fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
SuccDiffBounded (C : ℕ) (u : ℕ → ℕ) : Prop :=
∀ n : ℕ, u (n + 2) - u (n + 1) ≤ C • (u (n + 1) - u n) | def | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | SuccDiffBounded | A sequence `u` has the property that its ratio of successive differences is bounded
when there is a positive real number `C` such that, for all n ∈ ℕ,
(u (n + 2) - u (n + 1)) ≤ C * (u (n + 1) - u n) |
le_sum_schlomilch' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu : Monotone u) (n : ℕ) :
(∑ k ∈ Ico (u 0) (u n), f k) ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) := by
induction n with
| zero => simp
| succ n ihn =>
suffices (∑ k ∈ Ico (u n) (u (n + 1)), f k) ≤ (u (n + 1) - u n... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | le_sum_schlomilch' | null |
le_sum_condensed' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k ∈ Ico 1 (2 ^ n), f k) ≤ ∑ k ∈ range n, 2 ^ k • f (2 ^ k) := by
convert le_sum_schlomilch' hf (fun n => pow_pos zero_lt_two n)
(fun m n hm => pow_right_mono₀ one_le_two hm) n using 2
simp [pow_succ, mul_two] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | le_sum_condensed' | null |
le_sum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu : Monotone u) (n : ℕ) :
(∑ k ∈ range (u n), f k) ≤
∑ k ∈ range (u 0), f k + ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) := by
convert add_le_add_left (le_sum_schlomilch' hf h_pos hu n) (∑ k ∈ range (u 0), f k)
rw [← ... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | le_sum_schlomilch | null |
le_sum_condensed (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k ∈ range (2 ^ n), f k) ≤ f 0 + ∑ k ∈ range n, 2 ^ k • f (2 ^ k) := by
convert add_le_add_left (le_sum_condensed' hf n) (f 0)
rw [← sum_range_add_sum_Ico _ n.one_le_two_pow, sum_range_succ, sum_range_zero, zero_add] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | le_sum_condensed | null |
sum_schlomilch_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu : Monotone u) (n : ℕ) :
(∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1))) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k := by
induction n with
| zero => simp
| succ n ihn =>
suffices (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | sum_schlomilch_le' | null |
sum_condensed_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k ∈ range n, 2 ^ k • f (2 ^ (k + 1))) ≤ ∑ k ∈ Ico 2 (2 ^ n + 1), f k := by
convert sum_schlomilch_le' hf (fun n => pow_pos zero_lt_two n)
(fun m n hm => pow_right_mono₀ one_le_two hm) n using 2
simp [pow_succ, mul_two] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | sum_condensed_le' | null |
sum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(h_nonneg : ∀ n, 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) (n : ℕ) :
∑ k ∈ range (n + 1), (u (k + 1) - u k) • f (u k) ≤
(u 1 - u 0) • f (u 0) + C • ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k := by
rw [s... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | sum_schlomilch_le | null |
sum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k ∈ range (n + 1), 2 ^ k • f (2 ^ k)) ≤ f 1 + 2 • ∑ k ∈ Ico 2 (2 ^ n + 1), f k := by
convert add_le_add_left (nsmul_le_nsmul_right (sum_condensed_le' hf n) 2) (f 1)
simp [sum_range_succ', add_comm, pow_succ', mul_nsmul', sum_nsmul] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | sum_condensed_le | null |
le_tsum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu : StrictMono u) :
