{-# OPTIONS --rewriting #-}
module A201801.LPTTTypesLemmas where
open import A201801.Prelude
open import A201801.Category
open import A201801.Fin
open import A201801.FinLemmas
open import A201801.Vec
open import A201801.VecLemmas
open import A201801.S4TTTerms
open import A201801.S4TTTermsLemmas
open import A201801.LPTTTypes
id-TMREN : ∀ {d} → (A : Type d)
→ TMREN id A ≡ A
id-TMREN (ι P) = refl
id-TMREN (A ⊃ B) = _⊃_ & id-TMREN A ⊗ id-TMREN B
id-TMREN ([ M ] A) = [_]_ & id-MREN M ⊗ id-TMREN A
id-TMRENS : ∀ {d n} → (Ξ : Types d n)
→ TMRENS id Ξ ≡ Ξ
id-TMRENS ∙ = refl
id-TMRENS (Ξ , A) = _,_ & id-TMRENS Ξ ⊗ id-TMREN A
comp-TMREN : ∀ {d d′ d″} → (e₁ : d′ ≥ d) (e₂ : d″ ≥ d′) (A : Type d)
→ TMREN (e₁ ∘ e₂) A ≡ (TMREN e₂ ∘ TMREN e₁) A
comp-TMREN e₁ e₂ (ι P) = refl
comp-TMREN e₁ e₂ (A ⊃ B) = _⊃_ & comp-TMREN e₁ e₂ A ⊗ comp-TMREN e₁ e₂ B
comp-TMREN e₁ e₂ ([ M ] A) = [_]_ & comp-MREN e₁ e₂ M ⊗ comp-TMREN e₁ e₂ A
comp-TMRENS : ∀ {d d′ d″ n} → (e₁ : d′ ≥ d) (e₂ : d″ ≥ d′) (Ξ : Types d n)
→ TMRENS (e₁ ∘ e₂) Ξ ≡ (TMRENS e₂ ∘ TMRENS e₁) Ξ
comp-TMRENS e₁ e₂ ∙ = refl
comp-TMRENS e₁ e₂ (Ξ , A) = _,_ & comp-TMRENS e₁ e₂ Ξ ⊗ comp-TMREN e₁ e₂ A
𝐓𝐌𝐑𝐄𝐍 : Presheaf 𝐆𝐄𝐐 Type
𝐓𝐌𝐑𝐄𝐍 = record
{ ℱ = TMREN
; idℱ = funext! id-TMREN
; compℱ = \ e₁ e₂ → funext! (comp-TMREN e₁ e₂)
}
𝐓𝐌𝐑𝐄𝐍𝐒 : ∀ {n} → Presheaf 𝐆𝐄𝐐 (\ d → Types d n)
𝐓𝐌𝐑𝐄𝐍𝐒 = record
{ ℱ = TMRENS
; idℱ = funext! id-TMRENS
; compℱ = \ e₁ e₂ → funext! (comp-TMRENS e₁ e₂)
}
comp-TMWK-TMREN-drop : ∀ {d d′} → (e : d′ ≥ d) (A : Type d)
→ TMREN (drop e) A ≡ (TMWK ∘ TMREN e) A
comp-TMWK-TMREN-drop e A = (\ e′ → TMREN (drop e′) A) & rid∘ e ⁻¹
⋮ comp-TMREN e (drop id) A
comp-TMWK-TMREN-keep : ∀ {d d′} → (e : d′ ≥ d) (A : Type d)
→ (TMREN (keep e) ∘ TMWK) A ≡ (TMWK ∘ TMREN e) A
comp-TMWK-TMREN-keep e A = comp-TMREN (drop id) (keep e) A ⁻¹
⋮ (\ e′ → TMREN (drop e′) A) & (lid∘ e ⋮ rid∘ e ⁻¹)
⋮ comp-TMREN e (drop id) A
comp-TMWKS-TMRENS-keep : ∀ {d d′ n} → (e : d′ ≥ d) (Ξ : Types d n)
→ (TMRENS (keep e) ∘ TMWKS) Ξ ≡ (TMWKS ∘ TMRENS e) Ξ
