Research proof index

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...

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...

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...

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...

A power series which is differentiable everywhere, and in the limit as n → ∞ tends to ‖x‖. -/ def normPowerSeries {d} : ℕ → Space d → ℝ := fun n x => √(‖x‖ ^ 2 + 1/(n + 1)) lemma normPowerSeries_eq (n : ℕ) : normPowerSeries (d := d) n = fun x => √(‖x‖ ^ 2 + 1/(n + 1)) := rfl lemma normPowerSeries_eq_rpow {d} (n : ℕ) : normPowerSeries (d := d) n = fun...

The divergence of a matrix-valued spatial field. For a field T : Space d → Matrix (Fin d) (Fin d) ℝ, matrixDiv T is the vector field whose ith component is ∑ j, ∂[j] (fun x => T x i j) x. -/ noncomputable def matrixDiv (d : ℕ) (T : Space d → Matrix (Fin d) (Fin d) ℝ) : Space d → EuclideanSpace ℝ (Fin d) := fun x => WithLp.toLp 2 fun i => div (fun...