∑' k, f k ≤ ∑ k ∈ range (u 0), f k + ∑' k : ℕ, (u (k + 1) - u k) * f (u k) := by
rw [ENNReal.tsum_eq_iSup_nat' hu.tendsto_atTop]
refine iSup_le fun n =>
(Finset.le_sum_schlomilch hf h_pos hu.monoton... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | le_tsum_schlomilch | null |
le_tsum_condensed (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
∑' k, f k ≤ f 0 + ∑' k : ℕ, 2 ^ k * f (2 ^ k) := by
rw [ENNReal.tsum_eq_iSup_nat' (Nat.tendsto_pow_atTop_atTop_of_one_lt _root_.one_lt_two)]
refine iSup_le fun n => (Finset.le_sum_condensed hf n).trans (add_le_add_left ?_ _)
simp only [nsmul_eq_mul... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | le_tsum_condensed | null |
tsum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(h_nonneg : ∀ n, 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) :
∑' k : ℕ, (u (k + 1) - u k) * f (u k) ≤ (u 1 - u 0) * f (u 0) + C * ∑' k, f k := by
rw [ENNReal.tsum_eq_iSup_nat' (tendsto_atTop_mono Na... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | tsum_schlomilch_le | null |
tsum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) :
(∑' k : ℕ, 2 ^ k * f (2 ^ k)) ≤ f 1 + 2 * ∑' k, f k := by
rw [ENNReal.tsum_eq_iSup_nat' (tendsto_atTop_mono Nat.le_succ tendsto_id), two_mul, ← two_nsmul]
refine
iSup_le fun n =>
le_trans ?_
(add_le_add_left
(nsmul_le_nsmu... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | tsum_condensed_le | null |
summable_schlomilch_iff {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ≥0}
(hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m)
(h_pos : ∀ n, 0 < u n) (hu_strict : StrictMono u)
(hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u) :
(Summable fun k : ℕ => (u (k + 1) - (u k : ℝ≥0)) * f (u k)) ↔ Summable f := by
simp only [← t... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_schlomilch_iff | for a series of `NNReal` version. |
summable_condensed_iff {f : ℕ → ℝ≥0} (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
(Summable fun k : ℕ => (2 : ℝ≥0) ^ k * f (2 ^ k)) ↔ Summable f := by
have h_succ_diff : SuccDiffBounded 2 (2 ^ ·) := by
intro n
simp [pow_succ, mul_two, two_mul]
convert summable_schlomilch_iff hf (pow_pos zero_lt_two) (pow... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_condensed_iff | null |
summable_schlomilch_iff_of_nonneg {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ} (h_nonneg : ∀ n, 0 ≤ f n)
(hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu_strict : StrictMono u) (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u) :
(Summable fun k : ℕ => (u (k + 1) - (u k : ℝ)) * f (u k)) ↔ Summa... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_schlomilch_iff_of_nonneg | for series of nonnegative real numbers. |
summable_condensed_iff_of_nonneg {f : ℕ → ℝ} (h_nonneg : ∀ n, 0 ≤ f n)
(h_mono : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
(Summable fun k : ℕ => (2 : ℝ) ^ k * f (2 ^ k)) ↔ Summable f := by
have h_succ_diff : SuccDiffBounded 2 (2 ^ ·) := by
intro n
simp [pow_succ, mul_two, two_mul]
convert summable_schl... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_condensed_iff_of_nonneg | Cauchy condensation test for antitone series of nonnegative real numbers. |
@[simp]
summable_nat_rpow_inv {p : ℝ} :
Summable (fun n => ((n : ℝ) ^ p)⁻¹ : ℕ → ℝ) ↔ 1 < p := by
rcases le_or_gt 0 p with hp | hp
/- Cauchy condensation test applies only to antitone sequences, so we consider the
cases `0 ≤ p` and `p < 0` separately. -/
· rw [← summable_condensed_iff_of_nonneg]
