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tsum_biUnion_le {ι : Type*} (f : α → ℝ≥0∞) (s : Finset ι) (t : ι → Set α) :
∑' x : ⋃ i ∈ s, t i, f x ≤ ∑ i ∈ s, ∑' x : t i, f x :=
(tsum_biUnion_le_tsum f s.toSet t).trans_eq (Finset.tsum_subtype s fun i => ∑' x : t i, f 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
|
tsum_biUnion_le
| null |
tsum_iUnion_le {ι : Type*} [Fintype ι] (f : α → ℝ≥0∞) (t : ι → Set α) :
∑' x : ⋃ i, t i, f x ≤ ∑ i, ∑' x : t i, f x := by
rw [← tsum_fintype]
exact tsum_iUnion_le_tsum f t
|
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
| null |
tsum_union_le (f : α → ℝ≥0∞) (s t : Set α) :
∑' x : ↑(s ∪ t), f x ≤ ∑' x : s, f x + ∑' x : t, f x :=
calc ∑' x : ↑(s ∪ t), f x = ∑' x : ⋃ b, cond b s t, f x := tsum_congr_set_coe _ union_eq_iUnion
_ ≤ _ := by simpa using tsum_iUnion_le f (cond · s t)
open Classical in
|
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_union_le
| null |
tsum_eq_add_tsum_ite {f : β → ℝ≥0∞} (b : β) :
∑' x, f x = f b + ∑' x, ite (x = b) 0 (f x) :=
ENNReal.summable.tsum_eq_add_tsum_ite' 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_eq_add_tsum_ite
| null |
tsum_add_one_eq_top {f : ℕ → ℝ≥0∞} (hf : ∑' n, f n = ∞) (hf0 : f 0 ≠ ∞) :
∑' n, f (n + 1) = ∞ := by
rw [tsum_eq_zero_add' ENNReal.summable, add_eq_top] at hf
exact hf.resolve_left hf0
|
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_one_eq_top
| null |
finite_const_le_of_tsum_ne_top {ι : Type*} {a : ι → ℝ≥0∞} (tsum_ne_top : ∑' i, a i ≠ ∞)
{ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0) : { i : ι | ε ≤ a i }.Finite := by
by_contra h
have := Infinite.to_subtype h
refine tsum_ne_top (top_unique ?_)
calc ∞ = ∑' _ : { i | ε ≤ a i }, ε := (tsum_const_eq_top_of_ne_zero ε_ne_zero).symm
_ ≤ ∑' i, a i := ENNReal.summable.tsum_le_tsum_of_inj (↑)
Subtype.val_injective (fun _ _ => zero_le _) (fun i => i.2) 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
|
finite_const_le_of_tsum_ne_top
|
A sum of extended nonnegative reals which is finite can have only finitely many terms
above any positive threshold.
|
finset_card_const_le_le_of_tsum_le {ι : Type*} {a : ι → ℝ≥0∞} {c : ℝ≥0∞} (c_ne_top : c ≠ ∞)
(tsum_le_c : ∑' i, a i ≤ c) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0) :
∃ hf : { i : ι | ε ≤ a i }.Finite, #hf.toFinset ≤ c / ε := by
have hf : { i : ι | ε ≤ a i }.Finite :=
finite_const_le_of_tsum_ne_top (ne_top_of_le_ne_top c_ne_top tsum_le_c) ε_ne_zero
refine ⟨hf, (ENNReal.le_div_iff_mul_le (.inl ε_ne_zero) (.inr c_ne_top)).2 ?_⟩
calc #hf.toFinset * ε = ∑ _i ∈ hf.toFinset, ε := by rw [Finset.sum_const, nsmul_eq_mul]
_ ≤ ∑ i ∈ hf.toFinset, a i := Finset.sum_le_sum fun i => hf.mem_toFinset.1
_ ≤ ∑' i, a i := ENNReal.sum_le_tsum _
_ ≤ c := tsum_le_c
|
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
|
finset_card_const_le_le_of_tsum_le
|
Markov's inequality for `Finset.card` and `tsum` in `ℝ≥0∞`.
|
tsum_fiberwise (f : β → ℝ≥0∞) (g : β → γ) :
∑' x, ∑' b : g ⁻¹' {x}, f b = ∑' i, f i := by
apply HasSum.tsum_eq
let equiv := Equiv.sigmaFiberEquiv g
apply (equiv.hasSum_iff.mpr ENNReal.summable.hasSum).sigma
exact fun _ ↦ ENNReal.summable.hasSum_iff.mpr 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_fiberwise
| null |
tendsto_toReal_iff {ι} {fi : Filter ι} {f : ι → ℝ≥0∞} (hf : ∀ i, f i ≠ ∞) {x : ℝ≥0∞}
(hx : x ≠ ∞) : Tendsto (fun n => (f n).toReal) fi (𝓝 x.toReal) ↔ Tendsto f fi (𝓝 x) := by
lift f to ι → ℝ≥0 using hf
lift x to ℝ≥0 using hx
simp [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
|
tendsto_toReal_iff
| null |
tsum_coe_ne_top_iff_summable_coe {f : α → ℝ≥0} :
(∑' a, (f a : ℝ≥0∞)) ≠ ∞ ↔ Summable fun a => (f a : ℝ) := by
rw [NNReal.summable_coe]
exact tsum_coe_ne_top_iff_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_coe_ne_top_iff_summable_coe
| null |
tsum_coe_eq_top_iff_not_summable_coe {f : α → ℝ≥0} :
(∑' a, (f a : ℝ≥0∞)) = ∞ ↔ ¬Summable fun a => (f a : ℝ) :=
tsum_coe_ne_top_iff_summable_coe.not_right
|
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_top_iff_not_summable_coe
| null |
hasSum_toReal {f : α → ℝ≥0∞} (hsum : ∑' x, f x ≠ ∞) :
HasSum (fun x => (f x).toReal) (∑' x, (f x).toReal) := by
lift f to α → ℝ≥0 using ENNReal.ne_top_of_tsum_ne_top hsum
simp only [coe_toReal, ← NNReal.coe_tsum, NNReal.hasSum_coe]
exact (tsum_coe_ne_top_iff_summable.1 hsum).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
|
hasSum_toReal
| null |
summable_toReal {f : α → ℝ≥0∞} (hsum : ∑' x, f x ≠ ∞) : Summable fun x => (f x).toReal :=
(hasSum_toReal hsum).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_toReal
| null |
tsum_eq_toNNReal_tsum {f : β → ℝ≥0} : ∑' b, f b = (∑' b, (f b : ℝ≥0∞)).toNNReal := by
by_cases h : Summable f
· rw [← ENNReal.coe_tsum h, ENNReal.toNNReal_coe]
· have A := tsum_eq_zero_of_not_summable h
simp only [← ENNReal.tsum_coe_ne_top_iff_summable, Classical.not_not] at h
simp only [h, ENNReal.toNNReal_top, 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_toNNReal_tsum
| null |
exists_le_hasSum_of_le {f g : β → ℝ≥0} {r : ℝ≥0} (hgf : ∀ b, g b ≤ f b) (hfr : HasSum f r) :
∃ p ≤ r, HasSum g p :=
have : (∑' b, (g b : ℝ≥0∞)) ≤ r := by
refine hasSum_le (fun b => ?_) ENNReal.summable.hasSum (ENNReal.hasSum_coe.2 hfr)
exact ENNReal.coe_le_coe.2 (hgf _)
let ⟨p, Eq, hpr⟩ := ENNReal.le_coe_iff.1 this
⟨p, hpr, ENNReal.hasSum_coe.1 <| Eq ▸ 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
|
exists_le_hasSum_of_le
|
Comparison test of convergence of `ℝ≥0`-valued series.
