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330
FFOLCompleteness.agda
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330
FFOLCompleteness.agda
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{-# OPTIONS --prop --rewriting #-}
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open import PropUtil
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module FFOLCompleteness where
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open import Agda.Primitive
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open import FinitaryFirstOrderLogic
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open import ListUtil
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record Family : Set (lsuc (ℓ¹)) where
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field
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World : Set ℓ¹
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_≤_ : World → World → Prop
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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 : World → Set ℓ¹
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TM≤ : {w w' : World} → w ≤ w' → TM w → TM w'
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REL : (w : World) → TM w → TM w → Prop ℓ¹
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REL≤ : {w w' : World} → {t u : TM w} → (eq : w ≤ w') → REL w t u → REL w' (TM≤ eq t) (TM≤ eq u)
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infixr 10 _∘_
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Con = World → Set ℓ¹
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Sub : Con → Con → Set ℓ¹
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Sub Δ Γ = (w : World) → Δ w → Γ w
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_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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α ∘ β = λ w γ → α w (β w γ)
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id : {Γ : Con} → Sub Γ Γ
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id = λ w γ → γ
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◇ : Con -- The initial object of the category
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◇ = λ w → ⊤ₛ
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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ε w Γ = ttₛ
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-- Functor Con → Set called Tm
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Tm : Con → Set ℓ¹
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Tm Γ = (w : World) → (Γ w) → TM w
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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 w → λ γ → t w (σ w γ)
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-- Tm⁺
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_▹ₜ : Con → Con
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Γ ▹ₜ = λ w → (Γ w) × (TM w)
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πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ
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πₜ¹ σ = λ w → λ x → proj×₁ (σ w x)
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πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ
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πₜ² σ = λ w → λ x → proj×₂ (σ w x)
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_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ)
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σ ,ₜ t = λ w → λ x → (σ w x) ,× (t w x)
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-- Functor Con → Set called For
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For : Con → Set (lsuc ℓ¹)
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For Γ = (w : World) → (Γ w) → Prop ℓ¹
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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 [ σ ]f = λ w → λ x → F w (σ w x)
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-- Formulas with relation on terms
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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 w (t w γ) (u w γ)
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-- Proofs
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_⊢_ : (Γ : Con) → For Γ → Prop ℓ¹
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Γ ⊢ F = ∀ w (γ : Γ w) → F w γ
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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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prf [ σ ]p = λ w → λ γ → prf w (σ w γ)
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-- Equalities below are useless because Γ ⊢ F is in prop
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-- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf
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-- []p-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ α ∘ β ]p ≡ (prf [ β ]p) [ α ]p
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-- → Prop⁺
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_▹ₚ_ : (Γ : Con) → For Γ → Con
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Γ ▹ₚ F = λ w → (Γ w) ×'' (F w)
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πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
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πₚ¹ σ w δ = proj×''₁ (σ w δ)
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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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πₚ² σ w δ = proj×''₂ (σ w δ)
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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_ {F = F} σ pf w δ = (σ w δ) ,×'' pf w δ
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-- Implication
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_⇒_ : {Γ : Con} → For Γ → For Γ → For Γ
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F ⇒ G = λ w → λ γ → (∀ w' → w ≤ w' → (F w γ) → (G w γ))
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-- Forall
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∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ
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∀∀ F = λ w → λ γ → ∀ t → F w (γ ,× t)
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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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lam prf = λ w γ w' s h → prf w (γ ,×'' h)
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app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G
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app prf prf' = λ w γ → prf w γ w ≤refl (prf' w γ)
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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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∀i p w γ = λ t → p w (γ ,× t)
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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} w γ = p w γ (t w γ)
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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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; _∘_ = _∘_
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; ∘-ass = refl
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; id = id
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; idl = refl
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; idr = refl
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; ◇ = ◇
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; ε = ε
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; ε-u = refl
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; Tm = Tm
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; _[_]t = _[_]t
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; []t-id = refl
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; []t-∘ = refl
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; _▹ₜ = _▹ₜ
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; πₜ¹ = πₜ¹
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; πₜ² = πₜ²
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; _,ₜ_ = _,ₜ_
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; πₜ²∘,ₜ = refl
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; πₜ¹∘,ₜ = refl
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; ,ₜ∘πₜ = refl
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; ,ₜ∘ = refl
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; For = For
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; _[_]f = _[_]f
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; []f-id = refl
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; []f-∘ = refl
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; _⊢_ = _⊢_
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; _[_]p = _[_]p
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; _▹ₚ_ = _▹ₚ_
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; πₚ¹ = πₚ¹
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; πₚ² = πₚ²
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; _,ₚ_ = _,ₚ_
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; ,ₚ∘πₚ = refl
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; πₚ¹∘,ₚ = refl
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; ,ₚ∘ = refl
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; _⇒_ = _⇒_
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; []f-⇒ = refl
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; ∀∀ = ∀∀
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; []f-∀∀ = refl
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; lam = lam
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; app = app
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; ∀i = ∀i
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; ∀e = ∀e
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; R = R
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; R[] = refl
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}
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record Presheaf : Set (lsuc (ℓ¹)) where
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field
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World : Set ℓ¹
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Arr : World → World → Set ℓ¹ -- arrows
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id-a : {w : World} → Arr w w -- id arrow
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_∘a_ : {w w' w'' : World} → Arr w w' → Arr w' w'' → Arr w w'' -- arrow composition
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∘a-ass : {w w' w'' w''' : World}{a : Arr w w'}{b : Arr w' w''}{c : Arr w'' w'''}
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→ ((a ∘a b) ∘a c) ≡ (a ∘a (b ∘a c))
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idl-a : {w w' : World} → {a : Arr w w'} → (id-a {w}) ∘a a ≡ a
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idr-a : {w w' : World} → {a : Arr w w'} → a ∘a (id-a {w'}) ≡ a
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TM : World → Set ℓ¹
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TM≤ : {w w' : World} → Arr w w' → TM w' → TM w
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REL : (w : World) → TM w → TM w → Prop ℓ¹
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REL≤ : {w w' : World} → {t u : TM w'} → (eq : Arr w w') → REL w' t u → REL w (TM≤ eq t) (TM≤ eq u)
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infixr 10 _∘_
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Con = World → Set ℓ¹
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Sub : Con → Con → Set ℓ¹
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Sub Δ Γ = (w : World) → Δ w → Γ w
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_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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α ∘ β = λ w γ → α w (β w γ)
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id : {Γ : Con} → Sub Γ Γ
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id = λ w γ → γ
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◇ : Con -- The initial object of the category
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◇ = λ w → ⊤ₛ
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ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
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ε w Γ = ttₛ
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-- Functor Con → Set called Tm
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Tm : Con → Set ℓ¹
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Tm Γ = (w : World) → (Γ w) → TM w
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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 w → λ γ → t w (σ w γ)
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-- Tm⁺
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_▹ₜ : Con → Con
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Γ ▹ₜ = λ w → (Γ w) × (TM w)
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πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ
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πₜ¹ σ = λ w → λ x → proj×₁ (σ w x)
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πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ
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πₜ² σ = λ w → λ x → proj×₂ (σ w x)
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_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ)
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σ ,ₜ t = λ w → λ x → (σ w x) ,× (t w x)
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-- Functor Con → Set called For
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For : Con → Set (lsuc ℓ¹)
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For Γ = (w : World) → (Γ w) → Prop ℓ¹
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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 [ σ ]f = λ w → λ x → F w (σ w x)
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-- Formulas with relation on terms
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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 w (t w γ) (u w γ)
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-- Proofs
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_⊢_ : (Γ : Con) → For Γ → Prop ℓ¹
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Γ ⊢ F = ∀ w (γ : Γ w) → F w γ
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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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prf [ σ ]p = λ w → λ γ → prf w (σ w γ)
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-- Equalities below are useless because Γ ⊢ F is in prop
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-- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf
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-- []p-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ α ∘ β ]p ≡ (prf [ β ]p) [ α ]p
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-- → Prop⁺
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_▹ₚ_ : (Γ : Con) → For Γ → Con
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Γ ▹ₚ F = λ w → (Γ w) ×'' (F w)
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πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
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πₚ¹ σ w δ = proj×''₁ (σ w δ)
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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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πₚ² σ w δ = proj×''₂ (σ w δ)
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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_ {F = F} σ pf w δ = (σ w δ) ,×'' pf w δ
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-- Implication
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_⇒_ : {Γ : Con} → For Γ → For Γ → For Γ
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F ⇒ G = λ w → λ γ → (∀ w' → Arr w w' → (F w γ) → (G w γ))
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-- Forall
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∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ
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∀∀ F = λ w → λ γ → ∀ t → F w (γ ,× t)
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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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lam prf = λ w γ w' s h → prf w (γ ,×'' h)
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app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G
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app prf prf' = λ w γ → prf w γ w id-a (prf' w γ)
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-- Again, we don't write the _[_]p equalities as everything is in Prop
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vv
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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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∀i p w γ = λ t → p w (γ ,× t)
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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} w γ = p w γ (t w γ)
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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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; _∘_ = _∘_
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; ∘-ass = refl
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; id = id
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; idl = refl
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; idr = refl
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; ◇ = ◇
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; ε = ε
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; ε-u = refl
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; Tm = Tm
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; _[_]t = _[_]t
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; []t-id = refl
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; []t-∘ = refl
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; _▹ₜ = _▹ₜ
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; πₜ¹ = πₜ¹
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; πₜ² = πₜ²
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; _,ₜ_ = _,ₜ_
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; πₜ²∘,ₜ = refl
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; πₜ¹∘,ₜ = refl
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; ,ₜ∘πₜ = refl
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; ,ₜ∘ = refl
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; For = For
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; _[_]f = _[_]f
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; []f-id = refl
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; []f-∘ = refl
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; _⊢_ = _⊢_
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; _[_]p = _[_]p
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; _▹ₚ_ = _▹ₚ_
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; πₚ¹ = πₚ¹
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; πₚ² = πₚ²
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; _,ₚ_ = _,ₚ_
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; ,ₚ∘πₚ = refl
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; πₚ¹∘,ₚ = refl
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; ,ₚ∘ = refl
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; _⇒_ = _⇒_
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; []f-⇒ = refl
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; ∀∀ = ∀∀
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; []f-∀∀ = refl
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; lam = lam
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; app = app
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; ∀i = ∀i
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||||
; ∀e = ∀e
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||||
; R = R
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; R[] = refl
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}
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-- Completeness proof
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-- We first build our universal Kripke model
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module ComplenessProof where
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-- We have a model, we construct the Universal Presheaf model of this model
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import FFOLInitial as I
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UniversalPresheaf : Presheaf
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UniversalPresheaf = record
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{ World = I.Con
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; Arr = I.Sub
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; id-a = I.id
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; _∘a_ = λ σ σ' → σ' I.∘ σ
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; ∘a-ass = ≡sym I.∘-ass
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; idl-a = I.idr
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; idr-a = I.idl
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; TM = λ Γ → I.Tm (I.Con.t Γ)
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; TM≤ = λ σ t → t I.[ I.Sub.t σ ]t
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; REL = λ Γ t u → I.Pf Γ (I.r t u)
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; REL≤ = λ σ pf → (pf I.[ I.Sub.t σ ]pₜ) I.[ I.Sub.p σ ]p
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}
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||||
-- I.xx are from initial, xx are from up
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open Presheaf UniversalPresheaf
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||||
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-- Now we want to show universality of this model, that is
|
||||
-- if you have a proof in UP, you have the same in I.