· sim... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_nat_rpow_inv | Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, (n ^ p)⁻¹` converges
if and only if `1 < p`. |
summable_nat_rpow {p : ℝ} : Summable (fun n => (n : ℝ) ^ p : ℕ → ℝ) ↔ p < -1 := by
rcases neg_surjective p with ⟨p, rfl⟩
simp [rpow_neg] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_nat_rpow | null |
summable_one_div_nat_rpow {p : ℝ} :
Summable (fun n => 1 / (n : ℝ) ^ p : ℕ → ℝ) ↔ 1 < p := by
simp | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_one_div_nat_rpow | Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, 1 / n ^ p` converges
if and only if `1 < p`. |
@[simp]
summable_nat_pow_inv {p : ℕ} :
Summable (fun n => ((n : ℝ) ^ p)⁻¹ : ℕ → ℝ) ↔ 1 < p := by
simp only [← rpow_natCast, summable_nat_rpow_inv, Nat.one_lt_cast] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_nat_pow_inv | Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, (n ^ p)⁻¹` converges
if and only if `1 < p`. |
summable_one_div_nat_pow {p : ℕ} :
Summable (fun n => 1 / (n : ℝ) ^ p : ℕ → ℝ) ↔ 1 < p := by
simp only [one_div, Real.summable_nat_pow_inv] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_one_div_nat_pow | Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, 1 / n ^ p` converges
if and only if `1 < p`. |
summable_one_div_int_pow {p : ℕ} :
(Summable fun n : ℤ ↦ 1 / (n : ℝ) ^ p) ↔ 1 < p := by
refine ⟨fun h ↦ summable_one_div_nat_pow.mp (h.comp_injective Nat.cast_injective),
fun h ↦ .of_nat_of_neg (summable_one_div_nat_pow.mpr h)
(((summable_one_div_nat_pow.mpr h).mul_left <| 1 / (-1 : ℝ) ^ p).congr fun n ... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_one_div_int_pow | Summability of the `p`-series over `ℤ`. |
summable_abs_int_rpow {b : ℝ} (hb : 1 < b) :
Summable fun n : ℤ => |(n : ℝ)| ^ (-b) := by
apply Summable.of_nat_of_neg
on_goal 2 => simp_rw [Int.cast_neg, abs_neg]
all_goals
simp_rw [Int.cast_natCast, fun n : ℕ => abs_of_nonneg (n.cast_nonneg : 0 ≤ (n : ℝ))]
rwa [summable_nat_rpow, neg_lt_neg_iff] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_abs_int_rpow | null |
not_summable_natCast_inv : ¬Summable (fun n => n⁻¹ : ℕ → ℝ) := by
have : ¬Summable (fun n => ((n : ℝ) ^ 1)⁻¹ : ℕ → ℝ) :=
mt (summable_nat_pow_inv (p := 1)).1 (lt_irrefl 1)
simpa | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | not_summable_natCast_inv | Harmonic series is not unconditionally summable. |
not_summable_one_div_natCast : ¬Summable (fun n => 1 / n : ℕ → ℝ) := by
simpa only [inv_eq_one_div] using not_summable_natCast_inv | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | not_summable_one_div_natCast | Harmonic series is not unconditionally summable. |
tendsto_sum_range_one_div_nat_succ_atTop :
Tendsto (fun n => ∑ i ∈ Finset.range n, (1 / (i + 1) : ℝ)) atTop atTop := by
rw [← not_summable_iff_tendsto_nat_atTop_of_nonneg]
· exact_mod_cast mt (_root_.summable_nat_add_iff 1).1 not_summable_one_div_natCast
· exact fun i => by positivity | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | tendsto_sum_range_one_div_nat_succ_atTop | **Divergence of the Harmonic Series** |
@[simp]
summable_rpow_inv {p : ℝ} :
Summable (fun n => ((n : ℝ≥0) ^ p)⁻¹ : ℕ → ℝ≥0) ↔ 1 < p := by
simp [← NNReal.summable_coe]
@[simp] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_rpow_inv | null |
summable_rpow {p : ℝ} : Summable (fun n => (n : ℝ≥0) ^ p : ℕ → ℝ≥0) ↔ p < -1 := by
simp [← NNReal.summable_coe] | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_rpow | null |
