* Γ → Γ ⊢⁺ Γ'
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mon∈*⊢⁺ zero∈* = tt
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mon∈*⊢⁺ (next∈* x h) = ⟨ (zero x) , (mon∈*⊢⁺ h) ⟩
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-- The relation respects ⊆
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mon⊆⊢⁺ : Γ' ⊆ Γ → Γ ⊢⁺ Γ'
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mon⊆⊢⁺ h = mon∈*⊢⁺ (⊆→∈* h)
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-- The relation is reflexive
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refl⊢⁺ : Γ ⊢⁺ Γ
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refl⊢⁺ {[]} = tt
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refl⊢⁺ {x ∷ Γ} = ⟨ zero zero∈ , mon⊆⊢⁺ (next⊆ zero⊆) ⟩
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-- We can add hypotheses to to a proof
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addhyp⊢⁺ : Γ ∈* Γ' → Γ ⊢⁺ Γ'' → Γ' ⊢⁺ Γ''
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addhyp⊢⁺ {Γ'' = []} s h = tt
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addhyp⊢⁺ {Γ'' = x ∷ Γ''} s ⟨ Γx , ΓΓ'' ⟩ = ⟨ addhyp⊢ s Γx , addhyp⊢⁺ s ΓΓ'' ⟩
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-- The relation respects ⊢
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halftran⊢⁺ : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁺ Γ' → Γ' ⊢ F → Γ ⊢ F
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halftran⊢⁺ {Γ' = F ∷ Γ'} h⁺ (zero zero∈) = proj₁ h⁺
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halftran⊢⁺ {Γ' = F ∷ Γ'} h⁺ (zero (next∈ x)) = halftran⊢⁺ (proj₂ h⁺) (zero x)
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halftran⊢⁺ h⁺ (lam h) = lam (halftran⊢⁺ ⟨ (zero zero∈) , (addhyp⊢⁺ (right∈* refl∈*) h⁺) ⟩ h)
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halftran⊢⁺ h⁺ (app h h₁) = app (halftran⊢⁺ h⁺ h) (halftran⊢⁺ h⁺ h₁)
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halftran⊢⁺ h⁺ (andi hf hg) = andi (halftran⊢⁺ h⁺ hf) (halftran⊢⁺ h⁺ hg)
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halftran⊢⁺ h⁺ (ande₁ hfg) = ande₁ (halftran⊢⁺ h⁺ hfg)
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halftran⊢⁺ h⁺ (ande₂ hfg) = ande₂ (halftran⊢⁺ h⁺ hfg)
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halftran⊢⁺ h⁺ (true) = true
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halftran⊢⁺ h⁺ (∀i h) = ∀i (halftran⊢⁺ h⁺ h)
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halftran⊢⁺ h⁺ (∀e h {t}) = ∀e (halftran⊢⁺ h⁺ h)
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-- The relation is transitive
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tran⊢⁺ : {Γ Γ' Γ'' : Con} → Γ ⊢⁺ Γ' → Γ' ⊢⁺ Γ'' → Γ ⊢⁺ Γ''
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tran⊢⁺ {Γ'' = []} h h' = tt
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tran⊢⁺ {Γ'' = x ∷ Γ*} h h' = ⟨ halftran⊢⁺ h (proj₁ h') , tran⊢⁺ h (proj₂ h') ⟩
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{--- DEFINITIONS OF ⊢⁰ and ⊢* ---}
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-- ⊢⁰ are neutral forms
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-- ⊢* are normal forms
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data _⊢⁰_ : Con → Form → Prop
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data _⊢*_ : Con → Form → Prop
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data _⊢⁰_ where
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zero : A ∈ Γ → Γ ⊢⁰ A
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app : Γ ⊢⁰ (A ⇒ B) → Γ ⊢* A → Γ ⊢⁰ B
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ande₁ : Γ ⊢⁰ A ∧∧ B → Γ ⊢⁰ A
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ande₂ : Γ ⊢⁰ A ∧∧ B → Γ ⊢⁰ B
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∀e : {F : Term → Form} → Γ ⊢⁰ (∀∀ F) → ( {t : Term} → Γ ⊢⁰ (F t) )
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data _⊢*_ where
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neu⁰ : Γ ⊢⁰ Rel r ts → Γ ⊢* Rel r ts
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lam : (A ∷ Γ) ⊢* B → Γ ⊢* (A ⇒ B)
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andi : Γ ⊢* A → Γ ⊢* B → Γ ⊢* (A ∧∧ B)
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∀i : {F : Term → Form} → ({t : Term} → Γ ⊢* F t) → Γ ⊢* ∀∀ F
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true : Γ ⊢* ⊤⊤
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infix 5 _⊢⁰_
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infix 5 _⊢*_
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-- We can add hypotheses to a proof
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addhyp⊢⁰ : Γ ∈* Γ' → Γ ⊢⁰ A → Γ' ⊢⁰ A
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addhyp⊢* : Γ ∈* Γ' → Γ ⊢* A → Γ' ⊢* A
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addhyp⊢⁰ s (zero x) = zero (mon∈∈* x s)
