module Language.Base where

open import Data.List as List using (List)
open import Data.Nat using (ℕ; suc)
open import Data.Product using (Σ; _,_; proj₁)
open import Data.String as String using (String)
open import Data.Vec using (Vec; foldr; lookup)
open import Relation.Binary.PropositionalEquality using (_≡_; refl)

open import Lattice

data Expr : Set where
    _+_ : Expr → Expr → Expr
    _-_ : Expr → Expr → Expr
    `_ : String → Expr
    #_ : ℕ → Expr

data BasicStmt : Set where
    _←_ : String → Expr → BasicStmt
    noop : BasicStmt

infixr 2 _then_
infix 3 if_then_else_
infix 3 while_repeat_
data Stmt : Set where
    ⟨_⟩ : BasicStmt → Stmt
    _then_ : Stmt → Stmt → Stmt
    if_then_else_ : Expr → Stmt → Stmt → Stmt
    while_repeat_ : Expr → Stmt → Stmt

data _∈ᵉ_ : String → Expr → Set where
    in⁺₁ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₁ → k ∈ᵉ (e₁ + e₂)
    in⁺₂ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₂ → k ∈ᵉ (e₁ + e₂)
    in⁻₁ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₁ → k ∈ᵉ (e₁ - e₂)
    in⁻₂ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₂ → k ∈ᵉ (e₁ - e₂)
    here : ∀ {k : String} → k ∈ᵉ (` k)

data _∈ᵇ_ : String → BasicStmt → Set where
    in←₁ : ∀ {k : String} {e : Expr} → k ∈ᵇ (k ← e)
    in←₂ : ∀ {k k' : String} {e : Expr} → k ∈ᵉ e → k ∈ᵇ (k' ← e)

open import Lattice.MapSet (String._≟_)
    renaming
        ( MapSet to StringSet
        ; insert to insertˢ
        ; empty to emptyˢ
        ; singleton to singletonˢ
        ; _⊔_ to _⊔ˢ_
        ; `_ to `ˢ_
        ; _∈_ to _∈ˢ_
        ; ⊔-preserves-∈k₁ to ⊔ˢ-preserves-∈k₁
        ; ⊔-preserves-∈k₂ to ⊔ˢ-preserves-∈k₂
        )

Expr-vars : Expr → StringSet
Expr-vars (l + r) = Expr-vars l ⊔ˢ Expr-vars r
Expr-vars (l - r) = Expr-vars l ⊔ˢ Expr-vars r
Expr-vars (` s) = singletonˢ s
Expr-vars (# _) = emptyˢ

-- ∈-Expr-vars⇒∈ : ∀ {k : String} (e : Expr) → k ∈ˢ (Expr-vars e) → k ∈ᵉ e
-- ∈-Expr-vars⇒∈ {k} (e₁ + e₂) k∈vs
--     with Expr-Provenance k ((`ˢ (Expr-vars e₁)) ∪ (`ˢ (Expr-vars e₂))) k∈vs
-- ...   | in₁ (single k,tt∈vs₁) _ = (in⁺₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
-- ...   | in₂ _ (single k,tt∈vs₂) = (in⁺₂ (∈-Expr-vars⇒∈ e₂ (forget k,tt∈vs₂)))
-- ...   | bothᵘ (single k,tt∈vs₁) _ = (in⁺₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
-- ∈-Expr-vars⇒∈ {k} (e₁ - e₂) k∈vs
--     with Expr-Provenance k ((`ˢ (Expr-vars e₁)) ∪ (`ˢ (Expr-vars e₂))) k∈vs
-- ...   | in₁ (single k,tt∈vs₁) _ = (in⁻₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
-- ...   | in₂ _ (single k,tt∈vs₂) = (in⁻₂ (∈-Expr-vars⇒∈ e₂ (forget k,tt∈vs₂)))
-- ...   | bothᵘ (single k,tt∈vs₁) _ = (in⁻₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
-- ∈-Expr-vars⇒∈ {k} (` k) (RelAny.here refl) = here

-- ∈⇒∈-Expr-vars : ∀ {k : String} {e : Expr} → k ∈ᵉ e → k ∈ˢ (Expr-vars e)
-- ∈⇒∈-Expr-vars {k} {e₁ + e₂} (in⁺₁ k∈e₁) =
--     ⊔ˢ-preserves-∈k₁ {m₁ = Expr-vars e₁}
--                      {m₂ = Expr-vars e₂}
--                      (∈⇒∈-Expr-vars k∈e₁)
-- ∈⇒∈-Expr-vars {k} {e₁ + e₂} (in⁺₂ k∈e₂) =
--     ⊔ˢ-preserves-∈k₂ {m₁ = Expr-vars e₁}
--                      {m₂ = Expr-vars e₂}
--                      (∈⇒∈-Expr-vars k∈e₂)
-- ∈⇒∈-Expr-vars {k} {e₁ - e₂} (in⁻₁ k∈e₁) =
--     ⊔ˢ-preserves-∈k₁ {m₁ = Expr-vars e₁}
--                      {m₂ = Expr-vars e₂}
--                      (∈⇒∈-Expr-vars k∈e₁)
-- ∈⇒∈-Expr-vars {k} {e₁ - e₂} (in⁻₂ k∈e₂) =
--     ⊔ˢ-preserves-∈k₂ {m₁ = Expr-vars e₁}
--                      {m₂ = Expr-vars e₂}
--                      (∈⇒∈-Expr-vars k∈e₂)
-- ∈⇒∈-Expr-vars here = RelAny.here refl