summable_one_div_rpow {p : ℝ} :
Summable (fun n => 1 / (n : ℝ≥0) ^ p : ℕ → ℝ≥0) ↔ 1 < p := by
simp | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_one_div_rpow | null |
sum_Ioc_inv_sq_le_sub {k n : ℕ} (hk : k ≠ 0) (h : k ≤ n) :
(∑ i ∈ Ioc k n, ((i : α) ^ 2)⁻¹) ≤ (k : α)⁻¹ - (n : α)⁻¹ := by
refine Nat.le_induction ?_ ?_ n h
· simp only [Ioc_self, sum_empty, sub_self, le_refl]
intro n hn IH
rw [sum_Ioc_succ_top hn]
grw [IH]
push_cast
have A : 0 < (n : α) := by simpa us... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | sum_Ioc_inv_sq_le_sub | null |
sum_Ioo_inv_sq_le (k n : ℕ) : (∑ i ∈ Ioo k n, (i ^ 2 : α)⁻¹) ≤ 2 / (k + 1) :=
calc
(∑ i ∈ Ioo k n, ((i : α) ^ 2)⁻¹) ≤ ∑ i ∈ Ioc k (max (k + 1) n), ((i : α) ^ 2)⁻¹ := by
apply sum_le_sum_of_subset_of_nonneg
· intro x hx
simp only [mem_Ioo] at hx
simp only [hx, hx.2.le, mem_Ioc, le_max_i... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | sum_Ioo_inv_sq_le | null |
Real.not_summable_indicator_one_div_natCast {m : ℕ} (hm : m ≠ 0) (k : ZMod m) :
¬ Summable ({n : ℕ | (n : ZMod m) = k}.indicator fun n : ℕ ↦ (1 / n : ℝ)) := by
have : NeZero m := ⟨hm⟩ -- instance is needed below
rw [← summable_nat_add_iff 1] -- shift by one to avoid non-monotonicity at zero
have h (n : ℕ) : {... | lemma | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | Real.not_summable_indicator_one_div_natCast | The harmonic series restricted to a residue class is not summable. |
Real.summable_one_div_nat_add_rpow (a : ℝ) (s : ℝ) :
Summable (fun n : ℕ ↦ 1 / |n + a| ^ s) ↔ 1 < s := by
suffices ∀ (b c : ℝ), Summable (fun n : ℕ ↦ 1 / |n + b| ^ s) →
Summable (fun n : ℕ ↦ 1 / |n + c| ^ s) by
simp_rw [← summable_one_div_nat_rpow, Iff.intro (this a 0) (this 0 a), add_zero, Nat.abs_cast... | lemma | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | Real.summable_one_div_nat_add_rpow | null |
Real.summable_one_div_int_add_rpow (a : ℝ) (s : ℝ) :
Summable (fun n : ℤ ↦ 1 / |n + a| ^ s) ↔ 1 < s := by
simp_rw [summable_int_iff_summable_nat_and_neg, ← abs_neg (↑(-_ : ℤ) + a), neg_add,
Int.cast_neg, neg_neg, Int.cast_natCast, summable_one_div_nat_add_rpow, and_self] | lemma | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | Real.summable_one_div_int_add_rpow | null |
summable_pow_div_add {α : Type*} (x : α) [RCLike α] (q k : ℕ) (hq : 1 < q) :
Summable fun n : ℕ => ‖(x / (↑n + k) ^ q)‖ := by
simp_rw [norm_div]
apply Summable.const_div
simpa [hq, Nat.cast_add, one_div, norm_inv, norm_pow, RCLike.norm_natCast,
Real.summable_nat_pow_inv, iff_true]
using summable_nat... | theorem | Analysis | [
"Mathlib.Analysis.SpecialFunctions.Pow.NNReal",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity",
"Mathlib.Analysis.SumOverResidueClass"
] | Mathlib/Analysis/PSeries.lean | summable_pow_div_add | null |
Complex.summable_one_div_nat_cpow {p : ℂ} :
Summable (fun n : ℕ ↦ 1 / (n : ℂ) ^ p) ↔ 1 < re p := by
rw [← Real.summable_one_div_nat_rpow, ← summable_nat_add_iff 1 (G := ℝ),
← summable_nat_add_iff 1 (G := ℂ), ← summable_norm_iff]
simp only [norm_div, norm_one, ← ofReal_natCast, norm_cpow_eq_rpow_re_of_pos
... | lemma | Analysis | [
"Mathlib.Analysis.Normed.Module.FiniteDimension",
"Mathlib.Analysis.PSeries",
"Mathlib.LinearAlgebra.Complex.FiniteDimensional"
] | Mathlib/Analysis/PSeriesComplex.lean | Complex.summable_one_div_nat_cpow | null |
inner_self (a : ℍ) : ⟪a, a⟫ = normSq a :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | inner_self | null |