|
summable_of_le {f g : β → ℝ≥0} (hgf : ∀ b, g b ≤ f b) : Summable f → Summable g
| ⟨_r, hfr⟩ =>
let ⟨_p, _, hp⟩ := exists_le_hasSum_of_le hgf hfr
hp.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_of_le
|
Comparison test of convergence of `ℝ≥0`-valued series.
|
_root_.Summable.countable_support_nnreal (f : α → ℝ≥0) (h : Summable f) :
f.support.Countable := by
rw [← NNReal.summable_coe] at h
simpa [support] using h.countable_support
|
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_.Summable.countable_support_nnreal
|
Summable non-negative functions have countable support
|
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
rw [← ENNReal.hasSum_coe, ENNReal.hasSum_iff_tendsto_nat]
simp only [← ENNReal.coe_finset_sum]
exact ENNReal.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_iff_tendsto_nat
|
A series of non-negative real numbers converges to `r` in the sense of `HasSum` if and only if
the sequence of partial sum converges to `r`.
|
not_summable_iff_tendsto_nat_atTop {f : ℕ → ℝ≥0} :
¬Summable f ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop := by
constructor
· intro h
refine ((tendsto_of_monotone ?_).resolve_right h).comp ?_
exacts [Finset.sum_mono_set _, tendsto_finset_range]
· rintro hnat ⟨r, hr⟩
exact not_tendsto_nhds_of_tendsto_atTop hnat _ (hasSum_iff_tendsto_nat.1 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
|
not_summable_iff_tendsto_nat_atTop
| null |
summable_iff_not_tendsto_nat_atTop {f : ℕ → ℝ≥0} :
Summable f ↔ ¬Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop := by
rw [← not_iff_not, Classical.not_not, not_summable_iff_tendsto_nat_atTop]
|
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_iff_not_tendsto_nat_atTop
| null |
summable_of_sum_range_le {f : ℕ → ℝ≥0} {c : ℝ≥0}
(h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : Summable f := by
refine summable_iff_not_tendsto_nat_atTop.2 fun H => ?_
rcases exists_lt_of_tendsto_atTop H 0 c with ⟨n, -, hn⟩
exact lt_irrefl _ (hn.trans_le (h 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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
Mathlib/Topology/Instances/ENNReal/Lemmas.lean
|
summable_of_sum_range_le
| null |
tsum_le_of_sum_range_le {f : ℕ → ℝ≥0} {c : ℝ≥0}
(h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : ∑' n, f n ≤ c :=
(summable_of_sum_range_le h).tsum_le_of_sum_range_le 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_le_of_sum_range_le
| null |
tsum_comp_le_tsum_of_inj {β : Type*} {f : α → ℝ≥0} (hf : Summable f) {i : β → α}
(hi : Function.Injective i) : (∑' x, f (i x)) ≤ ∑' x, f x :=
(summable_comp_injective hf hi).tsum_le_tsum_of_inj i hi (fun _ _ => zero_le _) (fun _ => le_rfl)
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_comp_le_tsum_of_inj
| null |
summable_sigma {β : α → Type*} {f : (Σ x, β x) → ℝ≥0} :
Summable f ↔ (∀ x, Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' y, f ⟨x, y⟩ := by
constructor
· simp only [← NNReal.summable_coe, NNReal.coe_tsum]
exact fun h => ⟨h.sigma_factor, h.sigma⟩
· rintro ⟨h₁, h₂⟩
simpa only [← ENNReal.tsum_coe_ne_top_iff_summable, ENNReal.tsum_sigma',
ENNReal.coe_tsum (h₁ _)] using h₂
|
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_sigma
| null |
indicator_summable {f : α → ℝ≥0} (hf : Summable f) (s : Set α) :
Summable (s.indicator f) := by
classical
refine NNReal.summable_of_le (fun a => le_trans (le_of_eq (s.indicator_apply f a)) ?_) hf
split_ifs
· exact le_refl (f a)
· exact zero_le_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
|
indicator_summable
| null |
tsum_indicator_ne_zero {f : α → ℝ≥0} (hf : Summable f) {s : Set α} (h : ∃ a ∈ s, f a ≠ 0) :
(∑' x, (s.indicator f) x) ≠ 0 := fun h' =>
let ⟨a, ha, hap⟩ := h
hap ((Set.indicator_apply_eq_self.mpr (absurd ha)).symm.trans
((indicator_summable hf s).tsum_eq_zero_iff.1 h' a))
open Finset
|
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_indicator_ne_zero
| null |
tendsto_sum_nat_add (f : ℕ → ℝ≥0) : Tendsto (fun i => ∑' k, f (k + i)) atTop (𝓝 0) := by
rw [← tendsto_coe]
convert _root_.tendsto_sum_nat_add fun i => (f i : ℝ)
norm_cast
nonrec theorem hasSum_lt {f g : α → ℝ≥0} {sf sg : ℝ≥0} {i : α} (h : ∀ a : α, f a ≤ g a)
(hi : f i < g i) (hf : HasSum f sf) (hg : HasSum g sg) : sf < sg := by
have A : ∀ a : α, (f a : ℝ) ≤ g a := fun a => NNReal.coe_le_coe.2 (h a)
have : (sf : ℝ) < sg := hasSum_lt A (NNReal.coe_lt_coe.2 hi) (hasSum_coe.2 hf) (hasSum_coe.2 hg)
exact NNReal.coe_lt_coe.1 this
@[mono]
|
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_sum_nat_add
|
For `f : ℕ → ℝ≥0`, then `∑' k, f (k + i)` tends to zero. This does not require a summability
assumption on `f`, as otherwise all sums are zero.
