{-# OPTIONS --rewriting #-}
module A201801.IPLStandardDerivationsLemmas where
open import A201801.Prelude
open import A201801.Category
open import A201801.List
open import A201801.ListLemmas
open import A201801.AllList
open import A201801.AllListLemmas
open import A201801.IPLPropositions
open import A201801.IPLStandardDerivations
comp-ren-get : ∀ {Γ Γ′ Ξ A} → (η : Γ′ ⊇ Γ) (ξ : Γ ⊢ Ξ alltrue) (i : Ξ ∋ A)
→ get (rens η ξ) i ≡ (ren η ∘ get ξ) i
comp-ren-get η (ξ , 𝒟) zero = refl
comp-ren-get η (ξ , 𝒟) (suc i) = comp-ren-get η ξ i
comp-rens-gets : ∀ {Γ Γ′ Ξ Ξ′} → (η₁ : Γ′ ⊇ Γ) (ξ : Γ ⊢ Ξ′ alltrue) (η₂ : Ξ′ ⊇ Ξ)
→ gets (rens η₁ ξ) η₂ ≡ (rens η₁ ∘ gets ξ) η₂
comp-rens-gets η₁ ∙ done = refl
comp-rens-gets η₁ (ξ , 𝒟) (drop η₂) = comp-rens-gets η₁ ξ η₂
comp-rens-gets η₁ (ξ , 𝒟) (keep η₂) = (_, ren η₁ 𝒟) & comp-rens-gets η₁ ξ η₂
comp-lifts-gets : ∀ {Γ Ξ Ξ′ A} → (ξ : Γ ⊢ Ξ′ alltrue) (η : Ξ′ ⊇ Ξ)
→ gets (lifts {A} ξ) (keep η) ≡ (lifts ∘ gets ξ) η
comp-lifts-gets ξ η = (_, vz) & comp-rens-gets (drop id) ξ η
get-var : ∀ {Γ A} → (i : Γ ∋ A)
→ get ids i ≡ var i
get-var zero = refl
get-var (suc i) = comp-ren-get (drop id) ids i
⋮ wk & get-var i
⋮ (\ i′ → var (suc i′)) & id-ren∋ i
id-ren : ∀ {Γ A} → (𝒟 : Γ ⊢ A true)
→ ren id 𝒟 ≡ 𝒟
id-ren (var i) = var & id-ren∋ i
id-ren (lam 𝒟) = lam & id-ren 𝒟
id-ren (app 𝒟 ℰ) = app & id-ren 𝒟 ⊗ id-ren ℰ
id-rens : ∀ {Γ Ξ} → (ξ : Γ ⊢ Ξ alltrue)
→ rens id ξ ≡ ξ
id-rens ∙ = refl
id-rens (ξ , 𝒟) = _,_ & id-rens ξ ⊗ id-ren 𝒟
comp-ren : ∀ {Γ Γ′ Γ″ A} → (η₁ : Γ′ ⊇ Γ) (η₂ : Γ″ ⊇ Γ′) (𝒟 : Γ ⊢ A true)
→ ren (η₁ ∘ η₂) 𝒟 ≡ (ren η₂ ∘ ren η₁) 𝒟
comp-ren η₁ η₂ (var i) = var & comp-ren∋ η₁ η₂ i
comp-ren η₁ η₂ (lam 𝒟) = lam & comp-ren (keep η₁) (keep η₂) 𝒟
comp-ren η₁ η₂ (app 𝒟 ℰ) = app & comp-ren η₁ η₂ 𝒟 ⊗ comp-ren η₁ η₂ ℰ
comp-rens : ∀ {Γ Γ′ Γ″ Ξ} → (η₁ : Γ′ ⊇ Γ) (η₂ : Γ″ ⊇ Γ′) (ξ : Γ ⊢ Ξ alltrue)
→ rens (η₁ ∘ η₂) ξ ≡ (rens η₂ ∘ rens η₁) ξ
comp-rens η₁ η₂ ∙ = refl
comp-rens η₁ η₂ (ξ , 𝒟) = _,_ & comp-rens η₁ η₂ ξ ⊗ comp-ren η₁ η₂ 𝒟
𝐫𝐞𝐧 : ∀ {A} → Presheaf 𝐎𝐏𝐄 (_⊢ A true)
