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 ⌀ |
|---|---|---|---|---|---|---|
nhds_zero_basis : (𝓝 (0 : ℝ≥0∞)).HasBasis (fun a : ℝ≥0∞ => 0 < a) fun a => Iio a :=
nhds_bot_basis | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhds_zero_basis | null |
nhds_zero_basis_Iic : (𝓝 (0 : ℝ≥0∞)).HasBasis (fun a : ℝ≥0∞ => 0 < a) Iic :=
nhds_bot_basis_Iic
@[instance] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhds_zero_basis_Iic | null |
nhdsGT_coe_neBot {r : ℝ≥0} : (𝓝[>] (r : ℝ≥0∞)).NeBot :=
nhdsGT_neBot_of_exists_gt ⟨∞, ENNReal.coe_lt_top⟩
@[instance] theorem nhdsGT_zero_neBot : (𝓝[>] (0 : ℝ≥0∞)).NeBot := nhdsGT_coe_neBot
@[instance] theorem nhdsGT_one_neBot : (𝓝[>] (1 : ℝ≥0∞)).NeBot := nhdsGT_coe_neBot
@[instance] theorem nhdsGT_nat_neBot (n : ... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhdsGT_coe_neBot | null |
nhdsGT_ofNat_neBot (n : ℕ) [n.AtLeastTwo] : (𝓝[>] (OfNat.ofNat n : ℝ≥0∞)).NeBot :=
nhdsGT_coe_neBot
@[instance] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhdsGT_ofNat_neBot | null |
nhdsLT_neBot [NeZero x] : (𝓝[<] x).NeBot :=
nhdsWithin_Iio_self_neBot' ⟨0, NeZero.pos x⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhdsLT_neBot | null |
hasBasis_nhds_of_ne_top' (xt : x ≠ ∞) :
(𝓝 x).HasBasis (· ≠ 0) (fun ε => Icc (x - ε) (x + ε)) := by
rcases (zero_le x).eq_or_lt with rfl | x0
· simp_rw [zero_tsub, zero_add, ← bot_eq_zero, Icc_bot, ← bot_lt_iff_ne_bot]
exact nhds_bot_basis_Iic
· refine (nhds_basis_Ioo' ⟨_, x0⟩ ⟨_, xt.lt_top⟩).to_hasBasis... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | hasBasis_nhds_of_ne_top' | Closed intervals `Set.Icc (x - ε) (x + ε)`, `ε ≠ 0`, form a basis of neighborhoods of an
extended nonnegative real number `x ≠ ∞`. We use `Set.Icc` instead of `Set.Ioo` because this way the
statement works for `x = 0`. |
hasBasis_nhds_of_ne_top (xt : x ≠ ∞) :
(𝓝 x).HasBasis (0 < ·) (fun ε => Icc (x - ε) (x + ε)) := by
simpa only [pos_iff_ne_zero] using hasBasis_nhds_of_ne_top' xt | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | hasBasis_nhds_of_ne_top | null |
Icc_mem_nhds (xt : x ≠ ∞) (ε0 : ε ≠ 0) : Icc (x - ε) (x + ε) ∈ 𝓝 x :=
(hasBasis_nhds_of_ne_top' xt).mem_of_mem ε0 | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Icc_mem_nhds | null |
nhds_of_ne_top (xt : x ≠ ∞) : 𝓝 x = ⨅ ε > 0, 𝓟 (Icc (x - ε) (x + ε)) :=
(hasBasis_nhds_of_ne_top xt).eq_biInf | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhds_of_ne_top | null |
biInf_le_nhds : ∀ x : ℝ≥0∞, ⨅ ε > 0, 𝓟 (Icc (x - ε) (x + ε)) ≤ 𝓝 x
| ∞ => iInf₂_le_of_le 1 one_pos <| by
simpa only [← coe_one, top_sub_coe, top_add, Icc_self, principal_singleton] using pure_le_nhds _
| (x : ℝ≥0) => (nhds_of_ne_top coe_ne_top).ge | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | biInf_le_nhds | null |
protected tendsto_nhds_of_Icc {f : Filter α} {u : α → ℝ≥0∞} {a : ℝ≥0∞}
(h : ∀ ε > 0, ∀ᶠ x in f, u x ∈ Icc (a - ε) (a + ε)) : Tendsto u f (𝓝 a) := by
refine Tendsto.mono_right ?_ (biInf_le_nhds _)
simpa only [tendsto_iInf, tendsto_principal] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nhds_of_Icc | null |
protected tendsto_nhds {f : Filter α} {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha : a ≠ ∞) :
Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, u x ∈ Icc (a - ε) (a + ε) := by
simp only [nhds_of_ne_top ha, tendsto_iInf, tendsto_principal] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nhds | Characterization of neighborhoods for `ℝ≥0∞` numbers. See also `tendsto_order`
for a version with strict inequalities. |
protected tendsto_nhds_zero {f : Filter α} {u : α → ℝ≥0∞} :
Tendsto u f (𝓝 0) ↔ ∀ ε > 0, ∀ᶠ x in f, u x ≤ ε :=
nhds_zero_basis_Iic.tendsto_right_iff | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nhds_zero | null |
