{-# OPTIONS --without-K --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.Introductions.Pi {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped as U hiding (wk)
open import Definition.Untyped.Properties
open import Definition.Typed
open import Definition.Typed.Weakening as T hiding (wk; wkEq; wkTerm; wkEqTerm)
open import Definition.Typed.Properties
open import Definition.LogicalRelation
open import Definition.LogicalRelation.ShapeView
open import Definition.LogicalRelation.Weakening
open import Definition.LogicalRelation.Irrelevance
open import Definition.LogicalRelation.Properties
open import Definition.LogicalRelation.Substitution
open import Definition.LogicalRelation.Substitution.Weakening
open import Definition.LogicalRelation.Substitution.Properties
import Definition.LogicalRelation.Substitution.Irrelevance as S
open import Definition.LogicalRelation.Substitution.Introductions.Universe
open import Tools.Nat
open import Tools.Product
import Tools.PropositionalEquality as PE
Πᵛ : ∀ {F G Γ l}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l ⟩ F / [Γ])
→ Γ ∙ F ⊩ᵛ⟨ l ⟩ G / [Γ] ∙ [F]
→ Γ ⊩ᵛ⟨ l ⟩ Π F ▹ G / [Γ]
Πᵛ {F} {G} {Γ} {l} [Γ] [F] [G] {Δ = Δ} {σ = σ} ⊢Δ [σ] =
let [F]σ {σ′} [σ′] = [F] {σ = σ′} ⊢Δ [σ′]
[σF] = proj₁ ([F]σ [σ])
⊢F {σ′} [σ′] = escape (proj₁ ([F]σ {σ′} [σ′]))
⊢F≡F = escapeEq [σF] (reflEq [σF])
[G]σ {σ′} [σ′] = [G] {σ = liftSubst σ′} (⊢Δ ∙ ⊢F [σ′])
(liftSubstS {F = F} [Γ] ⊢Δ [F] [σ′])
⊢G {σ′} [σ′] = escape (proj₁ ([G]σ {σ′} [σ′]))
⊢G≡G = escapeEq (proj₁ ([G]σ [σ])) (reflEq (proj₁ ([G]σ [σ])))
⊢ΠF▹G = Π ⊢F [σ] ▹ ⊢G [σ]
[G]a : ∀ {ρ Δ₁} a ([ρ] : ρ ∷ Δ₁ ⊆ Δ) (⊢Δ₁ : ⊢ Δ₁)
([a] : Δ₁ ⊩⟨ l ⟩ a ∷ subst (ρ •ₛ σ) F
/ proj₁ ([F] ⊢Δ₁ (wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ])))
