91 lines
3.7 KiB
Lean4
91 lines
3.7 KiB
Lean4
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/-
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Port of `Language/Semantics.agda`.
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Correspondence:
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Value (↑ᶻ) ↦ Value.int
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Env ↦ Env (= List (String × Value))
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_∈_ (env lookup) ↦ Env.Mem
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_,_⇒ᵉ_ ↦ EvalExpr
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_,_⇒ᵇ_ ↦ EvalBasicStmt
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_,_⇒ᵇˢ_ ↦ EvalBasicStmts
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_,_⇒ˢ_ ↦ EvalStmt
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LatticeInterpretation:
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⟦_⟧ ↦ interp
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⟦⟧-respects-≈ ↦ (trivial with `=`; field dropped)
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⟦⟧-⊔-∨ ↦ interp_sup
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⟦⟧-⊓-∧ ↦ interp_inf
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(the `Utils` combinators `_⇒_`, `_∨_`, `_∧_` are inlined as plain logic)
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-/
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import Spa.Language.Base
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import Spa.Lattice
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namespace Spa
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inductive Value where
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| int (z : ℤ)
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deriving DecidableEq
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def Env : Type := List (String × Value)
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/-- Agda: `_∈_` on environments — lookup respecting shadowing. -/
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inductive Env.Mem : String × Value → Env → Prop
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| here (s : String) (v : Value) (ρ : Env) : Env.Mem (s, v) ((s, v) :: ρ)
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| there (s s' : String) (v v' : Value) (ρ : Env) :
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¬(s = s') → Env.Mem (s, v) ρ → Env.Mem (s, v) ((s', v') :: ρ)
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/-- Agda: `_,_⇒ᵉ_`. -/
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inductive EvalExpr : Env → Expr → Value → Prop
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| num (ρ : Env) (n : ℕ) : EvalExpr ρ (.num n) (.int n)
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| var (ρ : Env) (x : String) (v : Value) :
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Env.Mem (x, v) ρ → EvalExpr ρ (.var x) v
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| add (ρ : Env) (e₁ e₂ : Expr) (z₁ z₂ : ℤ) :
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EvalExpr ρ e₁ (.int z₁) → EvalExpr ρ e₂ (.int z₂) →
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EvalExpr ρ (.add e₁ e₂) (.int (z₁ + z₂))
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| sub (ρ : Env) (e₁ e₂ : Expr) (z₁ z₂ : ℤ) :
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EvalExpr ρ e₁ (.int z₁) → EvalExpr ρ e₂ (.int z₂) →
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EvalExpr ρ (.sub e₁ e₂) (.int (z₁ - z₂))
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/-- Agda: `_,_⇒ᵇ_`. -/
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inductive EvalBasicStmt : Env → BasicStmt → Env → Prop
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| noop (ρ : Env) : EvalBasicStmt ρ .noop ρ
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| assign (ρ : Env) (x : String) (e : Expr) (v : Value) :
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EvalExpr ρ e v → EvalBasicStmt ρ (.assign x e) ((x, v) :: ρ)
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/-- Agda: `_,_⇒ᵇˢ_`. -/
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inductive EvalBasicStmts : Env → List BasicStmt → Env → Prop
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| nil {ρ : Env} : EvalBasicStmts ρ [] ρ
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| cons {ρ₁ ρ₂ ρ₃ : Env} {bs : BasicStmt} {bss : List BasicStmt} :
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EvalBasicStmt ρ₁ bs ρ₂ → EvalBasicStmts ρ₂ bss ρ₃ →
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EvalBasicStmts ρ₁ (bs :: bss) ρ₃
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/-- Agda: `_,_⇒ˢ_`. -/
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inductive EvalStmt : Env → Stmt → Env → Prop
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| basic (ρ₁ ρ₂ : Env) (bs : BasicStmt) :
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EvalBasicStmt ρ₁ bs ρ₂ → EvalStmt ρ₁ (.basic bs) ρ₂
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| andThen (ρ₁ ρ₂ ρ₃ : Env) (s₁ s₂ : Stmt) :
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EvalStmt ρ₁ s₁ ρ₂ → EvalStmt ρ₂ s₂ ρ₃ →
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EvalStmt ρ₁ (.andThen s₁ s₂) ρ₃
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| ifTrue (ρ₁ ρ₂ : Env) (e : Expr) (z : ℤ) (s₁ s₂ : Stmt) :
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EvalExpr ρ₁ e (.int z) → ¬(z = 0) → EvalStmt ρ₁ s₁ ρ₂ →
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EvalStmt ρ₁ (.ifElse e s₁ s₂) ρ₂
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| ifFalse (ρ₁ ρ₂ : Env) (e : Expr) (s₁ s₂ : Stmt) :
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EvalExpr ρ₁ e (.int 0) → EvalStmt ρ₁ s₂ ρ₂ →
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EvalStmt ρ₁ (.ifElse e s₁ s₂) ρ₂
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| whileTrue (ρ₁ ρ₂ ρ₃ : Env) (e : Expr) (z : ℤ) (s : Stmt) :
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EvalExpr ρ₁ e (.int z) → ¬(z = 0) → EvalStmt ρ₁ s ρ₂ →
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EvalStmt ρ₂ (.whileLoop e s) ρ₃ →
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EvalStmt ρ₁ (.whileLoop e s) ρ₃
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| whileFalse (ρ : Env) (e : Expr) (s : Stmt) :
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EvalExpr ρ e (.int 0) →
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EvalStmt ρ (.whileLoop e s) ρ
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/-- Agda: `LatticeInterpretation`. -/
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structure LatticeInterpretation (L : Type*) [Lattice L] where
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interp : L → Value → Prop
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interp_sup : ∀ {l₁ l₂ : L} (v : Value),
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interp l₁ v ∨ interp l₂ v → interp (l₁ ⊔ l₂) v
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interp_inf : ∀ {l₁ l₂ : L} (v : Value),
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interp l₁ v ∧ interp l₂ v → interp (l₁ ⊓ l₂) v
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end Spa
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