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

open import Axiom.Extensionality.Propositional

-- Realizability of the PER model
--
-- Fundamentally, realizability states that if two values are related, then their
-- corresponding syntactic normal forms are equal up to any UMoT. More precisely,
-- realizability states that the following subsumption relations:
--
--       Bot ⊆ El i A ⊆ Top
--             𝕌 i    ⊆ TopT
--
-- Due to these subsumptions, we can always derive Top or TopT from El or 𝕌 and thus
-- obtain the equality we want.
module Mint.Semantics.Realizability (fext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′) where

open import Data.Nat.Induction
open import Lib

open import Mint.Semantics.Domain
open import Mint.Semantics.Evaluation
open import Mint.Semantics.PER
open import Mint.Semantics.Properties.Domain fext
open import Mint.Semantics.Properties.PER.Core fext
open import Mint.Semantics.Readback

mutual
  realizability-Re-Acc : ∀ {i} →
                         Acc (_<_) i →
                         (A≈A′ : A ≈ A′ ∈ 𝕌 i) →
                         c ≈ c′ ∈ Bot →
                         ↑ A c ≈ ↑ A′ c′ ∈ El i A≈A′
  realizability-Re-Acc <i (ne C≈C′)     c≈c′ = ne c≈c′
  realizability-Re-Acc <i N             c≈c′ = ne c≈c′
  realizability-Re-Acc <i (U j<i refl)  c≈c′
    rewrite 𝕌-wellfounded-≡-𝕌 _ j<i         = ne c≈c′
  realizability-Re-Acc {□ A} {□ A′} {c} {c′} {i} <i (□ A≈A′) c≈c′ n κ =
    record
    { ua    = unbox′ (A [ ins κ  ] [ ins vone n ]) n (c [ κ ])
    ; ub    = unbox′ (A′ [ ins κ  ] [ ins vone n ]) n (c′ [ κ ])
    ; ↘ua   = unbox∙ n
    ; ↘ub   = unbox∙ n
    ; ua≈ub = ua≈ub
    }
    where
      unbox[c[κ]]≈unbox[c′[κ]] : unbox n (mtran-c c κ) ≈ unbox n (mtran-c c′ κ) ∈ Bot
      unbox[c[κ]]≈unbox[c′[κ]] ns κ′
        with c≈c′ (ns ∥ O κ′ n) (κ ø κ′ ∥ n)
      ... | u , c↘u , c′↘u
          rewrite Dn-comp c κ (κ′ ∥ n)
                | Dn-comp c′ κ (κ′ ∥ n) = unbox (O κ′ n) u , Ru ns (O κ′ n) c↘u , Ru ns (O κ′ n) c′↘u

      ua≈ub : unbox′ (A [ ins κ  ] [ ins vone n ]) n (c [ κ ]) ≈ unbox′ (A′ [ ins κ  ] [ ins vone n ]) n (c′ [ κ ]) ∈ El i (A≈A′ (ins κ n))
      ua≈ub
        rewrite D-ins-ins A κ n
              | D-ins-ins A′ κ n = realizability-Re-Acc <i (A≈A′ (ins κ n)) unbox[c[κ]]≈unbox[c′[κ]]
  realizability-Re-Acc {A = Π A _ _} {A′ = Π A′ _ _} {c} {c′} <i (Π A≈A′ T≈T′) c≈c′ {a = b} {b = b′} κ b≈b′
    with T≈T′ κ b≈b′
  ...  | record
         { ⟦T⟧   = ⟦T⟧
         ; ⟦T′⟧  = ⟦T′⟧
         ; ↘⟦T⟧  = ↘⟦T⟧
         ; ↘⟦T′⟧ = ↘⟦T′⟧
         ; T≈T′  = T≈T′
         } =
      record
      { fa     = [ ⟦T⟧ ] c [ κ ] $′ ↓ (A [ κ ]) b
      ; fa′    = [ ⟦T′⟧ ] c′ [ κ ] $′ ↓ (A′ [ κ ]) b′
      ; ↘fa    = $∙ (A [ κ ]) (c [ κ ]) (↘⟦T⟧)
      ; ↘fa′   = $∙ (A′ [ κ ]) (c′ [ κ ]) (↘⟦T′⟧)
      ; fa≈fa′ = realizability-Re-Acc <i T≈T′ c[κ]$b≈c′[κ]$b′
      }
    where
      c[κ]$b≈c′[κ]$b′ : c [ κ ] $ ↓ (A [ κ ]) b ≈ c′ [ κ ] $ ↓ (A′ [ κ ]) b′ ∈ Bot
      c[κ]$b≈c′[κ]$b′ ns κ′
        with c≈c′ ns (κ ø κ′) | realizability-Rf-Acc <i (A≈A′ κ) b≈b′ ns κ′
      ...  | u , c↘u , c′↘u   | w , b↘w , b′↘w
          rewrite Dn-comp c κ κ′
                | Dn-comp c′ κ κ′ = u $ w , R$ ns c↘u b↘w , R$ ns c′↘u b′↘w