→ Σ (Δ₁ ⊩⟨ l ⟩ subst (consSubst (ρ •ₛ σ) a) G)
(λ [Aσ] →
{σ′ : Nat → Term} →
(Σ (Δ₁ ⊩ˢ tail σ′ ∷ Γ / [Γ] / ⊢Δ₁)
(λ [tailσ] →
Δ₁ ⊩⟨ l ⟩ head σ′ ∷ subst (tail σ′) F / proj₁ ([F] ⊢Δ₁ [tailσ]))) →
Δ₁ ⊩ˢ consSubst (ρ •ₛ σ) a ≡ σ′ ∷ Γ ∙ F /
[Γ] ∙ [F] / ⊢Δ₁ /
consSubstS {t = a} {A = F} [Γ] ⊢Δ₁ (wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ]) [F]
[a] →
Δ₁ ⊩⟨ l ⟩ subst (consSubst (ρ •ₛ σ) a) G ≡
subst σ′ G / [Aσ])
[G]a {ρ} a [ρ] ⊢Δ₁ [a] = ([G] {σ = consSubst (ρ •ₛ σ) a} ⊢Δ₁
(consSubstS {t = a} {A = F} [Γ] ⊢Δ₁
(wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ])
[F] [a]))
[G]a′ : ∀ {ρ Δ₁} a ([ρ] : ρ ∷ Δ₁ ⊆ Δ) (⊢Δ₁ : ⊢ Δ₁)
→ Δ₁ ⊩⟨ l ⟩ a ∷ subst (ρ •ₛ σ) F
/ proj₁ ([F] ⊢Δ₁ (wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ]))
→ Δ₁ ⊩⟨ l ⟩ U.wk (lift ρ) (subst (liftSubst σ) G) [ a ]
[G]a′ a ρ ⊢Δ₁ [a] = irrelevance′ (PE.sym (singleSubstWkComp a σ G))
(proj₁ ([G]a a ρ ⊢Δ₁ [a]))
in Π′ (subst σ F) (subst (liftSubst σ) G)
(idRed:*: ⊢ΠF▹G) (⊢F [σ]) (⊢G [σ]) (≅-Π-cong (⊢F [σ]) ⊢F≡F ⊢G≡G)
(λ ρ ⊢Δ₁ → wk ρ ⊢Δ₁ [σF])
(λ {ρ} {Δ₁} {a} [ρ] ⊢Δ₁ [a] →
let [a]′ = irrelevanceTerm′
(wk-subst F) (wk [ρ] ⊢Δ₁ [σF])
(proj₁ ([F] ⊢Δ₁ (wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ]))) [a]
in [G]a′ a [ρ] ⊢Δ₁ [a]′)
(λ {ρ} {Δ₁} {a} {b} [ρ] ⊢Δ₁ [a] [b] [a≡b] →
let [ρσ] = wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ]
[a]′ = irrelevanceTerm′
(wk-subst F) (wk [ρ] ⊢Δ₁ [σF])
(proj₁ ([F] ⊢Δ₁ [ρσ])) [a]
[b]′ = irrelevanceTerm′
(wk-subst F) (wk [ρ] ⊢Δ₁ [σF])
(proj₁ ([F] ⊢Δ₁ [ρσ])) [b]
[a≡b]′ = irrelevanceEqTerm′
(wk-subst F) (wk [ρ] ⊢Δ₁ [σF])
(proj₁ ([F] ⊢Δ₁ [ρσ])) [a≡b]
in irrelevanceEq″
(PE.sym (singleSubstWkComp a σ G))
(PE.sym (singleSubstWkComp b σ G))