tendsto_const_sub_nhds_zero_iff {l : Filter α} {f : α → ℝ≥0∞} {a : ℝ≥0∞} (ha : a ≠ ∞)
(hfa : ∀ n, f n ≤ a) :
Tendsto (fun n ↦ a - f n) l (𝓝 0) ↔ Tendsto (fun n ↦ f n) l (𝓝 a) := by
rw [ENNReal.tendsto_nhds_zero, ENNReal.tendsto_nhds ha]
refine ⟨fun h ε hε ↦ ?_, fun h ε hε ↦ ?_⟩
· filter_upwards [h ε hε]... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_const_sub_nhds_zero_iff | null |
protected tendsto_atTop [Nonempty β] [SemilatticeSup β] {f : β → ℝ≥0∞} {a : ℝ≥0∞}
(ha : a ≠ ∞) : Tendsto f atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, f n ∈ Icc (a - ε) (a + ε) :=
.trans (atTop_basis.tendsto_iff (hasBasis_nhds_of_ne_top ha)) (by simp only [true_and]; rfl) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_atTop | null |
protected tendsto_atTop_zero [Nonempty β] [SemilatticeSup β] {f : β → ℝ≥0∞} :
Tendsto f atTop (𝓝 0) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, f n ≤ ε :=
.trans (atTop_basis.tendsto_iff nhds_zero_basis_Iic) (by simp only [true_and]; rfl) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_atTop_zero | null |
tendsto_atTop_zero_iff_le_of_antitone {β : Type*} [Nonempty β] [SemilatticeSup β]
{f : β → ℝ≥0∞} (hf : Antitone f) :
Filter.Tendsto f Filter.atTop (𝓝 0) ↔ ∀ ε, 0 < ε → ∃ n : β, f n ≤ ε := by
rw [ENNReal.tendsto_atTop_zero]
refine ⟨fun h ↦ fun ε hε ↦ ?_, fun h ↦ fun ε hε ↦ ?_⟩
· obtain ⟨n, hn⟩ := h ε hε
... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_atTop_zero_iff_le_of_antitone | null |
tendsto_atTop_zero_iff_lt_of_antitone {β : Type*} [Nonempty β] [SemilatticeSup β]
{f : β → ℝ≥0∞} (hf : Antitone f) :
Filter.Tendsto f Filter.atTop (𝓝 0) ↔ ∀ ε, 0 < ε → ∃ n : β, f n < ε := by
rw [ENNReal.tendsto_atTop_zero_iff_le_of_antitone hf]
constructor <;> intro h ε hε
· obtain ⟨n, hn⟩ := h (min 1 (ε... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_atTop_zero_iff_lt_of_antitone | null |
tendsto_sub : ∀ {a b : ℝ≥0∞}, (a ≠ ∞ ∨ b ≠ ∞) →
Tendsto (fun p : ℝ≥0∞ × ℝ≥0∞ => p.1 - p.2) (𝓝 (a, b)) (𝓝 (a - b))
| ∞, ∞, h => by simp only [ne_eq, not_true_eq_false, or_self] at h
| ∞, (b : ℝ≥0), _ => by
rw [top_sub_coe, tendsto_nhds_top_iff_nnreal]
refine fun x => ((lt_mem_nhds <| @coe_lt_top (b + 1... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_sub | null |
protected Tendsto.sub {f : Filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hma : Tendsto ma f (𝓝 a)) (hmb : Tendsto mb f (𝓝 b)) (h : a ≠ ∞ ∨ b ≠ ∞) :
Tendsto (fun a => ma a - mb a) f (𝓝 (a - b)) :=
show Tendsto ((fun p : ℝ≥0∞ × ℝ≥0∞ => p.1 - p.2) ∘ fun a => (ma a, mb a)) f (𝓝 (a - b)) from
Ten... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Tendsto.sub | null |
protected tendsto_mul (ha : a ≠ 0 ∨ b ≠ ∞) (hb : b ≠ 0 ∨ a ≠ ∞) :
Tendsto (fun p : ℝ≥0∞ × ℝ≥0∞ => p.1 * p.2) (𝓝 (a, b)) (𝓝 (a * b)) := by
have ht : ∀ b : ℝ≥0∞, b ≠ 0 →
Tendsto (fun p : ℝ≥0∞ × ℝ≥0∞ => p.1 * p.2) (𝓝 (∞, b)) (𝓝 ∞) := fun b hb => by
refine tendsto_nhds_top_iff_nnreal.2 fun n => ?_
r... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_mul | null |
protected Tendsto.mul {f : Filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hma : Tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ ∞) (hmb : Tendsto mb f (𝓝 b))
(hb : b ≠ 0 ∨ a ≠ ∞) : Tendsto (fun a => ma a * mb a) f (𝓝 (a * b)) :=
show Tendsto ((fun p : ℝ≥0∞ × ℝ≥0∞ => p.1 * p.2) ∘ fun a => (ma a, mb a)) f (𝓝... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Tendsto.mul | null |
_root_.ContinuousOn.ennreal_mul [TopologicalSpace α] {f g : α → ℝ≥0∞} {s : Set α}
(hf : ContinuousOn f s) (hg : ContinuousOn g s) (h₁ : ∀ x ∈ s, f x ≠ 0 ∨ g x ≠ ∞)
(h₂ : ∀ x ∈ s, g x ≠ 0 ∨ f x ≠ ∞) : ContinuousOn (fun x => f x * g x) s := fun x hx =>