inner_def (a b : ℍ) : ⟪a, b⟫ = (a * star b).re :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | inner_def | null |
normSq_eq_norm_mul_self (a : ℍ) : normSq a = ‖a‖ * ‖a‖ := by
rw [← inner_self, real_inner_self_eq_norm_mul_norm] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | normSq_eq_norm_mul_self | null |
@[simp, norm_cast]
norm_coe (a : ℝ) : ‖(a : ℍ)‖ = ‖a‖ := by
rw [norm_eq_sqrt_real_inner, inner_self, normSq_coe, Real.sqrt_sq_eq_abs, Real.norm_eq_abs]
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | norm_coe | null |
nnnorm_coe (a : ℝ) : ‖(a : ℍ)‖₊ = ‖a‖₊ :=
Subtype.ext <| norm_coe a | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | nnnorm_coe | null |
norm_star (a : ℍ) : ‖star a‖ = ‖a‖ := by
simp_rw [norm_eq_sqrt_real_inner, inner_self, normSq_star] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | norm_star | null |
nnnorm_star (a : ℍ) : ‖star a‖₊ = ‖a‖₊ :=
Subtype.ext <| norm_star a | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | nnnorm_star | null |
@[coe] coeComplex (z : ℂ) : ℍ := ⟨z.re, z.im, 0, 0⟩ | def | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | coeComplex | Coercion from `ℂ` to `ℍ`. |
@[simp, norm_cast]
re_coeComplex (z : ℂ) : (z : ℍ).re = z.re :=
rfl
@[deprecated (since := "2025-08-31")] alias coeComplex_re := re_coeComplex
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | re_coeComplex | null |
imI_coeComplex (z : ℂ) : (z : ℍ).imI = z.im :=
rfl
@[deprecated (since := "2025-08-31")] alias coeComplex_imI := imI_coeComplex
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | imI_coeComplex | null |
imJ_coeComplex (z : ℂ) : (z : ℍ).imJ = 0 :=
rfl
@[deprecated (since := "2025-08-31")] alias coeComplex_imJ := imJ_coeComplex
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | imJ_coeComplex | null |
imK_coeComplex (z : ℂ) : (z : ℍ).imK = 0 :=
rfl
@[deprecated (since := "2025-08-31")] alias coeComplex_imK := imK_coeComplex
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | imK_coeComplex | null |
coeComplex_add (z w : ℂ) : ↑(z + w) = (z + w : ℍ) := by ext <;> simp
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | coeComplex_add | null |
coeComplex_mul (z w : ℂ) : ↑(z * w) = (z * w : ℍ) := by ext <;> simp
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | coeComplex_mul | null |
coeComplex_zero : ((0 : ℂ) : ℍ) = 0 :=
rfl
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | coeComplex_zero | null |
coeComplex_one : ((1 : ℂ) : ℍ) = 1 :=
rfl
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | coeComplex_one | null |
coe_real_complex_mul (r : ℝ) (z : ℂ) : (r • z : ℍ) = ↑r * ↑z := by ext <;> simp
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | coe_real_complex_mul | null |
coeComplex_coe (r : ℝ) : ((r : ℂ) : ℍ) = r :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | coeComplex_coe | null |
ofComplex : ℂ →ₐ[ℝ] ℍ where
toFun := (↑)
map_one' := rfl
map_zero' := rfl
map_add' := coeComplex_add
map_mul' := coeComplex_mul
commutes' _ := rfl
@[simp] | def | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | ofComplex | Coercion `ℂ →ₐ[ℝ] ℍ` as an algebra homomorphism. |
coe_ofComplex : ⇑ofComplex = coeComplex := rfl | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | coe_ofComplex | null |
norm_toLp_equivTuple (x : ℍ) : ‖WithLp.toLp 2 (equivTuple ℝ x)‖ = ‖x‖ := by
rw [norm_eq_sqrt_real_inner, norm_eq_sqrt_real_inner, inner_self, normSq_def', PiLp.inner_apply,
Fin.sum_univ_four]
simp_rw [RCLike.inner_apply, starRingEnd_apply, star_trivial, ← sq]
