{-# OPTIONS --without-K --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.Reducibility {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped
open import Definition.Untyped.Properties
open import Definition.Typed
open import Definition.LogicalRelation
open import Definition.LogicalRelation.Irrelevance
open import Definition.LogicalRelation.Substitution
open import Definition.LogicalRelation.Substitution.Properties
open import Tools.Product
reducibleᵛ : ∀ {A Γ l}
([Γ] : ⊩ᵛ Γ)
→ Γ ⊩ᵛ⟨ l ⟩ A / [Γ]
→ Γ ⊩⟨ l ⟩ A
reducibleᵛ [Γ] [A] =
let ⊢Γ = soundContext [Γ]
[id] = idSubstS [Γ]
in irrelevance′ (subst-id _) (proj₁ ([A] ⊢Γ [id]))
reducibleEqᵛ : ∀ {A B Γ l}
([Γ] : ⊩ᵛ Γ)
([A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ])
→ Γ ⊩ᵛ⟨ l ⟩ A ≡ B / [Γ] / [A]
→ Γ ⊩⟨ l ⟩ A ≡ B / reducibleᵛ [Γ] [A]
reducibleEqᵛ {A = A} [Γ] [A] [A≡B] =
let [σA] = reducibleᵛ {A = A} [Γ] [A]
⊢Γ = soundContext [Γ]
[id] = idSubstS [Γ]
in irrelevanceEq″ (subst-id _) (subst-id _)
(proj₁ ([A] ⊢Γ [id])) [σA] ([A≡B] ⊢Γ [id])
reducibleTermᵛ : ∀ {t A Γ l}
([Γ] : ⊩ᵛ Γ)
([A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ])
→ Γ ⊩ᵛ⟨ l ⟩ t ∷ A / [Γ] / [A]
→ Γ ⊩⟨ l ⟩ t ∷ A / reducibleᵛ [Γ] [A]
reducibleTermᵛ {A = A} [Γ] [A] [t] =
let [σA] = reducibleᵛ {A = A} [Γ] [A]
⊢Γ = soundContext [Γ]
[id] = idSubstS [Γ]
in irrelevanceTerm″ (subst-id _) (subst-id _)
(proj₁ ([A] ⊢Γ [id])) [σA] (proj₁ ([t] ⊢Γ [id]))
reducibleEqTermᵛ : ∀ {t u A Γ l}
([Γ] : ⊩ᵛ Γ)
([A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ])
→ Γ ⊩ᵛ⟨ l ⟩ t ≡ u ∷ A / [Γ] / [A]
→ Γ ⊩⟨ l ⟩ t ≡ u ∷ A / reducibleᵛ [Γ] [A]
reducibleEqTermᵛ {A = A} [Γ] [A] [t≡u] =
let [σA] = reducibleᵛ {A = A} [Γ] [A]
⊢Γ = soundContext [Γ]
[id] = idSubstS [Γ]
in irrelevanceEqTerm″ (subst-id _) (subst-id _) (subst-id _)
(proj₁ ([A] ⊢Γ [id])) [σA] ([t≡u] ⊢Γ [id])