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06_TASK_LEAN_PROOFS.md

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Task 5: Lean 4 Formal Proofs

Objective

Formalize Oblibeny semantics and prove the 7 key properties in Lean 4.

Deliverables

1. Syntax Formalization

inductive Expr where
  | int : Int → Expr
  | bool : Bool → Expr
  | var : String → Expr
  | boundedFor : String → Expr → Expr → List Expr → Expr
  | defunDeploy : String → List String → List Expr → Expr
  | app : Expr → List Expr → Expr

inductive Phase where
  | compile
  | deploy

2. Operational Semantics

inductive Step : Expr → Expr → Prop where
  | β_app : Step (Expr.app (Expr.defunDeploy name params body) args) (subst body params args)
  | bounded_for_step : Step (Expr.boundedFor var start end body) ...

3. Type System

inductive HasType : Context → Expr → Type → Prop where
  | t_int : HasType Γ (Expr.int n) Type.int
  | t_bounded_for :
      HasType Γ start Type.int →
      HasType Γ end Type.int →
      HasType (Γ.extend var Type.int) body τ →
      HasType Γ (Expr.boundedFor var start end body) τ

4. Property Proofs

Property 1: Phase Separation

theorem phase_separation_sound (p : Program) :
  deploy_phase p → ¬ contains_compile_construct p := by
  sorry -- To be proven

Property 2: Termination

theorem deploy_terminates (p : Program) (input : Input) :
  deploy_phase p → ∃ n, eval_steps p input n = some value := by
  sorry -- To be proven

Properties 3-7: Similar structure

5. Lemmas and Tactics

  • Custom tactics for proof automation
  • Common lemmas library
  • Proof strategies documentation

File Structure

lean-proofs/
├── lakefile.lean
├── Oblibeny/
│   ├── Syntax.lean
│   ├── Semantics.lean
│   ├── TypeSystem.lean
│   ├── Properties/
│   │   ├── PhaseSeparation.lean
│   │   ├── Termination.lean
│   │   ├── ResourceBounds.lean
│   │   ├── CapabilitySound.lean
│   │   ├── ObfuscationPreserves.lean
│   │   ├── CallGraphAcyclic.lean
│   │   └── MemorySafe.lean
│   └── Proofs/
│       ├── Lemmas.lean
│       └── Tactics.lean