Project-declaredLean 4.32.0 · mathlib@81a5d257

Rel Triple simulate Q run writer T of triples

OracleComp.ProgramLogic.Relational.relTriple_simulateQ_run_writerT_of_triples

program verification separation logic cryptography handler from unary

Plain-language statement

WriterT analogue of relTriple_simulateQ_run_of_triples (monoid variant). Given matching unary Std.Do.Triple specs for two WriterT-based handlers, a monoid-congruent writer relation R_writer (via hR_one and hR_mul), and a synchronization condition on per-query postconditions, derive a whole-program RelTriple on the full (output, writer) o...

Exact Lean statement

theorem relTriple_simulateQ_run_writerT_of_triples
    {ι : Type} {spec : OracleSpec.{0, 0} ι} [IsUniformSpec spec]
    {ω₁ ω₂ : Type} [Monoid ω₁] [Monoid ω₂]
    (impl₁ : QueryImpl spec (WriterT ω₁ (OracleComp spec₁)))
    (impl₂ : QueryImpl spec (WriterT ω₂ (OracleComp spec₂)))
    (R_writer : ω₁ → ω₂ → Prop)
    (hR_one : R_writer 1 1)
    (hR_mul : ∀ w₁ w₁' w₂ w₂', R_writer w₁ w₂ → R_writer w₁' w₂' →
      R_writer (w₁ * w₁') (w₂ * w₂'))
    (oa : OracleComp spec α)
    (Q₁ : ∀ (t : spec.Domain), spec.Range t → ω₁ → Prop)
    (Q₂ : ∀ (t : spec.Domain), spec.Range t → ω₂ → Prop)
    (h₁ : ∀ (t : spec.Domain), Std.Do.Triple
      (impl₁ t : WriterT ω₁ (OracleComp spec₁) (spec.Range t))
      (spred(fun s => ⌜s = 1⌝))
      (⇓a s' => ⌜Q₁ t a s'⌝))
    (h₂ : ∀ (t : spec.Domain), Std.Do.Triple
      (impl₂ t : WriterT ω₂ (OracleComp spec₂) (spec.Range t))
      (spred(fun s => ⌜s = 1⌝))
      (⇓a s' => ⌜Q₂ t a s'⌝))
    (hsync : ∀ (t : spec.Domain) a₁ w₁ a₂ w₂,
      Q₁ t a₁ w₁ → Q₂ t a₂ w₂ → a₁ = a₂ ∧ R_writer w₁ w₂) :
    RelTriple
      (simulateQ impl₁ oa).run
      (simulateQ impl₂ oa).run
      (fun p₁ p₂ => p₁.1 = p₂.1 ∧ R_writer p₁.2 p₂.2)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem relTriple_simulateQ_run_writerT_of_triples    {ι : Type} {spec : OracleSpec.{0, 0} ι} [IsUniformSpec spec]    {ω₁ ω₂ : Type} [Monoid ω₁] [Monoid ω₂]    (impl₁ : QueryImpl spec (WriterT ω₁ (OracleComp spec₁)))    (impl₂ : QueryImpl spec (WriterT ω₂ (OracleComp spec₂)))    (R_writer : ω₁ → ω₂ → Prop)    (hR_one : R_writer 1 1)    (hR_mul : ∀ w₁ w₁' w₂ w₂', R_writer w₁ w₂ → R_writer w₁' w₂' →      R_writer (w₁ * w₁') (w₂ * w₂'))    (oa : OracleComp spec α)    (Q₁ : ∀ (t : spec.Domain), spec.Range t → ω₁ → Prop)    (Q₂ : ∀ (t : spec.Domain), spec.Range t → ω₂ → Prop)    (h₁ : ∀ (t : spec.Domain), Std.Do.Triple      (impl₁ t : WriterT ω₁ (OracleComp spec₁) (spec.Range t))      (spred(fun s => ⌜s = 1⌝))      (⇓a s' => ⌜Q₁ t a s'⌝))    (h₂ : ∀ (t : spec.Domain), Std.Do.Triple      (impl₂ t : WriterT ω₂ (OracleComp spec₂) (spec.Range t))      (spred(fun s => ⌜s = 1⌝))      (⇓a s' => ⌜Q₂ t a s'⌝))    (hsync : ∀ (t : spec.Domain) a₁ w₁ a₂ w₂,      Q₁ t a₁ w₁ → Q₂ t a₂ w₂ → a₁ = a₂ ∧ R_writer w₁ w₂) :    RelTriple      (simulateQ impl₁ oa).run      (simulateQ impl₂ oa).run      (fun p₁ p₂ => p₁.1 = p₂.1 ∧ R_writer p₁.2 p₂.2) := by  refine relTriple_simulateQ_run_writerT    impl₁ impl₂ R_writer hR_one hR_mul oa ?_  intro t  refine relTriple_post_mono (relTriple_run_writerT_of_triple_monoid    (mx₁ := impl₁ t) (mx₂ := impl₂ t)    (s₁ := (1 : ω₁)) (s₂ := (1 : ω₂))    (P₁ := fun s => s = 1) (P₂ := fun s => s = 1)    (Q₁ := Q₁ t) (Q₂ := Q₂ t)    rfl rfl (h₁ t) (h₂ t)) ?_  rintro ⟨a₁, w₁⟩ ⟨a₂, w₂⟩ ⟨hQ₁, hQ₂⟩  rw [one_mul] at hQ₁ hQ₂  exact hsync t a₁ w₁ a₂ w₂ hQ₁ hQ₂

Project: VCVio
License: Apache-2.0
Commit: 2ceb2d825ee3
Source: VCVio/ProgramLogic/Relational/HandlerFromUnary.lean:219-256

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