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addhyp⊢⁰ s (app h h₁) = app (addhyp⊢⁰ s h) (addhyp⊢* s h₁)
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addhyp⊢⁰ s (ande₁ h) = ande₁ (addhyp⊢⁰ s h)
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addhyp⊢⁰ s (ande₂ h) = ande₂ (addhyp⊢⁰ s h)
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addhyp⊢⁰ s (∀e h {t}) = ∀e (addhyp⊢⁰ s h) {t}
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addhyp⊢* s (neu⁰ x) = neu⁰ (addhyp⊢⁰ s x)
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addhyp⊢* s (lam h) = lam (addhyp⊢* (both∈* s) h)
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addhyp⊢* s (andi h₁ h₂) = andi (addhyp⊢* s h₁) (addhyp⊢* s h₂)
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addhyp⊢* s true = true
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addhyp⊢* s (∀i h) = ∀i (addhyp⊢* s h)
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-- Extension of ⊢⁰ to contexts
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-- i.e. there is a neutral proof for any element
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_⊢⁰⁺_ : Con → Con → Prop
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Γ ⊢⁰⁺ [] = ⊤
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Γ ⊢⁰⁺ (F ∷ Γ') = (Γ ⊢⁰ F) ∧ (Γ ⊢⁰⁺ Γ')
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infix 5 _⊢⁰⁺_
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-- The relation respects ∈*
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mon∈*⊢⁰⁺ : Γ' ∈* Γ → Γ ⊢⁰⁺ Γ'
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mon∈*⊢⁰⁺ zero∈* = tt
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mon∈*⊢⁰⁺ (next∈* x h) = ⟨ (zero x) , (mon∈*⊢⁰⁺ h) ⟩
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-- The relation respects ⊆
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mon⊆⊢⁰⁺ : Γ' ⊆ Γ → Γ ⊢⁰⁺ Γ'
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mon⊆⊢⁰⁺ h = mon∈*⊢⁰⁺ (⊆→∈* h)
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-- This relation is reflexive
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refl⊢⁰⁺ : Γ ⊢⁰⁺ Γ
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refl⊢⁰⁺ {[]} = tt
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refl⊢⁰⁺ {x ∷ Γ} = ⟨ zero zero∈ , mon⊆⊢⁰⁺ (next⊆ zero⊆) ⟩
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-- A useful lemma, that we can add hypotheses
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addhyp⊢⁰⁺ : Γ ∈* Γ' → Γ ⊢⁰⁺ Γ'' → Γ' ⊢⁰⁺ Γ''
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addhyp⊢⁰⁺ {Γ'' = []} s h = tt
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addhyp⊢⁰⁺ {Γ'' = A ∷ Γ'} s ⟨ Γx , ΓΓ'' ⟩ = ⟨ addhyp⊢⁰ s Γx , addhyp⊢⁰⁺ s ΓΓ'' ⟩
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-- The relation preserves ⊢⁰ and ⊢*
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halftran⊢⁰⁺* : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁰⁺ Γ' → Γ' ⊢* F → Γ ⊢* F
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halftran⊢⁰⁺⁰ : {Γ Γ' : Con} → {F : Form} → Γ ⊢⁰⁺ Γ' → Γ' ⊢⁰ F → Γ ⊢⁰ F
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halftran⊢⁰⁺* h⁺ (neu⁰ x) = neu⁰ (halftran⊢⁰⁺⁰ h⁺ x)
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halftran⊢⁰⁺* h⁺ (lam h) = lam (halftran⊢⁰⁺* ⟨ zero zero∈ , addhyp⊢⁰⁺ (right∈* refl∈*) h⁺ ⟩ h)
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halftran⊢⁰⁺* h⁺ (andi h₁ h₂) = andi (halftran⊢⁰⁺* h⁺ h₁) (halftran⊢⁰⁺* h⁺ h₂)
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halftran⊢⁰⁺* h⁺ true = true
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halftran⊢⁰⁺* h⁺ (∀i h) = ∀i (halftran⊢⁰⁺* h⁺ h)
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halftran⊢⁰⁺⁰ {Γ' = x ∷ Γ'} h⁺ (zero zero∈) = proj₁ h⁺
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halftran⊢⁰⁺⁰ {Γ' = x ∷ Γ'} h⁺ (zero (next∈ h)) = halftran⊢⁰⁺⁰ (proj₂ h⁺) (zero h)
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halftran⊢⁰⁺⁰ h⁺ (app h h') = app (halftran⊢⁰⁺⁰ h⁺ h) (halftran⊢⁰⁺* h⁺ h')
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halftran⊢⁰⁺⁰ h⁺ (ande₁ h) = ande₁ (halftran⊢⁰⁺⁰ h⁺ h)
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halftran⊢⁰⁺⁰ h⁺ (ande₂ h) = ande₂ (halftran⊢⁰⁺⁰ h⁺ h)
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halftran⊢⁰⁺⁰ h⁺ (∀e h {t}) = ∀e (halftran⊢⁰⁺⁰ h⁺ h)
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-- The relation is transitive
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tran⊢⁰⁺ : {Γ Γ' Γ'' : Con} → Γ ⊢⁰⁺ Γ' → Γ' ⊢⁰⁺ Γ'' → Γ ⊢⁰⁺ Γ''
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tran⊢⁰⁺ {Γ'' = []} h h' = tt
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tran⊢⁰⁺ {Γ'' = x ∷ Γ} h h' = ⟨ halftran⊢⁰⁺⁰ h (proj₁ h') , tran⊢⁰⁺ h (proj₂ h') ⟩
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-}
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Reference in New Issue
Block a user