ENNReal.Tendsto.mul (hf x hx) (h₁ x hx) (hg x hx) (h₂ x hx) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | _root_.ContinuousOn.ennreal_mul | null |
_root_.Continuous.ennreal_mul [TopologicalSpace α] {f g : α → ℝ≥0∞} (hf : Continuous f)
(hg : Continuous g) (h₁ : ∀ x, f x ≠ 0 ∨ g x ≠ ∞) (h₂ : ∀ x, g x ≠ 0 ∨ f x ≠ ∞) :
Continuous fun x => f x * g x :=
continuous_iff_continuousAt.2 fun x =>
ENNReal.Tendsto.mul hf.continuousAt (h₁ x) hg.continuousAt (h₂ x... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | _root_.Continuous.ennreal_mul | null |
protected Tendsto.const_mul {f : Filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hm : Tendsto m f (𝓝 b)) (hb : b ≠ 0 ∨ a ≠ ∞) : Tendsto (fun b => a * m b) f (𝓝 (a * b)) :=
by_cases (fun (this : a = 0) => by simp [this, tendsto_const_nhds]) fun ha : a ≠ 0 =>
ENNReal.Tendsto.mul tendsto_const_nhds (Or.inl ha) hm hb | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Tendsto.const_mul | null |
protected Tendsto.mul_const {f : Filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hm : Tendsto m f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ ∞) : Tendsto (fun x => m x * b) f (𝓝 (a * b)) := by
simpa only [mul_comm] using ENNReal.Tendsto.const_mul hm ha | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Tendsto.mul_const | null |
tendsto_finset_prod_of_ne_top {ι : Type*} {f : ι → α → ℝ≥0∞} {x : Filter α} {a : ι → ℝ≥0∞}
(s : Finset ι) (h : ∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) (h' : ∀ i ∈ s, a i ≠ ∞) :
Tendsto (fun b => ∏ c ∈ s, f c b) x (𝓝 (∏ c ∈ s, a c)) := by
classical
induction s using Finset.induction with
| empty => simp [ten... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_finset_prod_of_ne_top | null |
protected continuousAt_const_mul {a b : ℝ≥0∞} (h : a ≠ ∞ ∨ b ≠ 0) :
ContinuousAt (a * ·) b :=
Tendsto.const_mul tendsto_id h.symm | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuousAt_const_mul | null |
protected continuousAt_mul_const {a b : ℝ≥0∞} (h : a ≠ ∞ ∨ b ≠ 0) :
ContinuousAt (fun x => x * a) b :=
Tendsto.mul_const tendsto_id h.symm
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuousAt_mul_const | null |
protected continuous_const_mul {a : ℝ≥0∞} (ha : a ≠ ∞) : Continuous (a * ·) :=
continuous_iff_continuousAt.2 fun _ => ENNReal.continuousAt_const_mul (Or.inl ha)
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_const_mul | null |
protected continuous_mul_const {a : ℝ≥0∞} (ha : a ≠ ∞) : Continuous fun x => x * a :=
continuous_iff_continuousAt.2 fun _ => ENNReal.continuousAt_mul_const (Or.inl ha)
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_mul_const | null |
protected continuous_div_const (c : ℝ≥0∞) (c_ne_zero : c ≠ 0) :
Continuous fun x : ℝ≥0∞ => x / c :=
ENNReal.continuous_mul_const <| ENNReal.inv_ne_top.2 c_ne_zero
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_div_const | null |
protected continuous_pow (n : ℕ) : Continuous fun a : ℝ≥0∞ => a ^ n := by
induction n with
| zero => simp [continuous_const]
| succ n IH =>
simp_rw [pow_add, pow_one, continuous_iff_continuousAt]
intro x
refine ENNReal.Tendsto.mul (IH.tendsto _) ?_ tendsto_id ?_ <;> by_cases H : x = 0
· simp only ... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_pow | null |
continuousOn_sub :
ContinuousOn (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) { p : ℝ≥0∞ × ℝ≥0∞ | p ≠ ⟨∞, ∞⟩ } := by
rw [ContinuousOn]
rintro ⟨x, y⟩ hp
simp only [Ne, Set.mem_setOf_eq, Prod.mk_inj] at hp
exact tendsto_nhdsWithin_of_tendsto_nhds (tendsto_sub (not_and_or.mp hp)) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuousOn_sub | null |
continuous_sub_left {a : ℝ≥0∞} (a_ne_top : a ≠ ∞) : Continuous (a - ·) := by
change Continuous (Function.uncurry Sub.sub ∘ (a, ·))
