I FINALLY HAVE A SYNTAX !!!!
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FFOLCompleteness.agda
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327
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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_≤_ : World → World → Set ℓ¹ -- arrows
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≤refl : {w : World} → w ≤ w -- id arrow
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≤tran : {w w' w'' : World} → w ≤ w' → w' ≤ w'' → w ≤ w'' -- arrow composition
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≤-ass : {w w' w'' w''' : World}{a : w ≤ w'}{b : w' ≤ w''}{c : w'' ≤ w'''}
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→ (≤tran (≤tran a b) c) ≡ (≤tran a (≤tran b c))
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≤-idl : {w w' : World} → {a : w ≤ w'} → ≤tran (≤refl {w}) a ≡ a
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≤-idr : {w w' : World} → {a : w ≤ w'} → ≤tran a (≤refl {w'}) ≡ a
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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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-- 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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open import FFOLInitial as I
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World : Set₁
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World = I.Con
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_≤_ : World → World → Set₁
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Γ ≤ Δ = I.Sub Γ Δ
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UP : Presheaf
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UP = record
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{ World = I.Con
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; _≤_ = I.Sub
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; ≤refl = I.id
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; ≤tran = λ σ σ' → σ' I.∘ σ
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; ≤-ass = λ {w}{w'}{w''}{w'''}{a}{b}{c} → ≡sym I.∘-ass
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; ≤-idl = I.idr
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; ≤-idr = I.idl
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; TM = λ Γ → I.Tm (Con.t Γ)
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; TM≤ = {!!}
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; REL = λ Γ t u → {!I.r t u!}
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; REL≤ = {!!}
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}
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163
FFOLInitial.agda
163
FFOLInitial.agda
@ -134,7 +134,8 @@ module FFOLInitial where
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∘ₜ-ass {α = α} {β} {γ ,ₜ x} = cong₂ _,ₜ_ ∘ₜ-ass (≡sym ([]t-∘ {t = x}))
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[]f-∀∀ : {A : For (Γₜ ▹t⁰)} → {σₜ : Subt Δₜ Γₜ} → (∀∀ A) [ σₜ ]f ≡ (∀∀ (A [ (σₜ ∘ₜ πₜ¹ idₜ) ,ₜ πₜ² idₜ ]f))
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[]f-∀∀ {A = A} = cong ∀∀ (cong (_[_]f A) (cong₂ _,ₜ_ (≡tran (cong wkₜσt (≡sym σ-idr)) (≡sym lem3)) refl))
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εₜ-u : {σₜ : Subt Γₜ ◇t} → σₜ ≡ εₜ
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εₜ-u {σₜ = εₜ} = refl
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data Conp : Cont → Set₁ -- pu tit in Prop
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variable
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@ -240,8 +241,6 @@ module FFOLInitial where
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∈ₚ*→Sub zero∈ₚ* = εₚ
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∈ₚ*→Sub (next∈ₚ* x s) = ∈ₚ*→Sub s ,ₚ var x
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idₚ : Subp {Δₜ} Δₚ Δₚ
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idₚ = ∈ₚ*→Sub refl∈ₚ*
|
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|
||||
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)
|
||||
@ -253,8 +252,23 @@ module FFOLInitial where
|
||||
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})
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
@ -282,7 +296,6 @@ module FFOLInitial where
|
||||
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
|
||||
@ -302,33 +315,118 @@ module FFOLInitial where
|
||||
_∘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} = ?
|
||||
idlₚ {Δₚ = Δₚ ▹p⁰ x} = ?