𝐫𝐞𝐧 = record
{ ℱ = ren
; idℱ = funext! id-ren
; compℱ = \ η₁ η₂ → funext! (comp-ren η₁ η₂)
}
𝐫𝐞𝐧𝐬 : ∀ {Ξ} → Presheaf 𝐎𝐏𝐄 (_⊢ Ξ alltrue)
𝐫𝐞𝐧𝐬 = record
{ ℱ = rens
; idℱ = funext! id-rens
; compℱ = \ η₁ η₂ → funext! (comp-rens η₁ η₂)
}
comp-wk-ren-keep : ∀ {Γ Γ′ A B} → (η : Γ′ ⊇ Γ) (𝒟 : Γ ⊢ A true)
→ (ren (keep {A = B} η) ∘ wk) 𝒟 ≡ (wk ∘ ren η) 𝒟
comp-wk-ren-keep η 𝒟 = comp-ren (drop id) (keep η) 𝒟 ⁻¹
⋮ (\ η′ → ren (drop η′) 𝒟) & (lid∘ η ⋮ rid∘ η ⁻¹)
⋮ comp-ren η (drop id) 𝒟
comp-wks-rens-keep : ∀ {Γ Γ′ Ξ A} → (η : Γ′ ⊇ Γ) (ξ : Γ ⊢ Ξ alltrue)
→ (rens (keep {A = A} η) ∘ wks) ξ ≡ (wks ∘ rens η) ξ
comp-wks-rens-keep η ∙ = refl
comp-wks-rens-keep η (ξ , 𝒟) = _,_ & comp-wks-rens-keep η ξ ⊗ comp-wk-ren-keep η 𝒟
comp-lifts-rens : ∀ {Γ Γ′ Ξ A} → (η : Γ′ ⊇ Γ) (ξ : Γ ⊢ Ξ alltrue)
→ (rens (keep {A = A} η) ∘ lifts) ξ ≡ (lifts ∘ rens η) ξ
comp-lifts-rens η ξ = (_, vz) & comp-wks-rens-keep η ξ
comp-sub-get : ∀ {Γ Ξ Ψ A} → (ξ : Γ ⊢ Ξ alltrue) (ψ : Ξ ⊢ Ψ alltrue) (i : Ψ ∋ A)
→ get (subs ξ ψ) i ≡ (sub ξ ∘ get ψ) i
comp-sub-get ξ (ψ , 𝒟) zero = refl
comp-sub-get ξ (ψ , ℰ) (suc i) = comp-sub-get ξ ψ i
comp-sub-ren : ∀ {Γ Ξ Ξ′ A} → (ξ : Γ ⊢ Ξ′ alltrue) (η : Ξ′ ⊇ Ξ) (𝒟 : Ξ ⊢ A true)
→ sub (gets ξ η) 𝒟 ≡ (sub ξ ∘ ren η) 𝒟
comp-sub-ren ξ η (var i) = comp-get-ren∋ ξ η i
comp-sub-ren ξ η (lam 𝒟) = lam & ( (\ ξ′ → sub ξ′ 𝒟) & comp-lifts-gets ξ η ⁻¹
⋮ comp-sub-ren (lifts ξ) (keep η) 𝒟
)
comp-sub-ren ξ η (app 𝒟 ℰ) = app & comp-sub-ren ξ η 𝒟 ⊗ comp-sub-ren ξ η ℰ
id-cons-wk-sub : ∀ {Γ Ξ A B} → (ξ : Γ ⊢ Ξ alltrue) (𝒟 : Γ ⊢ A true) (ℰ : Ξ ⊢ B true)
→ (sub (ξ , 𝒟) ∘ wk) ℰ ≡ sub ξ ℰ
id-cons-wk-sub ξ 𝒟 ℰ = comp-sub-ren (ξ , 𝒟) (drop id) ℰ ⁻¹
⋮ (\ ξ′ → sub ξ′ ℰ) & id-gets ξ
id-cons-wks-subs : ∀ {Γ Ξ Ψ A} → (ξ : Γ ⊢ Ξ alltrue) (𝒟 : Γ ⊢ A true) (ψ : Ξ ⊢ Ψ alltrue)
→ (subs (ξ , 𝒟) ∘ wks) ψ ≡ subs ξ ψ
id-cons-wks-subs ξ 𝒟 ∙ = refl
id-cons-wks-subs ξ 𝒟 (ψ , ℰ) = _,_ & id-cons-wks-subs ξ 𝒟 ψ ⊗ id-cons-wk-sub ξ 𝒟 ℰ
comp-ren-sub : ∀ {Γ Γ′ Ξ A} → (η : Γ′ ⊇ Γ) (ξ : Γ ⊢ Ξ alltrue) (𝒟 : Ξ ⊢ A true)
