EverythingHome
{-# OPTIONS --without-K --safe #-}

-- The overall properties of the Concise formulation which are used by later modules.
--
-- Many properties have been proved in the Full formulation. We can use the
-- equivalence between the Full and Concise formulation to bring existing conclusion
-- to this file so later modules can conveniently use these results.
module NonCumulative.Statics.Ascribed.Properties where

open import Lib
open import Data.Nat
open import Data.Nat.Properties
open import Relation.Binary using (PartialSetoid; IsPartialEquivalence)
import Relation.Binary.Reasoning.PartialSetoid as PS

-- open import MLTT.Statics.Syntax public
open import NonCumulative.Statics.Ascribed.Full
import NonCumulative.Statics.Ascribed.Presup as Presup
import NonCumulative.Statics.Ascribed.Refl as Refl
import NonCumulative.Statics.Ascribed.Misc as Misc
import NonCumulative.Statics.Ascribed.PER as PER
import NonCumulative.Statics.Ascribed.CtxEquiv as CtxEquiv
import NonCumulative.Statics.Ascribed.Properties.Contexts as Ctxₚ
import NonCumulative.Statics.Ascribed.Properties.Subst as Subₚ
open Misc
  using ( _[wk]*_)
  public

⊢≈-refl : ⊢ Γ →
          --------
          ⊢ Γ ≈ Γ
⊢≈-refl ⊢Γ = (Refl.≈-Ctx-refl ⊢Γ)

⊢≈-trans : ⊢ Γ ≈ Γ′ → ⊢ Γ′ ≈ Γ″ → ⊢ Γ ≈ Γ″
⊢≈-trans ⊢Γ≈Γ′ ⊢Γ′≈Γ′′ = PER.⊢≈-trans ⊢Γ≈Γ′ ⊢Γ′≈Γ′′

[]-cong-Se‴ : ∀ {i} → Δ ⊢ T ∶[ 1 + i ] Se i → Γ ⊢s σ ≈ σ′ ∶ Δ → Γ ⊢ T [ σ ] ≈ T [ σ′ ] ∶[ 1 + i ] Se i
[]-cong-Se‴ ⊢T σ≈σ′ = Misc.[]-cong-Se″ ⊢T (proj₁ (proj₂ (Presup.presup-s-≈ σ≈σ′))) σ≈σ′

-- A closed term cannot be neutral.

no-closed-Ne-gen : ∀ {i} →
                   Γ ⊢ t ∶[ i ] T →
                   Γ ≡ [] →
                   ----------------
                   ¬ (t ≡ Ne⇒Exp u)
no-closed-Ne-gen {_} {_} {_} {rec T z s u} {_} (N-E _ _ _ ⊢u) refl refl = no-closed-Ne-gen ⊢u refl refl
no-closed-Ne-gen {_} {_} {_} {u $ n} {_} (Λ-E _ _ ⊢u _ _) refl refl = no-closed-Ne-gen ⊢u refl refl
no-closed-Ne-gen {_} {_} {_} {unlift u} {_} (L-E _ _ ⊢u) refl refl = no-closed-Ne-gen ⊢u refl refl
no-closed-Ne-gen {_} {_} {_} {_} (conv ⊢u _) refl refl    = no-closed-Ne-gen ⊢u refl refl

no-closed-Ne : ∀ {i} → ¬ ([] ⊢ Ne⇒Exp u ∶[ i ] T)
no-closed-Ne ⊢u = no-closed-Ne-gen ⊢u refl refl

-- inversions of judgments

⊢I-inv : Γ ⊢s I ∶ Δ → ⊢ Γ ≈ Δ
⊢I-inv (s-I ⊢Γ)         = ⊢≈-refl ⊢Γ
⊢I-inv (s-conv ⊢I Δ′≈Δ) = ⊢≈-trans (⊢I-inv ⊢I) Δ′≈Δ