|
||||
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 = {!!}
|
||||
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 {σ = sub σₜ σₚ} = cong₂' sub σ-idl {!!}
|
||||
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 {σ = sub σₜ σₚ} = cong₂' sub σ-idr {!!}
|
||||
{-
|
||||
∘ₚ-ass :
|
||||
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 (Δₚ [ αₜ ]c) ((Ψₚ [ γₜ ∘ₜ βₜ ]c)[ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c)}
|
||||
{eq₂ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c)[ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]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)!}}
|
||||
{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)}
|
||||
→ (coe eq₁ (((coe eq₂ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ) ∘ₚ αₚ) ≡ (coe eq₃ (γₚ [ βₜ ∘ₜ αₜ ]σₚ)) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ) ∘ₚ αₚ))
|
||||
-}
|
||||
postulate ∘-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} = {!Subp (Con.p Ξ [ βₜ ∘ₜ αₜ ]c) (Con.p Ψ [ γₜ ∘ₜ (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
|
||||
|
||||
@ -380,18 +478,11 @@ module FFOLInitial where
|
||||
|
||||
,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
|
||||
--funlol : {Γₜ Δₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Ξₜ}{σₜ : Subt Γₜ Ξₜ}{δₜ : Subt Δₜ Γₜ}{δₚ : Subp Δₚ (Γₚ [ δₜ ]c)}{A : For Ξₜ}{prf : Pf (con Δₜ (Γₚ [ δₜ ]c)) ((A [ σₜ ∘ₜ δₜ ]f))} → Subp {Δₜ} (Γₚ [ δₜ ]c) ((Ξₚ [ σₜ ∘ₜ δₜ ]c) ▹p⁰ ((A [ σₜ ]f) [ δₜ ]f)) → Subp {Δₜ} (Δₚ) ((Ξₚ [ σₜ ∘ₜ δₜ ]c) ▹p⁰ (A [ σₜ ∘ₜ δₜ ]f))
|
||||
--funlol {Γₚ = Γₚ} {Ξₚ = Ξₚ} {σₜ = σₜ} {δₜ = δₜ} {δₚ = δₚ} {prf = prf} (ξ ,ₚ pf) = ((subst (λ X → Subp (Γₚ [ δₜ ]c) ((Ξₚ [ σₜ ∘ₜ δₜ ]c) ▹p⁰ X)) (≡sym []f-∘) ξ) ,ₚ ?) ∘ₚ δₚ
|
||||
postulate ,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 = Γ} {Δ = Δ} {σ = sub σₜ σₚ} {sub δₜ δₚ} {F = A} {prf} = cong (sub (σₜ ∘ₜ δₜ)) (cong {!funlol!}
|
||||
(substfpoly
|
||||
{P = λ X → Subp (Con.p Γ [ δₜ ]c) (X ▹p⁰ ((A [ σₜ ]f) [ δₜ ]f))}
|
||||
{R = λ X → Subp (Con.p Γ [ δₜ ]c) X}
|
||||
{eq = ≡sym []c-∘}
|
||||
{f = λ ξ → ξ ,ₚ (prf [ δₜ ]pₜ)}
|
||||
{x = σₚ [ δₜ ]σₚ}
|
||||
))
|
||||
-}
|
||||
,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
|
||||
@ -415,11 +506,11 @@ module FFOLInitial where
|
||||
; _∘_ = _∘_
|
||||
; ∘-ass = ∘-ass
|
||||
; id = id
|
||||
; idl = {!!}
|
||||
; idr = {!!}
|
||||
; idl = idl
|
||||
; idr = idr
|
||||
; ◇ = ◇
|
||||
; ε = sub εₜ εₚ
|
||||
; ε-u = {!!}
|
||||
; 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
|
||||
@ -438,8 +529,8 @@ module FFOLInitial where
|
||||
; []f-∘ = []f-∘
|
||||
; R = r
|
||||
; R[] = refl
|
||||
; _⊢_ = λ Γ A → Pf Γ A
|
||||
; _[_]p = λ {Γ}{Δ}{F} pf σ → (pf [ Sub.t σ ]pₜ) [ Sub.p σ ]p
|
||||
; _⊢_ = Pf
|
||||
; _[_]p = λ pf σ → (pf [ Sub.t σ ]pₜ) [ Sub.p σ ]p
|
||||
; _▹ₚ_ = _▹p_
|
||||
; πₚ¹ = πₚ¹*
|
||||
; πₚ² = πₚ²
|
||||
|
||||
@ -78,10 +78,16 @@ module PropUtil where
|
||||
≡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
|
||||
@ -93,6 +99,11 @@ module PropUtil where
|
||||
-- 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
|
||||
@ -111,15 +122,25 @@ module PropUtil where
|
||||
|
||||
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
|
||||
@ -141,6 +162,26 @@ module PropUtil where
|
||||
{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}
|
||||
@ -150,9 +191,43 @@ module PropUtil where
|
||||
|
||||
{-# 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} = {!!}
|
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-}
|
||||
|
||||
|
||||
|
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@ -184,6 +259,13 @@ module PropUtil where
|
||||
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
|
||||
@ -199,4 +281,14 @@ module PropUtil where
|
||||
proj×''₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Prop ℓ'} → (p : A ×'' B) → B (proj×''₁ p)
|
||||
proj×''₂ p = _×''_.b p
|
||||
|
||||
proj×ᵈ₁ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Set ℓ'} → (A ×ᵈ B) → A
|
||||
proj×ᵈ₁ p = _×ᵈ_.a p
|
||||
proj×ᵈ₂ : {ℓ ℓ' : Level}{A : Set ℓ}{B : A → Set ℓ'} → (p : A ×ᵈ B) → (B (proj×ᵈ₁ p))
|
||||
proj×ᵈ₂ p = _×ᵈ_.b p
|
||||
|
||||
|
||||
×≡ : {A : Set ℓ}{B : Set ℓ'}{a a' : A}{b b' : B} → a ≡ a' → b ≡ b' → a ,× b ≡ a' ,× b'
|
||||
×≡ refl refl = refl
|
||||
|
||||
×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'
|
||||
×ᵈ≡ {B = B} {a = a}{b = b} refl refl = cong₂' _,×ᵈ_ refl refl
|
||||
|
||||
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Reference in New Issue
Block a user