|
||||
|
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q : {Γ : Con}{A : For Γ} → Γ ⊢ A → I.Pf {!!} {!!}
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u : {Γ : Con}{A : For Γ} → I.Pf {!!} {!!} → Γ ⊢ A
|
||||
550
FFOLInitial.agda
Normal file
550
FFOLInitial.agda
Normal file
@ -0,0 +1,550 @@
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{-# OPTIONS --prop --rewriting #-}
|
||||
|
||||
open import PropUtil
|
||||
|
||||
module FFOLInitial where
|
||||
|
||||
open import FinitaryFirstOrderLogic
|
||||
open import Agda.Primitive
|
||||
open import ListUtil
|
||||
|
||||
-- First definition of terms and term contexts --
|
||||
data Cont : Set₁ where
|
||||
◇t : Cont
|
||||
_▹t⁰ : Cont → Cont
|
||||
variable
|
||||
Γₜ Δₜ Ξₜ : Cont
|
||||
data TmVar : Cont → Set₁ where
|
||||
tvzero : TmVar (Γₜ ▹t⁰)
|
||||
tvnext : TmVar Γₜ → TmVar (Γₜ ▹t⁰)
|
||||
|
||||
data Tm : Cont → Set₁ where
|
||||
var : TmVar Γₜ → Tm Γₜ
|
||||
|
||||
-- Now we can define formulæ
|
||||
data For : Cont → Set₁ where
|
||||
r : Tm Γₜ → Tm Γₜ → For Γₜ
|
||||
_⇒_ : For Γₜ → For Γₜ → For Γₜ
|
||||
∀∀ : For (Γₜ ▹t⁰) → For Γₜ
|
||||
|
||||
-- Then we define term substitutions, and the application of them on terms and formulæ
|
||||
data Subt : Cont → Cont → Set₁ where
|
||||
εₜ : Subt Γₜ ◇t
|
||||
_,ₜ_ : Subt Δₜ Γₜ → Tm Δₜ → Subt Δₜ (Γₜ ▹t⁰)
|
||||
|
||||
-- We subst on terms
|
||||
_[_]t : Tm Γₜ → Subt Δₜ Γₜ → Tm Δₜ
|
||||
var tvzero [ σ ,ₜ t ]t = t
|
||||
var (tvnext tv) [ σ ,ₜ t ]t = var tv [ σ ]t
|
||||
|
||||
-- We define liftings on term variables
|
||||
-- A term of n variables is a term of n+1 variables
|
||||
-- Same for a term array
|
||||
wkₜt : Tm Γₜ → Tm (Γₜ ▹t⁰)
|
||||
|
||||
wkₜt (var tv) = var (tvnext tv)
|
||||
|
||||
-- From a substition into n variables, we get a substitution into n+1 variables which don't use the last one
|
||||
wkₜσt : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) Γₜ
|
||||
wkₜσt εₜ = εₜ
|
||||
wkₜσt (σ ,ₜ t) = (wkₜσt σ) ,ₜ (wkₜt t)
|
||||
wkₜσt-wkₜt : {tv : TmVar Γₜ} → {σ : Subt Δₜ Γₜ} → wkₜt (var tv [ σ ]t) ≡ var tv [ wkₜσt σ ]t
|
||||
wkₜσt-wkₜt {tv = tvzero} {σ = σ ,ₜ x} = refl
|
||||
wkₜσt-wkₜt {tv = tvnext tv} {σ = σ ,ₜ x} = wkₜσt-wkₜt {tv = tv} {σ = σ}
|
||||
|
||||
-- From a substitution into n variables, we construct a substitution from n+1 variables to n+1 variables which maps it to itself
|
||||
-- i.e. 0 -> 0 and for all i ->(old) σ(i) we get i+1 -> σ(i)+1
|
||||
liftₜσ : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) (Γₜ ▹t⁰)
|
||||
liftₜσ σ = (wkₜσt σ) ,ₜ (var tvzero)
|
||||
|
||||
|
||||
-- We subst on formulæ
|
||||
_[_]f : For Γₜ → Subt Δₜ Γₜ → For Δₜ
|
||||
(r t u) [ σ ]f = r (t [ σ ]t) (u [ σ ]t)
|
||||
(A ⇒ B) [ σ ]f = (A [ σ ]f) ⇒ (B [ σ ]f)
|
||||
(∀∀ A) [ σ ]f = ∀∀ (A [ liftₜσ σ ]f)
|
||||
|
||||
-- We now can define identity on term substitutions
|
||||
idₜ : Subt Γₜ Γₜ
|
||||
idₜ {◇t} = εₜ
|
||||
idₜ {Γₜ ▹t⁰} = liftₜσ (idₜ {Γₜ})
|
||||
|
||||
_∘ₜ_ : Subt Δₜ Γₜ → Subt Ξₜ Δₜ → Subt Ξₜ Γₜ
|
||||
εₜ ∘ₜ β = εₜ
|
||||
(α ,ₜ x) ∘ₜ β = (α ∘ₜ β) ,ₜ (x [ β ]t)
|
||||
|
||||
|
||||
-- We have the access functions from the algebra, in restricted versions
|
||||
πₜ¹ : Subt Δₜ (Γₜ ▹t⁰) → Subt Δₜ Γₜ
|
||||
πₜ¹ (σₜ ,ₜ t) = σₜ
|
||||
πₜ² : Subt Δₜ (Γₜ ▹t⁰) → Tm Δₜ
|
||||
πₜ² (σₜ ,ₜ t) = t
|
||||
|
||||
-- And their equalities (the fact that there are reciprocical)
|
||||
πₜ²∘,ₜ : {σₜ : Subt Δₜ Γₜ} → {t : Tm Δₜ} → πₜ² (σₜ ,ₜ t) ≡ t
|
||||
πₜ²∘,ₜ = refl
|
||||
πₜ¹∘,ₜ : {σₜ : Subt Δₜ Γₜ} → {t : Tm Δₜ} → πₜ¹ (σₜ ,ₜ t) ≡ σₜ
|
||||
πₜ¹∘,ₜ = refl
|
||||
,ₜ∘πₜ : {σₜ : Subt Δₜ (Γₜ ▹t⁰)} → (πₜ¹ σₜ) ,ₜ (πₜ² σₜ) ≡ σₜ
|
||||
,ₜ∘πₜ {σₜ = σₜ ,ₜ t} = refl
|
||||
|
||||
-- We can also prove the substitution equalities
|
||||
[]t-id : {t : Tm Γₜ} → t [ idₜ {Γₜ} ]t ≡ t
|
||||
[]t-id {Γₜ ▹t⁰} {var tvzero} = refl
|
||||
[]t-id {Γₜ ▹t⁰} {var (tvnext tv)} = substP (λ t → t ≡ var (tvnext tv)) (wkₜσt-wkₜt {tv = tv} {σ = idₜ}) (substP (λ t → wkₜt t ≡ var (tvnext tv)) (≡sym ([]t-id {t = var tv})) refl)
|
||||
[]t-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → {t : Tm Γₜ} → t [ β ∘ₜ α ]t ≡ (t [ β ]t) [ α ]t
|
||||
[]t-∘ {α = α} {β = β ,ₜ t} {t = var tvzero} = refl
|
||||
[]t-∘ {α = α} {β = β ,ₜ t} {t = var (tvnext tv)} = []t-∘ {t = var tv}
|
||||
[]f-id : {F : For Γₜ} → F [ idₜ {Γₜ} ]f ≡ F
|
||||
[]f-id {F = r t u} = cong₂ r []t-id []t-id
|
||||
[]f-id {F = F ⇒ G} = cong₂ _⇒_ []f-id []f-id
|
||||
[]f-id {F = ∀∀ F} = cong ∀∀ []f-id
|
||||
wkₜσt-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → wkₜσt (β ∘ₜ α) ≡ (wkₜσt β ∘ₜ liftₜσ α)
|
||||
wkₜt[] : {α : Subt Δₜ Γₜ} → {t : Tm Γₜ} → wkₜt (t [ α ]t) ≡ (wkₜt t [ liftₜσ α ]t)
|
||||
wkₜσt-∘ {β = εₜ} = refl
|
||||
wkₜσt-∘ {β = β ,ₜ t} = cong₂ _,ₜ_ wkₜσt-∘ (wkₜt[] {t = t})
|
||||
wkₜt[] {α = α ,ₜ t} {var tvzero} = refl
|
||||
wkₜt[] {α = α ,ₜ t} {var (tvnext tv)} = wkₜt[] {t = var tv}
|
||||
liftₜσ-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → liftₜσ (β ∘ₜ α) ≡ (liftₜσ β) ∘ₜ (liftₜσ α)
|
||||
liftₜσ-∘ {α = α} {β = εₜ} = refl
|
||||
liftₜσ-∘ {α = α} {β = β ,ₜ t} = cong₂ _,ₜ_ (cong₂ _,ₜ_ wkₜσt-∘ (wkₜt[] {t = t})) refl
|
||||
[]f-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → {F : For Γₜ} → F [ β ∘ₜ α ]f ≡ (F [ β ]f) [ α ]f
|
||||
[]f-∘ {α = α} {β = β} {F = r t u} = cong₂ r ([]t-∘ {α = α} {β = β} {t = t}) ([]t-∘ {α = α} {β = β} {t = u})
|
||||
[]f-∘ {F = F ⇒ G} = cong₂ _⇒_ []f-∘ []f-∘
|
||||
[]f-∘ {F = ∀∀ F} = cong ∀∀ (≡tran (cong (λ σ → F [ σ ]f) liftₜσ-∘) []f-∘)
|
||||
R[] : {σ : Subt Δₜ Γₜ} → {t u : Tm Γₜ} → (r t u) [ σ ]f ≡ r (t [ σ ]t) (u [ σ ]t)
|
||||
R[] = refl
|
||||
lem3 : {α : Subt Γₜ Δₜ} → {β : Subt Ξₜ Γₜ} → α ∘ₜ (wkₜσt β) ≡ wkₜσt (α ∘ₜ β)
|
||||
lem3 {α = εₜ} = refl
|
||||
lem3 {α = α ,ₜ var tv} = cong₂ _,ₜ_ (lem3 {α = α}) (≡sym (wkₜσt-wkₜt {tv = tv}))
|
||||
wk[,] : {t : Tm Γₜ}{u : Tm Δₜ}{β : Subt Δₜ Γₜ} → (wkₜt t) [ β ,ₜ u ]t ≡ t [ β ]t
|
||||
wk[,] {t = var tvzero} = refl
|
||||
wk[,] {t = var (tvnext tv)} = refl
|
||||
wk∘, : {α : Subt Γₜ Δₜ}{β : Subt Ξₜ Γₜ}{t : Tm Ξₜ} → (wkₜσt α) ∘ₜ (β ,ₜ t) ≡ (α ∘ₜ β)
|
||||
wk∘, {α = εₜ} = refl
|
||||
wk∘, {α = α ,ₜ t} {β = β} = cong₂ _,ₜ_ (wk∘, {α = α}) (wk[,] {t = t} {β = β})
|
||||
σ-idl : {α : Subt Δₜ Γₜ} → idₜ ∘ₜ α ≡ α
|
||||
σ-idl {α = εₜ} = refl
|
||||
σ-idl {α = α ,ₜ x} = cong₂ _,ₜ_ (≡tran wk∘, σ-idl) refl
|
||||
σ-idr : {α : Subt Δₜ Γₜ} → α ∘ₜ idₜ ≡ α
|
||||
σ-idr {α = εₜ} = refl
|
||||
σ-idr {α = α ,ₜ x} = cong₂ _,ₜ_ σ-idr []t-id
|
||||
∘ₜ-ass : {Γₜ Δₜ Ξₜ Ψₜ : Cont}{α : Subt Γₜ Δₜ}{β : Subt Δₜ Ξₜ}{γ : Subt Ξₜ Ψₜ} → (γ ∘ₜ β) ∘ₜ α ≡ γ ∘ₜ (β ∘ₜ α)
|
||||
∘ₜ-ass {α = α} {β} {εₜ} = refl
|
||||
∘ₜ-ass {α = α} {β} {γ ,ₜ x} = cong₂ _,ₜ_ ∘ₜ-ass (≡sym ([]t-∘ {t = x}))
|
||||
[]f-∀∀ : {A : For (Γₜ ▹t⁰)} → {σₜ : Subt Δₜ Γₜ} → (∀∀ A) [ σₜ ]f ≡ (∀∀ (A [ (σₜ ∘ₜ πₜ¹ idₜ) ,ₜ πₜ² idₜ ]f))
|
||||
[]f-∀∀ {A = A} = cong ∀∀ (cong (_[_]f A) (cong₂ _,ₜ_ (≡tran (cong wkₜσt (≡sym σ-idr)) (≡sym lem3)) refl))
|
||||
εₜ-u : {σₜ : Subt Γₜ ◇t} → σₜ ≡ εₜ
|
||||
εₜ-u {σₜ = εₜ} = refl
|
||||
|
||||
data Conp : Cont → Set₁ -- pu tit in Prop
|
||||
variable
|
||||
Γₚ Γₚ' : Conp Γₜ
|
||||
Δₚ Δₚ' : Conp Δₜ
|
||||
Ξₚ : Conp Ξₜ
|
||||
|
||||
data Conp where
|
||||
◇p : Conp Γₜ
|
||||
_▹p⁰_ : Conp Γₜ → For Γₜ → Conp Γₜ
|
||||
|
||||
record Con : Set₁ where
|
||||
constructor con
|
||||
field
|
||||
t : Cont
|
||||
p : Conp t
|
||||
|
||||
◇ : Con
|
||||
◇ = con ◇t ◇p
|
||||
|
||||
|
||||
_▹p_ : (Γ : Con) → For (Con.t Γ) → Con
|
||||
Γ ▹p A = con (Con.t Γ) (Con.p Γ ▹p⁰ A)
|
||||
|
||||
variable
|
||||
Γ Δ Ξ : Con
|
||||
|
||||
|
||||
|
||||
-- We can add term, that will not be used in the formulæ already present
|
||||
-- (that's why we use wkₜσt)
|
||||
_▹tp : Conp Γₜ → Conp (Γₜ ▹t⁰)
|
||||
◇p ▹tp = ◇p
|
||||
(Γₚ ▹p⁰ A) ▹tp = (Γₚ ▹tp) ▹p⁰ (A [ wkₜσt idₜ ]f)
|
||||
|
||||
_▹t : Con → Con
|
||||
Γ ▹t = con ((Con.t Γ) ▹t⁰) (Con.p Γ ▹tp)
|
||||
|
||||
data PfVar : (Γ : Con) → For (Con.t Γ) → Set₁ where
|
||||
pvzero : {A : For (Con.t Γ)} → PfVar (Γ ▹p A) A
|
||||
pvnext : {A B : For (Con.t Γ)} → PfVar Γ A → PfVar (Γ ▹p B) A
|
||||
|
||||
data Pf : (Γ : Con) → For (Con.t Γ) → Prop₁ where
|
||||
var : {A : For (Con.t Γ)} → PfVar Γ A → Pf Γ A
|
||||
app : {A B : For (Con.t Γ)} → Pf Γ (A ⇒ B) → Pf Γ A → Pf Γ B
|
||||
lam : {A B : For (Con.t Γ)} → Pf (Γ ▹p A) B → Pf Γ (A ⇒ B)
|
||||
p∀∀e : {A : For ((Con.t Γ) ▹t⁰)} → {t : Tm (Con.t Γ)} → Pf Γ (∀∀ A) → Pf Γ (A [ idₜ ,ₜ t ]f)
|
||||
p∀∀i : {A : For (Con.t (Γ ▹t))} → Pf (Γ ▹t) A → Pf Γ (∀∀ A)
|
||||
|