rfl | lemma | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | norm_toLp_equivTuple | The norm of the components as a Euclidean vector equals the norm of the quaternion. |
@[simps apply symm_apply]
noncomputable linearIsometryEquivTuple : ℍ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin 4) :=
{ (QuaternionAlgebra.linearEquivTuple (-1 : ℝ) (0 : ℝ) (-1 : ℝ)).trans
(WithLp.linearEquiv 2 ℝ (Fin 4 → ℝ)).symm with
toFun := fun a => !₂[a.1, a.2, a.3, a.4]
invFun := fun a => ⟨a 0, a 1, a 2, a 3⟩
... | def | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | linearIsometryEquivTuple | `QuaternionAlgebra.linearEquivTuple` as a `LinearIsometryEquiv`. |
continuous_coe : Continuous (coe : ℝ → ℍ) :=
continuous_algebraMap ℝ ℍ
@[continuity] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | continuous_coe | null |
continuous_normSq : Continuous (normSq : ℍ → ℝ) := by
simpa [← normSq_eq_norm_mul_self] using
(continuous_norm.mul continuous_norm : Continuous fun q : ℍ => ‖q‖ * ‖q‖)
@[continuity] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | continuous_normSq | null |
continuous_re : Continuous fun q : ℍ => q.re :=
(continuous_apply 0).comp linearIsometryEquivTuple.continuous
@[continuity] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | continuous_re | null |
continuous_imI : Continuous fun q : ℍ => q.imI :=
(continuous_apply 1).comp linearIsometryEquivTuple.continuous
@[continuity] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | continuous_imI | null |
continuous_imJ : Continuous fun q : ℍ => q.imJ :=
(continuous_apply 2).comp linearIsometryEquivTuple.continuous
@[continuity] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | continuous_imJ | null |
continuous_imK : Continuous fun q : ℍ => q.imK :=
(continuous_apply 3).comp linearIsometryEquivTuple.continuous
@[continuity] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | continuous_imK | null |
continuous_im : Continuous fun q : ℍ => q.im := by
simpa only [← sub_re_self] using continuous_id.sub (continuous_coe.comp continuous_re) | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | continuous_im | null |
@[simp, norm_cast]
hasSum_coe {f : α → ℝ} {r : ℝ} : HasSum (fun a => (f a : ℍ)) (↑r : ℍ) ↔ HasSum f r :=
⟨fun h => by
simpa only using
h.map (show ℍ →ₗ[ℝ] ℝ from QuaternionAlgebra.reₗ _ _ _) continuous_re,
fun h => by simpa only using h.map (algebraMap ℝ ℍ) (continuous_algebraMap _ _)⟩
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | hasSum_coe | null |
summable_coe {f : α → ℝ} : (Summable fun a => (f a : ℍ)) ↔ Summable f := by
simpa only using
Summable.map_iff_of_leftInverse (algebraMap ℝ ℍ) (show ℍ →ₗ[ℝ] ℝ from
QuaternionAlgebra.reₗ _ _ _)
(continuous_algebraMap _ _) continuous_re re_coe
@[norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | summable_coe | null |
tsum_coe (f : α → ℝ) : (∑' a, (f a : ℍ)) = ↑(∑' a, f a) := by
by_cases hf : Summable f
· exact (hasSum_coe.mpr hf.hasSum).tsum_eq
· simp [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (summable_coe.not.mpr hf)] | theorem | Analysis | [
"Mathlib.Algebra.Quaternion",
"Mathlib.Analysis.InnerProductSpace.Continuous",
"Mathlib.Analysis.InnerProductSpace.PiL2",
"Mathlib.Topology.Algebra.Algebra"
] | Mathlib/Analysis/Quaternion.lean | tsum_coe | null |
Seminorm (𝕜 : Type*) (E : Type*) [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : ∀ (a : 𝕜) (x : E), toFun (a • x) = ‖a‖ * toFun x
attribute [nolint... | structure | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | Seminorm | A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. |
