{-# OPTIONS --without-K --safe #-}
module Mint.Statics.Properties.Substs where
open import Lib
open import Function.Base using ( flip )
open import Mint.Statics.Full
open import Mint.Statics.Misc
open import Mint.Statics.Refl
open import Mint.Statics.PER
wk,v0≈I : ⊢ T ∺ Γ →
T ∺ Γ ⊢s wk , v 0 ≈ I ∶ T ∺ Γ
wk,v0≈I ⊢TΓ@(⊢∺ _ ⊢T) = s-≈-sym (s-≈-trans (,-ext (s-I ⊢TΓ)) (,-cong (∘-I (s-wk ⊢TΓ)) ⊢T (≈-conv ([I] (vlookup ⊢TΓ here)) ([]-cong-Se″ ⊢T (s-wk ⊢TΓ) (s-≈-sym (∘-I (s-wk ⊢TΓ)))))))
qI≈I : ⊢ T ∺ Γ →
T ∺ Γ ⊢s q I ≈ I ∶ T ∺ Γ
qI≈I ⊢TΓ@(⊢∺ _ ⊢T) =
begin
q I
≈⟨ ,-cong (I-∘ (s-wk ⊢TΓ)) ⊢T (≈-refl (conv (vlookup ⊢TΓ here) ([]-cong-Se″ ⊢T (s-wk ⊢TΓ) (s-≈-sym (I-∘ (s-wk ⊢TΓ)))))) ⟩
(wk , v 0)
≈⟨ wk,v0≈I ⊢TΓ ⟩
I
∎
where
open SR
wk∘qσ≈σ∘wk : ∀ {i} →
⊢ Γ →
Δ ⊢ T ∶ Se i →
Γ ⊢s σ ∶ Δ →
(T [ σ ]) ∺ Γ ⊢s p (q σ) ≈ σ ∘ wk ∶ Δ
wk∘qσ≈σ∘wk {σ = σ} ⊢Γ ⊢T ⊢σ = p-, (s-∘ (s-wk ⊢T[σ]Γ) ⊢σ) ⊢T (conv (vlookup ⊢T[σ]Γ here) ([∘]-Se ⊢T ⊢σ (s-wk ⊢T[σ]Γ)))
where
⊢T[σ]Γ = ⊢∺ ⊢Γ (t[σ]-Se ⊢T ⊢σ)
wk∘[σ,t]≈σ : ⊢ T ∺ Δ →
Γ ⊢s σ ∶ Δ →
Γ ⊢ t ∶ T [ σ ] →
Γ ⊢s wk ∘ (σ , t) ≈ σ ∶ Δ
wk∘[σ,t]≈σ {σ = σ} {t = t} ⊢TΔ@(⊢∺ _ ⊢T) ⊢σ ⊢t =
begin
wk ∘ (σ , t)
≈⟨ p-, ⊢σ ⊢T ⊢t ⟩
σ
∎
where
open SR
wk∘wk∘,, : ∀ {i j} →
⊢ Δ →
Γ ⊢s σ ∶ Δ →
Δ ⊢ T ∶ Se i →
T ∺ Δ ⊢ S ∶ Se j →
Γ ⊢ t ∶ T [ σ ] →
Γ ⊢ s ∶ S [ σ , t ] →
Γ ⊢s wk ∘ wk ∘ ((σ , t) , s) ≈ σ ∶ Δ
wk∘wk∘,, ⊢Δ ⊢σ ⊢T ⊢S ⊢t ⊢s = s-≈-trans (∘-assoc ⊢wk ⊢wk′ (s-, ⊢σ,t ⊢S ⊢s))
(s-≈-trans (∘-cong (p-, ⊢σ,t ⊢S ⊢s) (s-≈-refl ⊢wk))
(p-, ⊢σ ⊢T ⊢t))
where ⊢TΔ = ⊢∺ ⊢Δ ⊢T
⊢STΔ = ⊢∺ ⊢TΔ ⊢S
⊢wk = s-wk ⊢TΔ
⊢wk′ = s-wk ⊢STΔ
⊢σ,t = s-, ⊢σ ⊢T ⊢t
module _ (⊢Γ : ⊢ Γ) (⊢TNΔ : ⊢ T ∺ N ∺ Δ) (⊢σ : Γ ⊢s σ ∶ Δ) where
private
