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

-- Definitions of semantic judgments for completeness
module NonCumulative.Completeness.LogRel where

open import Lib
open import NonCumulative.Semantics.Domain public
open import NonCumulative.Semantics.Evaluation public
open import NonCumulative.Semantics.PER public


record RelExp t ρ t′ ρ′ (R : Ty) : Set where
  field
    ⟦t⟧   : D
    ⟦t′⟧  : D
    ↘⟦t⟧  :  t  ρ  ⟦t⟧
    ↘⟦t′⟧ :  t′  ρ′  ⟦t′⟧
    t≈t′  : ⟦t⟧  ⟦t′⟧  R

infix 4 _⊨_≈_∶[_]_ _⊨_∶[_]_ _⊨s_≈_∶_ _⊨s_∶_

-- Two terms are related if their evaluations are related by the evaluation of their type.
_⊨_≈_∶[_]_ : Ctx  Exp  Exp    Typ  Set
Γ  t  t′ ∶[ i ] T = Σ ( Γ) λ ⊨Γ   {ρ ρ′} (ρ≈ρ′ : ρ  ρ′   ⊨Γ ⟧ρ)  Σ (RelTyp i T ρ T ρ′) λ rel  let open RelTyp rel in RelExp t ρ t′ ρ′ (El _ T≈T′)

_⊨_∶[_]_ : Ctx  Exp    Typ  Set
Γ  t ∶[ i ] T = Γ  t  t ∶[ i ] T


record RelSubst σ ρ δ ρ′ (R : Ev) : Set where
  field
    ⟦σ⟧  : Env
    ⟦δ⟧  : Env
    ↘⟦σ⟧ :  σ ⟧s ρ  ⟦σ⟧
    ↘⟦δ⟧ :  δ ⟧s ρ′  ⟦δ⟧
    σ≈δ  : ⟦σ⟧  ⟦δ⟧  R

-- Two substitutions are related if their evaluations are related.
_⊨s_≈_∶_ : Ctx  Subst  Subst  Ctx  Set
Γ ⊨s σ  σ′  Δ = Σ ( Γ) λ ⊨Γ  Σ ( Δ) λ ⊨Δ   {ρ ρ′} (ρ≈ρ′ : ρ  ρ′   ⊨Γ ⟧ρ)  RelSubst σ ρ σ′ ρ′  ⊨Δ ⟧ρ

_⊨s_∶_ : Ctx  Subst  Ctx  Set
Γ ⊨s σ  Δ = Γ ⊨s σ  σ  Δ

RelExp⇒RepTyp :  {i}  RelExp T ρ T′ ρ′ (𝕌 i)  RelTyp i T ρ T′ ρ′
RelExp⇒RepTyp rel = record
  { ⟦T⟧   = ⟦t⟧
  ; ⟦T′⟧  = ⟦t′⟧
  ; ↘⟦T⟧  = ↘⟦t⟧
  ; ↘⟦T′⟧ = ↘⟦t′⟧
  ; T≈T′  = t≈t′
  }
  where open RelExp rel