||||
|
||||
data Subp : {Δₜ : Cont} → Conp Δₜ → Conp Δₜ → Set₁ where
|
||||
εₚ : Subp Δₚ ◇p
|
||||
_,ₚ_ : {A : For Δₜ} → (σ : Subp Δₚ Δₚ') → Pf (con Δₜ Δₚ) A → Subp Δₚ (Δₚ' ▹p⁰ A)
|
||||
|
||||
|
||||
_[_]c : Conp Γₜ → Subt Δₜ Γₜ → Conp Δₜ
|
||||
◇p [ σₜ ]c = ◇p
|
||||
(Γₚ ▹p⁰ A) [ σₜ ]c = (Γₚ [ σₜ ]c) ▹p⁰ (A [ σₜ ]f)
|
||||
|
||||
[]c-id : Γₚ [ idₜ ]c ≡ Γₚ
|
||||
[]c-id {Γₚ = ◇p} = refl
|
||||
[]c-id {Γₚ = Γₚ ▹p⁰ x} = cong₂ _▹p⁰_ []c-id []f-id
|
||||
|
||||
[]c-∘ : {α : Subt Δₜ Ξₜ} {β : Subt Γₜ Δₜ} {Ξₚ : Conp Ξₜ} → Ξₚ [ α ∘ₜ β ]c ≡ (Ξₚ [ α ]c) [ β ]c
|
||||
[]c-∘ {α = α} {β = β} {◇p} = refl
|
||||
[]c-∘ {α = α} {β = β} {Ξₚ ▹p⁰ A} = cong₂ _▹p⁰_ []c-∘ []f-∘
|
||||
|
||||
|
||||
record Sub (Γ : Con) (Δ : Con) : Set₁ where
|
||||
constructor sub
|
||||
field
|
||||
t : Subt (Con.t Γ) (Con.t Δ)
|
||||
p : Subp {Con.t Γ} (Con.p Γ) ((Con.p Δ) [ t ]c)
|
||||
|
||||
-- An order on contexts, where we can only change positions
|
||||
infixr 5 _∈ₚ*_
|
||||
data _∈ₚ*_ : Conp Γₜ → Conp Γₜ → Set₁ where
|
||||
zero∈ₚ* : ◇p ∈ₚ* Γₚ
|
||||
next∈ₚ* : {A : For Δₜ} → PfVar (con Δₜ Δₚ) A → Δₚ' ∈ₚ* Δₚ → (Δₚ' ▹p⁰ A) ∈ₚ* Δₚ
|
||||
-- Allows to grow ∈ₚ* to the right
|
||||
right∈ₚ* :{A : For Δₜ} → Γₚ ∈ₚ* Δₚ → Γₚ ∈ₚ* (Δₚ ▹p⁰ A)
|
||||
right∈ₚ* zero∈ₚ* = zero∈ₚ*
|
||||
right∈ₚ* (next∈ₚ* x h) = next∈ₚ* (pvnext x) (right∈ₚ* h)
|
||||
both∈ₚ* : {A : For Γₜ} → Γₚ ∈ₚ* Δₚ → (Γₚ ▹p⁰ A) ∈ₚ* (Δₚ ▹p⁰ A)
|
||||
both∈ₚ* zero∈ₚ* = next∈ₚ* pvzero zero∈ₚ*
|
||||
both∈ₚ* (next∈ₚ* x h) = next∈ₚ* pvzero (next∈ₚ* (pvnext x) (right∈ₚ* h))
|
||||
refl∈ₚ* : Γₚ ∈ₚ* Γₚ
|
||||
refl∈ₚ* {Γₚ = ◇p} = zero∈ₚ*
|
||||
refl∈ₚ* {Γₚ = Γₚ ▹p⁰ x} = both∈ₚ* refl∈ₚ*
|
||||
|
||||
∈ₚ▹tp : {A : For Δₜ} → PfVar (con Δₜ Δₚ) A → PfVar (con Δₜ Δₚ ▹t) (A [ wkₜσt idₜ ]f)
|
||||
∈ₚ▹tp pvzero = pvzero
|
||||
∈ₚ▹tp (pvnext x) = pvnext (∈ₚ▹tp x)
|
||||
∈ₚ*▹tp : Γₚ ∈ₚ* Δₚ → (Γₚ ▹tp) ∈ₚ* (Δₚ ▹tp)
|
||||
∈ₚ*▹tp zero∈ₚ* = zero∈ₚ*
|
||||
∈ₚ*▹tp (next∈ₚ* x s) = next∈ₚ* (∈ₚ▹tp x) (∈ₚ*▹tp s)
|
||||
|
||||
mon∈ₚ∈ₚ* : {A : For Δₜ} → PfVar (con Δₜ Δₚ') A → Δₚ' ∈ₚ* Δₚ → PfVar (con Δₜ Δₚ) A
|
||||
mon∈ₚ∈ₚ* pvzero (next∈ₚ* x x₁) = x
|
||||
mon∈ₚ∈ₚ* (pvnext s) (next∈ₚ* x x₁) = mon∈ₚ∈ₚ* s x₁
|
||||
|
||||
∈ₚ*→Sub : Δₚ ∈ₚ* Δₚ' → Subp {Δₜ} Δₚ' Δₚ
|
||||
∈ₚ*→Sub zero∈ₚ* = εₚ
|
||||
∈ₚ*→Sub (next∈ₚ* x s) = ∈ₚ*→Sub s ,ₚ var x
|
||||
|
||||
|
||||
wkₚp : {A : For Δₜ} → Δₚ ∈ₚ* Δₚ' → Pf (con Δₜ Δₚ) A → Pf (con Δₜ Δₚ') A
|
||||
wkₚp s (var pv) = var (mon∈ₚ∈ₚ* pv s)
|
||||
wkₚp s (app pf pf₁) = app (wkₚp s pf) (wkₚp s pf₁)
|
||||
wkₚp s (lam {A = A} pf) = lam (wkₚp (both∈ₚ* s) pf)
|
||||
wkₚp s (p∀∀e pf) = p∀∀e (wkₚp s pf)
|
||||
wkₚp s (p∀∀i pf) = p∀∀i (wkₚp (∈ₚ*▹tp s) pf)
|
||||
lliftₚ : {Γₚ : Conp Δₜ} → Δₚ ∈ₚ* Δₚ' → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} Δₚ' Γₚ
|
||||
lliftₚ s εₚ = εₚ
|
||||
lliftₚ s (σₚ ,ₚ pf) = lliftₚ s σₚ ,ₚ wkₚp s pf
|
||||
|
||||
wkₚσt : {Δₜ : Cont} {Δₚ Γₚ : Conp Δₜ}{A : For Δₜ} → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} (Δₚ ▹p⁰ A) Γₚ
|
||||
wkₚσt εₚ = εₚ
|
||||
wkₚσt (σₚ ,ₚ pf) = (wkₚσt σₚ) ,ₚ wkₚp (right∈ₚ* refl∈ₚ*) pf
|
||||
--wkₜσt-wkₜt : {tv : TmVar Γₜ} → {σ : Subt Δₜ Γₜ} → wkₜt (var tv [ σ ]t) ≡ var tv [ wkₜσt σ ]t
|
||||
--wkₜσt-wkₜt {tv = tvzero} {σ = σ ,ₜ x} = refl
|
||||
--wkₜσt-wkₜt {tv = tvnext tv} {σ = σ ,ₜ x} = wkₜσt-wkₜt {tv = tv} {σ = σ}
|
||||
|
||||
-- From a substitution into n variables, we construct a substitution from n+1 variables to n+1 variables which maps it to itself
|
||||
-- i.e. 0 -> 0 and for all i ->(old) σ(i) we get i+1 -> σ(i)+1
|
||||
liftₚσ : {Δₜ : Cont}{Δₚ Γₚ : Conp Δₜ}{A : For Δₜ} → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} (Δₚ ▹p⁰ A) (Γₚ ▹p⁰ A)
|
||||
liftₚσ σ = (wkₚσt σ) ,ₚ (var pvzero)
|
||||
|
||||
idₚ : Subp {Δₜ} Δₚ Δₚ
|
||||
idₚ {Δₚ = ◇p} = εₚ
|
||||
idₚ {Δₚ = Δₚ ▹p⁰ x} = liftₚσ (idₚ {Δₚ = Δₚ})
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
lem7 : {σ : Subt Δₜ Γₜ} → ((Δₚ ▹tp) [ liftₜσ σ ]c) ≡ ((Δₚ [ σ ]c) ▹tp)
|
||||
lem7 {Δₚ = ◇p} = refl
|
||||
lem7 {Δₚ = Δₚ ▹p⁰ A} = cong₂ _▹p⁰_ lem7 (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (≡tran² (≡sym wkₜσt-∘) (cong wkₜσt (≡tran σ-idl (≡sym σ-idr))) (≡sym lem3))) []f-∘)
|
||||
lem8 : {σ : Subt Δₜ Γₜ} {t : Tm Γₜ} → ((wkₜσt σ ∘ₜ (idₜ ,ₜ (t [ σ ]t))) ,ₜ (t [ σ ]t)) ≡ ((idₜ ∘ₜ σ) ,ₜ (t [ σ ]t))
|
||||
lem8 = cong₂ _,ₜ_ (≡tran² wk∘, σ-idr (≡sym σ-idl)) refl
|
||||
|
||||
_[_]pvₜ : {A : For Δₜ} → PfVar (con Δₜ Δₚ) A → (σ : Subt Γₜ Δₜ) → PfVar (con Γₜ (Δₚ [ σ ]c)) (A [ σ ]f)
|
||||
_[_]pₜ : {A : For Δₜ} → Pf (con Δₜ Δₚ) A → (σ : Subt Γₜ Δₜ) → Pf (con Γₜ (Δₚ [ σ ]c)) (A [ σ ]f)
|
||||
pvzero [ σ ]pvₜ = pvzero
|
||||
pvnext pv [ σ ]pvₜ = pvnext (pv [ σ ]pvₜ)
|
||||
var pv [ σ ]pₜ = var (pv [ σ ]pvₜ)
|
||||
app pf pf' [ σ ]pₜ = app (pf [ σ ]pₜ) (pf' [ σ ]pₜ)
|
||||
lam pf [ σ ]pₜ = lam (pf [ σ ]pₜ)
|
||||
_[_]pₜ {Δₚ = Δₚ} {Γₜ = Γₜ} (p∀∀e {A = A} {t = t} pf) σ = substP (λ F → Pf (con Γₜ (Δₚ [ σ ]c)) F) (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (lem8 {t = t})) ([]f-∘)) (p∀∀e {t = t [ σ ]t} (pf [ σ ]pₜ))
|
||||
_[_]pₜ {Γₜ = Γₜ} (p∀∀i pf) σ = p∀∀i (substP (λ Ξₚ → Pf (con (Γₜ ▹t⁰) (Ξₚ)) _) lem7 (pf [ liftₜσ σ ]pₜ))
|
||||
|
||||
_[_]σₚ : Subp {Δₜ} Δₚ Δₚ' → (σ : Subt Γₜ Δₜ) → Subp {Γₜ} (Δₚ [ σ ]c) (Δₚ' [ σ ]c)
|
||||
εₚ [ σₜ ]σₚ = εₚ
|
||||
(σₚ ,ₚ pf) [ σₜ ]σₚ = (σₚ [ σₜ ]σₚ) ,ₚ (pf [ σₜ ]pₜ)
|
||||
|
||||
lem9 : (Δₚ [ wkₜσt idₜ ]c) ≡ (Δₚ ▹tp)
|
||||
lem9 {Δₚ = ◇p} = refl
|
||||
lem9 {Δₚ = Δₚ ▹p⁰ x} = cong₂ _▹p⁰_ lem9 refl
|
||||
wkₜσₚ : Subp {Δₜ} Δₚ' Δₚ → Subp {Δₜ ▹t⁰} (Δₚ' ▹tp) (Δₚ ▹tp)
|
||||
wkₜσₚ εₚ = εₚ
|
||||
wkₜσₚ {Δₜ = Δₜ} (_,ₚ_ {A = A} σₚ pf) = (wkₜσₚ σₚ) ,ₚ substP (λ Ξₚ → Pf (con (Δₜ ▹t⁰) Ξₚ) (A [ wkₜσt idₜ ]f)) lem9 (_[_]pₜ {Γₜ = Δₜ ▹t⁰} pf (wkₜσt idₜ))
|
||||
|
||||
_[_]p : {A : For Δₜ} → Pf (con Δₜ Δₚ) A → (σ : Subp {Δₜ} Δₚ' Δₚ) → Pf (con Δₜ Δₚ') A
|
||||
var pvzero [ σ ,ₚ pf ]p = pf
|
||||
var (pvnext pv) [ σ ,ₚ pf ]p = var pv [ σ ]p
|
||||
app pf pf₁ [ σ ]p = app (pf [ σ ]p) (pf₁ [ σ ]p)
|
||||
lam pf [ σ ]p = lam (pf [ lliftₚ (right∈ₚ* refl∈ₚ*) σ ,ₚ var pvzero ]p)
|
||||
p∀∀e pf [ σ ]p = p∀∀e (pf [ σ ]p)
|
||||
p∀∀i pf [ σ ]p = p∀∀i (pf [ wkₜσₚ σ ]p)
|
||||
|
||||
|
||||
_∘ₚ_ : {Γₚ Δₚ Ξₚ : Conp Δₜ} → Subp {Δₜ} Δₚ Ξₚ → Subp {Δₜ} Γₚ Δₚ → Subp {Δₜ} Γₚ Ξₚ
|
||||
εₚ ∘ₚ β = εₚ
|
||||
(α ,ₚ pf) ∘ₚ β = (α ∘ₚ β) ,ₚ (pf [ β ]p)
|
||||
ε-u : {Γₚ : Conp Γₜ} → {σ : Subp Γₚ ◇p} → σ ≡ εₚ {Δₚ = Γₚ}
|
||||
ε-u {σ = εₚ} = refl
|
||||
lemJ : {Δₜ : Cont}{Δₚ : Conp Δₜ}{A : For Δₜ} → Pf (con Δₜ Δₚ) A → (Pf (con Δₜ (Δₚ [ idₜ ]c)) (A [ idₜ ]f))
|
||||
lemJ {Δₜ}{Δₚ}{A} pf = substP (Pf (con Δₜ (Δₚ [ idₜ ]c))) (≡sym []f-id) (substP (λ Ξₚ → Pf (con Δₜ Ξₚ) A) (≡sym []c-id) pf)
|
||||
[]σₚ-id : {σₚ : Subp {Δₜ} Δₚ Δₚ'} → coe (cong₂ Subp []c-id []c-id) (σₚ [ idₜ ]σₚ) ≡ σₚ
|
||||
[]σₚ-id {σₚ = εₚ} = ε-u
|
||||
[]σₚ-id {Δₜ}{Δₚ}{Δₚ' ▹p⁰ A} {σₚ = σₚ ,ₚ x} = substP (λ ξ → coe (cong₂ Subp []c-id []c-id) (ξ ,ₚ (x [ idₜ ]pₜ)) ≡ (σₚ ,ₚ x)) (≡sym (coeshift ([]σₚ-id)))
|
||||
(≡sym (coeshift {eq = cong₂ Subp (≡sym []c-id) (≡sym []c-id)}
|
||||
(substfpoly'' {A = (Conp Δₜ) × (Conp Δₜ)}{P = λ W F → Subp (proj×₁ W) ((proj×₂ W) ▹p⁰ F)}{R = λ W → Subp (proj×₁ W) (proj×₂ W)}{Q = λ W F → Pf (con Δₜ (proj×₁ W)) F}{α = Δₚ ,× Δₚ'}{ε = A}{eq = ×≡ (≡sym []c-id) (≡sym []c-id)}{eq'' = ≡sym []f-id}{f = λ {W} {F} ξ pf → _,ₚ_ ξ pf}{x = σₚ}{y = x})))
|
||||
[]σₚ-∘ : {Ξₚ Ξₚ' : Conp Ξₜ}{α : Subt Δₜ Ξₜ} {β : Subt Γₜ Δₜ} {σₚ : Subp {Ξₜ} Ξₚ Ξₚ'}
|
||||
--{eq₅ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]c)}
|
||||
→ coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) ((σₚ [ α ]σₚ) [ β ]σₚ) ≡ σₚ [ α ∘ₜ β ]σₚ
|
||||
[]σₚ-∘ {σₚ = εₚ} = ε-u
|
||||
[]σₚ-∘ {Ξₜ}{Δₜ}{Γₜ}{Ξₚ}{Δₚ' ▹p⁰ A}{α}{β}{σₚ = σₚ ,ₚ pf} =
|
||||
substP (λ ξ → coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) (ξ ,ₚ ((pf [ α ]pₜ) [ β ]pₜ)) ≡ ((σₚ [ α ∘ₜ β ]σₚ) ,ₚ (pf [ α ∘ₜ β ]pₜ))) (≡sym (coeshift []σₚ-∘))
|
||||
(≡sym (coeshift {eq = cong₂ Subp []c-∘ []c-∘}
|
||||
(substfpoly''
|
||||
{A = (Conp Γₜ) × (Conp Γₜ)}
|
||||
{P = λ W F → Subp (proj×₁ W) ((proj×₂ W) ▹p⁰ F)}
|
||||
{R = λ W → Subp (proj×₁ W) (proj×₂ W)}
|
||||
{Q = λ W F → Pf (con Γₜ (proj×₁ W)) F}
|
||||
{eq = cong₂ _,×_ []c-∘ []c-∘}
|
||||
{eq'' = []f-∘ {α = β} {β = α} {F = A}}
|
||||
{f = λ {W} {F} ξ pf → _,ₚ_ ξ pf}{x = σₚ [ α ∘ₜ β ]σₚ}{y = pf [ α ∘ₜ β ]pₜ})
|
||||
))
|
||||
wkₚ∘, : {Δₜ : Cont}{Γₚ Δₚ Ξₚ : Conp Δₜ}{α : Subp {Δₜ} Γₚ Δₚ}{β : Subp {Δₜ} Ξₚ Γₚ}{A : For Δₜ}{pf : Pf (con Δₜ Ξₚ) A} → (wkₚσt α) ∘ₚ (β ,ₚ pf) ≡ (α ∘ₚ β)
|
||||
wkₚ∘, {α = εₚ} = refl
|
||||