(proj₁ ([G]a a [ρ] ⊢Δ₁ [a]′))
([G]a′ a [ρ] ⊢Δ₁ [a]′)
(proj₂ ([G]a a [ρ] ⊢Δ₁ [a]′)
([ρσ] , [b]′)
(reflSubst [Γ] ⊢Δ₁ [ρσ] , [a≡b]′)))
, (λ {σ′} [σ′] [σ≡σ′] →
let var0 = var (⊢Δ ∙ ⊢F [σ])
(PE.subst (λ x → zero ∷ x ∈ (Δ ∙ subst σ F))
(wk-subst F) here)
[wk1σ] = wk1SubstS [Γ] ⊢Δ (⊢F [σ]) [σ]
[wk1σ′] = wk1SubstS [Γ] ⊢Δ (⊢F [σ]) [σ′]
[wk1σ≡wk1σ′] = wk1SubstSEq [Γ] ⊢Δ (⊢F [σ]) [σ] [σ≡σ′]
[F][wk1σ] = proj₁ ([F] (⊢Δ ∙ ⊢F [σ]) [wk1σ])
[F][wk1σ′] = proj₁ ([F] (⊢Δ ∙ ⊢F [σ]) [wk1σ′])
var0′ = conv var0
(≅-eq (escapeEq [F][wk1σ]
(proj₂ ([F] (⊢Δ ∙ ⊢F [σ]) [wk1σ])
[wk1σ′] [wk1σ≡wk1σ′])))
in Π₌ _ _ (id (Π ⊢F [σ′] ▹ ⊢G [σ′]))
(≅-Π-cong (⊢F [σ])
(escapeEq (proj₁ ([F] ⊢Δ [σ]))
(proj₂ ([F] ⊢Δ [σ]) [σ′] [σ≡σ′]))
(escapeEq (proj₁ ([G]σ [σ])) (proj₂ ([G]σ [σ])
([wk1σ′] , neuTerm [F][wk1σ′] (var zero) var0′ (~-var var0′))
([wk1σ≡wk1σ′] , neuEqTerm [F][wk1σ]
(var zero) (var zero) var0 var0 (~-var var0)))))
(λ ρ ⊢Δ₁ → wkEq ρ ⊢Δ₁ [σF] (proj₂ ([F] ⊢Δ [σ]) [σ′] [σ≡σ′]))
(λ {ρ} {Δ₁} {a} [ρ] ⊢Δ₁ [a] →
let [ρσ] = wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ]
[ρσ′] = wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ′]
[a]′ = irrelevanceTerm′ (wk-subst F) (wk [ρ] ⊢Δ₁ [σF])
(proj₁ ([F] ⊢Δ₁ [ρσ])) [a]
[a]″ = convTerm₁ (proj₁ ([F] ⊢Δ₁ [ρσ]))
(proj₁ ([F] ⊢Δ₁ [ρσ′]))
(proj₂ ([F] ⊢Δ₁ [ρσ]) [ρσ′]
(wkSubstSEq [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ] [σ≡σ′]))
[a]′
[ρσa≡ρσ′a] = consSubstSEq {t = a} {A = F} [Γ] ⊢Δ₁
(wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ])
(wkSubstSEq [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ] [σ≡σ′])
[F] [a]′
in irrelevanceEq″ (PE.sym (singleSubstWkComp a σ G))
(PE.sym (singleSubstWkComp a σ′ G))
(proj₁ ([G]a a [ρ] ⊢Δ₁ [a]′))
([G]a′ a [ρ] ⊢Δ₁ [a]′)