[I]-inv : ∀ { i } → Γ ⊢ t [ I ] ∶[ i ] T → Γ ⊢ t ∶[ i ] T
[I]-inv (t[σ] ⊢t[I] ⊢sI) with ⊢t ← CtxEquiv.ctxeq-tm (Ctxₚ.⊢≈-sym (⊢I-inv ⊢sI)) ⊢t[I] = conv ⊢t (≈-sym ([I] (proj₂ (Presup.presup-tm ⊢t))))
[I]-inv (conv ⊢t[I] S≈T) = conv ([I]-inv ⊢t[I]) S≈T

[I]-≈ˡ : ∀ {i} → Γ ⊢ t [ I ] ≈ s ∶[ i ] T [ I ] →
         ------------------------------
         Γ ⊢ t ≈ s ∶[ i ] T
[I]-≈ˡ ≈s
  with _ , ⊢t[I] , _ , ⊢T[I] ← Presup.presup-≈ ≈s = ≈-conv (≈-trans (≈-sym ([I] ⊢t)) ≈s) ([I] ⊢T)
  where ⊢t = [I]-inv ⊢t[I]
        ⊢T = [I]-inv ⊢T[I]

[I]-≈ˡ-Se : ∀ {i} →
            Γ ⊢ T [ I ] ≈ S ∶[ 1 + i ] Se i →
            ----------------------------
            Γ ⊢ T ≈ S ∶[ 1 + i ] Se i
[I]-≈ˡ-Se ≈S
  with _ , ⊢T[I] , _ ← Presup.presup-≈ ≈S = ≈-trans (≈-sym ([I] ⊢T)) ≈S
  where ⊢T = [I]-inv ⊢T[I]

∘I-inv : Γ ⊢s σ ∘ I ∶ Δ → ∃ λ Δ′ → Γ ⊢s σ ∶ Δ′ × ⊢ Δ ≈ Δ′
∘I-inv (s-∘ ⊢I ⊢σ) = -, CtxEquiv.ctxeq-s (Ctxₚ.⊢≈-sym (⊢I-inv ⊢I)) ⊢σ , ⊢≈-refl (proj₂ (Presup.presup-s ⊢σ))
∘I-inv (s-conv ⊢σI Δ″≈Δ)
  with Δ′ , ⊢σ , Δ″≈Δ′ ← ∘I-inv ⊢σI = -, ⊢σ , ⊢≈-trans (Ctxₚ.⊢≈-sym Δ″≈Δ) Δ″≈Δ′

∘I-inv′ : Γ ⊢s σ ∘ I ∶ Δ → Γ ⊢s σ ∶ Δ
∘I-inv′ ⊢σI
  with _ , ⊢σ , Δ′≈Δ ← ∘I-inv ⊢σI = s-conv ⊢σ (Ctxₚ.⊢≈-sym Δ′≈Δ)

∷-cong″ : ∀ {i} → Γ ⊢ T ≈ T′ ∶[ 1 + i ] Se i → ⊢ (T ↙ i) ∷ Γ ≈ (T′ ↙ i) ∷ Γ
∷-cong″ T≈T′
 with ⊢Γ , ⊢T , ⊢T′ , _ ← Presup.presup-≈ T≈T′ = Refl.∷-cong′ ⊢Γ ⊢T ⊢T′ T≈T′

[∘]-N : Δ ⊢ t ∶[ 0 ] N → Γ ⊢s σ ∶ Δ → Γ′ ⊢s τ ∶ Γ → Γ′ ⊢ t [ σ ] [ τ ] ≈ t [ σ ∘ τ ] ∶[ 0 ] N
[∘]-N ⊢t ⊢σ ⊢τ = ≈-conv (≈-sym ([∘] ⊢τ ⊢σ ⊢t)) (N-[] (s-∘ ⊢τ ⊢σ))

Exp≈-isPER : ∀ {i} → IsPartialEquivalence (Γ ⊢_≈_∶[ i ] T)
Exp≈-isPER {Γ} {T} = record
  { sym   = ≈-sym
  ; trans = ≈-trans
  }

Exp≈-PER : ∀ {i} → Ctx → Typ → PartialSetoid _ _
Exp≈-PER {i} Γ T = record
  { Carrier              = Exp
  ; _≈_                  = Γ ⊢_≈_∶[ i ] T
  ; isPartialEquivalence = Exp≈-isPER
  }

module ER {i Γ T} = PS (Exp≈-PER {i} Γ T)