SeminormClass (F : Type*) (𝕜 E : outParam Type*) [SeminormedRing 𝕜] [AddGroup E]
[SMul 𝕜 E] [FunLike F E ℝ] : Prop extends AddGroupSeminormClass F E ℝ where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a ... | class | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | SeminormClass | `SeminormClass F 𝕜 E` states that `F` is a type of seminorms on the `𝕜`-module `E`.
You should extend this class when you extend `Seminorm`. |
Seminorm.of [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ)
(add_le : ∀ x y : E, f (x + y) ≤ f x + f y) (smul : ∀ (a : 𝕜) (x : E), f (a • x) = ‖a‖ * f x) :
Seminorm 𝕜 E where
toFun := f
map_zero' := by rw [← zero_smul 𝕜 (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smu... | def | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | Seminorm.of | Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing 𝕜`. |
Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (x), f (r • x) ≤ ‖r‖ * f x) :
Seminorm 𝕜 E :=
Seminorm.of f add_le fun r x => by
refine le_antisymm (smul_le r x) ?_
by_cases h : r = 0
... | def | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | Seminorm.ofSMulLE | Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. |
instFunLike : FunLike (Seminorm 𝕜 E) E ℝ where
coe f := f.toFun
coe_injective' f g h := by
rcases f with ⟨⟨_⟩⟩
rcases g with ⟨⟨_⟩⟩
congr | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instFunLike | null |
instSeminormClass : SeminormClass (Seminorm 𝕜 E) 𝕜 E where
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
@[ext] | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instSeminormClass | null |
ext {p q : Seminorm 𝕜 E} (h : ∀ x, (p : E → ℝ) x = q x) : p = q :=
DFunLike.ext p q h | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | ext | null |
instZero : Zero (Seminorm 𝕜 E) :=
⟨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }⟩
@[simp] | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instZero | null |
coe_zero : ⇑(0 : Seminorm 𝕜 E) = 0 :=
rfl
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_zero | null |
zero_apply (x : E) : (0 : Seminorm 𝕜 E) x = 0 :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | zero_apply | null |
instSMul [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] : SMul R (Seminorm 𝕜 E) where
smul r p :=
{ r • p.toAddGroupSeminorm with
toFun := fun x => r • p x
smul' := fun _ _ => by
simp only [← smul_one_smul ℝ≥0 r (_ : ℝ), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_com... | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instSMul | Any action on `ℝ` which factors through `ℝ≥0` applies to a seminorm. |
coe_smul [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : Seminorm 𝕜 E) :
⇑(r • p) = r • ⇑p :=
rfl
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_smul | null |
smul_apply [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : Seminorm 𝕜 E)
(x : E) : (r • p) x = r • p x :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | smul_apply | null |
instAdd : Add (Seminorm 𝕜 E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] } | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instAdd | null |
coe_add (p q : Seminorm 𝕜 E) : ⇑(p + q) = p + q :=
rfl
@[simp] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_add | null |
add_apply (p q : Seminorm 𝕜 E) (x : E) : (p + q) x = p x + q x :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | add_apply | null |
instAddMonoid : AddMonoid (Seminorm 𝕜 E) :=
DFunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instAddMonoid | null |
instAddCommMonoid : AddCommMonoid (Seminorm 𝕜 E) :=
DFunLike.coe_injective.addCommMonoid _ rfl coe_add fun _ _ => by rfl | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instAddCommMonoid | null |
instPartialOrder : PartialOrder (Seminorm 𝕜 E) :=
PartialOrder.lift _ DFunLike.coe_injective | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instPartialOrder | null |
instIsOrderedCancelAddMonoid : IsOrderedCancelAddMonoid (Seminorm 𝕜 E) :=
Function.Injective.isOrderedCancelAddMonoid DFunLike.coe coe_add .rfl | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instIsOrderedCancelAddMonoid | null |
instMulAction [Monoid R] [MulAction R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
MulAction R (Seminorm 𝕜 E) :=
DFunLike.coe_injective.mulAction _ (by intros; rfl)
variable (𝕜 E) | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instMulAction | null |
@[simps]
coeFnAddMonoidHom : AddMonoidHom (Seminorm 𝕜 E) (E → ℝ) where
toFun := (↑)
map_zero' := coe_zero
map_add' := coe_add | def | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coeFnAddMonoidHom | `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm 𝕜 E` is a module. |
coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom 𝕜 E) :=
show @Function.Injective (Seminorm 𝕜 E) (E → ℝ) (↑) from DFunLike.coe_injective
variable {𝕜 E} | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coeFnAddMonoidHom_injective | null |
instDistribMulAction [Monoid R] [DistribMulAction R ℝ] [SMul R ℝ≥0]
[IsScalarTower R ℝ≥0 ℝ] : DistribMulAction R (Seminorm 𝕜 E) :=
(coeFnAddMonoidHom_injective 𝕜 E).distribMulAction _ (by intros; rfl) | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instDistribMulAction | null |
instModule [Semiring R] [Module R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
Module R (Seminorm 𝕜 E) :=
(coeFnAddMonoidHom_injective 𝕜 E).module R _ (by intros; rfl) | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instModule | null |
instSup : Max (Seminorm 𝕜 E) where
max p q :=
{ p.toAddGroupSeminorm ⊔ q.toAddGroupSeminorm with
toFun := p ⊔ q
smul' := fun x v =>
(congr_arg₂ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp] | instance | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | instSup | null |
coe_sup (p q : Seminorm 𝕜 E) : ⇑(p ⊔ q) = (p : E → ℝ) ⊔ (q : E → ℝ) :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_sup | null |
sup_apply (p q : Seminorm 𝕜 E) (x : E) : (p ⊔ q) x = p x ⊔ q x :=
rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | sup_apply | null |
smul_sup [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : Seminorm 𝕜 E) :
r • (p ⊔ q) = r • p ⊔ r • q :=
have real.smul_max : ∀ x y : ℝ, r • max x y = max (r • x) (r • y) := fun x y => by
simpa only [← smul_eq_mul, ← NNReal.smul_def, smul_one_smul ℝ≥0 r (_ : ℝ)] using
mul_max_of_nonneg x ... | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | smul_sup | null |
coe_le_coe {p q : Seminorm 𝕜 E} : (p : E → ℝ) ≤ q ↔ p ≤ q :=
Iff.rfl
@[simp, norm_cast] | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_le_coe | null |
coe_lt_coe {p q : Seminorm 𝕜 E} : (p : E → ℝ) < q ↔ p < q :=
Iff.rfl | theorem | Analysis | [
"Mathlib.Algebra.Order.Pi",
"Mathlib.Analysis.Convex.Function",
"Mathlib.Analysis.LocallyConvex.Basic",
"Mathlib.Analysis.Normed.Module.Basic",
"Mathlib.Data.Real.Pointwise"
] | Mathlib/Analysis/Seminorm.lean | coe_lt_coe | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.