comp-TMWKS-TMRENS-keep e ∙ = refl
comp-TMWKS-TMRENS-keep e (Ξ , A) = _,_ & comp-TMWKS-TMRENS-keep e Ξ ⊗ comp-TMWK-TMREN-keep e A
comp-TMSUB-TMREN : ∀ {d n n′} → (τ : Terms* d n′) (e : n′ ≥ n) (A : Type n)
→ TMSUB (GETS τ e) A ≡ (TMSUB τ ∘ TMREN e) A
comp-TMSUB-TMREN τ e (ι P) = refl
comp-TMSUB-TMREN τ e (A ⊃ B) = _⊃_ & comp-TMSUB-TMREN τ e A ⊗ comp-TMSUB-TMREN τ e B
comp-TMSUB-TMREN τ e ([ M ] A) = [_]_ & comp-MSUB-MREN τ e M ⊗ comp-TMSUB-TMREN τ e A
id-cons-TMWK-TMSUB : ∀ {d n} → (τ : Terms* d n) (M : Term d zero) (A : Type n)
→ (TMSUB (τ , M) ∘ TMWK) A ≡ TMSUB τ A
id-cons-TMWK-TMSUB τ M N = comp-TMSUB-TMREN (τ , M) (drop id) N ⁻¹
⋮ (\ x′ → TMSUB x′ N) & id-GETS τ
id-cons-TMWKS-TMSUBS : ∀ {d n m} → (τ : Terms* d n) (M : Term d zero) (Ξ : Types n m)
→ (TMSUBS (τ , M) ∘ TMWKS) Ξ ≡ TMSUBS τ Ξ
id-cons-TMWKS-TMSUBS τ M ∙ = refl
id-cons-TMWKS-TMSUBS τ M (Ξ , A) = _,_ & id-cons-TMWKS-TMSUBS τ M Ξ ⊗ id-cons-TMWK-TMSUB τ M A
id-TMSUB : ∀ {d} → (A : Type d)
→ TMSUB MIDS* A ≡ A
id-TMSUB (ι P) = refl
id-TMSUB (A ⊃ B) = _⊃_ & id-TMSUB A ⊗ id-TMSUB B
id-TMSUB ([ M ] A) = [_]_ & id-MSUB M ⊗ id-TMSUB A
id-TMSUBS : ∀ {d n} → (Ξ : Types d n)
→ TMSUBS MIDS* Ξ ≡ Ξ
id-TMSUBS ∙ = refl
id-TMSUBS (Ξ , A) = _,_ & id-TMSUBS Ξ ⊗ id-TMSUB A
comp-TMSUB : ∀ {d n m} → (τ : Terms* d n) (υ : Terms* n m) (A : Type m)
→ TMSUB (MSUBS τ υ) A ≡ (TMSUB τ ∘ TMSUB υ) A
comp-TMSUB τ υ (ι P) = refl
comp-TMSUB τ υ (A ⊃ B) = _⊃_ & comp-TMSUB τ υ A ⊗ comp-TMSUB τ υ B
comp-TMSUB τ υ ([ M ] A) = [_]_ & comp-MSUB τ υ M ⊗ comp-TMSUB τ υ A
comp-TMSUBS : ∀ {d n m l} → (τ : Terms* d n) (υ : Terms* n m) (Ξ : Types m l)
→ TMSUBS (MSUBS τ υ) Ξ ≡ (TMSUBS τ ∘ TMSUBS υ) Ξ
comp-TMSUBS τ υ ∙ = refl
comp-TMSUBS τ υ (Ξ , A) = _,_ & comp-TMSUBS τ υ Ξ ⊗ comp-TMSUB τ υ A
𝐓𝐌𝐒𝐔𝐁 : Presheaf 𝐒𝟒𝐓𝐞𝐫𝐦𝐬* Type
𝐓𝐌𝐒𝐔𝐁 = record
{ ℱ = TMSUB
; idℱ = funext! id-TMSUB
; compℱ = \ υ τ → funext! (comp-TMSUB τ υ)
}
𝐓𝐌𝐒𝐔𝐁𝐒 : ∀ {n} → Presheaf 𝐒𝟒𝐓𝐞𝐫𝐦𝐬* (\ d → Types d n)
𝐓𝐌𝐒𝐔𝐁𝐒 = record
{ ℱ = TMSUBS
; idℱ = funext! id-TMSUBS
; compℱ = \ υ τ → funext! (comp-TMSUBS τ υ)
}