Substs≈-isPER : IsPartialEquivalence (Γ ⊢s_≈_∶ Δ)
Substs≈-isPER = record
  { sym   = s-≈-sym
  ; trans = s-≈-trans
  }

Substs≈-PER : Ctx → Ctx → PartialSetoid _ _
Substs≈-PER Γ Δ = record
  { Carrier              = Subst
  ; _≈_                  = Γ ⊢s_≈_∶ Δ
  ; isPartialEquivalence = Substs≈-isPER
  }

module SR {Γ Δ} = PS (Substs≈-PER Γ Δ)

q-cong : ∀ {i} → Γ ⊢s σ ≈ σ′ ∶ Δ → Δ ⊢ T ∶[ 1 + i ] Se i → (T [ σ ] ↙ i) ∷ Γ ⊢s q (T ↙ i) σ ≈ q (T ↙ i) σ′ ∶ (T ↙ i) ∷ Δ
q-cong {_} {σ} {σ′} {_} {T} σ≈σ′ ⊢T
  with ⊢Γ , ⊢σ , _ ← Presup.presup-s-≈ σ≈σ′ = ,-cong (∘-cong (wk-≈ ⊢TσΓ) σ≈σ′) ⊢T (Refl.≈-refl ⊢T) (Refl.≈-refl (conv (vlookup ⊢TσΓ here) (Misc.[∘]-Se ⊢T ⊢σ (s-wk ⊢TσΓ))))
  where open ER
        ⊢Tσ  = Misc.t[σ]-Se ⊢T ⊢σ
        ⊢TσΓ = ⊢∷ ⊢Γ ⊢Tσ

[]-q-∘-, : ∀ {i j} → (S ↙ i) ∷ Γ ⊢ T ∶[ 1 + j ] Se j → Δ ⊢s σ ∶ Γ → Δ′ ⊢s τ ∶ Δ → Δ′ ⊢ t ∶[ i ] S [ σ ] [ τ ] →
           Δ′ ⊢ T [ (σ ∘ τ) , t ∶ (S ↙ i) ] ≈ T [ q (S ↙ i) σ ] [ τ , t ∶ (sub S σ ↙ i) ] ∶[ 1 + j ] Se j
[]-q-∘-, {S} {T = T} {σ = σ} {τ = τ} {t = t} {i = i} {j} ⊢T ⊢σ ⊢τ ⊢t
   with ⊢∷ ⊢Γ ⊢S ← proj₁ (Presup.presup-tm ⊢T)
     | ⊢Δ′ , ⊢Δ ← Presup.presup-s ⊢τ =  begin
     T [ (σ ∘ τ) , t ∶ (S ↙ i) ] ≈⟨ []-cong-Se‴ ⊢T
                                                 (,-cong (s-≈-trans (∘-cong (s-≈-sym (p-, ⊢τ ⊢Sσ ⊢t)) (Refl.s-≈-refl ⊢σ))
                                                        (s-≈-sym (∘-assoc ⊢σ
                                                                          (s-wk ⊢SσΔ)
                                                                          (s-, ⊢τ ⊢Sσ ⊢t)))) ⊢S (Refl.≈-refl ⊢S) (≈-conv (≈-sym ([,]-v-ze ⊢τ ⊢Sσ ⊢t)) (Misc.[∘]-Se ⊢S ⊢σ ⊢τ) )) ⟩
     T [ (σ ∘ wk ∘ τ , t ∶ (sub S σ ↙ i)) , v 0 [ τ , t ∶ (sub S σ ↙ i)  ] ∶ (S ↙ i)  ] ≈˘⟨ []-cong-Se‴ ⊢T
                                                                                                           (,-∘ (s-∘ (s-wk ⊢SσΔ) ⊢σ) ⊢S (conv (vlookup ⊢SσΔ here) (Misc.[∘]-Se ⊢S ⊢σ (s-wk ⊢SσΔ))) (s-, ⊢τ ⊢Sσ ⊢t)) ⟩
     T [ q (S ↙ i) σ ∘ (τ , t ∶ (sub S σ ↙ i)) ]                      ≈˘⟨ Misc.[∘]-Se ⊢T ⊢qσ ⊢τ,t ⟩
     T [ q (S ↙ i) σ ] [ τ , t ∶ (sub S σ ↙ i) ] ∎
   where open ER
         ⊢qσ  = Misc.⊢q ⊢Δ ⊢σ ⊢S
         ⊢Sσ  = Misc.t[σ]-Se ⊢S ⊢σ
         ⊢τ,t = s-, ⊢τ ⊢Sσ ⊢t
         ⊢SσΔ = ⊢∷ ⊢Δ ⊢Sσ