refine continuousOn_sub.comp_continuous (.prodMk_right a) fun x => ?_
simp only [a_ne_top, Ne, mem_setOf_eq, Prod.mk_inj, false_and, not_false_iff] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_sub_left | null |
continuous_nnreal_sub {a : ℝ≥0} : Continuous fun x : ℝ≥0∞ => (a : ℝ≥0∞) - x :=
continuous_sub_left coe_ne_top | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_nnreal_sub | null |
continuousOn_sub_left (a : ℝ≥0∞) : ContinuousOn (a - ·) { x : ℝ≥0∞ | x ≠ ∞ } := by
rw [show (fun x => a - x) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨a, x⟩ by rfl]
apply continuousOn_sub.comp (by fun_prop)
rintro _ h (_ | _)
exact h none_eq_top | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuousOn_sub_left | null |
continuous_sub_right (a : ℝ≥0∞) : Continuous fun x : ℝ≥0∞ => x - a := by
by_cases a_infty : a = ∞
· simp [a_infty, continuous_const, tsub_eq_zero_of_le]
· rw [show (fun x => x - a) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨x, a⟩ by rfl]
apply continuousOn_sub.comp_continuous (by fun_prop)
intro... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_sub_right | null |
protected Tendsto.pow {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} {n : ℕ}
(hm : Tendsto m f (𝓝 a)) : Tendsto (fun x => m x ^ n) f (𝓝 (a ^ n)) :=
((ENNReal.continuous_pow n).tendsto a).comp hm | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Tendsto.pow | null |
le_of_forall_lt_one_mul_le {x y : ℝ≥0∞} (h : ∀ a < 1, a * x ≤ y) : x ≤ y := by
have : Tendsto (· * x) (𝓝[<] 1) (𝓝 (1 * x)) :=
(ENNReal.continuousAt_mul_const (Or.inr one_ne_zero)).mono_left inf_le_left
rw [one_mul] at this
exact le_of_tendsto this (eventually_nhdsWithin_iff.2 <| Eventually.of_forall h) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | le_of_forall_lt_one_mul_le | null |
inv_limsup {ι : Sort _} {x : ι → ℝ≥0∞} {l : Filter ι} :
(limsup x l)⁻¹ = liminf (fun i => (x i)⁻¹) l :=
OrderIso.invENNReal.limsup_apply | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | inv_limsup | null |
inv_liminf {ι : Sort _} {x : ι → ℝ≥0∞} {l : Filter ι} :
(liminf x l)⁻¹ = limsup (fun i => (x i)⁻¹) l :=
OrderIso.invENNReal.liminf_apply
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | inv_liminf | null |
protected continuous_zpow : ∀ n : ℤ, Continuous (· ^ n : ℝ≥0∞ → ℝ≥0∞)
| (n : ℕ) => mod_cast ENNReal.continuous_pow n
| .negSucc n => by simpa using (ENNReal.continuous_pow _).inv
@[simp] -- TODO: generalize to `[InvolutiveInv _] [ContinuousInv _]` | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_zpow | null |
protected tendsto_inv_iff {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} :
Tendsto (fun x => (m x)⁻¹) f (𝓝 a⁻¹) ↔ Tendsto m f (𝓝 a) :=
⟨fun h => by simpa only [inv_inv] using Tendsto.inv h, Tendsto.inv⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_inv_iff | null |
protected Tendsto.div {f : Filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hma : Tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) (hmb : Tendsto mb f (𝓝 b))
(hb : b ≠ ∞ ∨ a ≠ ∞) : Tendsto (fun a => ma a / mb a) f (𝓝 (a / b)) := by
apply Tendsto.mul hma _ (ENNReal.tendsto_inv_iff.2 hmb) _ <;> simp [ha, hb] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Tendsto.div | null |
protected Tendsto.const_div {f : Filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hm : Tendsto m f (𝓝 b)) (hb : b ≠ ∞ ∨ a ≠ ∞) : Tendsto (fun b => a / m b) f (𝓝 (a / b)) := by
apply Tendsto.const_mul (ENNReal.tendsto_inv_iff.2 hm)
simp [hb] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Tendsto.const_div | null |
protected Tendsto.div_const {f : Filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hm : Tendsto m f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) : Tendsto (fun x => m x / b) f (𝓝 (a / b)) := by
