statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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tsum_eq_top_of_eq_top : (∃ a, f a = ∞) → ∑' a, f a = ∞ | | ⟨a, ha⟩ := top_unique $ ha ▸ ennreal.le_tsum a | lemma | ennreal.tsum_eq_top_of_eq_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.le_tsum",
"top_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_top_of_tsum_ne_top {a : α → ℝ≥0∞} (tsum_ne_top : ∑' i, a i ≠ ∞) (j : α) :
a j < ∞ | begin
have key := not_imp_not.mpr ennreal.tsum_eq_top_of_eq_top,
simp only [not_exists] at key,
exact lt_top_iff_ne_top.mpr (key tsum_ne_top j),
end | lemma | ennreal.lt_top_of_tsum_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.tsum_eq_top_of_eq_top",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_top [nonempty α] : ∑' a : α, ∞ = ∞ | let ⟨a⟩ := ‹nonempty α› in ennreal.tsum_eq_top_of_eq_top ⟨a, rfl⟩ | lemma | ennreal.tsum_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.tsum_eq_top_of_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_const_eq_top_of_ne_zero {α : Type*} [infinite α] {c : ℝ≥0∞} (hc : c ≠ 0) :
(∑' (a : α), c) = ∞ | begin
have A : tendsto (λ (n : ℕ), (n : ℝ≥0∞) * c) at_top (𝓝 (∞ * c)),
{ apply ennreal.tendsto.mul_const tendsto_nat_nhds_top,
simp only [true_or, top_ne_zero, ne.def, not_false_iff] },
have B : ∀ (n : ℕ), (n : ℝ≥0∞) * c ≤ (∑' (a : α), c),
{ assume n,
rcases infinite.exists_subset_card_eq α n with ⟨s, ... | lemma | ennreal.tsum_const_eq_top_of_ne_zero | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.sum_le_tsum",
"ennreal.tendsto.mul_const",
"infinite",
"infinite.exists_subset_card_eq",
"le_of_tendsto'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_top_of_tsum_ne_top (h : ∑' a, f a ≠ ∞) (a : α) : f a ≠ ∞ | λ ha, h $ ennreal.tsum_eq_top_of_eq_top ⟨a, ha⟩ | lemma | ennreal.ne_top_of_tsum_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.tsum_eq_top_of_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_left : ∑'i, a * f i = a * ∑'i, f i | if h : ∀i, f i = 0 then by simp [h] else
let ⟨i, (hi : f i ≠ 0)⟩ := not_forall.mp h in
have sum_ne_0 : ∑'i, f i ≠ 0, from ne_of_gt $
calc 0 < f i : lt_of_le_of_ne (zero_le _) hi.symm
... ≤ ∑'i, f i : ennreal.le_tsum _,
have tendsto (λs:finset α, ∑ j in s, a * f j) at_top (𝓝 (a * ∑'i, f i)),
by rw [← show (*) a... | lemma | ennreal.tsum_mul_left | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.le_tsum",
"ennreal.tendsto.const_mul",
"finset",
"finset.mul_sum",
"has_sum.tsum_eq",
"tsum_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_right : (∑'i, f i * a) = (∑'i, f i) * a | by simp [mul_comm, ennreal.tsum_mul_left] | lemma | ennreal.tsum_mul_right | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.tsum_mul_left",
"mul_comm",
"tsum_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_const_smul {R} [has_smul R ℝ≥0∞] [is_scalar_tower 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) _ | lemma | ennreal.tsum_const_smul | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.tsum_mul_left",
"has_smul",
"is_scalar_tower",
"smul_one_mul",
"tsum_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_supr_eq {α : Type*} (a : α) {f : α → ℝ≥0∞} :
∑'b:α, (⨆ (h : a = b), f b) = f a | le_antisymm
(by rw [ennreal.tsum_eq_supr_sum]; exact supr_le (assume s,
calc (∑ b in s, ⨆ (h : a = b), f b) ≤ ∑ b in {a}, ⨆ (h : a = b), f b :
finset.sum_le_sum_of_ne_zero $ assume b _ hb,
suffices a = b, by simpa using this.symm,
classical.by_contradiction $ assume h,
by s... | lemma | ennreal.tsum_supr_eq | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.le_tsum",
"ennreal.tsum_eq_supr_sum",
"le_supr",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_iff_tendsto_nat {f : ℕ → ℝ≥0∞} (r : ℝ≥0∞) :
has_sum f r ↔ tendsto (λn:ℕ, ∑ i in finset.range n, f i) at_top (𝓝 r) | begin
refine ⟨has_sum.tendsto_sum_nat, assume h, _⟩,
rw [← supr_eq_of_tendsto _ h, ← ennreal.tsum_eq_supr_nat],
{ exact ennreal.summable.has_sum },
{ exact assume s t hst, finset.sum_le_sum_of_subset (finset.range_subset.2 hst) }
end | lemma | ennreal.has_sum_iff_tendsto_nat | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.tsum_eq_supr_nat",