[]-q-∘-,′ : ∀ {i j} → (S ↙ j) ∷ Γ ⊢ T ∶[ 1 + i ] Se i → Δ ⊢s σ ∶ Γ → Δ ⊢ t ∶[ j ] S [ σ ] →  Δ ⊢ T [ σ , t ∶ (S ↙ j) ] ≈ T [ q (S ↙ j) σ ] [| t ∶ (sub S σ ↙ j) ] ∶[ 1 + i ] Se i
[]-q-∘-,′ ⊢T ⊢σ ⊢t
  with ⊢∷ ⊢Γ ⊢S ← proj₁ (Presup.presup-tm ⊢T)
      | ⊢Δ , ⊢Γ ← Presup.presup-s ⊢σ = ≈-trans ([]-cong-Se‴ ⊢T (,-cong (s-≈-sym (∘-I ⊢σ)) ⊢S (Refl.≈-refl ⊢S) (Refl.≈-refl ⊢t)))
                                         ([]-q-∘-, ⊢T ⊢σ (s-I ⊢Δ) (conv ⊢t (≈-sym ([I] ⊢Sσ))))
  where ⊢qσ = Misc.⊢q ⊢Δ ⊢σ ⊢S
        ⊢Sσ = Misc.t[σ]-Se ⊢S ⊢σ

[]-,-∘ : ∀ {i j} → (T ↙ i) ∷ Γ ⊢ S ∶[ 1 + j ] Se j → Δ ⊢s σ ∶ Γ → Δ ⊢ t ∶[ i ] T [ σ ] → Δ′ ⊢s τ ∶ Δ → Δ′ ⊢ S [ σ , t ∶ T ↙ i ] [ τ ] ≈ S [ (σ ∘ τ) , t [ τ ] ∶ T ↙ i ] ∶[ 1 + j ] Se j
[]-,-∘ {T} {S = S} {σ = σ} {t} {τ = τ} {i = i} {j} ⊢S ⊢σ ⊢t ⊢τ
  with ⊢∷ ⊢Γ ⊢T ← proj₁ (Presup.presup-tm ⊢S) = begin
    S [ σ , t ∶ T ↙ i ] [ τ ]    ≈⟨ Misc.[∘]-Se ⊢S ⊢σ,t ⊢τ ⟩
    S [ σ , t ∶ T ↙ i  ∘ τ ]     ≈⟨ []-cong-Se‴ ⊢S (,-∘ ⊢σ ⊢T ⊢t ⊢τ) ⟩
    S [ (σ ∘ τ) , t [ τ ] ∶ T ↙ i ] ∎
  where
    open ER
    ⊢σ,t = s-, ⊢σ ⊢T ⊢t

I,∘≈, : ∀ {i} → Δ ⊢s σ ∶ Γ → Γ ⊢ T ∶[ 1 + i ] Se i → Γ ⊢ t ∶[ i ] T → Δ ⊢s (I , t ∶ T ↙ i) ∘ σ ≈ σ , t [ σ ] ∶ T ↙ i ∶ (T ↙ i) ∷ Γ
I,∘≈, ⊢σ ⊢T ⊢t = Subₚ.[I,t]∘σ≈σ,t[σ] (⊢∷ (proj₁ (Presup.presup-tm ⊢t)) ⊢T) ⊢σ ⊢t