BasicStmt-vars : BasicStmt → StringSet
BasicStmt-vars (x ← e) = (singletonˢ x) ⊔ˢ (Expr-vars e)
BasicStmt-vars noop = emptyˢ

Stmt-vars : Stmt → StringSet
Stmt-vars ⟨ bs ⟩ = BasicStmt-vars bs
Stmt-vars (s₁ then s₂) = (Stmt-vars s₁) ⊔ˢ (Stmt-vars s₂)
Stmt-vars (if e then s₁ else s₂) = ((Expr-vars e) ⊔ˢ (Stmt-vars s₁)) ⊔ˢ (Stmt-vars s₂)
Stmt-vars (while e repeat s) = (Expr-vars e) ⊔ˢ (Stmt-vars s)

-- ∈-Stmt-vars⇒∈ : ∀ {k : String} (s : Stmt) → k ∈ˢ (Stmt-vars s) → k ∈ᵇ s
-- ∈-Stmt-vars⇒∈ {k} (k' ← e) k∈vs
--     with Expr-Provenance k ((`ˢ (singletonˢ k')) ∪ (`ˢ (Expr-vars e))) k∈vs
-- ...   | in₁ (single (RelAny.here refl)) _ = in←₁
-- ...   | in₂ _ (single k,tt∈vs') = in←₂ (∈-Expr-vars⇒∈ e (forget k,tt∈vs'))
-- ...   | bothᵘ (single (RelAny.here refl)) _ = in←₁

-- ∈⇒∈-Stmt-vars : ∀ {k : String} {s : Stmt} → k ∈ᵇ s → k ∈ˢ (Stmt-vars s)
-- ∈⇒∈-Stmt-vars {k} {k ← e} in←₁ =
--     ⊔ˢ-preserves-∈k₁ {m₁ = singletonˢ k}
--                      {m₂ = Expr-vars e}
--                      (RelAny.here refl)
-- ∈⇒∈-Stmt-vars {k} {k' ← e} (in←₂ k∈e) =
--     ⊔ˢ-preserves-∈k₂ {m₁ = singletonˢ k'}
--                      {m₂ = Expr-vars e}
--                      (∈⇒∈-Expr-vars k∈e)

Stmts-vars : ∀ {n : ℕ} → Vec Stmt n → StringSet
Stmts-vars = foldr (λ n → StringSet)
                   (λ {k} stmt acc → (Stmt-vars stmt) ⊔ˢ acc) emptyˢ

-- ∈-Stmts-vars⇒∈ : ∀ {n : ℕ} {k : String} (ss : Vec Stmt n) →
--                  k ∈ˢ (Stmts-vars ss) → Σ (Fin n) (λ f → k ∈ᵇ lookup ss f)
-- ∈-Stmts-vars⇒∈ {suc n'} {k} (s ∷ ss') k∈vss
--     with Expr-Provenance k ((`ˢ (Stmt-vars s)) ∪ (`ˢ (Stmts-vars ss'))) k∈vss
-- ...   | in₁ (single k,tt∈vs) _ = (zero , ∈-Stmt-vars⇒∈ s (forget k,tt∈vs))
-- ...   | in₂ _ (single k,tt∈vss') =
--         let
--             (f' , k∈s') = ∈-Stmts-vars⇒∈ ss' (forget k,tt∈vss')
--         in
--             (suc f' , k∈s')
-- ...   | bothᵘ (single k,tt∈vs) _ = (zero , ∈-Stmt-vars⇒∈ s (forget k,tt∈vs))

-- ∈⇒∈-Stmts-vars : ∀ {n : ℕ} {k : String} {ss : Vec Stmt n} {f : Fin n} →
--                  k ∈ᵇ lookup ss f → k ∈ˢ (Stmts-vars ss)
-- ∈⇒∈-Stmts-vars {suc n} {k} {s ∷ ss'} {zero} k∈s =
--     ⊔ˢ-preserves-∈k₁ {m₁ = Stmt-vars s}
--                      {m₂ = Stmts-vars ss'}
--                      (∈⇒∈-Stmt-vars k∈s)
-- ∈⇒∈-Stmts-vars {suc n} {k} {s ∷ ss'} {suc f'} k∈ss' =
--     ⊔ˢ-preserves-∈k₂ {m₁ = Stmt-vars s}
--                      {m₂ = Stmts-vars ss'}
--                      (∈⇒∈-Stmts-vars {n} {k} {ss'} {f'} k∈ss')