wkₚ∘, {α = α ,ₚ pf} {β = β} {pf = pf'} = cong (λ ξ → ξ ,ₚ (pf [ β ]p)) wkₚ∘,
|
||||
idlₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → (idₚ {Δₚ = Δₚ}) ∘ₚ σₚ ≡ σₚ
|
||||
idlₚ {Δₚ = ◇p} {εₚ} = refl
|
||||
idlₚ {Δₚ = Δₚ ▹p⁰ pf} {σₚ ,ₚ pf'} = cong (λ ξ → ξ ,ₚ pf') (≡tran wkₚ∘, (idlₚ {σₚ = σₚ}))
|
||||
idrₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → σₚ ∘ₚ (idₚ {Δₚ = Γₚ}) ≡ σₚ
|
||||
idrₚ {σₚ = εₚ} = refl
|
||||
idrₚ {σₚ = σₚ ,ₚ prf} = cong (λ X → X ,ₚ prf) (idrₚ {σₚ = σₚ})
|
||||
wkₚ[] : {σₜ : Subt Γₜ Δₜ} {σₚ : Subp Δₚ Δₚ'} {A : For Δₜ} → (wkₚσt {A = A} σₚ) [ σₜ ]σₚ ≡ wkₚσt (σₚ [ σₜ ]σₚ)
|
||||
wkₚ[] {σₚ = εₚ} = refl
|
||||
wkₚ[] {σₚ = σₚ ,ₚ x} = cong (λ ξ → ξ ,ₚ _) (wkₚ[] {σₚ = σₚ})
|
||||
idₚ[] : {σₜ : Subt Γₜ Δₜ} → ((idₚ {Δₜ} {Δₚ}) [ σₜ ]σₚ) ≡ idₚ {Γₜ} {Δₚ [ σₜ ]c}
|
||||
idₚ[] {Δₚ = ◇p} = refl
|
||||
idₚ[] {Δₚ = Δₚ ▹p⁰ A} = cong (λ ξ → ξ ,ₚ var pvzero) (≡tran wkₚ[] (cong wkₚσt idₚ[]))
|
||||
|
||||
|
||||
id : Sub Γ Γ
|
||||
id {Γ} = sub idₜ (subst (Subp _) (≡sym []c-id) idₚ)
|
||||
_∘_ : Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
|
||||
sub αₜ αₚ ∘ sub βₜ βₚ = sub (αₜ ∘ₜ βₜ) (subst (Subp _) (≡sym []c-∘) (αₚ [ βₜ ]σₚ) ∘ₚ βₚ)
|
||||
idl : {Γ Δ : Con} {σ : Sub Γ Δ} → (id {Δ}) ∘ σ ≡ σ
|
||||
idl {Δ = Δ} {σ = sub σₜ σₚ} = cong₂' sub σ-idl (≡tran (substfpoly {α = ((Con.p Δ) [ idₜ ∘ₜ σₜ ]c)} {β = ((Con.p Δ) [ σₜ ]c)} {eq = cong (λ ξ → Con.p Δ [ ξ ]c) σ-idl} {f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ σₚ}) (≡tran (cong₂ _∘ₚ_ (≡tran³ coecoe-coe (substfpoly {eq = []c-id} {f = λ {Ξₚ} ξ → _[_]σₚ {Δₚ = Con.p Δ} {Δₚ' = Ξₚ} ξ σₜ}) (cong (λ ξ → ξ [ σₜ ]σₚ) coeaba) idₚ[]) refl) idlₚ))
|
||||
lemK : {Γ Δ : Con}{σₜ : Subt (Con.t Γ) (Con.t Δ)}{σₚ : Subp (Con.p Γ [ idₜ ]c) ((Con.p Δ [ σₜ ]c)[ idₜ ]c)}
|
||||
{eq1 : Subp (Con.p Γ) ((Con.p Δ [ σₜ ]c) [ idₜ ]c) ≡ Subp (Con.p Γ) (Con.p Δ [ σₜ ]c)}
|
||||
{eq2 : Subp (Con.p Γ) (Con.p Γ) ≡ Subp (Con.p Γ) (Con.p Γ [ idₜ ]c)}
|
||||
{eq3 : Subp (Con.p Γ [ idₜ ]c) ((Con.p Δ [ σₜ ]c)[ idₜ ]c) ≡ Subp (Con.p Γ) (Con.p Δ [ σₜ ]c)}
|
||||
→ coe eq1 (σₚ ∘ₚ coe eq2 idₚ)
|
||||
≡ (coe eq3 σₚ ∘ₚ idₚ)
|
||||
lemK {Γ}{Δ}{σₚ = σₚ}{eq1}{eq2}{eq3} = substP (λ X → coe eq1 (X ∘ₚ coe eq2 idₚ) ≡ (coe eq3 σₚ ∘ₚ idₚ)) (coeaba {eq1 = eq3}{eq2 = ≡sym eq3}) (coep∘ {p = λ {Γₚ}{Δₚ}{Ξₚ} x y → _∘ₚ_ {Δₚ = Γₚ} x y} {eq1 = refl}{eq2 = ≡sym []c-id}{eq3 = ≡sym []c-id})
|
||||
idr : {Γ Δ : Con} {σ : Sub Γ Δ} → σ ∘ (id {Γ}) ≡ σ
|
||||
idr {Γ} {Δ} {σ = sub σₜ σₚ} = cong₂' sub σ-idr (≡tran⁴ (cong (coe _) (≡sym (substfpoly {eq = ≡sym ([]c-∘ {α = σₜ} {β = idₜ}{Ξₚ = Con.p Δ})} {f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ (coe (cong (Subp (Con.p Γ)) (≡sym []c-id)) idₚ)} {x = σₚ [ idₜ ]σₚ}))) coecoe-coe lemK idrₚ []σₚ-id)
|
||||
∘ₚ-ass : {Γₚ Δₚ Ξₚ Ψₚ : Conp Γₜ}{αₚ : Subp Γₚ Δₚ}{βₚ : Subp Δₚ Ξₚ}{γₚ : Subp Ξₚ Ψₚ} → (γₚ ∘ₚ βₚ) ∘ₚ αₚ ≡ γₚ ∘ₚ (βₚ ∘ₚ αₚ)
|
||||
∘ₚ-ass {γₚ = εₚ} = refl
|
||||
∘ₚ-ass {αₚ = αₚ} {βₚ} {γₚ ,ₚ x} = cong (λ ξ → ξ ,ₚ (x [ βₚ ∘ₚ αₚ ]p)) ∘ₚ-ass
|
||||
|
||||
lemG' :
|
||||
{Γₜ Δₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Δₜ}{Ψₚ : Conp Δₜ}
|
||||
{αₜ : Subt Γₜ Δₜ}{γₚ : Subp Ξₚ Ψₚ}{βₚ : Subp Δₚ Ξₚ}{αₚ : Subp Γₚ (Δₚ [ αₜ ]c)}
|
||||
→ ((γₚ ∘ₚ βₚ) [ αₜ ]σₚ) ∘ₚ αₚ ≡ (γₚ [ αₜ ]σₚ) ∘ₚ ((βₚ [ αₜ ]σₚ) ∘ₚ αₚ)
|
||||
lemG' {γₚ = εₚ} = refl
|
||||
lemG' {αₜ = αₜ}{γₚ ,ₚ x}{βₚ}{αₚ} = cong (λ ξ → ξ ,ₚ (((x [ βₚ ]p) [ αₜ ]pₜ) [ αₚ ]p)) (lemG' {γₚ = γₚ})
|
||||
lemG :
|
||||
{Γₜ Δₜ Ξₜ Ψₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Ξₜ}{Ψₚ : Conp Ψₜ}
|
||||
{αₜ : Subt Γₜ Δₜ}{βₜ : Subt Δₜ Ξₜ}{γₜ : Subt Ξₜ Ψₜ}{γₚ : Subp Ξₚ (Ψₚ [ γₜ ]c)}{βₚ : Subp Δₚ (Ξₚ [ βₜ ]c)}{αₚ : Subp Γₚ (Δₚ [ αₜ ]c)}
|
||||
{eq₁ : Subp Γₚ (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c) ≡ Subp Γₚ (Ψₚ [ γₜ ∘ₜ (βₜ ∘ₜ αₜ) ]c)}
|
||||
{eq₂ : Subp (Δₚ [ αₜ ]c) ((Ψₚ [ γₜ ∘ₜ βₜ ]c) [ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c)}
|
||||
{eq₃ : Subp (Ξₚ [ βₜ ∘ₜ αₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ∘ₜ αₜ ]c) ≡ Subp (Ξₚ [ βₜ ∘ₜ αₜ ]c) (Ψₚ [ γₜ ∘ₜ (βₜ ∘ₜ αₜ) ]c)}
|
||||
{eq₄ : Subp (Δₚ [ αₜ ]c) ((Ξₚ [ βₜ ]c) [ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ξₚ [ βₜ ∘ₜ αₜ ]c)}
|
||||
{eq₅ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]c)}
|
||||
→ coe eq₁ ((coe eq₂ (((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ)) ∘ₚ αₚ) ≡ (coe eq₃ (γₚ [ βₜ ∘ₜ αₜ ]σₚ)) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ)
|
||||
lemG {Γₜ}{Δₜ}{Ξₜ}{Ψₜ}{Γₚ}{Δₚ}{Ξₚ}{Ψₚ}{αₜ = αₜ}{βₜ}{γₜ}{γₚ}{βₚ}{αₚ}{eq₁}{eq₂}{eq₃}{eq₄}{eq₅} =
|
||||
substP (λ X → coe eq₁ ((coe eq₂ (((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ)) ∘ₚ αₚ) ≡ (coe eq₃ X) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ)) []σₚ-∘ (
|
||||
≡tran⁵
|
||||
(cong (coe eq₁) (≡tran (
|
||||
≡sym (substfpoly
|
||||
{R = λ X → Subp (Δₚ [ αₜ ]c) X}
|
||||
{eq = ≡sym []c-∘}
|
||||
{f = λ ξ → ξ ∘ₚ αₚ}
|
||||
{x = ((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ}))
|
||||
(cong (coe (cong (λ z → Subp Γₚ z) (≡sym []c-∘)))
|
||||
(≡sym (substfpoly
|
||||
{R = λ X → Subp (Ξₚ [ βₜ ]c) X}
|
||||
{eq = ≡sym []c-∘}
|
||||
{f = λ ξ → ((ξ ∘ₚ βₚ) [ αₜ ]σₚ) ∘ₚ αₚ}
|
||||
{x = γₚ [ βₜ ]σₚ}
|
||||
)))
|
||||
))
|
||||
(≡tran coecoe-coe coecoe-coe)
|
||||
(cong (coe (≡tran (cong (λ z → Subp Γₚ (z [ αₜ ]c)) (≡sym []c-∘)) (≡tran (cong (λ z → Subp Γₚ z) (≡sym []c-∘)) eq₁))) lemG')
|
||||
(≡sym coecoe-coe)
|
||||
(cong (coe (cong (λ z → Subp Γₚ z) (≡sym []c-∘))) (substppoly
|
||||
{A = (Conp Γₜ) × (Conp Γₜ)}
|
||||
{R = λ W → Subp (proj×₁ W) (proj×₂ W)}
|
||||
{Q = λ W → Subp (Δₚ [ αₜ ]c) (proj×₁ W)}
|
||||
{eq = ×≡ (≡sym []c-∘) (≡sym []c-∘)}
|
||||
{f = λ x y → x ∘ₚ (y ∘ₚ αₚ)}
|
||||
{x = (γₚ [ βₜ ]σₚ) [ αₜ ]σₚ}
|
||||
{y = βₚ [ αₜ ]σₚ}
|
||||
))(substfpoly
|
||||
{R = λ X → Subp (Ξₚ [ βₜ ∘ₜ αₜ ]c) X}
|
||||
{eq = ≡sym []c-∘}
|
||||
{f = λ {τ} ξ → (ξ ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ))}
|
||||
{x = (coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) ((γₚ [ βₜ ]σₚ) [ αₜ ]σₚ))}
|
||||
))
|
||||
∘-ass : {Γ Δ Ξ Ψ : Con}{α : Sub Γ Δ}{β : Sub Δ Ξ}{γ : Sub Ξ Ψ} → (γ ∘ β) ∘ α ≡ γ ∘ (β ∘ α)
|
||||
∘-ass {Γ}{Δ}{Ξ}{Ψ}{α = sub αₜ αₚ} {β = sub βₜ βₚ} {γ = sub γₜ γₚ} = cong₂' sub ∘ₜ-ass lemG
|
||||
|
||||
-- SUB-ization
|
||||
|
||||
lemA : {σₜ : Subt Γₜ Δₜ}{t : Tm Γₜ} → (Γₚ ▹tp) [ σₜ ,ₜ t ]c ≡ Γₚ [ σₜ ]c
|
||||
lemA {Γₚ = ◇p} = refl
|
||||
lemA {Γₚ = Γₚ ▹p⁰ t} = cong₂ _▹p⁰_ lemA (≡tran (≡sym []f-∘) (cong (λ σ → t [ σ ]f) (≡tran wk∘, σ-idl)))
|
||||
πₜ¹* : {Γ Δ : Con} → Sub Δ (Γ ▹t) → Sub Δ Γ
|
||||
πₜ¹* (sub (σₜ ,ₜ t) σₚ) = sub σₜ (subst (Subp _) lemA σₚ)
|
||||
πₜ²* : {Γ Δ : Con} → Sub Δ (Γ ▹t) → Tm (Con.t Δ)
|
||||
πₜ²* (sub (σₜ ,ₜ t) σₚ) = t
|
||||
_,ₜ*_ : {Γ Δ : Con} → Sub Δ Γ → Tm (Con.t Δ) → Sub Δ (Γ ▹t)
|
||||
(sub σₜ σₚ) ,ₜ* t = sub (σₜ ,ₜ t) (subst (Subp _) (≡sym lemA) σₚ)
|
||||
πₜ²∘,ₜ* : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm (Con.t Δ)} → πₜ²* (σ ,ₜ* t) ≡ t
|
||||
πₜ²∘,ₜ* = refl
|
||||
πₜ¹∘,ₜ* : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm (Con.t Δ)} → πₜ¹* (σ ,ₜ* t) ≡ σ
|
||||
πₜ¹∘,ₜ* {Γ}{Δ}{σ}{t} = cong (sub (Sub.t σ)) coeaba
|
||||
,ₜ∘πₜ* : {Γ Δ : Con} → {σ : Sub Δ (Γ ▹t)} → (πₜ¹* σ) ,ₜ* (πₜ²* σ) ≡ σ
|
||||
,ₜ∘πₜ* {Γ} {Δ} {sub (σₜ ,ₜ t) σₚ} = cong (sub (σₜ ,ₜ t)) coeaba
|
||||
,ₜ∘* : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{t : Tm (Con.t Γ)} → (σ ,ₜ* t) ∘ δ ≡ (σ ∘ δ) ,ₜ* (t [ Sub.t δ ]t)
|
||||
lemE : {σₜ : Subt Γₜ Ξₜ}{σₚ : Subp Γₚ (Ξₚ [ σₜ ]c)} {δₜ : Subt Δₜ Γₜ} → (coe _ σₚ [ δₜ ]σₚ) ≡ coe _ (σₚ [ δₜ ]σₚ)
|
||||
lemE {δₜ = δₜ} = coecong {eq = refl} {eq' = refl} (λ ξₚ → ξₚ [ δₜ ]σₚ)
|
||||
lemF : {Γα Γβ : Conp Δₜ}{δₜ : Subt Δₜ Γₜ}{δₚ : Subp Δₚ (Γₚ [ δₜ ]c)} → (eq : Γβ ≡ Γα) → (ξ : Subp (Γₚ [ δₜ ]c) Γβ) → coe (cong (Subp Δₚ) eq) (ξ ∘ₚ δₚ) ≡ coe (cong (Subp _) eq) ξ ∘ₚ δₚ
|
||||
lemF refl ξ = ≡tran coerefl (cong₂ _∘ₚ_ (≡sym coerefl) refl)
|
||||
--lemG : {Γα Γβ : Conp Δₜ}{σₜ : Subt Γₜ Ξₜ}{δₜ : Subt Δₜ Γₜ} → (eq : Γβ ≡ Γα) → (ξ : Subp Γₚ (Ξₚ [ σₜ ]c)) → coe refl (ξ [ δₜ ]σₚ) ≡ (coe refl ξ) [ δₜ ]σₚ
|
||||
--lemG eq ε= {!!}
|
||||
substf : {ℓ ℓ' : Level}{A : Set ℓ}{P : A → Set ℓ'}{Q : A → Set ℓ'}{a b c d : A}{eqa : a ≡ a}{eqb : b ≡ b}{eqcd : c ≡ d}{eqdc : d ≡ c}{f : P a → P b}{g : P b → Q c}{x : P a} → g (subst P eqb (f (subst P eqa x))) ≡ subst Q eqdc (subst Q eqcd (g (f x)))
|
||||
substf {P = P} {Q = Q} {eqcd = refl} {f = f} {g = g} = ≡tran² (cong g (≡tran (substrefl {P = P} {e = refl}) (cong f (substrefl {P = P} {e = refl})))) (≡sym (substrefl {P = Q} {e = refl})) (≡sym (substrefl {P = Q} {e = refl}))
|
||||
|
||||