[]-I,-∘ : ∀ {i j} → (T ↙ i) ∷ Γ ⊢ S ∶[ 1 + j ] Se j → Δ ⊢s σ ∶ Γ → Γ ⊢ t ∶[ i ] T → Δ ⊢ S [| t ∶ T ↙ i ] [ σ ] ≈ S [ σ , t [ σ ] ∶ T ↙ i ] ∶[ 1 + j ] Se j
[]-I,-∘ {T = T} {S = S} {σ = σ} {t = t} {i = i} {j = j} ⊢S ⊢σ ⊢t
  with ⊢∷ ⊢Γ ⊢T ← proj₁ (Presup.presup-tm ⊢S) = begin
  S [| t ∶ T ↙ i ] [ σ ]    ≈⟨ Misc.[∘]-Se ⊢S ⊢I,t ⊢σ ⟩
  S [ (I , t ∶ T ↙ i) ∘ σ ] ≈⟨ []-cong-Se‴ ⊢S (I,∘≈, ⊢σ ⊢T ⊢t) ⟩
  S [ σ , t [ σ ] ∶ T ↙ i ]
  ∎
  where open ER
        ⊢I,t = Misc.⊢I,t ⊢Γ ⊢T ⊢t

[]-I,ze-∘ : ∀ {i} → (N ↙ 0) ∷ Γ ⊢ S ∶[ 1 + i ] Se i → Δ ⊢s σ ∶ Γ → Δ ⊢ S [| ze ∶ N ↙ 0 ] [ σ ] ≈ S [ σ , ze ∶ N ↙ 0 ] ∶[ 1 + i ] Se i
[]-I,ze-∘ {_} {S} {_} {σ} ⊢S ⊢σ
  with ⊢∷ ⊢Γ ⊢T ← proj₁ (Presup.presup-tm ⊢S) =
    begin
    S [| ze ∶ N ↙ 0 ] [ σ ]    ≈⟨ Misc.[∘]-Se ⊢S I,t ⊢σ ⟩
    S [ (I , ze ∶ N ↙ 0 ) ∘ σ ] ≈⟨ []-cong-Se‴ ⊢S (Subₚ.[I,ze]∘σ≈σ,ze ⊢Γ ⊢σ) ⟩
    S [ σ , ze ∶ N ↙ 0 ]  ∎
    where open ER
          I,t = Misc.⊢I,t ⊢Γ (N-wf ⊢Γ) (ze-I ⊢Γ)
---------------------
-- other easy helpers

p-∘ : ∀ {i} → Γ ⊢s σ ∶ (T ↙ i) ∷ Δ →
      Γ′ ⊢s τ ∶ Γ →
      ------------------------------
      Γ′ ⊢s p (σ ∘ τ) ≈ p σ ∘ τ ∶ Δ
p-∘ ⊢σ ⊢τ = s-≈-sym (∘-assoc (s-wk (proj₂ (Presup.presup-s ⊢σ))) ⊢σ ⊢τ)


-- q related properties
module _ {i} {j : ℕ} (⊢σ : Γ ⊢s σ ∶ Δ)
         (⊢T : Δ ⊢ T ∶[ 1 + i ] Se i)
         (⊢τ : Δ′ ⊢s τ ∶ Γ)
         (⊢t : Δ′ ⊢ t ∶[ i ] T [ σ ] [ τ ]) where
  private
    ⊢Γ   = proj₁ (Presup.presup-s ⊢σ)
    ⊢Tσ  = Misc.t[σ]-Se ⊢T ⊢σ
    ⊢TσΓ = ⊢∷ ⊢Γ ⊢Tσ
    ⊢wk  = s-wk ⊢TσΓ
    ⊢σwk = s-∘ ⊢wk ⊢σ
    ⊢qσ  = Misc.⊢q ⊢Γ ⊢σ ⊢T
    ⊢τ,t = s-, ⊢τ ⊢Tσ ⊢t

    eq = begin
      σ ∘ wk ∘ (τ , t ∶  T [ σ ] ↙ i) ≈⟨ ∘-assoc ⊢σ ⊢wk ⊢τ,t ⟩
      σ ∘ (wk ∘ (τ , t ∶ T [ σ ] ↙ i)) ≈⟨ ∘-cong (p-, ⊢τ ⊢Tσ ⊢t) (Refl.s-≈-refl ⊢σ) ⟩
      σ ∘ τ ∎
      where open SR