(proj₂ ([G]a a [ρ] ⊢Δ₁ [a]′)
(wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ′] , [a]″)
[ρσa≡ρσ′a])))
Π-congᵛ : ∀ {F G H E Γ l}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l ⟩ F / [Γ])
([G] : Γ ∙ F ⊩ᵛ⟨ l ⟩ G / [Γ] ∙ [F])
([H] : Γ ⊩ᵛ⟨ l ⟩ H / [Γ])
([E] : Γ ∙ H ⊩ᵛ⟨ l ⟩ E / [Γ] ∙ [H])
([F≡H] : Γ ⊩ᵛ⟨ l ⟩ F ≡ H / [Γ] / [F])
([G≡E] : Γ ∙ F ⊩ᵛ⟨ l ⟩ G ≡ E / [Γ] ∙ [F] / [G])
→ Γ ⊩ᵛ⟨ l ⟩ Π F ▹ G ≡ Π H ▹ E / [Γ] / Πᵛ {F} {G} [Γ] [F] [G]
Π-congᵛ {F} {G} {H} {E} [Γ] [F] [G] [H] [E] [F≡H] [G≡E] {σ = σ} ⊢Δ [σ] =
let [ΠFG] = Πᵛ {F} {G} [Γ] [F] [G]
[σΠFG] = proj₁ ([ΠFG] ⊢Δ [σ])
_ , Π F′ G′ D′ ⊢F′ ⊢G′ A≡A′ [F]′ [G]′ G-ext′ = extractMaybeEmb (Π-elim [σΠFG])
[σF] = proj₁ ([F] ⊢Δ [σ])
⊢σF = escape [σF]
[σG] = proj₁ ([G] (⊢Δ ∙ ⊢σF) (liftSubstS {F = F} [Γ] ⊢Δ [F] [σ]))
⊢σH = escape (proj₁ ([H] ⊢Δ [σ]))
⊢σE = escape (proj₁ ([E] (⊢Δ ∙ ⊢σH) (liftSubstS {F = H} [Γ] ⊢Δ [H] [σ])))
⊢σF≡σH = escapeEq [σF] ([F≡H] ⊢Δ [σ])
⊢σG≡σE = escapeEq [σG] ([G≡E] (⊢Δ ∙ ⊢σF) (liftSubstS {F = F} [Γ] ⊢Δ [F] [σ]))
in Π₌ (subst σ H)
(subst (liftSubst σ) E)
(id (Π ⊢σH ▹ ⊢σE))
(≅-Π-cong ⊢σF ⊢σF≡σH ⊢σG≡σE)
(λ ρ ⊢Δ₁ → let [ρσ] = wkSubstS [Γ] ⊢Δ ⊢Δ₁ ρ [σ]
in irrelevanceEq″ (PE.sym (wk-subst F))
(PE.sym (wk-subst H))
(proj₁ ([F] ⊢Δ₁ [ρσ]))
([F]′ ρ ⊢Δ₁)
([F≡H] ⊢Δ₁ [ρσ]))
(λ {ρ} {Δ} {a} [ρ] ⊢Δ₁ [a] →
let [ρσ] = wkSubstS [Γ] ⊢Δ ⊢Δ₁ [ρ] [σ]
[a]′ = irrelevanceTerm′ (wk-subst F)
([F]′ [ρ] ⊢Δ₁)
(proj₁ ([F] ⊢Δ₁ [ρσ])) [a]
[aρσ] = consSubstS {t = a} {A = F} [Γ] ⊢Δ₁ [ρσ] [F] [a]′
in irrelevanceEq″ (PE.sym (singleSubstWkComp a σ G))
(PE.sym (singleSubstWkComp a σ E))
(proj₁ ([G] ⊢Δ₁ [aρσ]))
([G]′ [ρ] ⊢Δ₁ [a])
([G≡E] ⊢Δ₁ [aρσ]))
Πᵗᵛ : ∀ {F G Γ} ([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ ¹ ⟩ F / [Γ])
([U] : Γ ∙ F ⊩ᵛ⟨ ¹ ⟩ U / [Γ] ∙ [F])