⊢-split : ∀ {T Γ} → ⊢ T ∺ Γ → Γ ⊢ T × ⊢ Γ
⊢-split (⊢∺ ⊢Γ ⊢T) = (-, ⊢T) , ⊢Γ
lvl = proj₁ (proj₁ (⊢-split ⊢TNΔ))
⊢T = proj₂ (proj₁ (⊢-split ⊢TNΔ))
⊢NΔ = proj₂ (⊢-split ⊢TNΔ)
⊢N = proj₂ (proj₁ (⊢-split ⊢NΔ))
⊢Δ = proj₂ (⊢-split ⊢NΔ)
⊢NΓ = ⊢∺ ⊢Γ (N-wf 0 ⊢Γ)
⊢qσ = ⊢q-N ⊢Γ ⊢Δ ⊢σ
⊢q[qσ] = ⊢q ⊢NΓ ⊢qσ ⊢T
⊢T[qσ] = t[σ]-Se ⊢T ⊢qσ
⊢T[qσ]NΓ = ⊢∺ ⊢NΓ ⊢T[qσ]
⊢wk∘wk = (s-∘ (s-wk ⊢TNΔ) (s-wk ⊢NΔ))
⊢wk∘wk′ = (s-∘ (s-wk ⊢T[qσ]NΓ) (s-wk ⊢NΓ))
⊢v1 = ⊢vn∶N L.[ T ] ⊢TNΔ refl
⊢v1′ = ⊢vn∶N L.[ T [ q σ ] ] ⊢T[qσ]NΓ refl
⊢su[v1] = conv-N-[]-sym (su-I ⊢v1) ⊢wk∘wk
⊢su[v1]′ = conv-N-[]-sym (su-I ⊢v1′) ⊢wk∘wk′
⊢[wk∘wk],su[v1]′ = ⊢[wk∘wk],su[v1] ⊢TNΔ
⊢[wk∘wk],su[v1]′′ = ⊢[wk∘wk],su[v1] ⊢T[qσ]NΓ
⊢qσ∘wk = s-∘ (s-wk ⊢T[qσ]NΓ) ⊢qσ
⊢σ∘wk = s-∘ (s-wk ⊢NΓ) ⊢σ
⊢σ∘wk∘wk = s-∘ (s-wk ⊢T[qσ]NΓ) ⊢σ∘wk
[wk∘wk]∘q[qσ]≈σ∘[wk∘wk]-TN : (T [ q σ ]) ∺ N ∺ Γ ⊢s wk ∘ wk ∘ q (q σ) ≈ σ ∘ (wk ∘ wk) ∶ Δ
[wk∘wk]∘q[qσ]≈σ∘[wk∘wk]-TN =
begin
wk ∘ wk ∘ q (q σ)
≈⟨ ∘-assoc (s-wk ⊢NΔ) (s-wk ⊢TNΔ) ⊢q[qσ] ⟩
wk ∘ (wk ∘ q (q σ))
≈⟨ ∘-cong (wk∘qσ≈σ∘wk ⊢NΓ ⊢T ⊢qσ) (s-≈-refl (s-wk ⊢NΔ)) ⟩
wk ∘ ((q σ) ∘ wk)
≈˘⟨ ∘-assoc (s-wk ⊢NΔ) ⊢qσ (s-wk ⊢T[qσ]NΓ) ⟩
wk ∘ (q σ) ∘ wk
≈⟨ ∘-cong (s-≈-refl (s-conv (s-wk ⊢T[qσ]NΓ) (∺-cong′ ⊢Γ (N-wf 0 ⊢Γ) (t[σ]-Se (N-wf 0 ⊢Δ) ⊢σ) (≈-sym (N-[] 0 ⊢σ))))) (wk∘qσ≈σ∘wk ⊢Γ ⊢N ⊢σ) ⟩
σ ∘ wk ∘ wk
≈⟨ ∘-assoc ⊢σ (s-wk ⊢NΓ) (s-wk ⊢T[qσ]NΓ) ⟩
σ ∘ (wk ∘ wk)
∎
where open SR
wk∘wk∘qqσ≈σ∘wk∘[wkwk,suv1] : (T [ q σ ]) ∺ N ∺ Γ ⊢s wk ∘ wk ∘ q (q σ) ≈ σ ∘ wk ∘ ((wk ∘ wk) , su (v 1)) ∶ Δ
wk∘wk∘qqσ≈σ∘wk∘[wkwk,suv1] =
begin
wk ∘ wk ∘ q (q σ)
≈⟨ [wk∘wk]∘q[qσ]≈σ∘[wk∘wk]-TN ⟩
σ ∘ (wk ∘ wk)
≈˘⟨ ∘-cong (wk∘[σ,t]≈σ ⊢NΓ ⊢wk∘wk′ ⊢su[v1]′) (s-≈-refl ⊢σ) ⟩
