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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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 : ℕ) : (𝓝[>] (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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 ?_ fun ε ε0 => ?_
· rintro ⟨a, b⟩ ⟨ha, hb⟩
rcases exists_between (tsub_pos_of_lt ha) with ⟨ε, ε0, hε⟩
rcases lt_iff_exists_add_pos_lt.1 hb with ⟨δ, δ0, hδ⟩
refine ⟨min ε δ, (lt_min ε0 (coe_pos.2 δ0)).ne', Icc_subset_Ioo ?_ ?_⟩
· exact lt_tsub_comm.2 ((min_le_left _ _).trans_lt hε)
· exact (add_le_add_left (min_le_right _ _) _).trans_lt hδ
· exact ⟨(x - ε, x + ε), ⟨ENNReal.sub_lt_self xt x0.ne' ε0,
lt_add_right xt ε0⟩, Ioo_subset_Icc_self⟩
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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] with n hn
refine ⟨?_, (hfa n).trans (le_add_right le_rfl)⟩
rw [tsub_le_iff_right] at hn ⊢
rwa [add_comm]
· filter_upwards [h ε hε] with n hn
have hN_left := hn.1
rw [tsub_le_iff_right] at hN_left ⊢
rwa [add_comm]
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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
exact ⟨n, hn n le_rfl⟩
· obtain ⟨n, hn⟩ := h ε hε
exact ⟨n, fun m hm ↦ (hf hm).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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 / 2))
(lt_min_iff.mpr ⟨zero_lt_one, (ENNReal.div_pos_iff.mpr ⟨ne_of_gt hε, ENNReal.ofNat_ne_top⟩)⟩)
· refine ⟨n, hn.trans_lt ?_⟩
by_cases hε_top : ε = ∞
· rw [hε_top]
exact (min_le_left _ _).trans_lt ENNReal.one_lt_top
refine (min_le_right _ _).trans_lt ?_
rw [ENNReal.div_lt_iff (Or.inr hε.ne') (Or.inr hε_top)]
conv_lhs => rw [← mul_one ε]
rw [ENNReal.mul_lt_mul_left hε.ne' hε_top]
simp
· obtain ⟨n, hn⟩ := h ε hε
exact ⟨n, hn.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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 + x)).prod_nhds
(ge_mem_nhds <| coe_lt_coe.2 <| lt_add_one b)).mono fun y hy => ?_
rw [lt_tsub_iff_left]
calc y.2 + x ≤ ↑(b + 1) + x := add_le_add_right hy.2 _
_ < y.1 := hy.1
| (a : ℝ≥0), ∞, _ => by
rw [sub_top]
refine (tendsto_pure.2 ?_).mono_right (pure_le_nhds _)
exact ((gt_mem_nhds <| coe_lt_coe.2 <| lt_add_one a).prod_nhds
(lt_mem_nhds <| @coe_lt_top (a + 1))).mono fun x hx =>
tsub_eq_zero_iff_le.2 (hx.1.trans hx.2).le
| (a : ℝ≥0), (b : ℝ≥0), _ => by
simp only [nhds_coe_coe, tendsto_map'_iff, ← ENNReal.coe_sub, Function.comp_def, tendsto_coe]
exact continuous_sub.tendsto (a, b)
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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
Tendsto.comp (ENNReal.tendsto_sub h) (hma.prodMk_nhds hmb)
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 => ?_
rcases lt_iff_exists_nnreal_btwn.1 (pos_iff_ne_zero.2 hb) with ⟨ε, hε, hεb⟩
have : ∀ᶠ c : ℝ≥0∞ × ℝ≥0∞ in 𝓝 (∞, b), ↑n / ↑ε < c.1 ∧ ↑ε < c.2 :=
(lt_mem_nhds <| div_lt_top coe_ne_top hε.ne').prod_nhds (lt_mem_nhds hεb)
refine this.mono fun c hc => ?_
exact (ENNReal.div_mul_cancel hε.ne' coe_ne_top).symm.trans_lt (mul_lt_mul hc.1 hc.2)
induction a with
| top => simp only [ne_eq, or_false, not_true_eq_false] at hb; simp [ht b hb, top_mul hb]
| coe a =>
induction b with
| top =>
simp only [ne_eq, or_false, not_true_eq_false] at ha
simpa [Function.comp_def, mul_comm, mul_top ha]
using (ht a ha).comp (continuous_swap.tendsto (ofNNReal a, ∞))
| coe b =>
simp only [nhds_coe_coe, ← coe_mul, tendsto_coe, tendsto_map'_iff, Function.comp_def,
tendsto_mul]
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 (a * b)) from
Tendsto.comp (ENNReal.tendsto_mul ha hb) (hma.prodMk_nhds hmb)
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 [tendsto_const_nhds]
| insert _ _ has IH =>
simp only [Finset.prod_insert has]