|
hasSum_strict_mono {f g : α → ℝ≥0} {sf sg : ℝ≥0} (hf : HasSum f sf) (hg : HasSum g sg)
(h : f < g) : sf < sg :=
let ⟨hle, _i, hi⟩ := Pi.lt_def.mp h
hasSum_lt hle hi hf hg
|
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_strict_mono
| null |
tsum_lt_tsum {f g : α → ℝ≥0} {i : α} (h : ∀ a : α, f a ≤ g a) (hi : f i < g i)
(hg : Summable g) : ∑' n, f n < ∑' n, g n :=
hasSum_lt h hi (summable_of_le h hg).hasSum hg.hasSum
@[mono]
|
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_lt_tsum
| null |
tsum_strict_mono {f g : α → ℝ≥0} (hg : Summable g) (h : f < g) : ∑' n, f n < ∑' n, g n :=
let ⟨hle, _i, hi⟩ := Pi.lt_def.mp h
tsum_lt_tsum hle hi hg
|
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_strict_mono
| null |
tsum_pos {g : α → ℝ≥0} (hg : Summable g) (i : α) (hi : 0 < g i) : 0 < ∑' b, g b := by
rw [← tsum_zero]
exact tsum_lt_tsum (fun a => zero_le _) hi hg
open Classical in
|
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_pos
| null |
tsum_eq_add_tsum_ite {f : α → ℝ≥0} (hf : Summable f) (i : α) :
∑' x, f x = f i + ∑' x, ite (x = i) 0 (f x) := by
refine (NNReal.summable_of_le (fun i' => ?_) hf).tsum_eq_add_tsum_ite' i
rw [Function.update_apply]
split_ifs <;> simp only [zero_le', le_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_eq_add_tsum_ite
| null |
tsum_toNNReal_eq {f : α → ℝ≥0∞} (hf : ∀ a, f a ≠ ∞) :
(∑' a, f a).toNNReal = ∑' a, (f a).toNNReal :=
(congr_arg ENNReal.toNNReal (tsum_congr fun x => (coe_toNNReal (hf x)).symm)).trans
NNReal.tsum_eq_toNNReal_tsum.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_toNNReal_eq
| null |
tsum_toReal_eq {f : α → ℝ≥0∞} (hf : ∀ a, f a ≠ ∞) :
(∑' a, f a).toReal = ∑' a, (f a).toReal := by
simp only [ENNReal.toReal, tsum_toNNReal_eq hf, NNReal.coe_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_toReal_eq
| null |
tendsto_sum_nat_add (f : ℕ → ℝ≥0∞) (hf : ∑' i, f i ≠ ∞) :
Tendsto (fun i => ∑' k, f (k + i)) atTop (𝓝 0) := by
lift f to ℕ → ℝ≥0 using ENNReal.ne_top_of_tsum_ne_top hf
replace hf : Summable f := tsum_coe_ne_top_iff_summable.1 hf
simp only [← ENNReal.coe_tsum, NNReal.summable_nat_add _ hf, ← ENNReal.coe_zero]
exact mod_cast NNReal.tendsto_sum_nat_add f
|
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_sum_nat_add
| null |
tsum_le_of_sum_range_le {f : ℕ → ℝ≥0∞} {c : ℝ≥0∞}
(h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : ∑' n, f n ≤ c :=
ENNReal.summable.tsum_le_of_sum_range_le 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_le_of_sum_range_le
| null |
hasSum_lt {f g : α → ℝ≥0∞} {sf sg : ℝ≥0∞} {i : α} (h : ∀ a : α, f a ≤ g a) (hi : f i < g i)
(hsf : sf ≠ ∞) (hf : HasSum f sf) (hg : HasSum g sg) : sf < sg := by
by_cases hsg : sg = ∞
· exact hsg.symm ▸ lt_of_le_of_ne le_top hsf
· have hg' : ∀ x, g x ≠ ∞ := ENNReal.ne_top_of_tsum_ne_top (hg.tsum_eq.symm ▸ hsg)
lift f to α → ℝ≥0 using fun x =>
ne_of_lt (lt_of_le_of_lt (h x) <| lt_of_le_of_ne le_top (hg' x))
lift g to α → ℝ≥0 using hg'
lift sf to ℝ≥0 using hsf
lift sg to ℝ≥0 using hsg
simp only [coe_le_coe, coe_lt_coe] at h hi ⊢
exact NNReal.hasSum_lt h hi (ENNReal.hasSum_coe.1 hf) (ENNReal.hasSum_coe.1 hg)
|
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_lt
| null |
tsum_lt_tsum {f g : α → ℝ≥0∞} {i : α} (hfi : tsum f ≠ ∞) (h : ∀ a : α, f a ≤ g a)
(hi : f i < g i) : ∑' x, f x < ∑' x, g x :=
hasSum_lt h hi hfi ENNReal.summable.hasSum 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_lt_tsum
| null |
tsum_comp_le_tsum_of_inj {β : Type*} {f : α → ℝ} (hf : Summable f) (hn : ∀ a, 0 ≤ f a)
{i : β → α} (hi : Function.Injective i) : tsum (f ∘ i) ≤ tsum f := by
lift f to α → ℝ≥0 using hn
rw [NNReal.summable_coe] at hf
simpa only [Function.comp_def, ← NNReal.coe_tsum] using NNReal.tsum_comp_le_tsum_of_inj hf hi
|
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_inj
| null |
Summable.of_nonneg_of_le {f g : β → ℝ} (hg : ∀ b, 0 ≤ g b) (hgf : ∀ b, g b ≤ f b)
(hf : Summable f) : Summable g := by
lift f to β → ℝ≥0 using fun b => (hg b).trans (hgf b)
lift g to β → ℝ≥0 using hg
rw [NNReal.summable_coe] at hf ⊢
exact NNReal.summable_of_le (fun b => NNReal.coe_le_coe.1 (hgf b)) 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.of_nonneg_of_le
|
Comparison test of convergence of series of non-negative real numbers.
