import Mathlib.MeasureTheory.Order.Lattice
import Mathlib.Probability.Moments.SubGaussian
import Mathlib.Probability.Moments.MGFAnalytic
import Mathlib.Analysis.Convex.Integral

/-! # Standalone extraction for `ProbabilityTheory.HasSubexponentialMGF.congr_identDistrib`
Definitions are copied verbatim; theorem proofs are replaced by `sorry`.
Auto-generated by ChallengeGen. -/

set_option quotPrecheck false

-- Namespace stubs (so later `open`s resolve).
namespace Finset
end Finset
namespace MeasureTheory
end MeasureTheory
namespace ENNReal
end ENNReal
namespace ProbabilityTheory
end ProbabilityTheory
namespace ProbabilityTheory.HasSubexponentialMGF
end ProbabilityTheory.HasSubexponentialMGF

-- ═══ ForMathlib.MeasureTheory.Order.Lattice ═══
section
open Finset
variable {α δ : Type*} [MeasurableSpace δ] [SemilatticeInf α] {m : MeasurableSpace α} [MeasurableInf₂ α]
attribute [to_dual existing] MeasurableInf₂
end

-- ═══ ForMathlib.Probability.Moments.SubExponential ═══
section
open MeasureTheory Real Finset
open scoped ENNReal NNReal Topology
namespace ProbabilityTheory
variable {Ω : Type*} {m mΩ : MeasurableSpace Ω} {μ : Measure Ω} {X : Ω → ℝ} {V b : ℝ}

/-- `X` has a sub-exponential moment generating function with parameters `(V, b)`: for every
`t` with `b * |t| ≤ 1`, `exp (t * X)` is integrable and `mgf X μ t ≤ exp (V * t ^ 2 / 2)`.
For `b = 0` this is `HasSubgaussianMGF X V μ`.

This is equivalent to `Kernel.HasSubexponentialMGF X V b (Kernel.const Unit μ) (Measure.dirac ())`,
as proved in `hasSubexponentialMGF_iff_kernel`. -/
structure HasSubexponentialMGF (X : Ω → ℝ) (V b : ℝ) (μ : Measure Ω := sorry) :
    Prop where
  integrable_exp_mul : ∀ t : ℝ, b * |t| ≤ 1 → Integrable (fun ω ↦ exp (t * X ω)) μ
  mgf_le : ∀ t : ℝ, b * |t| ≤ 1 → mgf X μ t ≤ exp (V * t ^ 2 / 2)

namespace HasSubexponentialMGF

lemma congr_identDistrib {Ω' : Type*} {mΩ' : MeasurableSpace Ω'} {μ' : Measure Ω'}
    {Y : Ω' → ℝ} (hX : HasSubexponentialMGF X V b μ) (hXY : IdentDistrib X Y μ μ') :
    HasSubexponentialMGF Y V b μ' := sorry

end HasSubexponentialMGF
end ProbabilityTheory
end