  realizability-Rf-Acc : ∀ {i} →
                         Acc (_<_) i →
                         (A≈A′ : A ≈ A′ ∈ 𝕌 i) →
                         a ≈ a′ ∈ El i A≈A′ →
                         ↓ A a ≈ ↓ A′ a′ ∈ Top
  realizability-Rf-Acc <i       (ne C≈C′)     (ne c≈c′) ns κ
    with c≈c′ ns κ
  ...  | u , c↘u , c′↘u                                      = ne u , Rne ns c↘u , Rne ns c′↘u
  realizability-Rf-Acc <i       N             ze        ns κ = ze   , Rze ns     , Rze ns
  realizability-Rf-Acc <i       N             (su a≈a′) ns κ
    with realizability-Rf-Acc <i N a≈a′ ns κ
  ... | w , a↘w , a′↘w                                       = su w , Rsu ns a↘w , Rsu ns a′↘w
  realizability-Rf-Acc <i       N             (ne c≈c′) ns κ
    with c≈c′ ns κ
  ...  | u , c↘u , c′↘u                                      = ne u , RN ns c↘u  , RN ns c′↘u
  realizability-Rf-Acc (acc <i) (U j<i refl)  a≈a′
    rewrite 𝕌-wellfounded-≡-𝕌 _ j<i                         = realizability-Rty-Acc (<i j<i) a≈a′
  realizability-Rf-Acc {A = □ A} {A′ = □ A′} <i (□ A≈A′) a≈a′ ns κ
    with a≈a′  κ
  ...  | record
         { ua    = ua
         ; ub    = ua′
         ; ↘ua   = ↘ua
         ; ↘ub   = ↘ua′
         ; ua≈ub = ua≈ua′
         }
      with realizability-Rf-Acc <i (A≈A′ (ins κ )) ua≈ua′ ( ∷⁺ ns) vone
  ...    | w , ua↘w , ua′↘w
        rewrite D-ap-vone (A [ ins κ  ])
              | D-ap-vone (A′ [ ins κ  ])
              | D-ap-vone ua
              | D-ap-vone ua′                                = box w , R□ ns ↘ua ua↘w , R□ ns ↘ua′ ua′↘w
  realizability-Rf-Acc <i       (Π A≈A′ T≈T′) a≈a′     ns κ
    with realizability-Re-Acc <i (A≈A′ κ) (Bot-l (head ns))
  ...  | z≈z
      with a≈a′ κ z≈z
  ...    | record
           { fa     = fa
           ; fa′    = fa′
           ; ↘fa    = ↘fa
           ; ↘fa′   = ↘fa′
           ; fa≈fa′ = fa≈fa′
           }
        with T≈T′ κ z≈z
  ...      | record
             { ⟦T⟧   = ⟦T⟧
             ; ⟦T′⟧  = ⟦T′⟧
             ; ↘⟦T⟧  = ↘⟦T⟧
             ; ↘⟦T′⟧ = ↘⟦T′⟧
             ; T≈T′  = T≈T′
             }
          with realizability-Rf-Acc <i T≈T′ fa≈fa′ (inc ns) vone
  ...        | w , fa↘w , fa′↘w
            rewrite D-ap-vone fa
                  | D-ap-vone fa′
                  | D-ap-vone ⟦T⟧
                  | D-ap-vone ⟦T′⟧                           = Λ w , RΛ ns ↘fa ↘⟦T⟧ fa↘w , RΛ ns ↘fa′ ↘⟦T′⟧ fa′↘w