|
Summable.toNNReal {f : α → ℝ} (hf : Summable f) : Summable fun n => (f n).toNNReal := by
apply NNReal.summable_coe.1
refine .of_nonneg_of_le (fun n => NNReal.coe_nonneg _) (fun n => ?_) hf.abs
simp only [le_abs_self, Real.coe_toNNReal', max_le_iff, abs_nonneg, and_self_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
|
Summable.toNNReal
| null |
Summable.tsum_ofReal_lt_top {f : α → ℝ} (hf : Summable f) : ∑' i, .ofReal (f i) < ∞ := by
unfold ENNReal.ofReal
rw [lt_top_iff_ne_top, ENNReal.tsum_coe_ne_top_iff_summable]
exact hf.toNNReal
|
lemma
|
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.tsum_ofReal_lt_top
| null |
Summable.tsum_ofReal_ne_top {f : α → ℝ} (hf : Summable f) : ∑' i, .ofReal (f i) ≠ ∞ :=
hf.tsum_ofReal_lt_top.ne
|
lemma
|
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.tsum_ofReal_ne_top
| null |
_root_.Summable.countable_support_ennreal {f : α → ℝ≥0∞} (h : ∑' (i : α), f i ≠ ∞) :
f.support.Countable := by
lift f to α → ℝ≥0 using ENNReal.ne_top_of_tsum_ne_top h
simpa [support] using (ENNReal.tsum_coe_ne_top_iff_summable.1 h).countable_support_nnreal
|
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_.Summable.countable_support_ennreal
|
Finitely summable non-negative functions have countable support
|
hasSum_iff_tendsto_nat_of_nonneg {f : ℕ → ℝ} (hf : ∀ i, 0 ≤ f i) (r : ℝ) :
HasSum f r ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop (𝓝 r) := by
lift f to ℕ → ℝ≥0 using hf
simp only [HasSum, ← NNReal.coe_sum, NNReal.tendsto_coe']
exact exists_congr fun hr => NNReal.hasSum_iff_tendsto_nat
|
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_of_nonneg
|
A series of non-negative real numbers converges to `r` in the sense of `HasSum` if and only if
the sequence of partial sum converges to `r`.
|
ENNReal.ofReal_tsum_of_nonneg {f : α → ℝ} (hf_nonneg : ∀ n, 0 ≤ f n) (hf : Summable f) :
ENNReal.ofReal (∑' n, f n) = ∑' n, ENNReal.ofReal (f n) := by
simp_rw [ENNReal.ofReal, ENNReal.tsum_coe_eq (NNReal.hasSum_real_toNNReal_of_nonneg hf_nonneg 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
|
ENNReal.ofReal_tsum_of_nonneg
| null |
edist_ne_top_of_mem_ball {a : β} {r : ℝ≥0∞} (x y : ball a r) : edist x.1 y.1 ≠ ∞ :=
ne_of_lt <|
calc
edist x y ≤ edist a x + edist a y := edist_triangle_left x.1 y.1 a
_ < r + r := by rw [edist_comm a x, edist_comm a y]; exact ENNReal.add_lt_add x.2 y.2
_ ≤ ∞ := le_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
|
edist_ne_top_of_mem_ball
|
In an emetric ball, the distance between points is everywhere finite
|
metricSpaceEMetricBall (a : β) (r : ℝ≥0∞) : MetricSpace (ball a r) :=
EMetricSpace.toMetricSpace edist_ne_top_of_mem_ball
|
def
|
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
|
metricSpaceEMetricBall
|
Each ball in an extended metric space gives us a metric space, as the edist
is everywhere finite.
|
nhds_eq_nhds_emetric_ball (a x : β) (r : ℝ≥0∞) (h : x ∈ ball a r) :
𝓝 x = map ((↑) : ball a r → β) (𝓝 ⟨x, h⟩) :=
(map_nhds_subtype_coe_eq_nhds _ <| IsOpen.mem_nhds EMetric.isOpen_ball 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
|
nhds_eq_nhds_emetric_ball
| null |
tendsto_iff_edist_tendsto_0 {l : Filter β} {f : β → α} {y : α} :
Tendsto f l (𝓝 y) ↔ Tendsto (fun x => edist (f x) y) l (𝓝 0) := by
simp only [EMetric.nhds_basis_eball.tendsto_right_iff, EMetric.mem_ball,
@tendsto_order ℝ≥0∞ β _ _, forall_prop_of_false ENNReal.not_lt_zero, forall_const, true_and]
|
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_iff_edist_tendsto_0
| null |
EMetric.cauchySeq_iff_le_tendsto_0 [Nonempty β] [SemilatticeSup β] {s : β → α} :
CauchySeq s ↔ ∃ b : β → ℝ≥0∞, (∀ n m N : β, N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧
Tendsto b atTop (𝓝 0) := EMetric.cauchySeq_iff.trans <| by
constructor
· intro hs
/- `s` is Cauchy sequence. Let `b n` be the diameter of the set `s '' Set.Ici n`. -/
refine ⟨fun N => EMetric.diam (s '' Ici N), fun n m N hn hm => ?_, ?_⟩
· exact EMetric.edist_le_diam_of_mem (mem_image_of_mem _ hn) (mem_image_of_mem _ hm)
· refine ENNReal.tendsto_nhds_zero.2 fun ε ε0 => ?_
rcases hs ε ε0 with ⟨N, hN⟩
refine (eventually_ge_atTop N).mono fun n hn => EMetric.diam_le ?_
rintro _ ⟨k, hk, rfl⟩ _ ⟨l, hl, rfl⟩
exact (hN _ (hn.trans hk) _ (hn.trans hl)).le
· rintro ⟨b, ⟨b_bound, b_lim⟩⟩ ε εpos
have : ∀ᶠ n in atTop, b n < ε := b_lim.eventually (gt_mem_nhds εpos)
rcases this.exists with ⟨N, hN⟩
refine ⟨N, fun m hm n hn => ?_⟩
calc edist (s m) (s n) ≤ b N := b_bound m n N hm hn
_ < ε := 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
|
EMetric.cauchySeq_iff_le_tendsto_0
|
Yet another metric characterization of Cauchy sequences on integers. This one is often the
most efficient.