σ ∘ (wk ∘ ((wk ∘ wk), su (v 1)))
≈˘⟨ ∘-assoc ⊢σ (s-wk ⊢NΓ) ⊢[wk∘wk],su[v1]′′ ⟩
(σ ∘ wk ∘ ((wk ∘ wk) , su (v 1)))
∎
where open SR
suv1[qqσ]≈v0[wkwk,suv1] : (T [ q σ ]) ∺ N ∺ Γ ⊢ su (v 1) [ q (q σ) ] ≈ v 0 [ (wk ∘ wk) , su (v 1) ] ∶ N
suv1[qqσ]≈v0[wkwk,suv1] =
begin
su (v 1) [ q (q σ) ]
≈⟨ su-[] ⊢q[qσ] ⊢v1 ⟩
su (v 1 [ q (q σ) ])
≈⟨ su-cong (≈-conv ([,]-v-su ⊢qσ∘wk ⊢T (conv (vlookup ⊢T[qσ]NΓ here) ([∘]-Se ⊢T ⊢qσ (s-wk ⊢T[qσ]NΓ))) here) (N-[][] 0 (s-wk ⊢NΔ) ⊢qσ∘wk)) ⟩
su (v 0 [ q σ ∘ wk ])
≈⟨ su-cong ([]-cong-N″ (⊢vn∶N [] ⊢NΔ refl) ⊢qσ∘wk (,-∘ ⊢σ∘wk (N-wf 0 ⊢Δ) (conv (⊢vn∶N L.[] ⊢NΓ refl) (≈-sym (N-[] 0 ⊢σ∘wk))) (s-wk ⊢T[qσ]NΓ))) ⟩
su (v 0 [ (σ ∘ wk ∘ wk) , v 0 [ wk ] ])
≈⟨ su-cong (≈-conv-N-[] ([,]-v-ze ⊢σ∘wk∘wk (N-wf 0 ⊢Δ) (conv (t[σ] (conv (⊢vn∶N L.[] ⊢NΓ refl) (≈-sym (N-[] 0 ⊢σ∘wk))) (s-wk ⊢T[qσ]NΓ)) ([∘]-Se (N-wf 0 ⊢Δ) ⊢σ∘wk (s-wk ⊢T[qσ]NΓ)))) ⊢σ∘wk∘wk) ⟩
su (v 0 [ wk ])
≈⟨ su-cong (≈-conv (≈-trans ([wk] ⊢T[qσ]NΓ here) (≈-sym ([I] (⊢vn∶T[wk]suc[n] {Ψ = L.[ T [ q σ ] ]} ⊢T[qσ]NΓ refl)))) (N-[][] 0 (s-wk ⊢NΓ) (s-wk ⊢T[qσ]NΓ))) ⟩
su (v 1 [ I ])
≈⟨ su-cong ([I] ⊢v1′) ⟩
su (v 1)
≈˘⟨ ≈-conv-N-[] ([,]-v-ze ⊢wk∘wk′ (N-wf 0 ⊢Γ) ⊢su[v1]′) ⊢wk∘wk′ ⟩
v 0 [ (wk ∘ wk) , su (v 1) ]
∎
where open ER
[wkwk,suv1]∘qqσ≈qσ∘[wkwk,suv1] : (T [ q σ ]) ∺ N ∺ Γ ⊢s ((wk ∘ wk) , su (v 1)) ∘ q (q σ) ≈ q σ ∘ ((wk ∘ wk) , su (v 1)) ∶ N ∺ Δ
[wkwk,suv1]∘qqσ≈qσ∘[wkwk,suv1] =
begin
((wk ∘ wk) , su (v 1)) ∘ q (q σ)
≈⟨ ,-∘ ⊢wk∘wk (N-wf 0 ⊢Δ) ⊢su[v1] ⊢q[qσ] ⟩
(wk ∘ wk ∘ q (q σ)) , su (v 1) [ q (q σ) ]
≈⟨ ,-cong wk∘wk∘qqσ≈σ∘wk∘[wkwk,suv1] (N-wf 0 ⊢Δ) (≈-conv-N-[]-sym suv1[qqσ]≈v0[wkwk,suv1] (s-∘ ⊢q[qσ] ⊢wk∘wk)) ⟩