  realizability-Rty-Acc : ∀ {i} →
                          Acc (_<_) i →
                          (A≈A′ : A ≈ A′ ∈ 𝕌 i) →
                          ↓ (U i) A ≈ ↓ (U i) A′ ∈ Top
  realizability-Rty-Acc <i (ne C≈C′)     ns κ
    with C≈C′ ns κ
  ...  | V , C↘V , C′↘V                       = ne V , RU ns (Rne ns C↘V) , RU ns (Rne ns C′↘V)
  realizability-Rty-Acc <i N             ns κ = N    , RU ns (RN ns)      , RU ns (RN ns)
  realizability-Rty-Acc <i (U j<i refl)  ns κ = Se _ , RU ns (RU ns)      , RU ns (RU ns)
  realizability-Rty-Acc {A = □ A} {A′ = □ A′} <i (□ A≈A′) ns κ
    with realizability-Rty-Acc <i (A≈A′ (ins κ )) ( ∷⁺ ns) vone
  ...  | W , RU _ A↘W , RU _ A′↘W
      rewrite D-ap-vone (A [ ins κ  ])
            | D-ap-vone (A′ [ ins κ  ])      = □ W  , RU ns (R□ ns A↘W)  , RU ns (R□ ns A′↘W)
  realizability-Rty-Acc {A = Π A _ _} {A′ = Π A′ _ _} <i (Π A≈A′ T≈T′) ns κ
    with realizability-Re-Acc <i (A≈A′ κ) (Bot-l (head ns))
  ...  | z≈z
      with T≈T′ κ z≈z
  ...    | record
           { ⟦T⟧   = ⟦T⟧
           ; ⟦T′⟧  = ⟦T′⟧
           ; ↘⟦T⟧  = ↘⟦T⟧
           ; ↘⟦T′⟧ = ↘⟦T′⟧
           ; T≈T′  = T≈T′
           }
        with realizability-Rty-Acc <i (A≈A′ κ) ns vone
  ...      | W , RU _ A↘W , RU _ A′↘W
          with realizability-Rty-Acc <i T≈T′ (inc ns) vone
  ...        | w , RU _ T↘w , RU _ T′↘w
            rewrite D-ap-vone (A [ κ ])
                  | D-ap-vone (A′ [ κ ])
                  | D-ap-vone ⟦T⟧
                  | D-ap-vone ⟦T′⟧                          = Π W w , RU ns (RΠ ns A↘W ↘⟦T⟧ T↘w) , RU ns (RΠ ns A′↘W ↘⟦T′⟧ T′↘w)

realizability-Re : ∀ {i} (A≈A′ : A ≈ A′ ∈ 𝕌 i) →
                   (c ≈ c′ ∈ Bot → ↑ A c ≈ ↑ A′ c′ ∈ El i A≈A′)
realizability-Re A≈A′ = realizability-Re-Acc (<-wellFounded _) A≈A′

realizability-Rf : ∀ {i} (A≈A′ : A ≈ A′ ∈ 𝕌 i) →
                   (a ≈ a′ ∈ El i A≈A′ → ↓ A a ≈ ↓ A′ a′ ∈ Top)
realizability-Rf A≈A′ = realizability-Rf-Acc (<-wellFounded _) A≈A′

realizability-Rty : ∀ {i} (A≈A′ : A ≈ A′ ∈ 𝕌 i) →
                    A ≈ A′ ∈ TopT
realizability-Rty A≈A′ ns κ
  with realizability-Rty-Acc (<-wellFounded _) A≈A′ ns κ
... | W , RU .ns ↘W , RU .ns ↘W′ = W , ↘W , ↘W′