  q∘,≈∘, : Δ′ ⊢s q (T ↙ i) σ ∘ (τ , t ∶ T [ σ ] ↙ i) ≈ (σ ∘ τ) , t ∶ T ↙ i ∶ (T ↙ i) ∷ Δ
  q∘,≈∘, = begin
    q (T ↙ i) σ ∘ (τ , t ∶ T [ σ ] ↙ i)   ≈⟨ ,-∘ ⊢σwk ⊢T (conv (vlookup ⊢TσΓ here) (Misc.[∘]-Se ⊢T ⊢σ ⊢wk)) ⊢τ,t ⟩
    (σ ∘ wk ∘ (τ , t ∶ T [ σ ] ↙ i)) , v 0 [ τ , t ∶ T [ σ ] ↙ i ] ∶ T ↙ i ≈⟨ ,-cong eq ⊢T (Refl.≈-refl ⊢T) (≈-conv ([,]-v-ze ⊢τ ⊢Tσ ⊢t)
                                                                                    ( ≈-trans (Misc.[∘]-Se ⊢T ⊢σ ⊢τ) (≈-sym ([]-cong-Se‴ ⊢T eq)) )) ⟩
    (σ ∘ τ) , t ∶ T ↙ i
    ∎
    where open SR


  []-q-, : (T ↙ i) ∷ Δ ⊢ s ∶[ j ] S →
           Δ′ ⊢ s [ q (T ↙ i) σ ] [ τ , t ∶ T [ σ ] ↙ i ] ≈ s [ (σ ∘ τ) , t ∶ T ↙ i ] ∶[ j ] S [ (σ ∘ τ) , t ∶ T ↙ i ]
  []-q-, {s} ⊢s
    with _ , ⊢S ← Presup.presup-tm ⊢s = begin
      s [ q (T ↙ i) σ ] [ τ , t ∶ T [ σ ] ↙ i ] ≈˘⟨ ≈-conv ([∘] ⊢τ,t ⊢qσ ⊢s) ([]-cong-Se‴ ⊢S q∘,≈∘,) ⟩
      s [ (q (T ↙ i) σ) ∘ ( τ , t ∶ T [ σ ] ↙ i) ] ≈⟨ ≈-conv ([]-cong (Refl.≈-refl ⊢s) q∘,≈∘,) ([]-cong-Se‴ ⊢S q∘,≈∘,) ⟩
      s [ (σ ∘ τ) , t ∶ T ↙ i ]
    ∎
    where open ER

module _ (⊢σ : Δ ⊢s σ ∶ Γ) (⊢τ : Δ′ ⊢s τ ∶ Δ) where
  private
    ⊢Δ  = proj₁ (Presup.presup-s ⊢σ)
    ⊢Γ  = proj₂ (Presup.presup-s ⊢σ)
    ⊢Δ′ = proj₁ (Presup.presup-s ⊢τ)
    ⊢τσ = s-∘ ⊢τ ⊢σ

  q∘q : ∀ {i} → Γ ⊢ T ∶[ 1 + i ] Se i → (T [ σ ∘ τ ] ↙ i) ∷ Δ′ ⊢s q (T ↙ i) σ ∘ q ( T [ σ ] ↙ i ) τ ≈ q (T ↙ i) (σ ∘ τ) ∶ (T ↙ i) ∷ Γ
  q∘q {T} {i} ⊢T = begin
      q (T ↙ i) σ ∘ q ( T [ σ ] ↙ i ) τ ≈⟨ q∘,≈∘, {j = 0} ⊢σ ⊢T (s-∘ ⊢wk ⊢τ) ⊢v0 ⟩
      (σ ∘ (τ ∘ wk)) , v 0 ∶ T ↙ i ≈⟨ ,-cong (s-≈-sym στwk≈) ⊢T (Refl.≈-refl ⊢T)
                                              (Refl.≈-refl (conv (vlookup ⊢Δ′T[στ] here)
                                              T[στ]wk≈T[στwk])) ⟩
      q (T ↙ i) (σ ∘ τ)
    ∎
    where
      open SR
      Δ′⊢T[στ] = Misc.t[σ]-Se ⊢T ⊢τσ
      ⊢Δ′T[στ] = ⊢∷ ⊢Δ′ Δ′⊢T[στ]
      ⊢wk = s-wk ⊢Δ′T[στ]
      στwk≈ = ∘-assoc ⊢σ ⊢τ ⊢wk
      T[στ]wk≈T[στwk] = ≈-trans (Misc.[∘]-Se ⊢T ⊢τσ ⊢wk) ([]-cong-Se‴ ⊢T στwk≈)
      ⊢v0 = conv (vlookup ⊢Δ′T[στ] here)
                 (≈-trans T[στ]wk≈T[στwk] (≈-sym (Misc.[∘]-Se ⊢T ⊢σ (s-∘ ⊢wk ⊢τ))))