(σ ∘ wk ∘ ((wk ∘ wk) , su (v 1))) , v 0 [ (wk ∘ wk) , su (v 1) ]
≈˘⟨ ,-∘ ⊢σ∘wk (N-wf 0 ⊢Δ) (conv (⊢vn∶N [] ⊢NΓ refl) (≈-sym (N-[] 0 ⊢σ∘wk))) ⊢[wk∘wk],su[v1]′′ ⟩
q σ ∘ ((wk ∘ wk) , su (v 1))
∎
where open SR
rec-β-su-T-swap : (T [ q σ ]) ∺ N ∺ Γ ⊢ T [ ((wk ∘ wk) , su (v 1)) ] [ q (q σ) ] ≈ T [ q σ ] [ ((wk ∘ wk) , su (v 1)) ] ∶ Se lvl
rec-β-su-T-swap = ≈-trans ([∘]-Se ⊢T ⊢[wk∘wk],su[v1]′ ⊢q[qσ])
(≈-trans ([]-cong-Se″ ⊢T (s-∘ ⊢q[qσ] (s-, ⊢wk∘wk ⊢N ⊢su[v1])) [wkwk,suv1]∘qqσ≈qσ∘[wkwk,suv1])
(≈-sym ([∘]-Se ⊢T ⊢qσ ⊢[wk∘wk],su[v1]′′)))
[I,t]∘σ≈σ,t[σ] : ⊢ T ∺ Δ →
Γ ⊢s σ ∶ Δ →
Δ ⊢ t ∶ T →
Γ ⊢s (I , t) ∘ σ ≈ σ , t [ σ ] ∶ T ∺ Δ
[I,t]∘σ≈σ,t[σ] {σ = σ} {t = t} (⊢∺ ⊢Δ ⊢T) ⊢σ ⊢t =
begin
(I , t) ∘ σ
≈⟨ ,-∘ (s-I ⊢Δ) ⊢T (conv ⊢t (≈-sym ([I] ⊢T))) ⊢σ ⟩
(I ∘ σ) , t [ σ ]
≈⟨ ,-cong (I-∘ ⊢σ) ⊢T (≈-refl (conv (t[σ] ⊢t ⊢σ) ([]-cong-Se″ ⊢T ⊢σ (s-≈-sym (I-∘ ⊢σ))))) ⟩
σ , t [ σ ]
∎
where
open SR
[I,ze]∘σ≈σ,ze : ⊢ Δ →
Γ ⊢s σ ∶ Δ →
Γ ⊢s (I , ze) ∘ σ ≈ σ , ze ∶ N ∺ Δ
[I,ze]∘σ≈σ,ze {σ = σ} ⊢Δ ⊢σ =
begin
(I , ze) ∘ σ
≈⟨ [I,t]∘σ≈σ,t[σ] (⊢∺ ⊢Δ (N-wf 0 ⊢Δ)) ⊢σ (ze-I ⊢Δ) ⟩
σ , ze [ σ ]
≈⟨ ,-cong (s-≈-refl ⊢σ) (N-wf 0 ⊢Δ) (≈-conv-N-[]-sym (ze-[] ⊢σ) ⊢σ) ⟩
σ , ze
∎
where
open SR
qσ∘[I,t]≈σ,t : ∀ {i} →
⊢ Γ →
Δ ⊢ T ∶ Se i →
Γ ⊢s σ ∶ Δ →
Γ ⊢ t ∶ T [ σ ] →
Γ ⊢s q σ ∘ (I , t) ≈ σ , t ∶ T ∺ Δ
qσ∘[I,t]≈σ,t {Γ = Γ} {Δ = Δ} {σ = σ} {t = t} ⊢Γ ⊢T ⊢σ ⊢t =
begin
((σ ∘ wk) , v 0) ∘ (I , t)
≈⟨ ,-∘ (s-∘ (s-wk (⊢∺ ⊢Γ (conv (t[σ] ⊢T ⊢σ) (Se-[] _ ⊢σ)))) ⊢σ) ⊢T (conv (vlookup ⊢T[σ]Γ here) ([∘]-Se ⊢T ⊢σ (s-wk ⊢T[σ]Γ))) (⊢I,t ⊢Γ ⊢T[σ] ⊢t) ⟩
(σ ∘ wk ∘ (I , t)) , v 0 [ I , t ]
