Project-declaredLean 4.32.0 · mathlib@81a5d257

Of Real tv Dist bind left le const

ofReal_tvDist_bind_left_le_const

program verification separation logic cryptography tvdist

Plain-language statement

ℝ≥0∞ form of tvDist_bind_left_le_const, matching the quantitative APIs: a per-a bound ENNReal.ofReal (tvDist (f a) (g a)) ≤ ε on the support of mx lifts through the shared bind.

Exact Lean statement

theorem ofReal_tvDist_bind_left_le_const
    {m : Type u → Type v} [Monad m] [LawfulMonad m] [MonadLiftT m PMF] [LawfulMonadLiftT m PMF]
    [MonadLiftT m SetM] [EvalDistCompatible m]
    {α β : Type u}
    (mx : m α) (f g : α → m β) (ε : ℝ≥0∞)
    (hfg : ∀ a, a ∈ support mx → ENNReal.ofReal (tvDist (f a) (g a)) ≤ ε) :
    ENNReal.ofReal (tvDist (mx >>= f) (mx >>= g)) ≤ ε

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem ofReal_tvDist_bind_left_le_const    {m : Type u → Type v} [Monad m] [LawfulMonad m] [MonadLiftT m PMF] [LawfulMonadLiftT m PMF]    [MonadLiftT m SetM] [EvalDistCompatible m]    {α β : Type u}    (mx : m α) (f g : α → m β) (ε : ℝ≥0∞)    (hfg : ∀ a, a ∈ support mx → ENNReal.ofReal (tvDist (f a) (g a)) ≤ ε) :    ENNReal.ofReal (tvDist (mx >>= f) (mx >>= g)) ≤ ε := by  classical  by_cases htop : ε = (⊤ : ℝ≥0∞)  · simp [htop]  · have hreal : tvDist (mx >>= f) (mx >>= g) ≤ ε.toReal :=      tvDist_bind_left_le_const mx f g ε.toReal        (fun a ha => (ENNReal.ofReal_le_iff_le_toReal htop).mp (hfg a ha))    rw [← ENNReal.ofReal_toReal htop]    exact ENNReal.ofReal_le_ofReal hreal

Project: VCVio
License: Apache-2.0
Commit: 2ceb2d825ee3
Source: VCVio/EvalDist/TVDist.lean:278-292

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