,ₜ∘* {Γ} {Δ} {Ξ} {sub σₜ σₚ} {sub δₜ δₚ} {t} = cong (sub ((σₜ ∘ₜ δₜ) ,ₜ (t [ δₜ ]t)))
|
||||
(substfgpoly
|
||||
{P = Subp {Con.t Δ} (Con.p Δ)}
|
||||
{Q = Subp {Con.t Δ} ((Con.p Γ) [ δₜ ]c)}
|
||||
{R = Subp {Con.t Γ} (Con.p Γ)}
|
||||
{F = λ X → X [ δₜ ]c}
|
||||
{eq₁ = ≡sym lemA}
|
||||
{eq₂ = ≡sym []c-∘}
|
||||
{eq₃ = ≡sym []c-∘}
|
||||
{eq₄ = ≡sym lemA}
|
||||
{g = λ σₚ → σₚ ∘ₚ δₚ}
|
||||
{f = λ σₚ → σₚ [ δₜ ]σₚ}
|
||||
{x = σₚ})
|
||||
|
||||
πₚ¹* : {Γ Δ : Con} {A : For (Con.t Γ)} → Sub Δ (Γ ▹p A) → Sub Δ Γ
|
||||
πₚ¹* (sub σₜ (σₚ ,ₚ pf)) = sub σₜ σₚ
|
||||
πₚ² : {Γ Δ : Con} {F : For (Con.t Γ)} (σ : Sub Δ (Γ ▹p F)) → Pf Δ (F [ Sub.t (πₚ¹* σ) ]f)
|
||||
πₚ² (sub σₜ (σₚ ,ₚ pf)) = pf
|
||||
_,ₚ*_ : {Γ Δ : Con} {F : For (Con.t Γ)} (σ : Sub Δ Γ) → Pf Δ (F [ Sub.t σ ]f) → Sub Δ (Γ ▹p F)
|
||||
sub σₜ σₚ ,ₚ* pf = sub σₜ (σₚ ,ₚ pf)
|
||||
|
||||
,ₚ∘πₚ : {Γ Δ : Con} → {F : For (Con.t Γ)} → {σ : Sub Δ (Γ ▹p F)} → (πₚ¹* σ) ,ₚ* (πₚ² σ) ≡ σ
|
||||
,ₚ∘πₚ {σ = sub σₜ (σₚ ,ₚ p)} = refl
|
||||
,ₚ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{F : For (Con.t Ξ)}{prf : Pf Γ (F [ Sub.t σ ]f)}
|
||||
→ (σ ,ₚ* prf) ∘ δ ≡ (σ ∘ δ) ,ₚ* (substP (λ F → Pf Δ F) (≡sym []f-∘) ((prf [ Sub.t δ ]pₜ) [ Sub.p δ ]p))
|
||||
,ₚ∘ {Γ}{Δ}{Ξ}{σ = sub σₜ σₚ} {sub δₜ δₚ} {F = A} {prf} = cong (sub (σₜ ∘ₜ δₜ)) (cong (λ ξ → ξ ∘ₚ δₚ)
|
||||
(substfpoly⁴ {P = λ W → Subp (Con.p Γ [ δₜ ]c) ((proj×₁ W) ▹p⁰ (proj×₂ W))}{R = λ W → Subp (Con.p Γ [ δₜ ]c) (proj×₁ W)}{Q = λ W → Pf (con (Con.t Δ) (Con.p Γ [ δₜ ]c)) (proj×₂ W)}{α = ((Con.p Ξ [ σₜ ]c) [ δₜ ]c) ,× ((A [ σₜ ]f) [ δₜ ]f)}{eq = cong₂ _,×_ (≡sym []c-∘) (≡sym []f-∘)}{f = λ ξ p → ξ ,ₚ p} {x = σₚ [ δₜ ]σₚ}{y = prf [ δₜ ]pₜ})) --
|
||||
|
||||
--_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹t)
|
||||
--πₜ²∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ² (σ ,ₜ t) ≡ t
|
||||
--πₜ¹∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ¹ (σ ,ₜ t) ≡ σ
|
||||
--,ₜ∘πₜ : {Γ Δ : Con} → {σ : Sub Δ (Γ ▹ₜ)} → (πₜ¹ σ) ,ₜ (πₜ² σ) ≡ σ
|
||||
--,ₜ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{t : Tm Γ} → (σ ,ₜ t) ∘ δ ≡ (σ ∘ δ) ,ₜ (t [ δ ]t)
|
||||
|
||||
-- lemB : ∀{ℓ}{A : Set ℓ}{ℓ'}{P : A → Set ℓ'}{a a' : A}{e : a ≡ a'}{p : P a}{p' : P a'} → p' ≡ p → subst P e p' ≡ p
|
||||
|
||||
lemC : {σₜ : Subt Δₜ Γₜ}{t : Tm Δₜ} → (Γₚ ▹tp) [ σₜ ,ₜ t ]c ≡ Γₚ [ σₜ ]c
|
||||
lemC {Γₚ = ◇p} = refl
|
||||
lemC {Γₚ = Γₚ ▹p⁰ x} = cong₂ _▹p⁰_ lemC (≡tran (≡sym []f-∘) (cong (λ σ → x [ σ ]f) (≡tran wk∘, σ-idl)))
|
||||
|
||||
lemD : {A : For (Con.t Γ)}{σ : Sub Δ (Γ ▹p A)} → Sub.t (πₚ¹* σ) ≡ Sub.t σ
|
||||
lemD {σ = sub σₜ (σₚ ,ₚ pf)} = refl
|
||||
|
||||
|
||||
imod : FFOL {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc lzero}
|
||||
imod = record
|
||||
{ Con = Con
|
||||
; Sub = Sub
|
||||
; _∘_ = _∘_
|
||||
; ∘-ass = ∘-ass
|
||||
; id = id
|
||||
; idl = idl
|
||||
; idr = idr
|
||||
; ◇ = ◇
|
||||
; ε = sub εₜ εₚ
|
||||
; ε-u = cong₂' sub εₜ-u ε-u
|
||||
; Tm = λ Γ → Tm (Con.t Γ)
|
||||
; _[_]t = λ t σ → t [ Sub.t σ ]t
|
||||
; []t-id = []t-id
|
||||
; []t-∘ = λ {Γ}{Δ}{Ξ}{α}{β}{t} → []t-∘ {α = Sub.t α} {β = Sub.t β} {t = t}
|
||||
; _▹ₜ = _▹t
|
||||
; πₜ¹ = πₜ¹*
|
||||
; πₜ² = πₜ²*
|
||||
; _,ₜ_ = _,ₜ*_
|
||||
; πₜ²∘,ₜ = refl
|
||||
; πₜ¹∘,ₜ = πₜ¹∘,ₜ*
|
||||
; ,ₜ∘πₜ = ,ₜ∘πₜ*
|
||||
; ,ₜ∘ = ,ₜ∘*
|
||||
; For = λ Γ → For (Con.t Γ)
|
||||
; _[_]f = λ A σ → A [ Sub.t σ ]f
|
||||
; []f-id = []f-id
|
||||
; []f-∘ = []f-∘
|
||||
; R = r
|
||||
; R[] = refl
|
||||
; _⊢_ = Pf
|
||||
; _[_]p = λ pf σ → (pf [ Sub.t σ ]pₜ) [ Sub.p σ ]p
|
||||
; _▹ₚ_ = _▹p_
|
||||
; πₚ¹ = πₚ¹*
|
||||
; πₚ² = πₚ²
|
||||
; _,ₚ_ = _,ₚ*_
|
||||
; ,ₚ∘πₚ = ,ₚ∘πₚ
|
||||
; πₚ¹∘,ₚ = refl
|
||||
; ,ₚ∘ = λ {Γ}{Δ}{Ξ}{σ}{δ}{F}{prf} → ,ₚ∘ {prf = prf}
|
||||
; _⇒_ = _⇒_
|
||||
; []f-⇒ = refl
|
||||
; ∀∀ = ∀∀
|
||||
; []f-∀∀ = []f-∀∀
|
||||
; lam = λ {Γ}{F}{G} pf → substP (λ H → Pf Γ (F ⇒ H)) (≡tran (cong (_[_]f G) (lemD {σ = id})) []f-id) (lam pf)
|
||||
; app = app
|
||||
; ∀i = p∀∀i
|
||||
; ∀e = λ {Γ} {F} pf {t} → p∀∀e pf
|
||||
}
|
||||
|
||||
319
FinitaryFirstOrderLogic.agda
Normal file
319
FinitaryFirstOrderLogic.agda
Normal file
@ -0,0 +1,319 @@
|
||||
{-# OPTIONS --prop --rewriting #-}
|
||||
|
||||
open import PropUtil
|
||||
|
||||
module FinitaryFirstOrderLogic where
|
||||
|
||||
open import Agda.Primitive
|
||||
open import ListUtil
|
||||
|
||||
variable
|
||||
ℓ¹ ℓ² ℓ³ ℓ⁴ ℓ⁵ : Level
|
||||
|
||||
record FFOL : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴ ⊔ ℓ⁵)) where
|
||||
infixr 10 _∘_
|
||||
infixr 5 _⊢_
|
||||
field
|
||||
Con : Set ℓ¹
|
||||
Sub : Con → Con → Set ℓ⁵ -- It makes a category
|
||||
_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
|
||||
∘-ass : {Γ Δ Ξ Ψ : Con}{α : Sub Γ Δ}{β : Sub Δ Ξ}{γ : Sub Ξ Ψ} → (γ ∘ β) ∘ α ≡ γ ∘ (β ∘ α)
|
||||
id : {Γ : Con} → Sub Γ Γ
|
||||
idl : {Γ Δ : Con} {σ : Sub Γ Δ} → (id {Δ}) ∘ σ ≡ σ
|
||||
idr : {Γ Δ : Con} {σ : Sub Γ Δ} → σ ∘ (id {Γ}) ≡ σ
|
||||
◇ : Con -- The initial object of the category
|
||||
ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
|
||||
ε-u : {Γ : Con} → {σ : Sub Γ ◇} → σ ≡ ε {Γ}
|
||||
|
||||
-- Functor Con → Set called Tm
|
||||
Tm : Con → Set ℓ²
|
||||
_[_]t : {Γ Δ : Con} → Tm Γ → Sub Δ Γ → Tm Δ -- The functor's action on morphisms
|
||||
[]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
|
||||
|
||||
-- Tm : Set+
|
||||
_▹ₜ : Con → Con
|
||||
πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ
|
||||
πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ
|
||||
_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ)
|
||||
πₜ²∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ² (σ ,ₜ t) ≡ t
|
||||
πₜ¹∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ¹ (σ ,ₜ t) ≡ σ
|
||||
,ₜ∘πₜ : {Γ Δ : Con} → {σ : Sub Δ (Γ ▹ₜ)} → (πₜ¹ σ) ,ₜ (πₜ² σ) ≡ σ
|
||||
,ₜ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{t : Tm Γ} → (σ ,ₜ t) ∘ δ ≡ (σ ∘ δ) ,ₜ (t [ δ ]t)
|
||||
|
||||
-- Functor Con → Set called For
|
||||
For : Con → Set ℓ³
|
||||
_[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ -- The functor's action on morphisms
|
||||
[]f-id : {Γ : Con} → {F : For Γ} → F [ id {Γ} ]f ≡ F
|
||||
[]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
|
||||
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
|
||||
_[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms
|
||||
-- Equalities below are useless because Γ ⊢ F is in prop
|
||||
-- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf
|
||||
-- []p-∘ : {Γ Δ Ξ : Con} → {α : Sub Ξ Δ} → {β : Sub Δ Γ} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ α ∘ β ]p ≡ (prf [ β ]p) [ α ]p
|
||||
|
||||
-- → Prop⁺
|
||||
_▹ₚ_ : (Γ : Con) → For Γ → Con
|
||||
πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
|
||||
πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f)
|
||||
_,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F)
|
||||
-- Equalities below are useless because Γ ⊢ F is in Prop
|
||||
,ₚ∘πₚ : {Γ Δ : Con} → {F : For Γ} → {σ : Sub Δ (Γ ▹ₚ F)} → (πₚ¹ σ) ,ₚ (πₚ² σ) ≡ σ
|
||||
πₚ¹∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For Γ} → {prf : Δ ⊢ (F [ σ ]f)} → πₚ¹ (σ ,ₚ prf) ≡ σ
|
||||
-- πₚ²∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For Γ} → {prf : Δ ⊢ (F [ σ ]f)} → πₚ² (σ ,ₚ prf) ≡ prf
|
||||
,ₚ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{F : For Ξ}{prf : Γ ⊢ (F [ σ ]f)} → (σ ,ₚ prf) ∘ δ ≡ (σ ∘ δ) ,ₚ (substP (λ F → Δ ⊢ F) (≡sym []f-∘) (prf [ δ ]p))
|
||||
|
||||
|
||||
-- Implication
|
||||
_⇒_ : {Γ : Con} → For Γ → For Γ → For Γ
|
||||
[]f-⇒ : {Γ Δ : Con} → {F G : For Γ} → {σ : Sub Δ Γ} → (F ⇒ G) [ σ ]f ≡ (F [ σ ]f) ⇒ (G [ σ ]f)
|
||||
|
||||
-- Forall
|
||||
∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ
|
||||
[]f-∀∀ : {Γ Δ : Con} → {F : For (Γ ▹ₜ)} → {σ : Sub Δ Γ} → (∀∀ F) [ σ ]f ≡ (∀∀ (F [ (σ ∘ πₜ¹ id) ,ₜ πₜ² id ]f))
|
||||
|
||||
-- Lam & App
|
||||
lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G)
|
||||
app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G
|
||||
-- Again, we don't write the _[_]p equalities as everything is in Prop
|
||||
|
||||
-- ∀i and ∀e
|
||||
∀i : {Γ : Con} → {F : For (Γ ▹ₜ)} → (Γ ▹ₜ) ⊢ F → Γ ⊢ (∀∀ F)
|
||||
∀e : {Γ : Con} → {F : For (Γ ▹ₜ)} → Γ ⊢ (∀∀ F) → {t : Tm Γ} → Γ ⊢ ( F [(id {Γ}) ,ₜ t ]f)
|
||||
|
||||
|
||||
-- Examples
|
||||
-- Proof utils
|
||||
forall-in : {Γ Δ : Con} {σ : Sub Γ Δ} {A : For (Δ ▹ₜ)} → Γ ⊢ ∀∀ (A [ (σ ∘ πₜ¹ id) ,ₜ πₜ² id ]f) → Γ ⊢ (∀∀ A [ σ ]f)
|
||||
forall-in {Γ = Γ} f = substP (λ F → Γ ⊢ F) (≡sym ([]f-∀∀)) f