|
continuous_of_le_add_edist {f : α → ℝ≥0∞} (C : ℝ≥0∞) (hC : C ≠ ∞)
(h : ∀ x y, f x ≤ f y + C * edist x y) : Continuous f := by
refine continuous_iff_continuousAt.2 fun x => ENNReal.tendsto_nhds_of_Icc fun ε ε0 => ?_
rcases ENNReal.exists_nnreal_pos_mul_lt hC ε0.ne' with ⟨δ, δ0, hδ⟩
rw [mul_comm] at hδ
filter_upwards [EMetric.closedBall_mem_nhds x (ENNReal.coe_pos.2 δ0)] with y hy
refine ⟨tsub_le_iff_right.2 <| (h x y).trans ?_, (h y x).trans ?_⟩ <;>
refine add_le_add_left (le_trans (mul_le_mul_left' ?_ _) hδ.le) _
exacts [EMetric.mem_closedBall'.1 hy, EMetric.mem_closedBall.1 hy]
|
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_of_le_add_edist
| null |
continuous_edist : Continuous fun p : α × α => edist p.1 p.2 := by
apply continuous_of_le_add_edist 2 (by decide)
rintro ⟨x, y⟩ ⟨x', y'⟩
calc
edist x y ≤ edist x x' + edist x' y' + edist y' y := edist_triangle4 _ _ _ _
_ = edist x' y' + (edist x x' + edist y y') := by simp only [edist_comm]; ac_rfl
_ ≤ edist x' y' + (edist (x, y) (x', y') + edist (x, y) (x', y')) := by
gcongr <;> apply_rules [le_max_left, le_max_right]
_ = edist x' y' + 2 * edist (x, y) (x', y') := by rw [← mul_two, mul_comm]
@[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_edist
| null |
Continuous.edist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
(hg : Continuous g) : Continuous fun b => edist (f b) (g b) :=
continuous_edist.comp (hf.prodMk hg :)
|
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.edist
| null |
Filter.Tendsto.edist {f g : β → α} {x : Filter β} {a b : α} (hf : Tendsto f x (𝓝 a))
(hg : Tendsto g x (𝓝 b)) : Tendsto (fun x => edist (f x) (g x)) x (𝓝 (edist a b)) :=
(continuous_edist.tendsto (a, b)).comp (hf.prodMk_nhds hg)
|
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
|
Filter.Tendsto.edist
| null |
cauchySeq_of_edist_le_of_summable {f : ℕ → α} (d : ℕ → ℝ≥0)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : Summable d) : CauchySeq f := by
refine EMetric.cauchySeq_iff_NNReal.2 fun ε εpos ↦ ?_
replace hd : CauchySeq fun n : ℕ ↦ ∑ x ∈ Finset.range n, d x :=
let ⟨_, H⟩ := hd
H.tendsto_sum_nat.cauchySeq
refine (Metric.cauchySeq_iff'.1 hd ε (NNReal.coe_pos.2 εpos)).imp fun N hN n hn ↦ ?_
specialize hN n hn
rw [dist_nndist, NNReal.nndist_eq, ← Finset.sum_range_add_sum_Ico _ hn, add_tsub_cancel_left,
NNReal.coe_lt_coe, max_lt_iff] at hN
rw [edist_comm]
refine lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun _ _ ↦ hf _) ?_
exact mod_cast hN.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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
Mathlib/Topology/Instances/ENNReal/Lemmas.lean
|
cauchySeq_of_edist_le_of_summable
|
If the extended distance between consecutive points of a sequence is estimated
by a summable series of `NNReal`s, then the original sequence is a Cauchy sequence.
|
cauchySeq_of_edist_le_of_tsum_ne_top {f : ℕ → α} (d : ℕ → ℝ≥0∞)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : tsum d ≠ ∞) : CauchySeq f := by
lift d to ℕ → NNReal using fun i => ENNReal.ne_top_of_tsum_ne_top hd i
rw [ENNReal.tsum_coe_ne_top_iff_summable] at hd
exact cauchySeq_of_edist_le_of_summable d hf hd
|
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
|
cauchySeq_of_edist_le_of_tsum_ne_top
| null |
EMetric.isClosed_closedBall {a : α} {r : ℝ≥0∞} : IsClosed (closedBall a r) :=
isClosed_le (continuous_id.edist continuous_const) continuous_const
@[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
|
EMetric.isClosed_closedBall
| null |
EMetric.diam_closure (s : Set α) : diam (closure s) = diam s := by
refine le_antisymm (diam_le fun x hx y hy => ?_) (diam_mono subset_closure)
have : edist x y ∈ closure (Iic (diam s)) :=
map_mem_closure₂ continuous_edist hx hy fun x hx y hy => edist_le_diam_of_mem hx hy
rwa [closure_Iic] at this
@[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
|
EMetric.diam_closure
| null |
Metric.diam_closure {α : Type*} [PseudoMetricSpace α] (s : Set α) :
Metric.diam (closure s) = diam s := by simp only [Metric.diam, EMetric.diam_closure]
|
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
|
Metric.diam_closure
| null |
isClosed_setOf_lipschitzOnWith {α β} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : ℝ≥0)
(s : Set α) : IsClosed { f : α → β | LipschitzOnWith K f s } := by
simp only [LipschitzOnWith, setOf_forall]
refine isClosed_biInter fun x _ => isClosed_biInter fun y _ => isClosed_le ?_ ?_
exacts [.edist (continuous_apply x) (continuous_apply y), continuous_const]
|
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
|
isClosed_setOf_lipschitzOnWith
| null |
isClosed_setOf_lipschitzWith {α β} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : ℝ≥0) :
IsClosed { f : α → β | LipschitzWith K f } := by
simp only [← lipschitzOnWith_univ, isClosed_setOf_lipschitzOnWith]
|
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
|
isClosed_setOf_lipschitzWith
| null |
ediam_eq {s : Set ℝ} (h : Bornology.IsBounded s) :
EMetric.diam s = ENNReal.ofReal (sSup s - sInf s) := by
rcases eq_empty_or_nonempty s with (rfl | hne)
· simp
refine le_antisymm (Metric.ediam_le_of_forall_dist_le fun x hx y hy => ?_) ?_
· exact Real.dist_le_of_mem_Icc (h.subset_Icc_sInf_sSup hx) (h.subset_Icc_sInf_sSup hy)
· apply ENNReal.ofReal_le_of_le_toReal
rw [← Metric.diam, ← Metric.diam_closure]
calc sSup s - sInf s ≤ dist (sSup s) (sInf s) := le_abs_self _
_ ≤ Metric.diam (closure s) := dist_le_diam_of_mem h.closure (csSup_mem_closure hne h.bddAbove)
(csInf_mem_closure hne h.bddBelow)
|
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
|
ediam_eq
|
For a bounded set `s : Set ℝ`, its `EMetric.diam` is equal to `sSup s - sInf s` reinterpreted as
`ℝ≥0∞`.