→ sub (rens η ξ) 𝒟 ≡ (ren η ∘ sub ξ) 𝒟
comp-ren-sub η ξ (var i) = comp-ren-get η ξ i
comp-ren-sub η ξ (lam 𝒟) = lam & ( (\ ξ′ → sub ξ′ 𝒟) & comp-lifts-rens η ξ ⁻¹
⋮ comp-ren-sub (keep η) (lifts ξ) 𝒟
)
comp-ren-sub η ξ (app 𝒟 ℰ) = app & comp-ren-sub η ξ 𝒟 ⊗ comp-ren-sub η ξ ℰ
comp-rens-subs : ∀ {Γ Γ′ Ξ Ψ} → (η : Γ′ ⊇ Γ) (ξ : Γ ⊢ Ξ alltrue) (ψ : Ξ ⊢ Ψ alltrue)
→ subs (rens η ξ) ψ ≡ (rens η ∘ subs ξ) ψ
comp-rens-subs η ξ ∙ = refl
comp-rens-subs η ξ (ψ , 𝒟) = _,_ & comp-rens-subs η ξ ψ ⊗ comp-ren-sub η ξ 𝒟
comp-lifts-subs : ∀ {Γ Ξ Ψ A} → (ξ : Γ ⊢ Ξ alltrue) (ψ : Ξ ⊢ Ψ alltrue)
→ (subs (lifts {A} ξ) ∘ lifts) ψ ≡ (lifts ∘ subs ξ) ψ
comp-lifts-subs ξ ψ = (_, vz) & ( id-cons-wks-subs (wks ξ) vz ψ
⋮ comp-rens-subs (drop id) ξ ψ
)
id-sub : ∀ {Γ A} → (𝒟 : Γ ⊢ A true)
→ sub ids 𝒟 ≡ 𝒟
id-sub (var i) = get-var i
id-sub (lam 𝒟) = lam & id-sub 𝒟
id-sub (app 𝒟 ℰ) = app & id-sub 𝒟 ⊗ id-sub ℰ
comp-sub : ∀ {Γ Ξ Ψ A} → (ξ : Γ ⊢ Ξ alltrue) (ψ : Ξ ⊢ Ψ alltrue) (𝒟 : Ψ ⊢ A true)
→ sub (subs ξ ψ) 𝒟 ≡ (sub ξ ∘ sub ψ) 𝒟
comp-sub ξ ψ (var i) = comp-sub-get ξ ψ i
comp-sub ξ ψ (lam 𝒟) = lam & ( (\ ξ′ → sub ξ′ 𝒟) & comp-lifts-subs ξ ψ ⁻¹
⋮ comp-sub (lifts ξ) (lifts ψ) 𝒟
)
comp-sub ξ ψ (app 𝒟 ℰ) = app & comp-sub ξ ψ 𝒟 ⊗ comp-sub ξ ψ ℰ
lid-subs : ∀ {Γ Ξ} → (ξ : Γ ⊢ Ξ alltrue)
→ subs ids ξ ≡ ξ
lid-subs ∙ = refl
lid-subs (ξ , 𝒟) = _,_ & lid-subs ξ ⊗ id-sub 𝒟
rid-subs : ∀ {Γ Ξ} → (ξ : Γ ⊢ Ξ alltrue)
→ subs ξ ids ≡ ξ
rid-subs ∙ = refl
rid-subs (ξ , 𝒟) = (_, 𝒟) & ( id-cons-wks-subs ξ 𝒟 ids
⋮ rid-subs ξ
)
assoc-subs : ∀ {Γ Ξ Ψ Φ} → (ξ : Γ ⊢ Ξ alltrue) (ψ : Ξ ⊢ Ψ alltrue) (φ : Ψ ⊢ Φ alltrue)
→ subs (subs ξ ψ) φ ≡ (subs ξ ∘ subs ψ) φ
assoc-subs ξ ψ ∙ = refl
assoc-subs ξ ψ (φ , 𝒟) = _,_ & assoc-subs ξ ψ φ ⊗ comp-sub ξ ψ 𝒟
instance
𝐈𝐏𝐋 : Category (List Form) _⊢_alltrue
𝐈𝐏𝐋 = record
{ id = ids
; _∘_ = flip subs
; lid∘ = rid-subs
; rid∘ = lid-subs
; assoc∘ = \ φ ψ ξ → assoc-subs ξ ψ φ ⁻¹
}
𝐬𝐮𝐛 : ∀ {A} → Presheaf 𝐈𝐏𝐋 (_⊢ A true)
𝐬𝐮𝐛 = record
{ ℱ = sub
; idℱ = funext! id-sub
; compℱ = \ ψ ξ → funext! (comp-sub ξ ψ)
}