|
||||
wkₜ : {Γ : Con} → Sub (Γ ▹ₜ) Γ
|
||||
wkₜ = πₜ¹ id
|
||||
0ₜ : {Γ : Con} → Tm (Γ ▹ₜ)
|
||||
0ₜ = πₜ² id
|
||||
1ₜ : {Γ : Con} → Tm (Γ ▹ₜ ▹ₜ)
|
||||
1ₜ = πₜ² (πₜ¹ id)
|
||||
wkₚ : {Γ : Con} {A : For Γ} → Sub (Γ ▹ₚ A) Γ
|
||||
wkₚ = πₚ¹ id
|
||||
0ₚ : {Γ : Con} {A : For Γ} → Γ ▹ₚ A ⊢ A [ πₚ¹ id ]f
|
||||
0ₚ = πₚ² id
|
||||
|
||||
-- Examples
|
||||
ex0 : {A : For ◇} → ◇ ⊢ (A ⇒ A)
|
||||
ex0 {A = A} = lam 0ₚ
|
||||
{-
|
||||
ex1 : {A : For (◇ ▹ₜ)} → ◇ ⊢ ((∀∀ A) ⇒ (∀∀ A))
|
||||
-- πₚ¹ id is adding an unused variable (syntax's llift)
|
||||
ex1 {A = A} = lam (forall-in (∀i (substP (λ σ → ((◇ ▹ₚ ∀∀ A) ▹ₜ) ⊢ (A [ σ ]f)) {!!} {!!})))
|
||||
-- (∀ x ∀ y . A(y,x)) ⇒ ∀ x ∀ y . A(x,y)
|
||||
-- translation is (∀ ∀ A(0,1)) => (∀ ∀ A(1,0))
|
||||
ex1' : {A : For (◇ ▹ₜ ▹ₜ)} → ◇ ⊢ ((∀∀ (∀∀ A)) ⇒ ∀∀ (∀∀ ( A [ (ε ,ₜ 0ₜ) ,ₜ 1ₜ ]f)))
|
||||
ex1' = {!!}
|
||||
-- (A ⇒ ∀ x . B(x)) ⇒ ∀ x . A ⇒ B(x)
|
||||
ex2 : {A : For ◇} → {B : For (◇ ▹ₜ)} → ◇ ⊢ ((A ⇒ (∀∀ B)) ⇒ (∀∀ ((A [ wkₜ ]f) ⇒ B)))
|
||||
ex2 = {!!}
|
||||
-- ∀ x y . A(x,y) ⇒ ∀ x . A(x,x)
|
||||
-- For simplicity, I swiched positions of parameters of A (somehow...)
|
||||
ex3 : {A : For (◇ ▹ₜ ▹ₜ)} → ◇ ⊢ ((∀∀ (∀∀ A)) ⇒ (∀∀ (A [ id ,ₜ 0ₜ ]f)))
|
||||
ex3 = {!!}
|
||||
-- ∀ x . A (x) ⇒ ∀ x y . A(x)
|
||||
ex4 : {A : For (◇ ▹ₜ)} → ◇ ⊢ ((∀∀ A) ⇒ (∀∀ (∀∀ (A [ ε ,ₜ 1ₜ ]f))))
|
||||
ex4 = {!!}
|
||||
-- (((∀ x . A (x)) ⇒ B)⇒ B) ⇒ ∀ x . ((A (x) ⇒ B) ⇒ B)
|
||||
ex5 : {A : For (◇ ▹ₜ)} → {B : For ◇} → ◇ ⊢ ((((∀∀ A) ⇒ B) ⇒ B) ⇒ (∀∀ ((A ⇒ (B [ wkₜ ]f)) ⇒ (B [ wkₜ ]f))))
|
||||
ex5 = {!!}
|
||||
-}
|
||||
|
||||
record Tarski : Set₁ where
|
||||
field
|
||||
TM : Set
|
||||
REL : TM → TM → Prop
|
||||
infixr 10 _∘_
|
||||
Con = Set
|
||||
Sub : Con → Con → Set
|
||||
Sub Γ Δ = (Γ → Δ) -- It makes a posetal category
|
||||
_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
|
||||
f ∘ g = λ x → f (g x)
|
||||
id : {Γ : Con} → Sub Γ Γ
|
||||
id = λ x → x
|
||||
ε : {Γ : Con} → Sub Γ ⊤ₛ -- The morphism from the initial to any object
|
||||
ε Γ = ttₛ
|
||||
ε-u : {Γ : Con} → {σ : Sub Γ ⊤ₛ} → σ ≡ ε {Γ}
|
||||
ε-u = refl
|
||||
|
||||
-- Functor Con → Set called Tm
|
||||
Tm : Con → Set
|
||||
Tm Γ = Γ → TM
|
||||
_[_]t : {Γ Δ : Con} → Tm Γ → Sub Δ Γ → Tm Δ -- The functor's action on morphisms
|
||||
t [ σ ]t = λ γ → t (σ γ)
|
||||
[]t-id : {Γ : Con} → {x : Tm Γ} → x [ id {Γ} ]t ≡ x
|
||||
[]t-id = refl
|
||||
[]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
|
||||
[]t-∘ {α = α} {β} {t} = refl {_} {_} {λ z → t (β (α z))}
|
||||
|
||||
_[_]tz : {Γ Δ : Con} → {n : Nat} → Array (Tm Γ) n → Sub Δ Γ → Array (Tm Δ) n
|
||||
tz [ σ ]tz = map (λ s → s [ σ ]t) tz
|
||||
[]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
|
||||
[]tz-∘ {tz = zero} = refl
|
||||
[]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
|
||||
[]tz-id : {Γ : Con} → {n : Nat} → {tz : Array (Tm Γ) n} → tz [ id ]tz ≡ tz
|
||||
[]tz-id {tz = zero} = refl
|
||||
[]tz-id {tz = next x tz} = substP (λ tz' → next x tz' ≡ next x tz) (≡sym ([]tz-id {tz = tz})) refl
|
||||
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
|
||||
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
|
||||
|
||||
-- Tm⁺
|
||||
_▹ₜ : Con → Con
|
||||
Γ ▹ₜ = Γ × TM
|
||||
πₜ¹ : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Sub Δ Γ
|
||||
πₜ¹ σ = λ x → proj×₁ (σ x)
|
||||
πₜ² : {Γ Δ : Con} → Sub Δ (Γ ▹ₜ) → Tm Δ
|
||||
πₜ² σ = λ x → proj×₂ (σ x)
|
||||
_,ₜ_ : {Γ Δ : Con} → Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹ₜ)
|
||||
σ ,ₜ t = λ x → (σ x) ,× (t x)
|
||||
πₜ²∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ² (σ ,ₜ t) ≡ t
|
||||
πₜ²∘,ₜ {σ = σ} {t} = refl {a = t}
|
||||
πₜ¹∘,ₜ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm Δ} → πₜ¹ (σ ,ₜ t) ≡ σ
|
||||
πₜ¹∘,ₜ = refl
|
||||
,ₜ∘πₜ : {Γ Δ : Con} → {σ : Sub Δ (Γ ▹ₜ)} → (πₜ¹ σ) ,ₜ (πₜ² σ) ≡ σ
|
||||
,ₜ∘πₜ = refl
|
||||
,ₜ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{t : Tm Γ} → (σ ,ₜ t) ∘ δ ≡ (σ ∘ δ) ,ₜ (t [ δ ]t)
|
||||
,ₜ∘ = refl
|
||||
|
||||
-- Functor Con → Set called For
|
||||
For : Con → Set₁
|
||||
For Γ = Γ → Prop
|
||||
_[_]f : {Γ Δ : Con} → For Γ → Sub Δ Γ → For Δ
|
||||
F [ σ ]f = λ x → F (σ x)
|
||||
[]f-id : {Γ : Con} → {F : For Γ} → F [ id {Γ} ]f ≡ F
|
||||
[]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
|
||||
|
||||
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
|
||||
_[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f)
|
||||
prf [ σ ]p = λ γ → prf (σ γ)
|
||||
-- Two rules are irrelevent beccause Γ ⊢ F is in Prop
|
||||
|
||||
-- → Prop⁺
|
||||
_▹ₚ_ : (Γ : Con) → For Γ → Con
|
||||
Γ ▹ₚ F = Γ ×'' F
|
||||
πₚ¹ : {Γ Δ : Con} → {F : For Γ} → Sub Δ (Γ ▹ₚ F) → Sub Δ Γ
|
||||
πₚ¹ σ δ = proj×''₁ (σ δ)
|
||||
πₚ² : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ (Γ ▹ₚ F)) → Δ ⊢ (F [ πₚ¹ σ ]f)
|
||||
πₚ² σ δ = proj×''₂ (σ δ)
|
||||
_,ₚ_ : {Γ Δ : Con} → {F : For Γ} → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) → Sub Δ (Γ ▹ₚ F)
|
||||
_,ₚ_ {F = F} σ pf δ = (σ δ) ,×'' pf δ
|
||||
,ₚ∘πₚ : {Γ Δ : Con} → {F : For Γ} → {σ : Sub Δ (Γ ▹ₚ F)} → (πₚ¹ σ) ,ₚ (πₚ² σ) ≡ σ
|
||||
,ₚ∘πₚ = refl
|
||||
πₚ¹∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For Γ} → {prf : Δ ⊢ (F [ σ ]f)} → πₚ¹ {Γ} {Δ} {F} (σ ,ₚ prf) ≡ σ
|
||||
πₚ¹∘,ₚ = refl
|
||||
,ₚ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{F : For Ξ}{prf : Γ ⊢ (F [ σ ]f)} →
|
||||
(_,ₚ_ {F = F} σ prf) ∘ δ ≡ (σ ∘ δ) ,ₚ (substP (λ F → Δ ⊢ F) (≡sym ([]f-∘ {α = δ} {β = σ} {F = F})) (prf [ δ ]p))
|
||||
,ₚ∘ {Γ} {Δ} {Ξ} {σ} {δ} {F} {prf} = refl
|
||||
|
||||
-- Implication
|
||||
_⇒_ : {Γ : Con} → For Γ → For Γ → For Γ
|
||||
F ⇒ G = λ γ → (F γ) → (G γ)
|
||||
[]f-⇒ : {Γ Δ : Con} → {F G : For Γ} → {σ : Sub Δ Γ} → (F ⇒ G) [ σ ]f ≡ (F [ σ ]f) ⇒ (G [ σ ]f)
|
||||
[]f-⇒ = refl
|
||||
|
||||
-- Forall
|
||||
∀∀ : {Γ : Con} → For (Γ ▹ₜ) → For Γ
|
||||
∀∀ {Γ} F = λ (γ : Γ) → (∀ (t : TM) → F (γ ,× t))
|
||||
[]f-∀∀ : {Γ Δ : Con} → {F : For (Γ ▹ₜ)} → {σ : Sub Δ Γ} → (∀∀ F) [ σ ]f ≡ (∀∀ (F [ (σ ∘ πₜ¹ id) ,ₜ πₜ² id ]f))
|
||||
[]f-∀∀ {Γ} {Δ} {F} {σ} = refl
|
||||
|
||||
-- Lam & App
|
||||
lam : {Γ : Con} → {F : For Γ} → {G : For Γ} → (Γ ▹ₚ F) ⊢ (G [ πₚ¹ id ]f) → Γ ⊢ (F ⇒ G)
|
||||
lam pf = λ γ x → pf (γ ,×'' x)
|
||||
app : {Γ : Con} → {F G : For Γ} → Γ ⊢ (F ⇒ G) → Γ ⊢ F → Γ ⊢ G
|
||||
app pf pf' = λ γ → pf γ (pf' γ)
|
||||
-- Again, we don't write the _[_]p equalities as everything is in Prop
|
||||
|
||||
-- ∀i and ∀e
|
||||
∀i : {Γ : Con} → {F : For (Γ ▹ₜ)} → (Γ ▹ₜ) ⊢ F → Γ ⊢ (∀∀ F)
|
||||
∀i p γ = λ t → p (γ ,× t)
|
||||
∀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
|
||||
tod = record
|
||||
{ Con = Con
|
||||
; Sub = Sub
|
||||
; _∘_ = _∘_
|
||||
; ∘-ass = refl
|
||||
; id = id
|
||||
; idl = refl
|
||||
; idr = refl
|
||||
; ◇ = ⊤ₛ
|
||||
; ε = ε
|
||||
; ε-u = refl
|
||||
; Tm = Tm
|
||||
; _[_]t = _[_]t
|
||||
; []t-id = []t-id
|
||||
; []t-∘ = λ {Γ} {Δ} {Ξ} {α} {β} {t} → []t-∘ {Γ} {Δ} {Ξ} {α} {β} {t}
|
||||
; _▹ₜ = _▹ₜ
|
||||
; πₜ¹ = πₜ¹
|
||||
; πₜ² = πₜ²
|
||||
; _,ₜ_ = _,ₜ_
|
||||
; πₜ²∘,ₜ = λ {Γ} {Δ} {σ} → πₜ²∘,ₜ {Γ} {Δ} {σ}
|
||||
; πₜ¹∘,ₜ = λ {Γ} {Δ} {σ} {t} → πₜ¹∘,ₜ {Γ} {Δ} {σ} {t}
|
||||
; ,ₜ∘πₜ = ,ₜ∘πₜ
|
||||
; ,ₜ∘ = λ {Γ} {Δ} {Ξ} {σ} {δ} {t} → ,ₜ∘ {Γ} {Δ} {Ξ} {σ} {δ} {t}
|
||||
; For = For
|
||||
; _[_]f = _[_]f
|
||||
; []f-id = []f-id
|
||||
; []f-∘ = λ {Γ} {Δ} {Ξ} {α} {β} {F} → []f-∘ {Γ} {Δ} {Ξ} {α} {β} {F}
|
||||
; _⊢_ = _⊢_
|
||||
; _[_]p = _[_]p
|
||||
; _▹ₚ_ = _▹ₚ_
|
||||
; πₚ¹ = πₚ¹
|
||||
; πₚ² = πₚ²
|
||||
; _,ₚ_ = _,ₚ_
|
||||
; ,ₚ∘πₚ = ,ₚ∘πₚ
|
||||
; πₚ¹∘,ₚ = λ {Γ} {Δ} {F} {σ} {p} → πₚ¹∘,ₚ {Γ} {Δ} {F} {σ} {p}
|
||||
; ,ₚ∘ = λ {Γ} {Δ} {Ξ} {σ} {δ} {F} {prf} → ,ₚ∘ {Γ} {Δ} {Ξ} {σ} {δ} {F} {prf}
|
||||
; _⇒_ = _⇒_
|
||||
; []f-⇒ = λ {Γ} {F} {G} {σ} → []f-⇒ {Γ} {F} {G} {σ}
|
||||
; ∀∀ = ∀∀
|
||||
; []f-∀∀ = λ {Γ} {Δ} {F} {σ} → []f-∀∀ {Γ} {Δ} {F} {σ}
|
||||
; lam = lam
|
||||
; app = app
|
||||
; ∀i = ∀i
|
||||
; ∀e = ∀e
|
||||
; R = R
|
||||
; R[] = λ {Γ} {Δ} {σ} {t} {u} → R[] {Γ} {Δ} {σ} {t} {u}
|
||||
}
|
||||
|
||||
|
||||
-- (∀ x ∀ y . A(x,y)) ⇒ ∀ y ∀ x . A(y,x)
|
||||
-- both sides are ∀ ∀ A (0,1)
|
||||
ex1 : {A : For (⊤ₛ ▹ₜ ▹ₜ)} → ⊤ₛ ⊢ ((∀∀ (∀∀ A)) ⇒ (∀∀ (∀∀ A)))
|
||||
ex1 _ hyp = hyp
|
||||
-- (A ⇒ ∀ x . B(x)) ⇒ ∀ x . A ⇒ B(x)
|
||||
ex2 : {A : For ⊤ₛ} → {B : For (⊤ₛ ▹ₜ)} → ⊤ₛ ⊢ ((A ⇒ (∀∀ B)) ⇒ (∀∀ ((A [ πₜ¹ id ]f) ⇒ B)))
|
||||
ex2 _ h t h' = h h' t
|
||||
-- ∀ x y . A(x,y) ⇒ ∀ x . A(x,x)
|
||||
-- For simplicity, I swiched positions of parameters of A (somehow...)