apply Tendsto.mul_const hm
simp [ha] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | Tendsto.div_const | null |
protected tendsto_inv_nat_nhds_zero : Tendsto (fun n : ℕ => (n : ℝ≥0∞)⁻¹) atTop (𝓝 0) :=
ENNReal.inv_top ▸ ENNReal.tendsto_inv_iff.2 tendsto_nat_nhds_top | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_inv_nat_nhds_zero | null |
protected tendsto_coe_sub {b : ℝ≥0∞} :
Tendsto (fun b : ℝ≥0∞ => ↑r - b) (𝓝 b) (𝓝 (↑r - b)) :=
continuous_nnreal_sub.tendsto _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_coe_sub | null |
exists_countable_dense_no_zero_top :
∃ s : Set ℝ≥0∞, s.Countable ∧ Dense s ∧ 0 ∉ s ∧ ∞ ∉ s := by
obtain ⟨s, s_count, s_dense, hs⟩ :
∃ s : Set ℝ≥0∞, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∉ s) ∧ ∀ x, IsTop x → x ∉ s :=
exists_countable_dense_no_bot_top ℝ≥0∞
exact ⟨s, s_count, s_dense, fun h => hs.1 0 ... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | exists_countable_dense_no_zero_top | null |
exists_frequently_lt_of_liminf_ne_top {ι : Type*} {l : Filter ι} {x : ι → ℝ}
(hx : liminf (fun n => (Real.nnabs (x n) : ℝ≥0∞)) l ≠ ∞) : ∃ R, ∃ᶠ n in l, x n < R := by
by_contra h
simp_rw [not_exists, not_frequently, not_lt] at h
refine hx (ENNReal.eq_top_of_forall_nnreal_le fun r => le_limsInf_of_le (by isBoun... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | exists_frequently_lt_of_liminf_ne_top | null |
exists_frequently_lt_of_liminf_ne_top' {ι : Type*} {l : Filter ι} {x : ι → ℝ}
(hx : liminf (fun n => (Real.nnabs (x n) : ℝ≥0∞)) l ≠ ∞) : ∃ R, ∃ᶠ n in l, R < x n := by
by_contra h
simp_rw [not_exists, not_frequently, not_lt] at h
refine hx (ENNReal.eq_top_of_forall_nnreal_le fun r => le_limsInf_of_le (by isBou... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | exists_frequently_lt_of_liminf_ne_top' | null |
exists_upcrossings_of_not_bounded_under {ι : Type*} {l : Filter ι} {x : ι → ℝ}
(hf : liminf (fun i => (Real.nnabs (x i) : ℝ≥0∞)) l ≠ ∞)
(hbdd : ¬IsBoundedUnder (· ≤ ·) l fun i => |x i|) :
∃ a b : ℚ, a < b ∧ (∃ᶠ i in l, x i < a) ∧ ∃ᶠ i in l, ↑b < x i := by
rw [isBoundedUnder_le_abs, not_and_or] at hbdd
o... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | exists_upcrossings_of_not_bounded_under | null |
@[norm_cast]
protected hasSum_coe {f : α → ℝ≥0} {r : ℝ≥0} :
HasSum (fun a => (f a : ℝ≥0∞)) ↑r ↔ HasSum f r := by
simp only [HasSum, ← coe_finset_sum, tendsto_coe] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | hasSum_coe | null |
protected tsum_coe_eq {f : α → ℝ≥0} (h : HasSum f r) : (∑' a, (f a : ℝ≥0∞)) = r :=
(ENNReal.hasSum_coe.2 h).tsum_eq | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_coe_eq | null |
protected coe_tsum {f : α → ℝ≥0} : Summable f → ↑(tsum f) = ∑' a, (f a : ℝ≥0∞)
| ⟨r, hr⟩ => by rw [hr.tsum_eq, ENNReal.tsum_coe_eq hr] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | coe_tsum | null |
protected hasSum : HasSum f (⨆ s : Finset α, ∑ a ∈ s, f a) :=
tendsto_atTop_iSup fun _ _ => Finset.sum_le_sum_of_subset
@[simp] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | hasSum | null |
protected summable : Summable f :=
⟨_, ENNReal.hasSum⟩
macro_rules | `(tactic| gcongr_discharger) => `(tactic| apply ENNReal.summable) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | summable | null |
tsum_coe_ne_top_iff_summable {f : β → ℝ≥0} : (∑' b, (f b : ℝ≥0∞)) ≠ ∞ ↔ Summable f := by
refine ⟨fun h => ?_, fun h => ENNReal.coe_tsum h ▸ ENNReal.coe_ne_top⟩
lift ∑' b, (f b : ℝ≥0∞) to ℝ≥0 using h with a ha
refine ⟨a, ENNReal.hasSum_coe.1 ?_⟩
rw [ha]
exact ENNReal.summable.hasSum | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_coe_ne_top_iff_summable | null |
protected tsum_eq_iSup_sum : ∑' a, f a = ⨆ s : Finset α, ∑ a ∈ s, f a :=