|
diam_eq {s : Set ℝ} (h : Bornology.IsBounded s) : Metric.diam s = sSup s - sInf s := by
rw [Metric.diam, Real.ediam_eq h, ENNReal.toReal_ofReal]
exact sub_nonneg.2 (Real.sInf_le_sSup s h.bddBelow h.bddAbove)
@[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
|
diam_eq
|
For a bounded set `s : Set ℝ`, its `Metric.diam` is equal to `sSup s - sInf s`.
|
ediam_Ioo (a b : ℝ) : EMetric.diam (Ioo a b) = ENNReal.ofReal (b - a) := by
rcases le_or_gt b a with (h | h)
· simp [h]
· rw [Real.ediam_eq (isBounded_Ioo _ _), csSup_Ioo h, csInf_Ioo 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
|
ediam_Ioo
| null |
ediam_Icc (a b : ℝ) : EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) := by
rcases le_or_gt a b with (h | h)
· rw [Real.ediam_eq (isBounded_Icc _ _), csSup_Icc h, csInf_Icc h]
· simp [h, h.le]
@[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
|
ediam_Icc
| null |
ediam_Ico (a b : ℝ) : EMetric.diam (Ico a b) = ENNReal.ofReal (b - a) :=
le_antisymm (ediam_Icc a b ▸ diam_mono Ico_subset_Icc_self)
(ediam_Ioo a b ▸ diam_mono Ioo_subset_Ico_self)
@[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
|
ediam_Ico
| null |
ediam_Ioc (a b : ℝ) : EMetric.diam (Ioc a b) = ENNReal.ofReal (b - a) :=
le_antisymm (ediam_Icc a b ▸ diam_mono Ioc_subset_Icc_self)
(ediam_Ioo a b ▸ diam_mono Ioo_subset_Ioc_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
|
ediam_Ioc
| null |
diam_Icc {a b : ℝ} (h : a ≤ b) : Metric.diam (Icc a b) = b - a := by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 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
|
diam_Icc
| null |
diam_Ico {a b : ℝ} (h : a ≤ b) : Metric.diam (Ico a b) = b - a := by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 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
|
diam_Ico
| null |
diam_Ioc {a b : ℝ} (h : a ≤ b) : Metric.diam (Ioc a b) = b - a := by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 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
|
diam_Ioc
| null |
diam_Ioo {a b : ℝ} (h : a ≤ b) : Metric.diam (Ioo a b) = b - a := by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 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
|
diam_Ioo
| null |
edist_le_tsum_of_edist_le_of_tendsto {f : ℕ → α} (d : ℕ → ℝ≥0∞)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
edist (f n) a ≤ ∑' m, d (n + m) := by
refine le_of_tendsto (tendsto_const_nhds.edist ha) (mem_atTop_sets.2 ⟨n, fun m hnm => ?_⟩)
change edist _ _ ≤ _
refine le_trans (edist_le_Ico_sum_of_edist_le hnm fun _ _ => hf _) ?_
rw [Finset.sum_Ico_eq_sum_range]
exact ENNReal.summable.sum_le_tsum _ (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
|
edist_le_tsum_of_edist_le_of_tendsto
|
If `edist (f n) (f (n+1))` is bounded above by a function `d : ℕ → ℝ≥0∞`,
then the distance from `f n` to the limit is bounded by `∑'_{k=n}^∞ d k`.
|
edist_le_tsum_of_edist_le_of_tendsto₀ {f : ℕ → α} (d : ℕ → ℝ≥0∞)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n) {a : α} (ha : Tendsto f atTop (𝓝 a)) :
edist (f 0) a ≤ ∑' m, d m := by simpa using edist_le_tsum_of_edist_le_of_tendsto d hf ha 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.ProperSpace.Real",
"Mathlib.Topology.Metrizable.Uniformity"
] |
Mathlib/Topology/Instances/ENNReal/Lemmas.lean
|
edist_le_tsum_of_edist_le_of_tendsto₀
|
If `edist (f n) (f (n+1))` is bounded above by a function `d : ℕ → ℝ≥0∞`,
then the distance from `f 0` to the limit is bounded by `∑'_{k=0}^∞ d k`.
|
noncomputable truncateToReal (t x : ℝ≥0∞) : ℝ := (min t x).toReal
|
def
|
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
|
truncateToReal
|
With truncation level `t`, the truncated cast `ℝ≥0∞ → ℝ` is given by `x ↦ (min t x).toReal`.
Unlike `ENNReal.toReal`, this cast is continuous and monotone when `t ≠ ∞`.
|
truncateToReal_eq_toReal {t x : ℝ≥0∞} (t_ne_top : t ≠ ∞) (x_le : x ≤ t) :
truncateToReal t x = x.toReal := by
have x_lt_top : x < ∞ := lt_of_le_of_lt x_le t_ne_top.lt_top
have obs : min t x ≠ ∞ := by
simp_all only [ne_eq, min_eq_top, false_and, not_false_eq_true]
exact (ENNReal.toReal_eq_toReal obs x_lt_top.ne).mpr (min_eq_right x_le)
|
lemma
|
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
|
truncateToReal_eq_toReal
| null |
truncateToReal_le {t : ℝ≥0∞} (t_ne_top : t ≠ ∞) {x : ℝ≥0∞} :
truncateToReal t x ≤ t.toReal := by
rw [truncateToReal]
gcongr
exacts [t_ne_top, min_le_left t x]
|
lemma
|
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
|
truncateToReal_le
| null |
truncateToReal_nonneg {t x : ℝ≥0∞} : 0 ≤ truncateToReal t x := toReal_nonneg
|
lemma
|
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
|
truncateToReal_nonneg
| null |
monotone_truncateToReal {t : ℝ≥0∞} (t_ne_top : t ≠ ∞) : Monotone (truncateToReal t) := by
intro x y x_le_y
simp only [truncateToReal]
gcongr
exact ne_top_of_le_ne_top t_ne_top (min_le_left _ _)
|
lemma
|
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
|
monotone_truncateToReal
|
The truncated cast `ENNReal.truncateToReal t : ℝ≥0∞ → ℝ` is monotone when `t ≠ ∞`.