|
||||
ex3 : {A : For (⊤ₛ ▹ₜ ▹ₜ)} → ⊤ₛ ⊢ ((∀∀ (∀∀ A)) ⇒ (∀∀ (A [ id ,ₜ (πₜ² id) ]f)))
|
||||
ex3 _ h t = h t t
|
||||
-- ∀ x . A (x) ⇒ ∀ x y . A(x)
|
||||
ex4 : {A : For (⊤ₛ ▹ₜ)} → ⊤ₛ ⊢ ((∀∀ A) ⇒ (∀∀ (∀∀ (A [ ε ,ₜ (πₜ² (πₜ¹ id)) ]f))))
|
||||
ex4 {A} ◇◇ x t t' = x t
|
||||
-- (((∀ x . A (x)) ⇒ B)⇒ B) ⇒ ∀ x . ((A (x) ⇒ B) ⇒ B)
|
||||
ex5 : {A : For (⊤ₛ ▹ₜ)} → {B : For ⊤ₛ} → ⊤ₛ ⊢ ((((∀∀ A) ⇒ B) ⇒ B) ⇒ (∀∀ ((A ⇒ (B [ πₜ¹ id ]f)) ⇒ (B [ πₜ¹ id ]f))))
|
||||
ex5 ◇◇ h t h' = h (λ h'' → h' (h'' t))
|
||||
@ -1,8 +1,14 @@
|
||||
{-# OPTIONS --prop #-}
|
||||
{-# OPTIONS --prop --rewriting #-}
|
||||
|
||||
module ListUtil where
|
||||
|
||||
open import Data.List using (List; _∷_; []) public
|
||||
|
||||
infixr 5 _∷_
|
||||
data List : (T : Set₀) → Set where
|
||||
[] : {T : Set₀} → List T
|
||||
_∷_ : {T : Set₀} → T → List T → List T
|
||||
|
||||
{-# BUILTIN LIST List #-}
|
||||
|
||||
private
|
||||
variable
|
||||
@ -138,4 +144,17 @@ module ListUtil where
|
||||
|
||||
⊆→∈* : L ⊆ L' → L ∈* L'
|
||||
⊆→∈* h = ⊂⁺→∈* (⊂→⊂⁺ (⊆→⊂ h))
|
||||
|
||||
open import PropUtil using (Nat; zero; succ)
|
||||
open import Agda.Primitive
|
||||
variable
|
||||
ℓ : Level
|
||||
|
||||
data Array (T : Set ℓ) : Nat → Set ℓ where
|
||||
zero : Array T zero
|
||||
next : {n : Nat} → T → Array T n → Array T (succ n)
|
||||
|
||||
map : {T U : Set ℓ} → (T → U) → {n : Nat} → Array T n → Array U n
|
||||
map f zero = zero
|
||||
map f (next t a) = next (f t) (map f a)
|
||||
|
||||
|
||||
246
PropUtil.agda
246
PropUtil.agda
@ -1,7 +1,15 @@
|
||||
{-# OPTIONS --prop #-}
|
||||
{-# OPTIONS --prop --rewriting #-}
|
||||
|
||||
module PropUtil where
|
||||
|
||||
open import Agda.Primitive
|
||||
variable ℓ ℓ' : Level
|
||||
|
||||
data ⊥ₛ : Set where
|
||||
record ⊤ₛ : Set ℓ where
|
||||
constructor ttₛ
|
||||
|
||||
|
||||
-- ⊥ is a data with no constructor
|
||||
-- ⊤ is a record with one always-available constructor
|
||||
data ⊥ : Prop where
|
||||
@ -50,3 +58,239 @@ module PropUtil where
|
||||
infixr 200 _$_
|
||||
_$_ : {T U : Prop} → (T → U) → T → U
|
||||
h $ t = h t
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
postulate _≈_ : ∀{ℓ}{A : Set ℓ}(a : A) → A → Set ℓ
|
||||
infix 3 _≡_
|
||||
data _≡_ {ℓ}{A : Set ℓ}(a : A) : A → Prop ℓ where
|
||||
refl : a ≡ a
|
||||
{-# BUILTIN REWRITE _≡_ #-}
|
||||
|
||||
≡sym : {ℓ : Level} → {A : Set ℓ}→ {a a' : A} → a ≡ a' → a' ≡ a
|
||||
≡sym refl = refl
|
||||
|
||||
|
||||
≡tran : {ℓ : Level} {A : Set ℓ} → {a a' a'' : A} → a ≡ a' → a' ≡ a'' → a ≡ a''
|
||||
≡tran² : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₀ ≡ a₃
|
||||
≡tran³ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₀ ≡ a₄
|
||||
≡tran⁴ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₀ ≡ a₅
|
||||
≡tran⁵ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ a₆ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₅ ≡ a₆ → a₀ ≡ a₆
|
||||
≡tran⁶ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ a₆ a₇ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₅ ≡ a₆ → a₆ ≡ a₇ → a₀ ≡ a₇
|
||||
≡tran⁷ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₅ ≡ a₆ → a₆ ≡ a₇ → a₇ ≡ a₈ → a₀ ≡ a₈
|
||||
≡tran refl refl = refl
|
||||
≡tran² refl refl refl = refl
|
||||
≡tran³ refl refl refl refl = refl
|
||||
≡tran⁴ refl refl refl refl refl = refl
|
||||
≡tran⁵ refl refl refl refl refl refl = refl
|
||||
≡tran⁶ refl refl refl refl refl refl refl = refl
|
||||
≡tran⁷ refl refl refl refl refl refl refl refl = refl
|
||||
|
||||
cong : {ℓ ℓ' : Level}{A : Set ℓ}{B : Set ℓ'} → (f : A → B) → {a a' : A} → a ≡ a' → f a ≡ f a'
|
||||
cong f refl = refl
|
||||
cong₂ : {ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : Set ℓ'}{C : Set ℓ''} → (f : A → B → C) → {a a' : A} {b b' : B} → a ≡ a' → b ≡ b' → f a b ≡ f a' b'
|
||||
cong₂ f refl refl = refl
|
||||
cong₃ : {ℓ ℓ' ℓ'' ℓ''' : Level}{A : Set ℓ}{B : Set ℓ'}{C : Set ℓ''}{D : Set ℓ'''} → (f : A → B → C → D) → {a a' : A} {b b' : B}{c c' : C} → a ≡ a' → b ≡ b' → c ≡ c' → f a b c ≡ f a' b' c'
|
||||
cong₃ f refl refl refl = refl
|
||||
|
||||
-- We can make a proof-irrelevant substitution
|
||||
substP : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Prop ℓ'){a a' : A} → a ≡ a' → P a → P a'
|
||||
substP P refl h = h
|
||||
substPP : ∀{ℓ}{A B : Set ℓ}{Q : A → Prop ℓ}{ℓ'}(P : {k : A} → Q k → Prop ℓ'){a a' : A}{x : Q a}
|
||||
→ (eq : a ≡ a') → P x → P (substP Q eq x)
|
||||
substPP P refl h = h
|
||||
substP² : ∀{ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : Set ℓ'}(P : A → B → Prop ℓ''){a a' : A}{b b' : B} → a ≡ a' → b ≡ b' → P a b → P a' b'
|
||||
substP² P refl refl p = p
|
||||
|
||||
|
||||
postulate coe : ∀{ℓ}{A B : Set ℓ} → A ≡ B → A → B
|
||||
postulate coerefl : ∀{ℓ}{A : Set ℓ}{eq : A ≡ A}{a : A} → coe eq a ≡ a
|
||||
|
||||
postulate ≡fun : {ℓ ℓ' : Level} → {A : Set ℓ} → {B : Set ℓ'} → {f g : A → B} → ((x : A) → (f x ≡ g x)) → f ≡ g
|
||||
postulate ≡fun' : {ℓ ℓ' : Level} → {A : Set ℓ} → {B : A → Set ℓ'} → {f g : (a : A) → B a} → ((x : A) → (f x ≡ g x)) → f ≡ g
|
||||
|
||||
coeaba : {ℓ : Level}{A B : Set ℓ}{eq1 : A ≡ B}{eq2 : B ≡ A}{a : A} → coe eq2 (coe eq1 a) ≡ a
|
||||
coeaba {eq1 = refl} = ≡tran coerefl coerefl
|
||||
|
||||
coefgcong : {ℓ : Level}{A B C D : Set ℓ}{aa : A ≡ A}{dd : D ≡ D}{cb : C ≡ B}{f : B → A}{g : D → C}{x : D} → f (coe cb (g (coe dd x))) ≡ coe aa (f (coe cb (g x)))
|
||||
coefgcong {cb = refl} {f} {g} = ≡tran (cong f (cong (coe _) (cong g coerefl))) (≡sym coerefl)
|
||||
|
||||
coecong : {ℓ : Level}{A B : Set ℓ}{eq : B ≡ B}{eq' : A ≡ A}(f : A → B){x : A} → (f (coe eq' x)) ≡ (coe eq (f x))
|
||||
|
||||
coecong f = ≡tran (cong f coerefl) (≡sym coerefl)
|
||||
|
||||
coecoe-coe : {ℓ : Level}{A B C : Set ℓ}{eq1 : B ≡ A}{eq2 : C ≡ B}{x : C} → coe eq1 (coe eq2 x) ≡ coe (≡tran eq2 eq1) x
|
||||
coecoe-coe {eq1 = refl} {refl} = coerefl
|
||||
|
||||
coe-f : {ℓ : Level}{A B C D : Set ℓ} → (A → B) → A ≡ C → B ≡ D → C → D
|
||||
coe-f f ac bd x = coe bd (f (coe (≡sym ac) x))
|
||||
coewtf : {ℓ : Level}{A B C D E F G H : Set ℓ}{ab : A ≡ B}{cd : C ≡ D}{ef : E ≡ F}{gh : G ≡ H}{f : F → B}{g : H → E}{x : G} →
|
||||
{fd : F ≡ D} → f (coe ef (g (coe gh x))) ≡ coe ab ((coe-f f fd (≡sym ab)) (coe cd ((coe-f g (≡sym gh) (≡tran² ef fd (≡sym cd))) x)))
|
||||
coewtf {ab = refl} {refl} {refl} {refl} {f} {g} {fd = refl} = ≡tran (cong f (cong (coe _) (≡sym coeaba))) (≡sym coeaba)
|
||||
|
||||
coeshift : {ℓ : Level}{A B : Set ℓ}{x : A} {y : B} {eq : A ≡ B} → coe eq x ≡ y → x ≡ coe (≡sym eq) y
|
||||
coeshift {eq=refl} refl = ≡sym coeaba
|
||||
|
||||
subst : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Set ℓ'){a a' : A} → a ≡ a' → P a → P a'
|
||||
subst P eq p = coe (cong P eq) p
|
||||
subst² : ∀{ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : Set ℓ'}(P : A → B → Set ℓ''){a a' : A}{b b' : B} → a ≡ a' → b ≡ b' → P a b → P a' b'
|
||||
subst² P eq eq' p = coe (cong₂ P eq eq') p
|
||||
subst¹P : ∀{ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : Prop ℓ'}(P : A → B → Set ℓ''){a a' : A}{b : B} → a ≡ a' → P a b → P a' b
|
||||
subst¹P P {b = b} eq p = coe (cong (λ x → P x b) eq) p
|
||||
|
||||
--{-# REWRITE transprefl #-}
|
||||
|
||||
coereflrefl : {ℓ : Level}{A : Set ℓ}{eq eq' : A ≡ A}{a : A} → coe eq (coe eq' a) ≡ a
|
||||
coereflrefl = ≡tran coerefl coerefl
|
||||
|
||||
substrefl : ∀{ℓ}{A : Set ℓ}{ℓ'}{P : A → Set ℓ'}{a : A}{e : a ≡ a}{p : P a} → subst P e p ≡ p
|
||||
substrefl = coerefl
|
||||
--{-# REWRITE substrefl #-}
|
||||
substreflrefl : {ℓ ℓ' : Level}{A : Set ℓ}{P : A → Set ℓ'}{a : A}{e : a ≡ a}{p : P a} → subst P e (subst P e p) ≡ p
|
||||
substreflrefl {P = P} {a} {e} {p} = ≡tran (substrefl {P = P} {a = a} {e = e} {p = subst P e p}) (substrefl {P = P} {a = a} {e = e} {p = p})
|
||||
|
||||
cong₂' : {ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : A → Set ℓ'}{C : Set ℓ''} → (f : (a : A) → B a → C) → {a a' : A} {b : B a} {b' : B a'} → (eq : a ≡ a') → subst B eq b ≡ b' → f a b ≡ f a' b'
|
||||
cong₂' f {a} refl refl = cong (f a) (≡sym coerefl)
|
||||
|
||||
congsubst : {ℓ ℓ' : Level}{A : Set ℓ}{P : A → Set ℓ'}{a a' : A}{e : a ≡ a}{p : P a}{p' : P a} → p ≡ p' → subst P e p ≡ subst P e p'
|
||||
congsubst {P = P} {e = refl} h = cong (subst P refl) h
|
||||
|
||||
substfpoly : {ℓ ℓ' : Level}{A : Set ℓ}{P R : A → Set ℓ'}{α β : A}
|
||||
{eq : α ≡ β} {f : {ξ : A} → R ξ → P ξ} {x : R α}
|
||||
→ coe (cong P eq) (f {α} x) ≡ f (coe (cong R eq) x)
|
||||
substfpoly {eq = refl} {f} = ≡tran coerefl (cong f (≡sym coerefl))
|
||||
|
||||
substppoly : {ℓ ℓ' ℓ'' ℓ''' : Level}{A : Set ℓ}{P : A → Set ℓ'}{R : A → Set ℓ''}{Q : A → Set ℓ'''}{α β : A}