ENNReal.hasSum.tsum_eq | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_eq_iSup_sum | null |
protected tsum_eq_iSup_sum' {ι : Type*} (s : ι → Finset α) (hs : ∀ t, ∃ i, t ⊆ s i) :
∑' a, f a = ⨆ i, ∑ a ∈ s i, f a := by
rw [ENNReal.tsum_eq_iSup_sum]
symm
change ⨆ i : ι, (fun t : Finset α => ∑ a ∈ t, f a) (s i) = ⨆ s : Finset α, ∑ a ∈ s, f a
exact (Finset.sum_mono_set f).iSup_comp_eq hs | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_eq_iSup_sum' | null |
protected tsum_sigma {β : α → Type*} (f : ∀ a, β a → ℝ≥0∞) :
∑' p : Σ a, β a, f p.1 p.2 = ∑' (a) (b), f a b :=
ENNReal.summable.tsum_sigma' fun _ => ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_sigma | null |
protected tsum_sigma' {β : α → Type*} (f : (Σ a, β a) → ℝ≥0∞) :
∑' p : Σ a, β a, f p = ∑' (a) (b), f ⟨a, b⟩ :=
ENNReal.summable.tsum_sigma' fun _ => ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_sigma' | null |
protected tsum_biUnion' {ι : Type*} {S : Set ι} {f : α → ENNReal} {t : ι → Set α}
(h : S.PairwiseDisjoint t) : ∑' x : ⋃ i ∈ S, t i, f x = ∑' (i : S), ∑' (x : t i), f x := by
simp [← ENNReal.tsum_sigma, ← (Set.biUnionEqSigmaOfDisjoint h).tsum_eq] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_biUnion' | null |
protected tsum_biUnion {ι : Type*} {f : α → ENNReal} {t : ι → Set α}
(h : Set.univ.PairwiseDisjoint t) : ∑' x : ⋃ i, t i, f x = ∑' (i) (x : t i), f x := by
nth_rw 2 [← tsum_univ]
rw [← ENNReal.tsum_biUnion' h, Set.biUnion_univ] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_biUnion | null |
protected tsum_prod {f : α → β → ℝ≥0∞} : ∑' p : α × β, f p.1 p.2 = ∑' (a) (b), f a b :=
ENNReal.summable.tsum_prod' fun _ => ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_prod | null |
protected tsum_prod' {f : α × β → ℝ≥0∞} : ∑' p : α × β, f p = ∑' (a) (b), f (a, b) :=
ENNReal.summable.tsum_prod' fun _ => ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_prod' | null |
protected tsum_comm {f : α → β → ℝ≥0∞} : ∑' a, ∑' b, f a b = ∑' b, ∑' a, f a b :=
ENNReal.summable.tsum_comm' (fun _ => ENNReal.summable) fun _ => ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_comm | null |
protected tsum_add : ∑' a, (f a + g a) = ∑' a, f a + ∑' a, g a :=
ENNReal.summable.tsum_add ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_add | null |
protected tsum_le_tsum (h : ∀ a, f a ≤ g a) : ∑' a, f a ≤ ∑' a, g a :=
ENNReal.summable.tsum_le_tsum h ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_le_tsum | null |
protected sum_le_tsum {f : α → ℝ≥0∞} (s : Finset α) : ∑ x ∈ s, f x ≤ ∑' x, f x :=
ENNReal.summable.sum_le_tsum s (fun _ _ => zero_le _) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | sum_le_tsum | null |
protected tsum_eq_iSup_nat' {f : ℕ → ℝ≥0∞} {N : ℕ → ℕ} (hN : Tendsto N atTop atTop) :
∑' i : ℕ, f i = ⨆ i : ℕ, ∑ a ∈ Finset.range (N i), f a :=
ENNReal.tsum_eq_iSup_sum' _ fun t =>
let ⟨n, hn⟩ := t.exists_nat_subset_range
let ⟨k, _, hk⟩ := exists_le_of_tendsto_atTop hN 0 n
⟨k, Finset.Subset.trans hn (... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_eq_iSup_nat' | null |
protected tsum_eq_iSup_nat {f : ℕ → ℝ≥0∞} :
∑' i : ℕ, f i = ⨆ i : ℕ, ∑ a ∈ Finset.range i, f a :=
ENNReal.tsum_eq_iSup_sum' _ Finset.exists_nat_subset_range | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_eq_iSup_nat | null |
protected tsum_eq_liminf_sum_nat {f : ℕ → ℝ≥0∞} :
∑' i, f i = liminf (fun n => ∑ i ∈ Finset.range n, f i) atTop :=
ENNReal.summable.hasSum.tendsto_sum_nat.liminf_eq.symm | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_eq_liminf_sum_nat | null |
protected tsum_eq_limsup_sum_nat {f : ℕ → ℝ≥0∞} :
∑' i, f i = limsup (fun n => ∑ i ∈ Finset.range n, f i) atTop :=
ENNReal.summable.hasSum.tendsto_sum_nat.limsup_eq.symm | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_eq_limsup_sum_nat | null |
