------------------------------------------------------------------------
-- Agda Helium
--
-- Base definitions for the axiomatic semantics
------------------------------------------------------------------------

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

open import Helium.Data.Pseudocode.Algebra using (RawPseudocode)

module Helium.Semantics.Axiomatic.Core
  {b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃}
  (rawPseudocode : RawPseudocode b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃)
  where

private
  open module C = RawPseudocode rawPseudocode

open import Data.Bool as Bool using (Bool)
open import Data.Fin as Fin using (Fin; zero; suc)
open import Data.Fin.Patterns
open import Data.Nat as ℕ using (ℕ; suc)
import Data.Nat.Induction as Natᵢ
import Data.Nat.Properties as ℕₚ
open import Data.Product as × using (_×_; _,_; uncurry)
open import Data.Sum using (_⊎_)
open import Data.Unit using (⊤)
open import Data.Vec as Vec using (Vec; []; _∷_; _++_; lookup)
open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
open import Function using (_on_)
open import Helium.Data.Pseudocode.Core
open import Helium.Data.Pseudocode.Properties
import Induction.WellFounded as Wf
open import Level using (_⊔_; Lift; lift)
import Relation.Binary.Construct.On as On
open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; cong₂)
open import Relation.Nullary using (Dec; does; yes; no)
open import Relation.Nullary.Decidable.Core using (True; toWitness; map′)
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Unary using (_⊆_)

private
  variable
    t t′     : Type
    m n      : ℕ
    Γ Δ Σ ts : Vec Type m

⟦_⟧ₜ  : Type → Set (b₁ ⊔ i₁ ⊔ r₁)
⟦_⟧ₜ′ : Vec Type n → Set (b₁ ⊔ i₁ ⊔ r₁)

⟦ bool ⟧ₜ       = Lift (b₁ ⊔ i₁ ⊔ r₁) Bool
⟦ int ⟧ₜ        = Lift (b₁ ⊔ r₁) ℤ
⟦ fin n ⟧ₜ      = Lift (b₁ ⊔ i₁ ⊔ r₁) (Fin n)
⟦ real ⟧ₜ       = Lift (b₁ ⊔ i₁) ℝ
⟦ bit ⟧ₜ        = Lift (i₁ ⊔ r₁) Bit
⟦ bits n ⟧ₜ     = Vec ⟦ bit ⟧ₜ n
⟦ tuple n ts ⟧ₜ = ⟦ ts ⟧ₜ′ 
⟦ array t n ⟧ₜ  = Vec ⟦ t ⟧ₜ n

⟦ [] ⟧ₜ′     = Lift (b₁ ⊔ i₁ ⊔ r₁) ⊤
⟦ t ∷ ts ⟧ₜ′ = ⟦ t ⟧ₜ × ⟦ ts ⟧ₜ′

fetch : ∀ i → ⟦ Γ ⟧ₜ′ → ⟦ lookup Γ i ⟧ₜ
fetch {Γ = _ ∷ _} 0F      (x , _)  = x
fetch {Γ = _ ∷ _} (suc i) (_ , xs) = fetch i xs

Transform : Vec Type m → Type → Set (b₁ ⊔ i₁ ⊔ r₁)
Transform ts t = ⟦ ts ⟧ₜ′ → ⟦ t ⟧ₜ

Predicate : Vec Type m → Set (b₁ ⊔ i₁ ⊔ r₁)
Predicate ts =  ⟦ ts ⟧ₜ′ → Bool

--   data HoareTriple {n Γ m Δ} : Assertion Σ {n} Γ {m} Δ → Statement Γ → Assertion Σ Γ Δ → Set (b₁ ⊔ i₁ ⊔ r₁) where
--     _∙_ : ∀ {P Q R s₁ s₂} → HoareTriple P s₁ Q → HoareTriple Q s₂ R → HoareTriple P (s₁ ∙ s₂) R
--     skip : ∀ {P} → HoareTriple P skip P

--     assign : ∀ {P t ref canAssign e} → HoareTriple (asrtSubst P (toWitness canAssign) (ℰ e)) (_≔_ {t = t} ref {canAssign} e) P
--     declare : ∀ {t P Q e s} → HoareTriple (P ∧ equal (var 0F) (termWknVar (ℰ e))) s (asrtWknVar Q) → HoareTriple (asrtElimVar P (ℰ e)) (declare {t = t} e s) Q
--     invoke : ∀ {m ts P Q s e} → HoareTriple (assignMetas Δ ts (All.tabulate var) (asrtAddVars P)) s (asrtAddVars Q) → HoareTriple (assignMetas Δ ts (All.tabulate (λ i → ℰ (All.lookup i e))) (asrtAddVars P)) (invoke {m = m} {ts} (s ∙end) e) (asrtAddVars Q)
--     if : ∀ {P Q e s₁ s₂} → HoareTriple (P ∧ equal (ℰ e) (𝒦 (Bool.true ′b))) s₁ Q → HoareTriple (P ∧ equal (ℰ e) (𝒦 (Bool.false ′b))) s₂ Q → HoareTriple P (if e then s₁ else s₂) Q
--     for : ∀ {m} {I : Assertion Σ Γ (fin (suc m) ∷ Δ)} {s} → HoareTriple (some (asrtWknVar (asrtWknMetaAt 1F I) ∧ equal (meta 1F) (var 0F) ∧ equal (meta 0F) (func (λ { _ (lift x ∷ []) → lift (Fin.inject₁ x) }) (meta 1F ∷ [])))) s (some (asrtWknVar (asrtWknMetaAt 1F I) ∧ equal (meta 0F) (func (λ { _ (lift x ∷ []) → lift (Fin.suc x) }) (meta 1F ∷ [])))) → HoareTriple (some (I ∧ equal (meta 0F) (func (λ _ _ → lift 0F) []))) (for m s) (some (I ∧ equal (meta 0F) (func (λ _ _ → lift (Fin.fromℕ m)) [])))

--     consequence : ∀ {P₁ P₂ Q₁ Q₂ s} → ⟦ P₁ ⟧ₐ ⊆ ⟦ P₂ ⟧ₐ → HoareTriple P₂ s Q₂ → ⟦ Q₂ ⟧ₐ ⊆ ⟦ Q₁ ⟧ₐ → HoareTriple P₁ s Q₁
--     some : ∀ {t P Q s} → HoareTriple P s Q → HoareTriple (some {t = t} P) s (some Q)
--     frame : ∀ {P Q R s} → HoareTriple P s Q → HoareTriple (P * R) s (Q * R)