apply Tendsto.mul (h _ (Finset.mem_insert_self _ _))
· right
exact prod_ne_top fun i hi => h' _ (Finset.mem_insert_of_mem hi)
· exact IH (fun i hi => h _ (Finset.mem_insert_of_mem hi)) fun i hi =>
h' _ (Finset.mem_insert_of_mem hi)
· exact Or.inr (h' _ (Finset.mem_insert_self _ _))
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 [H, zero_ne_top, Ne, or_true, not_false_iff]
· exact Or.inl fun h => H (pow_eq_zero h)
· simp only [H, pow_eq_top_iff, zero_ne_top, false_or, not_true, Ne,
not_false_iff, false_and]
· simp only [H, true_or, Ne, 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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 x
simp only [a_infty, Ne, mem_setOf_eq, Prod.mk_inj, and_false, 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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 (by simp) h, fun h => hs.2 ∞ (by simp) 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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 isBoundedDefault) ?_)
simp only [eventually_map, ENNReal.coe_le_coe]
filter_upwards [h r] with i hi using hi.trans (le_abs_self (x i))
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 isBoundedDefault) ?_)
simp only [eventually_map, ENNReal.coe_le_coe]
filter_upwards [h (-r)] with i hi using(le_neg.1 hi).trans (neg_le_abs _)
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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
obtain hbdd | hbdd := hbdd
· obtain ⟨R, hR⟩ := exists_frequently_lt_of_liminf_ne_top hf
obtain ⟨q, hq⟩ := exists_rat_gt R
refine ⟨q, q + 1, (lt_add_iff_pos_right _).2 zero_lt_one, ?_, ?_⟩
· refine fun hcon => hR ?_
filter_upwards [hcon] with x hx using not_lt.2 (lt_of_lt_of_le hq (not_lt.1 hx)).le
· simp only [IsBoundedUnder, IsBounded, eventually_map, not_exists] at hbdd
refine fun hcon => hbdd ↑(q + 1) ?_
filter_upwards [hcon] with x hx using not_lt.1 hx
· obtain ⟨R, hR⟩ := exists_frequently_lt_of_liminf_ne_top' hf
obtain ⟨q, hq⟩ := exists_rat_lt R
refine ⟨q - 1, q, (sub_lt_self_iff _).2 zero_lt_one, ?_, ?_⟩
· simp only [IsBoundedUnder, IsBounded, eventually_map, not_exists] at hbdd
refine fun hcon => hbdd ↑(q - 1) ?_
filter_upwards [hcon] with x hx using not_lt.1 hx
· refine fun hcon => hR ?_
filter_upwards [hcon] with x hx using not_lt.2 ((not_lt.1 hx).trans hq.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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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 (Finset.range_mono hk)⟩
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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≥0∞) * c ≤ ∑' _ : α, c := fun n => by
rcases Infinite.exists_subset_card_eq α n with ⟨s, hs⟩
simpa [hs] using @ENNReal.sum_le_tsum α (fun _ => c) s
simpa [hc] using le_of_tendsto' A B
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.summable.hasSum (Or.inl hf)
exact HasSum.tsum_eq 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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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_of_subset (Finset.range_subset_range.2 hst)
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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_zero, tendsto_coe]
exact NNReal.tendsto_cofinite_zero_of_summable (summable_toNNReal_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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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]
exact NNReal.summable_comp_injective (tsum_coe_ne_top_iff_summable.1 hf) Subtype.coe_injective
|
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
Mathlib/Topology/Instances/ENNReal/Lemmas.lean
|
tsum_biUnion_le_tsum
| null |
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