→ Γ ⊩ᵛ⟨ ¹ ⟩ F ∷ U / [Γ] / Uᵛ [Γ]
→ Γ ∙ F ⊩ᵛ⟨ ¹ ⟩ G ∷ U / [Γ] ∙ [F] / (λ {Δ} {σ} → [U] {Δ} {σ})
→ Γ ⊩ᵛ⟨ ¹ ⟩ Π F ▹ G ∷ U / [Γ] / Uᵛ [Γ]
Πᵗᵛ {F} {G} {Γ} [Γ] [F] [U] [Fₜ] [Gₜ] {Δ = Δ} {σ = σ} ⊢Δ [σ] =
let [liftσ] = liftSubstS {F = F} [Γ] ⊢Δ [F] [σ]
⊢F = escape (proj₁ ([F] ⊢Δ [σ]))
⊢Fₜ = escapeTerm (U′ ⁰ 0<1 ⊢Δ) (proj₁ ([Fₜ] ⊢Δ [σ]))
⊢F≡Fₜ = escapeTermEq (U′ ⁰ 0<1 ⊢Δ)
(reflEqTerm (U′ ⁰ 0<1 ⊢Δ) (proj₁ ([Fₜ] ⊢Δ [σ])))
⊢Gₜ = escapeTerm (proj₁ ([U] (⊢Δ ∙ ⊢F) [liftσ]))
(proj₁ ([Gₜ] (⊢Δ ∙ ⊢F) [liftσ]))
⊢G≡Gₜ = escapeTermEq (proj₁ ([U] (⊢Δ ∙ ⊢F) [liftσ]))
(reflEqTerm (proj₁ ([U] (⊢Δ ∙ ⊢F) [liftσ]))
(proj₁ ([Gₜ] (⊢Δ ∙ ⊢F) [liftσ])))
[F]₀ = univᵛ {F} [Γ] (Uᵛ [Γ]) [Fₜ]
[Gₜ]′ = S.irrelevanceTerm {A = U} {t = G}
(_∙_ {A = F} [Γ] [F]) (_∙_ {A = F} [Γ] [F]₀)
(λ {Δ} {σ} → [U] {Δ} {σ})
(λ {Δ} {σ} → Uᵛ (_∙_ {A = F} [Γ] [F]₀) {Δ} {σ})
[Gₜ]
[G]₀ = univᵛ {G} (_∙_ {A = F} [Γ] [F]₀)
(λ {Δ} {σ} → Uᵛ (_∙_ {A = F} [Γ] [F]₀) {Δ} {σ})
(λ {Δ} {σ} → [Gₜ]′ {Δ} {σ})
[ΠFG] = (Πᵛ {F} {G} [Γ] [F]₀ [G]₀) ⊢Δ [σ]
in Uₜ (Π subst σ F ▹ subst (liftSubst σ) G) (idRedTerm:*: (Π ⊢Fₜ ▹ ⊢Gₜ))
Π (≅ₜ-Π-cong ⊢F ⊢F≡Fₜ ⊢G≡Gₜ) (proj₁ [ΠFG])
, (λ {σ′} [σ′] [σ≡σ′] →
let [liftσ′] = liftSubstS {F = F} [Γ] ⊢Δ [F] [σ′]
[wk1σ] = wk1SubstS [Γ] ⊢Δ ⊢F [σ]
[wk1σ′] = wk1SubstS [Γ] ⊢Δ ⊢F [σ′]
var0 = conv (var (⊢Δ ∙ ⊢F)
(PE.subst (λ x → zero ∷ x ∈ (Δ ∙ subst σ F))
(wk-subst F) here))
(≅-eq (escapeEq (proj₁ ([F] (⊢Δ ∙ ⊢F) [wk1σ]))
(proj₂ ([F] (⊢Δ ∙ ⊢F) [wk1σ]) [wk1σ′]
(wk1SubstSEq [Γ] ⊢Δ ⊢F [σ] [σ≡σ′]))))
[liftσ′]′ = [wk1σ′]
, neuTerm (proj₁ ([F] (⊢Δ ∙ ⊢F) [wk1σ′])) (var zero)
var0 (~-var var0)
⊢F′ = escape (proj₁ ([F] ⊢Δ [σ′]))
⊢Fₜ′ = escapeTerm (U′ ⁰ 0<1 ⊢Δ) (proj₁ ([Fₜ] ⊢Δ [σ′]))
⊢Gₜ′ = escapeTerm (proj₁ ([U] (⊢Δ ∙ ⊢F′) [liftσ′]))