protected le_tsum (a : α) : f a ≤ ∑' a, f a :=
ENNReal.summable.le_tsum' a
@[simp] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | le_tsum | null |
protected tsum_eq_zero : ∑' i, f i = 0 ↔ ∀ i, f i = 0 :=
ENNReal.summable.tsum_eq_zero_iff | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_eq_zero | null |
protected tsum_eq_top_of_eq_top : (∃ a, f a = ∞) → ∑' a, f a = ∞
| ⟨a, ha⟩ => top_unique <| ha ▸ ENNReal.le_tsum a | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_eq_top_of_eq_top | null |
protected lt_top_of_tsum_ne_top {a : α → ℝ≥0∞} (tsum_ne_top : ∑' i, a i ≠ ∞) (j : α) :
a j < ∞ := by
contrapose! tsum_ne_top with h
exact ENNReal.tsum_eq_top_of_eq_top ⟨j, top_unique h⟩
@[simp] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | lt_top_of_tsum_ne_top | null |
protected tsum_top [Nonempty α] : ∑' _ : α, ∞ = ∞ :=
let ⟨a⟩ := ‹Nonempty α›
ENNReal.tsum_eq_top_of_eq_top ⟨a, rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_top | null |
tsum_const_eq_top_of_ne_zero {α : Type*} [Infinite α] {c : ℝ≥0∞} (hc : c ≠ 0) :
∑' _ : α, c = ∞ := by
have A : Tendsto (fun n : ℕ => (n : ℝ≥0∞) * c) atTop (𝓝 (∞ * c)) := by
apply ENNReal.Tendsto.mul_const tendsto_nat_nhds_top
simp only [true_or, top_ne_zero, Ne, not_false_iff]
have B : ∀ n : ℕ, (n : ℝ≥... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_const_eq_top_of_ne_zero | null |
protected ne_top_of_tsum_ne_top (h : ∑' a, f a ≠ ∞) (a : α) : f a ≠ ∞ := fun ha =>
h <| ENNReal.tsum_eq_top_of_eq_top ⟨a, ha⟩ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | ne_top_of_tsum_ne_top | null |
protected tsum_mul_left : ∑' i, a * f i = a * ∑' i, f i := by
by_cases hf : ∀ i, f i = 0
· simp [hf]
· rw [← ENNReal.tsum_eq_zero] at hf
have : Tendsto (fun s : Finset α => ∑ j ∈ s, a * f j) atTop (𝓝 (a * ∑' i, f i)) := by
simp only [← Finset.mul_sum]
exact ENNReal.Tendsto.const_mul ENNReal.summa... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_mul_left | null |
protected tsum_mul_right : ∑' i, f i * a = (∑' i, f i) * a := by
simp [mul_comm, ENNReal.tsum_mul_left] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_mul_right | null |
protected tsum_const_smul {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (a : R) :
∑' i, a • f i = a • ∑' i, f i := by
simpa only [smul_one_mul] using @ENNReal.tsum_mul_left _ (a • (1 : ℝ≥0∞)) _
@[simp] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_const_smul | null |
tsum_iSup_eq {α : Type*} (a : α) {f : α → ℝ≥0∞} : (∑' b : α, ⨆ _ : a = b, f b) = f a :=
(tsum_eq_single a fun _ h => by simp [h.symm]).trans <| by simp | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_iSup_eq | null |
hasSum_iff_tendsto_nat {f : ℕ → ℝ≥0∞} (r : ℝ≥0∞) :
HasSum f r ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop (𝓝 r) := by
refine ⟨HasSum.tendsto_sum_nat, fun h => ?_⟩
rw [← iSup_eq_of_tendsto _ h, ← ENNReal.tsum_eq_iSup_nat]
· exact ENNReal.summable.hasSum
· exact fun s t hst => Finset.sum_le_sum_... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | hasSum_iff_tendsto_nat | null |
tendsto_nat_tsum (f : ℕ → ℝ≥0∞) :
Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop (𝓝 (∑' n, f n)) := by
rw [← hasSum_iff_tendsto_nat]
exact ENNReal.summable.hasSum | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nat_tsum | null |
toNNReal_apply_of_tsum_ne_top {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' i, f i ≠ ∞) (x : α) :
(((ENNReal.toNNReal ∘ f) x : ℝ≥0) : ℝ≥0∞) = f x :=
coe_toNNReal <| ENNReal.ne_top_of_tsum_ne_top hf _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | toNNReal_apply_of_tsum_ne_top | null |
summable_toNNReal_of_tsum_ne_top {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' i, f i ≠ ∞) :
