{-# OPTIONS --without-K --safe #-}
open import Level
open import Axiom.Extensionality.Propositional
module Cumulative.Completeness (fext : Extensionality 0ℓ (suc 0ℓ)) where
open import Lib
open import Cumulative.Completeness.Fundamental fext
open import Cumulative.Semantics.Domain
open import Cumulative.Semantics.Evaluation
open import Cumulative.Semantics.Properties.PER fext
open import Cumulative.Semantics.Readback
open import Cumulative.Semantics.Realizability fext
open import Cumulative.Statics
completeness : Γ ⊢ t ≈ t′ ∶ T →
∃ λ w → NbE Γ t T w × NbE Γ t′ T w
completeness {Γ} t≈t′
with ⊨Γ , _ , t≈t′ ← fundamental-t≈t′ t≈t′
with _ , _ , ↘ρ , ↘ρ′ , ρ≈ρ′ ← InitEnvs-related ⊨Γ
with t≈t′ ρ≈ρ′
... | record { ⟦T⟧ = ⟦T⟧ ; ⟦T′⟧ = ⟦T′⟧ ; ↘⟦T⟧ = ↘⟦T⟧ ; ↘⟦T′⟧ = ↘⟦T′⟧ ; T≈T′ = T≈T′ }
, record { ⟦t⟧ = ⟦t⟧ ; ⟦t′⟧ = ⟦t′⟧ ; ↘⟦t⟧ = ↘⟦t⟧ ; ↘⟦t′⟧ = ↘⟦t′⟧ ; t≈t′ = t≈t′ }
with _ , ↓⟦t⟧ , ↓⟦t′⟧ ← El⊆Top T≈T′ t≈t′ (len Γ)
= _
, record
{ init = ↘ρ
; nbe = record
{ ↘⟦t⟧ = ↘⟦t⟧
; ↘⟦T⟧ = ↘⟦T⟧
; ↓⟦t⟧ = ↓⟦t⟧
}
}
, record
{ init = ↘ρ′
; nbe = record
{ ↘⟦t⟧ = ↘⟦t′⟧
; ↘⟦T⟧ = ↘⟦T′⟧
; ↓⟦t⟧ = ↓⟦t′⟧
}
}