|
@[fun_prop]
continuous_truncateToReal {t : ℝ≥0∞} (t_ne_top : t ≠ ∞) : Continuous (truncateToReal t) := by
apply continuousOn_toReal.comp_continuous (by fun_prop)
simp [t_ne_top]
|
lemma
|
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_truncateToReal
|
The truncated cast `ENNReal.truncateToReal t : ℝ≥0∞ → ℝ` is continuous when `t ≠ ∞`.
|
limsup_sub_const (F : Filter ι) (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :
Filter.limsup (fun i ↦ f i - c) F = Filter.limsup f F - c := by
rcases F.eq_or_neBot with rfl | _
· simp only [limsup_bot, bot_eq_zero', zero_le, tsub_eq_zero_of_le]
· exact (Monotone.map_limsSup_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ x - c)
(fun _ _ h ↦ tsub_le_tsub_right h c) (continuous_sub_right c).continuousAt).symm
|
lemma
|
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
|
limsup_sub_const
| null |
liminf_sub_const (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :
Filter.liminf (fun i ↦ f i - c) F = Filter.liminf f F - c :=
(Monotone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ x - c)
(fun _ _ h ↦ tsub_le_tsub_right h c) (continuous_sub_right c).continuousAt).symm
|
lemma
|
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
|
liminf_sub_const
| null |
limsup_const_sub (F : Filter ι) (f : ι → ℝ≥0∞) {c : ℝ≥0∞} (c_ne_top : c ≠ ∞) :
Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F := by
rcases F.eq_or_neBot with rfl | _
· simp only [limsup_bot, bot_eq_zero', liminf_bot, le_top, tsub_eq_zero_of_le]
· exact (Antitone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ c - x)
(fun _ _ h ↦ tsub_le_tsub_left h c) (continuous_sub_left c_ne_top).continuousAt).symm
|
lemma
|
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
|
limsup_const_sub
| null |
liminf_const_sub (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) {c : ℝ≥0∞} (c_ne_top : c ≠ ∞) :
Filter.liminf (fun i ↦ c - f i) F = c - Filter.limsup f F :=
(Antitone.map_limsSup_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ c - x)
(fun _ _ h ↦ tsub_le_tsub_left h c) (continuous_sub_left c_ne_top).continuousAt).symm
|
lemma
|
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
|
liminf_const_sub
| null |
le_limsup_mul : limsup u f * liminf v f ≤ limsup (u * v) f :=
mul_le_of_forall_lt fun a a_u b b_v ↦ (le_limsup_iff).2 fun c c_ab ↦
Frequently.mono (Frequently.and_eventually ((frequently_lt_of_lt_limsup) a_u)
((eventually_lt_of_lt_liminf) b_v)) fun _ ab_x ↦ c_ab.trans (mul_lt_mul ab_x.1 ab_x.2)
|
lemma
|
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_limsup_mul
| null |
limsup_mul_le' (h : limsup u f ≠ 0 ∨ limsup v f ≠ ∞) (h' : limsup u f ≠ ∞ ∨ limsup v f ≠ 0) :
limsup (u * v) f ≤ limsup u f * limsup v f := by
refine le_mul_of_forall_lt h h' fun a a_u b b_v ↦ (limsup_le_iff).2 fun c c_ab ↦ ?_
filter_upwards [eventually_lt_of_limsup_lt a_u, eventually_lt_of_limsup_lt b_v] with x a_x b_x
exact (mul_lt_mul a_x b_x).trans c_ab
|
lemma
|
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
|
limsup_mul_le'
|
See also `ENNReal.limsup_mul_le`.
|
le_liminf_mul : liminf u f * liminf v f ≤ liminf (u * v) f := by
refine mul_le_of_forall_lt fun a a_u b b_v ↦ (le_liminf_iff).2 fun c c_ab ↦ ?_
filter_upwards [eventually_lt_of_lt_liminf a_u, eventually_lt_of_lt_liminf b_v] with x a_x b_x
exact c_ab.trans (mul_lt_mul a_x b_x)
|
lemma
|
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_liminf_mul
| null |
liminf_mul_le (h : limsup u f ≠ 0 ∨ liminf v f ≠ ∞) (h' : limsup u f ≠ ∞ ∨ liminf v f ≠ 0) :
liminf (u * v) f ≤ limsup u f * liminf v f :=
le_mul_of_forall_lt h h' fun a a_u b b_v ↦ (liminf_le_iff).2 fun c c_ab ↦
Frequently.mono (((frequently_lt_of_liminf_lt) b_v).and_eventually
((eventually_lt_of_limsup_lt) a_u)) fun _ ab_x ↦ (mul_lt_mul ab_x.2 ab_x.1).trans c_ab
|
lemma
|
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
|
liminf_mul_le
| null |
liminf_toReal_eq [NeBot f] {b : ℝ≥0∞} (b_ne_top : b ≠ ∞) (le_b : ∀ᶠ i in f, u i ≤ b) :
f.liminf (fun i ↦ (u i).toReal) = (f.liminf u).toReal := by
have liminf_le : f.liminf u ≤ b := by
apply liminf_le_of_le ⟨0, by simp⟩
intro y h
obtain ⟨i, hi⟩ := (Eventually.and h le_b).exists
exact hi.1.trans hi.2
have aux : ∀ᶠ i in f, (u i).toReal = ENNReal.truncateToReal b (u i) := by
filter_upwards [le_b] with i i_le_b
simp only [truncateToReal_eq_toReal b_ne_top i_le_b]
have aux' : (f.liminf u).toReal = ENNReal.truncateToReal b (f.liminf u) := by
rw [truncateToReal_eq_toReal b_ne_top liminf_le]
simp_rw [liminf_congr aux, aux']
have key := Monotone.map_liminf_of_continuousAt (F := f) (monotone_truncateToReal b_ne_top) u
(continuous_truncateToReal b_ne_top).continuousAt
(IsBoundedUnder.isCoboundedUnder_ge ⟨b, by simpa only [eventually_map] using le_b⟩)
⟨0, Eventually.of_forall (by simp)⟩
rw [key]
rfl
|
lemma
|
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
|
liminf_toReal_eq
|
If `u : ι → ℝ≥0∞` is bounded, then we have `liminf (toReal ∘ u) = toReal (liminf u)`.