≈⟨ ,-cong σ∘wk∘[I,s] ⊢T (≈-refl (conv (t[σ] (vlookup ⊢T[σ]Γ here) (⊢I,t ⊢Γ ⊢T[σ] ⊢t)) (≈-trans ([]-cong-Se′ ([∘]-Se ⊢T ⊢σ (s-wk ⊢T[σ]Γ)) (⊢I,t ⊢Γ ⊢T[σ] ⊢t)) ([∘]-Se ⊢T (s-∘ (s-wk ⊢T[σ]Γ) ⊢σ) (⊢I,t ⊢Γ ⊢T[σ] ⊢t))))) ⟩
σ , v 0 [ I , t ]
≈⟨ ,-cong (s-≈-refl ⊢σ) ⊢T (≈-conv ([,]-v-ze (s-I ⊢Γ) ⊢T[σ] ⊢t′) ([I] ⊢T[σ])) ⟩
σ , t
∎
where
open SR
⊢T[σ] = t[σ]-Se ⊢T ⊢σ
⊢T[σ]Γ = ⊢∺ ⊢Γ ⊢T[σ]
⊢t′ = conv ⊢t (≈-sym ([I] ⊢T[σ]))
σ∘wk∘[I,s] : Γ ⊢s σ ∘ wk ∘ (I , t) ≈ σ ∶ Δ
σ∘wk∘[I,s] =
begin
σ ∘ wk ∘ (I , t)
≈⟨ ∘-assoc ⊢σ (s-wk ⊢T[σ]Γ) (⊢I,t ⊢Γ ⊢T[σ] ⊢t) ⟩
σ ∘ (wk ∘ (I , t))
≈⟨ ∘-cong (wk∘[σ,t]≈σ ⊢T[σ]Γ (s-I ⊢Γ) ⊢t′) (s-≈-refl ⊢σ) ⟩
σ ∘ I
≈⟨ ∘-I ⊢σ ⟩
σ
∎
qσ∘[I,ze]≈σ,ze : ⊢ Γ →
⊢ Δ →
Γ ⊢s σ ∶ Δ →
Γ ⊢s q σ ∘ (I , ze) ≈ σ , ze ∶ N ∺ Δ
qσ∘[I,ze]≈σ,ze {Γ = Γ} {Δ = Δ} {σ = σ} ⊢Γ ⊢Δ ⊢σ =
begin
q σ ∘ (I , ze)
≈⟨ qσ∘[I,t]≈σ,t ⊢Γ (N-wf 0 ⊢Δ) ⊢σ (conv-N-[]-sym (ze-I ⊢Γ) ⊢σ) ⟩
σ , ze
∎
where
open SR
q[wk]∘[σ,v0[σ]]≈σ : ⊢ T ∺ Δ →
Γ ⊢s σ ∶ T ∺ Δ →
Γ ⊢s q wk ∘ (σ , v 0 [ σ ]) ≈ σ ∶ T ∺ Δ
q[wk]∘[σ,v0[σ]]≈σ {σ = σ} ⊢TΔ@(⊢∺ _ ⊢T) ⊢σ =
begin
q wk ∘ (σ , v 0 [ σ ])
≈˘⟨ ∘-cong ([I,t]∘σ≈σ,t[σ] (⊢∺ ⊢TΔ ⊢T[wk]) ⊢σ ⊢v0) (s-≈-refl ⊢q[wk]) ⟩
q wk ∘ ((I , v 0) ∘ σ)
≈˘⟨ ∘-assoc ⊢q[wk] (s-, (s-I ⊢TΔ) ⊢T[wk] (conv ⊢v0 (≈-sym ([I] ⊢T[wk])))) ⊢σ ⟩
(q wk ∘ (I , v 0)) ∘ σ
≈⟨ ∘-cong (s-≈-refl ⊢σ) (qσ∘[I,t]≈σ,t ⊢TΔ ⊢T ⊢wk′ ⊢v0) ⟩
(wk , v 0) ∘ σ
≈⟨ ∘-cong (s-≈-refl ⊢σ) (wk,v0≈I ⊢TΔ) ⟩
I ∘ σ
≈⟨ I-∘ ⊢σ ⟩
σ
∎
where
open SR
⊢v0 = vlookup ⊢TΔ here
⊢wk′ = s-wk ⊢TΔ
⊢T[wk] = t[σ]-Se ⊢T ⊢wk′
⊢q[wk] = ⊢q ⊢TΔ ⊢wk′ ⊢T
q[wk]∘[I,v0]≈I : ⊢ T ∺ Γ →
T ∺ Γ ⊢s q wk ∘ (I , v 0) ≈ I ∶ T ∺ Γ
q[wk]∘[I,v0]≈I ⊢TΓ@(⊢∺ _ ⊢T) =
begin
q wk ∘ (I , v 0)
≈⟨ qσ∘[I,t]≈σ,t ⊢TΓ ⊢T ⊢wk′ ⊢v0 ⟩
(wk , v 0)
≈⟨ wk,v0≈I ⊢TΓ ⟩
I
∎
where
open SR
⊢wk′ = s-wk ⊢TΓ
⊢v0 = vlookup ⊢TΓ here