|
||||
{eq : α ≡ β}{f : {ξ : A} → R ξ → Q ξ → P ξ} {x : R α} {y : Q α}
|
||||
→ coe (cong P eq) (f {α} x y) ≡ f {β} (coe (cong R eq) x) (coe (cong Q eq) y)
|
||||
substppoly {eq = refl} {f}{x}{y} = ≡tran coerefl (cong₂ f (≡sym coerefl) (≡sym coerefl))
|
||||
|
||||
substfpoly' : {ℓ ℓ' ℓ'' : Level}{A B : Set ℓ}{P R : A → Set ℓ'}{Q : B → Prop ℓ''}{α β : A}{γ δ : B}
|
||||
{eq : α ≡ β}{eq' : γ ≡ δ} {f : {ξ : A}{ι : B} → R ξ → Q ι → P ξ} {x : R α} {y : Q γ}
|
||||
→ coe (cong P eq) (f {α} {γ} x y) ≡ f {β} {δ} (coe (cong R eq) x) (substP Q eq' y)
|
||||
substfpoly' {eq = refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl
|
||||
substfpoly⁴ : {ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{P R : A → Set ℓ'}{Q : A → Prop ℓ''}{α β : A}
|
||||
{eq : α ≡ β} {f : {ξ : A} → R ξ → Q ξ → P ξ} {x : R α} {y : Q α}
|
||||
→ coe (cong P eq) (f {α} x y) ≡ f {β} (coe (cong R eq) x) (substP Q eq y)
|
||||
substfpoly⁴ {eq = refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl
|
||||
substfpoly³ : {ℓ ℓ' ℓ'' ℓ''' : Level}{A B C : Set ℓ}{R : A → Set ℓ'}{Q : B → Prop ℓ''}{P : C → Set ℓ'''}{α β : A}{γ δ : B}{ε φ : C}
|
||||
{eq : α ≡ β}{eq' : γ ≡ δ}{eq'' : ε ≡ φ} {f : {ξ : A}{ι : B}{τ : C} → R ξ → Q ι → P τ} {x : R α} {y : Q γ}
|
||||
→ coe (cong P eq'') (f {α} {γ} {ε} x y) ≡ f {β} {δ} {φ} (coe (cong R eq) x) (substP Q eq' y)
|
||||
substfpoly³ {eq = refl} {refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl
|
||||
substfpoly'' : {ℓ ℓ' ℓ'' : Level}{A C : Set ℓ}{P : A → C → Set ℓ'}{R : A → Set ℓ'}{Q : A → C → Prop ℓ''}{α β : A}{ε φ : C}
|
||||
{eq : α ≡ β}{eq'' : ε ≡ φ} {f : {ξ : A}{κ : C} → R ξ → Q ξ κ → P ξ κ} {x : R α} {y : Q α ε}
|
||||
→ coe (cong₂ P eq eq'') (f {α} {ε} x y) ≡ f {β} {φ} (coe (cong R eq) x) (substP (λ X → Q X φ) eq (substP (Q α) eq'' y))
|
||||
substfpoly'' {eq = refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl
|
||||
|
||||
substfgpoly : {ℓ ℓ' : Level}{A B : Set ℓ} {P Q : A → Set ℓ'} {R : B → Set ℓ'} {F : B → A} {α β : A} {ε φ : B}
|
||||
{eq₁ : α ≡ β} {eq₂ : F ε ≡ α} {eq₃ : F φ ≡ β} {eq₄ : ε ≡ φ}
|
||||
{g : {a : A} → Q a → P a} {f : {b : B} → R b → Q (F b)} {x : R ε}
|
||||
→ g {β} (subst Q eq₃ (f {φ} (subst R eq₄ x))) ≡ subst P eq₁ (g {α} (subst Q eq₂ (f {ε} x)))
|
||||
substfgpoly {P = P} {Q} {R} {eq₁ = refl} {refl} {refl} {refl} {g} {f} = ≡tran³ (cong g (substrefl {P = Q} {e = refl})) (cong g (cong f (substrefl {P = R} {e = refl}))) (cong g (≡sym (substrefl {P = Q} {e = refl}))) (≡sym (substrefl {P = P} {e = refl}))
|
||||
|
||||
{-# BUILTIN EQUALITY _≡_ #-}
|
||||
|
||||
coep² : {ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level} {A : Set ℓ₁} {R : A → Set ℓ₂}{T : A → Set ℓ₃}{S : A → Set ℓ₄}{α β : A}
|
||||
{p : {ξ : A} → R ξ → T ξ → S ξ}{x : R α}{y : T α}{eq : α ≡ β}
|
||||
→ subst S (≡sym eq) (p {β} (subst R eq x) (subst T eq y)) ≡ p {α} x y
|
||||
coep² {S = S}{p = p}{x}{y}{refl} = ≡tran (substrefl {P = S} {e = refl}) (cong₂ p (substrefl {a = x} {e = refl}) (substrefl {a = y} {e = refl}))
|
||||
coep²'' : {ℓ ℓ' : Level} {A : Set ℓ} {R S : A → Set ℓ'}{T : A → Prop ℓ'}{α β : A}
|
||||
{p : {ξ : A} → R ξ → T ξ → S ξ}{x : R α}{y : T α}{eq : α ≡ β}
|
||||
→ subst S (≡sym eq) (p {β} (subst R eq x) (substP T eq y)) ≡ p {α} x y
|
||||
coep²'' {S = S}{p = p}{x}{y}{refl} = ≡tran (substrefl {P = S} {e = refl}) (cong (λ X → p X y) (substrefl {a = x} {e = refl}))
|
||||
coep²' : {ℓ ℓ' : Level} {A : Set ℓ} {R T S : A → Set ℓ'}{α β : A}
|
||||
{p : {ξ : A} → R ξ → T ξ → S ξ}{x : R β}{y : T α}{eq : α ≡ β}
|
||||
→ subst S (≡sym eq) (p {β} x (subst T eq y)) ≡ p {α} (subst R (≡sym eq) x) y
|
||||
coep²' {S = S}{p = p}{x}{y}{refl} = ≡tran (substrefl {P = S} {e = refl}) (cong₂ p (≡sym (substrefl {a = x} {e = refl})) (substrefl {a = y} {e = refl}))
|
||||
|
||||
coep∘ : {ℓ ℓ' : Level}{A : Set ℓ} {R : A → A → Set ℓ'} {α β γ δ ε φ : A}
|
||||
{p : {x y z : A} → R x y → R z x → R z y}{x : R β γ}{y : R α β}
|
||||
{eq1 : α ≡ δ} {eq2 : β ≡ ε} {eq3 : γ ≡ φ} →
|
||||
coe (cong₂ R (≡sym eq1) (≡sym eq3)) (p (coe (cong₂ R eq2 eq3) x) (coe (cong₂ R eq1 eq2) y)) ≡ p x y
|
||||
coep∘ {p = p}{eq1 = refl}{refl}{refl} = ≡tran coerefl (cong₂ p coerefl coerefl)
|
||||
coep∘-helper = λ {ℓ ℓ' ℓ'' : Level}{B : Set ℓ}{A : B → Set ℓ''} {R : (b : B) → A b → A b → Set ℓ'}
|
||||
{b₁ b₂ : B} {α γ : A b₁} {δ φ : A b₂}
|
||||
{eq0 : b₁ ≡ b₂}{eqa : subst A eq0 α ≡ δ}{eqb : subst A eq0 γ ≡ φ}
|
||||
→ (≡tran² (cong (R b₂ δ) (≡sym eqb)) (cong (λ X → R b₂ X (subst A eq0 γ)) (≡sym eqa)) (≡tran (≡sym (substrefl {P = λ X → Set ℓ'}{a = R b₂ (subst A eq0 α) (subst A eq0 γ)}{e = refl})) (coep² {p = λ {t} x y → R t x y}{eq = eq0})))
|
||||
coep∘-helper-coe : {ℓ ℓ' ℓ'' : Level}{B : Set ℓ}{A : B → Set ℓ''} {R : (b : B) → A b → A b → Set ℓ'}
|
||||
{b₁ b₂ : B} {α γ : A b₁} {δ φ : A b₂}
|
||||
{eq0 : b₁ ≡ b₂}{eqa : subst A eq0 α ≡ δ}{eqb : subst A eq0 γ ≡ φ} → {a : R b₂ δ φ}{a' : R b₁ α γ} → coe (coep∘-helper {eq0 = eq0} {eqa = eqa} {eqb = eqb}) a ≡ a
|
||||
coep∘-helper-coe {eq0 = refl}{refl}{refl} = coerefl
|
||||
{-coep∘' : {ℓ ℓ' ℓ'' : Level}{B : Set ℓ}{A : B → Set ℓ''} {R : (b : B) → A b → A b → Set ℓ'}
|
||||
{b₁ b₂ : B} {α β γ : A b₁} {δ ε φ : A b₂}
|
||||
{p : {b : B}{x y z : A b} → R b x y → R b z x → R b z y}{x : R b₁ β γ}{y : R b₁ α β}
|
||||
{eq0 : b₁ ≡ b₂}{eq1 : subst A eq0 α ≡ δ} {eq2 : subst A eq0 β ≡ ε} {eq3 : subst A eq0 γ ≡ φ}
|
||||
{eq4 : R b₂ δ φ ≡ R b₁ α γ}{eq5 : R b₂ ε φ ≡ R b₁ β γ}{eq6 : R b₂ δ ε ≡ R b₁ α β}
|
||||
→ coe eq4
|
||||
(p {b₂} {ε} {φ} {δ} (coe (≡sym (eq5)) x) (coe (≡sym (
|
||||
eq6
|
||||
)) y)) ≡ p {b₁} {β} {γ} {α} x y
|
||||
--coep∘' {p = p} {x} {y} {eq0 = refl} {refl} {refl} {refl} {eq4} = {!!}
|
||||
-}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
data Nat : Set where
|
||||
zero : Nat
|
||||
succ : Nat → Nat
|
||||
|
||||
{-# BUILTIN NATURAL Nat #-}
|
||||
|
||||
record _×_ (A : Set ℓ) (B : Set ℓ') : Set (ℓ ⊔ ℓ') where
|
||||
constructor _,×_
|
||||
field
|
||||
a : A
|
||||
b : B
|
||||
|
||||
record _×'_ (A : Set ℓ) (B : Prop ℓ') : Set (ℓ ⊔ ℓ') where
|
||||
constructor _,×'_
|
||||
field
|
||||
a : A
|
||||
b : B
|
||||
|
||||
record _×''_ (A : Set ℓ) (B : A → Prop ℓ') : Set (ℓ ⊔ ℓ') where
|
||||
constructor _,×''_
|
||||
field
|
||||
a : A
|
||||
b : B a
|
||||
|
||||
record _×ᵈ_ (A : Set ℓ) (B : A → Set ℓ') : Set (ℓ ⊔ ℓ') where
|
||||
constructor _,×ᵈ_
|
||||
field
|
||||
a : A
|
||||
b : B a
|
||||
|
||||
|
||||
proj×₁ : {ℓ ℓ' : Level}{A : Set ℓ}{B : Set ℓ'} → (A × B) → A
|
||||
proj×₁ p = _×_.a p
|
||||
proj×₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : Set ℓ'} → (A × B) → B
|
||||
proj×₂ p = _×_.b p
|
||||
|
||||
proj×'₁ : {ℓ ℓ' : Level}{A : Set ℓ}{B : Prop ℓ'} → (A ×' B) → A
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proj×'₁ p = _×'_.a p
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proj×'₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : Prop ℓ'} → (A ×' B) → B
|
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proj×'₂ p = _×'_.b p
|
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proj×''₁ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Prop ℓ'} → (A ×'' B) → A
|
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proj×''₁ p = _×''_.a p
|
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proj×''₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Prop ℓ'} → (p : A ×'' B) → B (proj×''₁ p)
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proj×''₂ p = _×''_.b p
|
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proj×ᵈ₁ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Set ℓ'} → (A ×ᵈ B) → A
|
||||
proj×ᵈ₁ p = _×ᵈ_.a p
|
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proj×ᵈ₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Set ℓ'} → (p : A ×ᵈ B) → (B (proj×ᵈ₁ p))
|
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proj×ᵈ₂ p = _×ᵈ_.b p
|
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|
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|
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×≡ : {A : Set ℓ}{B : Set ℓ'}{a a' : A}{b b' : B} → a ≡ a' → b ≡ b' → a ,× b ≡ a' ,× b'
|
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×≡ refl refl = refl
|
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|
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×ᵈ≡ : {A : Set ℓ}{B : A → Set ℓ'}{a a' : A}{b : B a}{b' : B a'} → (eq : a ≡ a') → subst B eq b ≡ b' → a ,×ᵈ b ≡ a' ,×ᵈ b'
|
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×ᵈ≡ {B = B} {a = a}{b = b} refl refl = cong₂' _,×ᵈ_ refl refl
|
||||
|
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Reference in New Issue
Block a user