(proj₁ ([Gₜ] (⊢Δ ∙ ⊢F′) [liftσ′]))
⊢F≡F′ = escapeTermEq (U′ ⁰ 0<1 ⊢Δ)
(proj₂ ([Fₜ] ⊢Δ [σ]) [σ′] [σ≡σ′])
⊢G≡G′ = escapeTermEq (proj₁ ([U] (⊢Δ ∙ ⊢F) [liftσ]))
(proj₂ ([Gₜ] (⊢Δ ∙ ⊢F) [liftσ]) [liftσ′]′
(liftSubstSEq {F = F} [Γ] ⊢Δ [F] [σ] [σ≡σ′]))
[ΠFG]′ = (Πᵛ {F} {G} [Γ] [F]₀ [G]₀) ⊢Δ [σ′]
in Uₜ₌ (Π subst σ F ▹ subst (liftSubst σ) G)
(Π subst σ′ F ▹ subst (liftSubst σ′) G)
(idRedTerm:*: (Π ⊢Fₜ ▹ ⊢Gₜ))
(idRedTerm:*: (Π ⊢Fₜ′ ▹ ⊢Gₜ′))
Π Π (≅ₜ-Π-cong ⊢F ⊢F≡F′ ⊢G≡G′)
(proj₁ [ΠFG]) (proj₁ [ΠFG]′) (proj₂ [ΠFG] [σ′] [σ≡σ′]))
Π-congᵗᵛ : ∀ {F G H E Γ}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ ¹ ⟩ F / [Γ])
([H] : Γ ⊩ᵛ⟨ ¹ ⟩ H / [Γ])
([UF] : Γ ∙ F ⊩ᵛ⟨ ¹ ⟩ U / [Γ] ∙ [F])
([UH] : Γ ∙ H ⊩ᵛ⟨ ¹ ⟩ U / [Γ] ∙ [H])
([F]ₜ : Γ ⊩ᵛ⟨ ¹ ⟩ F ∷ U / [Γ] / Uᵛ [Γ])
([G]ₜ : Γ ∙ F ⊩ᵛ⟨ ¹ ⟩ G ∷ U / [Γ] ∙ [F]
/ (λ {Δ} {σ} → [UF] {Δ} {σ}))
([H]ₜ : Γ ⊩ᵛ⟨ ¹ ⟩ H ∷ U / [Γ] / Uᵛ [Γ])
([E]ₜ : Γ ∙ H ⊩ᵛ⟨ ¹ ⟩ E ∷ U / [Γ] ∙ [H]
/ (λ {Δ} {σ} → [UH] {Δ} {σ}))
([F≡H]ₜ : Γ ⊩ᵛ⟨ ¹ ⟩ F ≡ H ∷ U / [Γ] / Uᵛ [Γ])
([G≡E]ₜ : Γ ∙ F ⊩ᵛ⟨ ¹ ⟩ G ≡ E ∷ U / [Γ] ∙ [F]
/ (λ {Δ} {σ} → [UF] {Δ} {σ}))
→ Γ ⊩ᵛ⟨ ¹ ⟩ Π F ▹ G ≡ Π H ▹ E ∷ U / [Γ] / Uᵛ [Γ]
Π-congᵗᵛ {F} {G} {H} {E}
[Γ] [F] [H] [UF] [UH] [F]ₜ [G]ₜ [H]ₜ [E]ₜ [F≡H]ₜ [G≡E]ₜ {Δ} {σ} ⊢Δ [σ] =
let ⊢F = escape (proj₁ ([F] ⊢Δ [σ]))
⊢H = escape (proj₁ ([H] ⊢Δ [σ]))
[liftFσ] = liftSubstS {F = F} [Γ] ⊢Δ [F] [σ]
[liftHσ] = liftSubstS {F = H} [Γ] ⊢Δ [H] [σ]
[F]ᵤ = univᵛ {F} [Γ] (Uᵛ [Γ]) [F]ₜ
[G]ᵤ₁ = univᵛ {G} {l′ = ⁰} (_∙_ {A = F} [Γ] [F])
(λ {Δ} {σ} → [UF] {Δ} {σ}) [G]ₜ
[G]ᵤ = S.irrelevance {A = G} (_∙_ {A = F} [Γ] [F])
(_∙_ {A = F} [Γ] [F]ᵤ) [G]ᵤ₁
[H]ᵤ = univᵛ {H} [Γ] (Uᵛ [Γ]) [H]ₜ
[E]ᵤ = S.irrelevance {A = E} (_∙_ {A = H} [Γ] [H]) (_∙_ {A = H} [Γ] [H]ᵤ)
(univᵛ {E} {l′ = ⁰} (_∙_ {A = H} [Γ] [H])