|
limsup_toReal_eq [NeBot f] {b : ℝ≥0∞} (b_ne_top : b ≠ ∞) (le_b : ∀ᶠ i in f, u i ≤ b) :
f.limsup (fun i ↦ (u i).toReal) = (f.limsup u).toReal := by
have aux : ∀ᶠ i in f, (u i).toReal = ENNReal.truncateToReal b (u i) := by
filter_upwards [le_b] with i i_le_b
simp only [truncateToReal_eq_toReal b_ne_top i_le_b]
have aux' : (f.limsup u).toReal = ENNReal.truncateToReal b (f.limsup u) := by
rw [truncateToReal_eq_toReal b_ne_top (limsup_le_of_le ⟨0, by simp⟩ le_b)]
simp_rw [limsup_congr aux, aux']
have key := Monotone.map_limsup_of_continuousAt (F := f) (monotone_truncateToReal b_ne_top) u
(continuous_truncateToReal b_ne_top).continuousAt
⟨b, by simpa only [eventually_map] using le_b⟩
(IsBoundedUnder.isCoboundedUnder_le ⟨0, Eventually.of_forall (by simp)⟩)
rw [key]
rfl
@[simp, norm_cast]
|
lemma
|
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
|
limsup_toReal_eq
|
If `u : ι → ℝ≥0∞` is bounded, then we have `liminf (toReal ∘ u) = toReal (liminf u)`.
|
ofNNReal_limsup {u : ι → ℝ≥0} (hf : f.IsBoundedUnder (· ≤ ·) u) :
limsup u f = limsup (fun i ↦ (u i : ℝ≥0∞)) f := by
refine eq_of_forall_nnreal_iff fun r ↦ ?_
rw [coe_le_coe, le_limsup_iff, le_limsup_iff]
simp [forall_ennreal]
@[simp, norm_cast]
|
lemma
|
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
|
ofNNReal_limsup
| null |
ofNNReal_liminf {u : ι → ℝ≥0} (hf : f.IsCoboundedUnder (· ≥ ·) u) :
liminf u f = liminf (fun i ↦ (u i : ℝ≥0∞)) f := by
refine eq_of_forall_nnreal_iff fun r ↦ ?_
rw [coe_le_coe, le_liminf_iff, le_liminf_iff]
simp [forall_ennreal]
|
lemma
|
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
|
ofNNReal_liminf
| null |
liminf_add_of_right_tendsto_zero {u : Filter ι} {g : ι → ℝ≥0∞} (hg : u.Tendsto g (𝓝 0))
(f : ι → ℝ≥0∞) : u.liminf (f + g) = u.liminf f := by
refine le_antisymm ?_ <| liminf_le_liminf <| .of_forall <| by simp
refine liminf_le_of_le (by isBoundedDefault) fun b hb ↦ ?_
rw [Filter.le_liminf_iff']
rintro a hab
filter_upwards [hb, ENNReal.tendsto_nhds_zero.1 hg _ <| lt_min (tsub_pos_of_lt hab) one_pos]
with i hfg hg
exact ENNReal.le_of_add_le_add_right (hg.trans_lt <| by simp).ne <|
(add_le_of_le_tsub_left_of_le hab.le <| hg.trans <| min_le_left ..).trans hfg
|
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
|
liminf_add_of_right_tendsto_zero
| null |
liminf_add_of_left_tendsto_zero {u : Filter ι} {f : ι → ℝ≥0∞} (hf : u.Tendsto f (𝓝 0))
(g : ι → ℝ≥0∞) : u.liminf (f + g) = u.liminf g := by
rw [add_comm, liminf_add_of_right_tendsto_zero 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
|
liminf_add_of_left_tendsto_zero
| null |
limsup_add_of_right_tendsto_zero {u : Filter ι} {g : ι → ℝ≥0∞} (hg : u.Tendsto g (𝓝 0))
(f : ι → ℝ≥0∞) : u.limsup (f + g) = u.limsup f := by
refine le_antisymm ?_ <| limsup_le_limsup <| .of_forall <| by simp
refine le_limsup_of_le (by isBoundedDefault) fun b hb ↦ ?_
rw [Filter.limsup_le_iff']
rintro a hba
filter_upwards [hb, ENNReal.tendsto_nhds_zero.1 hg _ <| tsub_pos_of_lt hba] with i hf hg
calc f i + g i
_ ≤ b + g i := by gcongr
_ ≤ a := add_le_of_le_tsub_left_of_le hba.le hg
|
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
|
limsup_add_of_right_tendsto_zero
| null |
limsup_add_of_left_tendsto_zero {u : Filter ι} {f : ι → ℝ≥0∞} (hf : u.Tendsto f (𝓝 0))
(g : ι → ℝ≥0∞) : u.limsup (f + g) = u.limsup g := by
rw [add_comm, limsup_add_of_right_tendsto_zero 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
|
limsup_add_of_left_tendsto_zero
| null |
isEmbedding_coe : IsEmbedding ((↑) : ℝ → EReal) :=
coe_strictMono.isEmbedding_of_ordConnected <| by rw [range_coe_eq_Ioo]; exact ordConnected_Ioo
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
isEmbedding_coe
| null |
isOpenEmbedding_coe : IsOpenEmbedding ((↑) : ℝ → EReal) :=
⟨isEmbedding_coe, by simp only [range_coe_eq_Ioo, isOpen_Ioo]⟩
@[norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
isOpenEmbedding_coe
| null |
tendsto_coe {α : Type*} {f : Filter α} {m : α → ℝ} {a : ℝ} :
Tendsto (fun a => (m a : EReal)) f (𝓝 ↑a) ↔ Tendsto m f (𝓝 a) :=
isEmbedding_coe.tendsto_nhds_iff.symm
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
tendsto_coe
| null |
_root_.continuous_coe_real_ereal : Continuous ((↑) : ℝ → EReal) :=
isEmbedding_coe.continuous
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Inv",
"Mathlib.Topology.Semicontinuous"
] |
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
_root_.continuous_coe_real_ereal
| null |
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