  q∘q-N : N₀ ∷ Δ′ ⊢s q N₀ σ ∘ q N₀ τ ≈ q N₀ (σ ∘ τ) ∶ N₀ ∷ Γ
  q∘q-N = begin
      q N₀ σ ∘ q N₀ τ ≈⟨ ∘-cong (,-cong (Refl.s-≈-refl ⊢τwk) Δ⊢N ≈N (Refl.≈-refl (conv (vlookup ⊢NΔ′ here) (≈-trans (N-[] ⊢wk′) (≈-sym (N-[] ⊢τwk))))))
                          (Refl.s-≈-refl (s-, ⊢σwk Γ⊢N (conv (vlookup ⊢NΔ here) (≈-trans (N-[] ⊢wk) (≈-sym (N-[] ⊢σwk)))))) ⟩
      q N₀ σ ∘ q (N [ σ ] ↙ 0) τ ≈⟨ s-≈-conv (CtxEquiv.ctxeq-s-≈ (∷-cong″ (N-[] ⊢τσ)) (q∘q Γ⊢N)) (⊢≈-refl (⊢∷ ⊢Γ Γ⊢N)) ⟩
      q N₀ (σ ∘ τ)
    ∎
    where
      open SR
      Γ⊢N = N-wf ⊢Γ
      Δ⊢N = N-wf ⊢Δ
      ≈N   = ≈-sym (N-[] ⊢σ)
      ⊢NΔ = ⊢∷ ⊢Δ (N-wf ⊢Δ)
      ⊢NΔ′ = ⊢∷ ⊢Δ′ (N-wf ⊢Δ′)
      ⊢wk = s-wk ⊢NΔ
      ⊢wk′ = s-wk ⊢NΔ′
      ⊢σwk = s-∘ ⊢wk ⊢σ
      ⊢τwk = s-∘ ⊢wk′ ⊢τ

module NatTyping {i} (⊢T : (N ↙ 0) ∷ Γ ⊢ T ∶[ 1 + i ] Se i) (⊢σ : Δ ⊢s σ ∶ Γ) where

  ⊢Δ     = proj₁ (Presup.presup-s ⊢σ)
  ⊢Γ     = proj₂ (Presup.presup-s ⊢σ)
  ⊢qσ    = Misc.⊢q-N ⊢Δ ⊢Γ ⊢σ
  Γ⊢N    = N-wf ⊢Γ
  Δ⊢N    = N-wf ⊢Δ
  ⊢qqσ   = Misc.⊢q (⊢∷ ⊢Δ Δ⊢N) ⊢qσ ⊢T
  ⊢Tqσ   = Misc.t[σ]-Se ⊢T ⊢qσ
  ⊢NΓ    = ⊢∷ ⊢Γ Γ⊢N
  ⊢TNΓ   = ⊢∷ ⊢NΓ ⊢T
  ⊢NΔ    = ⊢∷ ⊢Δ Δ⊢N
  ⊢TqσNΔ = ⊢∷ ⊢NΔ ⊢Tqσ
  ⊢wk    = s-wk ⊢NΓ
  ⊢wk′   = s-wk ⊢TNΓ
  ⊢wkwk  = s-∘ ⊢wk′ ⊢wk