Norm smul nat pow
Space.IsDistBounded.norm_smul_nat_pow
Project documentation
The boundedness condition on a function Space d → F for it to form a distribution. -/ @[fun_prop] def IsDistBounded {d : ℕ} (f : Space d → F) : Prop := AEStronglyMeasurable (fun x => f x) volume ∧ ∃ n, ∃ c : Fin n → ℝ, ∃ g : Fin n → Space d, ∃ p : Fin n → ℤ, (∀ i, 0 ≤ c i) ∧ (∀ i, - (d - 1 : ℕ) ≤ p i) ∧ ∀ x, ‖f x‖ ≤ ∑ i, c i * ‖x + g i‖ ^ p i namespace...
Exact Lean statement
lemma norm_smul_nat_pow {d} (p : ℕ) (c : Space d) :
IsDistBounded (fun x => ‖x‖ * ‖x + c‖ ^ p)Formal artifact
Lean source
lemma norm_smul_nat_pow {d} (p : ℕ) (c : Space d) : IsDistBounded (fun x => ‖x‖ * ‖x + c‖ ^ p) := by refine IsDistBounded.mono (f := fun x => ‖x‖ * (‖x‖ + ‖c‖) ^ p) ?_ (AEMeasurable.aestronglyMeasurable (by fun_prop)) fun x => ?_ · simp only [add_pow, Finset.mul_sum, ← mul_assoc] refine IsDistBounded.sum_fun fun i _ => mul_const_fun (mul_const_fun ?_ _) _ simpa [pow_succ'] using IsDistBounded.nat_pow (d := d) (i + 1) · simp [norm_mul, norm_pow, Real.norm_eq_abs] rw [abs_of_nonneg (by positivity)] gcongr exact norm_add_le x c- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/IsDistBounded.lean:818-828
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