{-# OPTIONS --without-K --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution {{eqrel : EqRelSet}} where
open import Definition.Untyped
open import Definition.LogicalRelation
open import Definition.Typed
open import Tools.Nat
open import Tools.Product
open import Tools.Unit
import Tools.PropositionalEquality as PE
mutual
data ⊩ᵛ_ : Con Term → Set where
ε : ⊩ᵛ ε
_∙_ : ∀ {Γ A l} ([Γ] : ⊩ᵛ Γ) → Γ ⊩ᵛ⟨ l ⟩ A / [Γ]
→ ⊩ᵛ Γ ∙ A
_⊩ᵛ⟨_⟩_/_ : (Γ : Con Term) (l : TypeLevel) (A : Term) → ⊩ᵛ Γ → Set
Γ ⊩ᵛ⟨ l ⟩ A / [Γ] = ∀ {Δ σ} (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ)
→ Σ (Δ ⊩⟨ l ⟩ subst σ A)
(λ [Aσ] → ∀ {σ′} ([σ′] : Δ ⊩ˢ σ′ ∷ Γ / [Γ] / ⊢Δ)
→ ([σ≡σ′] : Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ])
→ Δ ⊩⟨ l ⟩ subst σ A ≡ subst σ′ A / [Aσ])
_⊩ˢ_∷_/_/_ : (Δ : Con Term) (σ : Subst) (Γ : Con Term) ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ)
→ Set
Δ ⊩ˢ σ ∷ .ε / ε / ⊢Δ = ⊤
Δ ⊩ˢ σ ∷ .(Γ ∙ A) / (_∙_ {Γ} {A} {l} [Γ] [A]) / ⊢Δ =
Σ (Δ ⊩ˢ tail σ ∷ Γ / [Γ] / ⊢Δ) λ [tailσ] →
(Δ ⊩⟨ l ⟩ head σ ∷ subst (tail σ) A / proj₁ ([A] ⊢Δ [tailσ]))
_⊩ˢ_≡_∷_/_/_/_ : (Δ : Con Term) (σ σ′ : Subst) (Γ : Con Term) ([Γ] : ⊩ᵛ Γ)
(⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) → Set
Δ ⊩ˢ σ ≡ σ′ ∷ .ε / ε / ⊢Δ / [σ] = ⊤
Δ ⊩ˢ σ ≡ σ′ ∷ .(Γ ∙ A) / (_∙_ {Γ} {A} {l} [Γ] [A]) / ⊢Δ / [σ] =
(Δ ⊩ˢ tail σ ≡ tail σ′ ∷ Γ / [Γ] / ⊢Δ / proj₁ [σ]) ×
(Δ ⊩⟨ l ⟩ head σ ≡ head σ′ ∷ subst (tail σ) A / proj₁ ([A] ⊢Δ (proj₁ [σ])))
_⊩ᵛ⟨_⟩_∷_/_/_ : (Γ : Con Term) (l : TypeLevel) (t A : Term) ([Γ] : ⊩ᵛ Γ)
([A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ]) → Set
Γ ⊩ᵛ⟨ l ⟩ t ∷ A / [Γ] / [A] =
∀ {Δ σ} (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) →
Σ (Δ ⊩⟨ l ⟩ subst σ t ∷ subst σ A / proj₁ ([A] ⊢Δ [σ])) λ [tσ] →
∀ {σ′} → Δ ⊩ˢ σ′ ∷ Γ / [Γ] / ⊢Δ → Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ]
→ Δ ⊩⟨ l ⟩ subst σ t ≡ subst σ′ t ∷ subst σ A / proj₁ ([A] ⊢Δ [σ])
_⊩ᵛ⟨_⟩_≡_/_/_ : (Γ : Con Term) (l : TypeLevel) (A B : Term) ([Γ] : ⊩ᵛ Γ)
([A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ]) → Set
Γ ⊩ᵛ⟨ l ⟩ A ≡ B / [Γ] / [A] =
∀ {Δ σ} (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ)
→ Δ ⊩⟨ l ⟩ subst σ A ≡ subst σ B / proj₁ ([A] ⊢Δ [σ])
_⊩ᵛ⟨_⟩_≡_∷_/_/_ : (Γ : Con Term) (l : TypeLevel) (t u A : Term) ([Γ] : ⊩ᵛ Γ)
([A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ]) → Set
Γ ⊩ᵛ⟨ l ⟩ t ≡ u ∷ A / [Γ] / [A] =
∀ {Δ σ} → (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ)
→ Δ ⊩⟨ l ⟩ subst σ t ≡ subst σ u ∷ subst σ A / proj₁ ([A] ⊢Δ [σ])
record [_⊩ᵛ⟨_⟩_≡_∷_/_] (Γ : Con Term) (l : TypeLevel)
(t u A : Term) ([Γ] : ⊩ᵛ Γ) : Set where
constructor modelsTermEq
field
[A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ]
[t] : Γ ⊩ᵛ⟨ l ⟩ t ∷ A / [Γ] / [A]
[u] : Γ ⊩ᵛ⟨ l ⟩ u ∷ A / [Γ] / [A]
[t≡u] : Γ ⊩ᵛ⟨ l ⟩ t ≡ u ∷ A / [Γ] / [A]
_⊩ᵛ_⇒_∷_/_ : (Γ : Con Term) (t u A : Term) ([Γ] : ⊩ᵛ Γ) → Set
Γ ⊩ᵛ t ⇒ u ∷ A / [Γ] = ∀ {Δ σ} (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ)
→ Δ ⊢ subst σ t ⇒ subst σ u ∷ subst σ A