(λ {Δ} {σ} → [UH] {Δ} {σ}) [E]ₜ)
[F≡H]ᵤ = univEqᵛ {F} {H} [Γ] (Uᵛ [Γ]) [F]ᵤ [F≡H]ₜ
[G≡E]ᵤ = S.irrelevanceEq {A = G} {B = E} (_∙_ {A = F} [Γ] [F])
(_∙_ {A = F} [Γ] [F]ᵤ) [G]ᵤ₁ [G]ᵤ
(univEqᵛ {G} {E} (_∙_ {A = F} [Γ] [F])
(λ {Δ} {σ} → [UF] {Δ} {σ}) [G]ᵤ₁ [G≡E]ₜ)
ΠFGₜ = Π escapeTerm {l = ¹} (U′ ⁰ 0<1 ⊢Δ) (proj₁ ([F]ₜ ⊢Δ [σ]))
▹ escapeTerm (proj₁ ([UF] (⊢Δ ∙ ⊢F) [liftFσ]))
(proj₁ ([G]ₜ (⊢Δ ∙ ⊢F) [liftFσ]))
ΠHEₜ = Π escapeTerm {l = ¹} (U′ ⁰ 0<1 ⊢Δ) (proj₁ ([H]ₜ ⊢Δ [σ]))
▹ escapeTerm (proj₁ ([UH] (⊢Δ ∙ ⊢H) [liftHσ]))
(proj₁ ([E]ₜ (⊢Δ ∙ ⊢H) [liftHσ]))
in Uₜ₌ (Π subst σ F ▹ subst (liftSubst σ) G)
(Π subst σ H ▹ subst (liftSubst σ) E)
(idRedTerm:*: ΠFGₜ) (idRedTerm:*: ΠHEₜ)
Π Π
(≅ₜ-Π-cong ⊢F (escapeTermEq (U′ ⁰ 0<1 ⊢Δ) ([F≡H]ₜ ⊢Δ [σ]))
(escapeTermEq (proj₁ ([UF] (⊢Δ ∙ ⊢F) [liftFσ]))
([G≡E]ₜ (⊢Δ ∙ ⊢F) [liftFσ])))
(proj₁ (Πᵛ {F} {G} [Γ] [F]ᵤ [G]ᵤ ⊢Δ [σ]))
(proj₁ (Πᵛ {H} {E} [Γ] [H]ᵤ [E]ᵤ ⊢Δ [σ]))
(Π-congᵛ {F} {G} {H} {E} [Γ] [F]ᵤ [G]ᵤ [H]ᵤ [E]ᵤ [F≡H]ᵤ [G≡E]ᵤ ⊢Δ [σ])
▹▹ᵛ : ∀ {F G Γ l}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l ⟩ F / [Γ])
→ Γ ⊩ᵛ⟨ l ⟩ G / [Γ]
→ Γ ⊩ᵛ⟨ l ⟩ F ▹▹ G / [Γ]
▹▹ᵛ {F} {G} [Γ] [F] [G] =
Πᵛ {F} {wk1 G} [Γ] [F] (wk1ᵛ {G} {F} [Γ] [F] [G])
▹▹-congᵛ : ∀ {F F′ G G′ Γ l}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l ⟩ F / [Γ])
([F′] : Γ ⊩ᵛ⟨ l ⟩ F′ / [Γ])
([F≡F′] : Γ ⊩ᵛ⟨ l ⟩ F ≡ F′ / [Γ] / [F])
([G] : Γ ⊩ᵛ⟨ l ⟩ G / [Γ])
([G′] : Γ ⊩ᵛ⟨ l ⟩ G′ / [Γ])
([G≡G′] : Γ ⊩ᵛ⟨ l ⟩ G ≡ G′ / [Γ] / [G])
→ Γ ⊩ᵛ⟨ l ⟩ F ▹▹ G ≡ F′ ▹▹ G′ / [Γ] / ▹▹ᵛ {F} {G} [Γ] [F] [G]
▹▹-congᵛ {F} {F′} {G} {G′} [Γ] [F] [F′] [F≡F′] [G] [G′] [G≡G′] =
Π-congᵛ {F} {wk1 G} {F′} {wk1 G′} [Γ]
[F] (wk1ᵛ {G} {F} [Γ] [F] [G])
[F′] (wk1ᵛ {G′} {F′} [Γ] [F′] [G′])
[F≡F′] (wk1Eqᵛ {G} {G′} {F} [Γ] [F] [G] [G≡G′])