Summable (ENNReal.toNNReal ∘ f) := by
simpa only [← tsum_coe_ne_top_iff_summable, toNNReal_apply_of_tsum_ne_top hf] using hf | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | summable_toNNReal_of_tsum_ne_top | null |
tendsto_cofinite_zero_of_tsum_ne_top {α} {f : α → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
Tendsto f cofinite (𝓝 0) := by
have f_ne_top : ∀ n, f n ≠ ∞ := ENNReal.ne_top_of_tsum_ne_top hf
have h_f_coe : f = fun n => ((f n).toNNReal : ENNReal) :=
funext fun n => (coe_toNNReal (f_ne_top n)).symm
rw [h_f_coe, ← @coe_zer... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_cofinite_zero_of_tsum_ne_top | null |
tendsto_atTop_zero_of_tsum_ne_top {f : ℕ → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
Tendsto f atTop (𝓝 0) := by
rw [← Nat.cofinite_eq_atTop]
exact tendsto_cofinite_zero_of_tsum_ne_top hf | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_atTop_zero_of_tsum_ne_top | null |
tendsto_tsum_compl_atTop_zero {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
Tendsto (fun s : Finset α => ∑' b : { x // x ∉ s }, f b) atTop (𝓝 0) := by
lift f to α → ℝ≥0 using ENNReal.ne_top_of_tsum_ne_top hf
convert ENNReal.tendsto_coe.2 (NNReal.tendsto_tsum_compl_atTop_zero f)
rw [ENNReal.coe_tsum]
ex... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_tsum_compl_atTop_zero | The sum over the complement of a finset tends to `0` when the finset grows to cover the whole
space. This does not need a summability assumption, as otherwise all sums are zero. |
protected tsum_apply {ι α : Type*} {f : ι → α → ℝ≥0∞} {x : α} :
(∑' i, f i) x = ∑' i, f i x :=
tsum_apply <| Pi.summable.mpr fun _ => ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_apply | null |
tsum_sub {f : ℕ → ℝ≥0∞} {g : ℕ → ℝ≥0∞} (h₁ : ∑' i, g i ≠ ∞) (h₂ : g ≤ f) :
∑' i, (f i - g i) = ∑' i, f i - ∑' i, g i :=
have : ∀ i, f i - g i + g i = f i := fun i => tsub_add_cancel_of_le (h₂ i)
ENNReal.eq_sub_of_add_eq h₁ <| by simp only [← ENNReal.tsum_add, this] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_sub | null |
tsum_comp_le_tsum_of_injective {f : α → β} (hf : Injective f) (g : β → ℝ≥0∞) :
∑' x, g (f x) ≤ ∑' y, g y :=
ENNReal.summable.tsum_le_tsum_of_inj f hf (fun _ _ => zero_le _) (fun _ => le_rfl)
ENNReal.summable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_comp_le_tsum_of_injective | null |
tsum_le_tsum_comp_of_surjective {f : α → β} (hf : Surjective f) (g : β → ℝ≥0∞) :
∑' y, g y ≤ ∑' x, g (f x) :=
calc ∑' y, g y = ∑' y, g (f (surjInv hf y)) := by simp only [surjInv_eq hf]
_ ≤ ∑' x, g (f x) := tsum_comp_le_tsum_of_injective (injective_surjInv hf) _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_le_tsum_comp_of_surjective | null |
tsum_mono_subtype (f : α → ℝ≥0∞) {s t : Set α} (h : s ⊆ t) :
∑' x : s, f x ≤ ∑' x : t, f x :=
tsum_comp_le_tsum_of_injective (inclusion_injective h) _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_mono_subtype | null |
tsum_iUnion_le_tsum {ι : Type*} (f : α → ℝ≥0∞) (t : ι → Set α) :
∑' x : ⋃ i, t i, f x ≤ ∑' i, ∑' x : t i, f x :=
calc ∑' x : ⋃ i, t i, f x ≤ ∑' x : Σ i, t i, f x.2 :=
tsum_le_tsum_comp_of_surjective (sigmaToiUnion_surjective t) _
_ = ∑' i, ∑' x : t i, f x := ENNReal.tsum_sigma' _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_iUnion_le_tsum | null |
tsum_biUnion_le_tsum {ι : Type*} (f : α → ℝ≥0∞) (s : Set ι) (t : ι → Set α) :
∑' x : ⋃ i ∈ s, t i, f x ≤ ∑' i : s, ∑' x : t i, f x :=
calc ∑' x : ⋃ i ∈ s, t i, f x = ∑' x : ⋃ i : s, t i, f x := tsum_congr_set_coe _ <| by simp
_ ≤ ∑' i : s, ∑' x : t i, f x := tsum_iUnion_le_tsum _ _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tsum_biUnion_le_tsum | null |
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