"finset.range",
"has_sum",
"supr_eq_of_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nat_tsum (f : ℕ → ℝ≥0∞) :
tendsto (λn:ℕ, ∑ i in finset.range n, f i) at_top (𝓝 (∑' n, f n)) | by { rw ← has_sum_iff_tendsto_nat, exact ennreal.summable.has_sum } | lemma | ennreal.tendsto_nat_tsum | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nnreal_apply_of_tsum_ne_top {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' i, f i ≠ ∞) (x : α) :
(((ennreal.to_nnreal ∘ f) x : ℝ≥0) : ℝ≥0∞) = f x | coe_to_nnreal $ ennreal.ne_top_of_tsum_ne_top hf _ | lemma | ennreal.to_nnreal_apply_of_tsum_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.ne_top_of_tsum_ne_top",
"ennreal.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_to_nnreal_of_tsum_ne_top {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' i, f i ≠ ∞) :
summable (ennreal.to_nnreal ∘ f) | by simpa only [←tsum_coe_ne_top_iff_summable, to_nnreal_apply_of_tsum_ne_top hf] using hf | lemma | ennreal.summable_to_nnreal_of_tsum_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_nnreal",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cofinite_zero_of_tsum_ne_top {α} {f : α → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
tendsto f cofinite (𝓝 0) | begin
have f_ne_top : ∀ n, f n ≠ ∞, from ennreal.ne_top_of_tsum_ne_top hf,
have h_f_coe : f = λ n, ((f n).to_nnreal : ennreal),
from funext (λ n, (coe_to_nnreal (f_ne_top n)).symm),
rw [h_f_coe, ←@coe_zero, tendsto_coe],
exact nnreal.tendsto_cofinite_zero_of_summable (summable_to_nnreal_of_tsum_ne_top hf),
... | lemma | ennreal.tendsto_cofinite_zero_of_tsum_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal",
"ennreal.ne_top_of_tsum_ne_top",
"nnreal.tendsto_cofinite_zero_of_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_zero_of_tsum_ne_top {f : ℕ → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
tendsto f at_top (𝓝 0) | by { rw ←nat.cofinite_eq_at_top, exact tendsto_cofinite_zero_of_tsum_ne_top hf } | lemma | ennreal.tendsto_at_top_zero_of_tsum_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_tsum_compl_at_top_zero {α : Type*} {f : α → ℝ≥0∞} (hf : ∑' x, f x ≠ ∞) :
tendsto (λ (s : finset α), ∑' b : {x // x ∉ s}, f b) at_top (𝓝 0) | begin
lift f to α → ℝ≥0 using ennreal.ne_top_of_tsum_ne_top hf,
convert ennreal.tendsto_coe.2 (nnreal.tendsto_tsum_compl_at_top_zero f),
ext1 s,
rw ennreal.coe_tsum,
exact nnreal.summable_comp_injective (tsum_coe_ne_top_iff_summable.1 hf) subtype.coe_injective
end | lemma | ennreal.tendsto_tsum_compl_at_top_zero | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.coe_tsum",
"ennreal.ne_top_of_tsum_ne_top",
"finset",
"lift",
"nnreal.summable_comp_injective",
"nnreal.tendsto_tsum_compl_at_top_zero",
"subtype.coe_injective",
"tendsto_tsum_compl_at_top_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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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 $ λ _, ennreal.summable | lemma | ennreal.tsum_apply | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.summable",
"tsum_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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) | begin
have h₃: ∑' i, (f i - g i) = ∑' i, (f i - g i + g i) - ∑' i, g i,
{ rw [ennreal.tsum_add, ennreal.add_sub_cancel_right h₁]},
have h₄:(λ i, (f i - g i) + (g i)) = f,
{ ext n, rw tsub_add_cancel_of_le (h₂ n)},
rw h₄ at h₃, apply h₃,
end | lemma | ennreal.tsum_sub | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.add_sub_cancel_right",
"ennreal.tsum_add",
"tsub_add_cancel_of_le",
"tsum_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mono_subtype (f : α → ℝ≥0∞) {s t : set α} (h : s ⊆ t) :
∑' (x : s), f x ≤ ∑' (x : t), f x | begin
simp only [tsum_subtype],
apply ennreal.tsum_le_tsum,
exact indicator_le_indicator_of_subset h (λ _, zero_le _),
end | lemma | ennreal.tsum_mono_subtype | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.tsum_le_tsum",
"tsum_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 : s ∪ (t \ s)), f x :
by { apply tsum_congr_subtype, rw union_diff_self }
... = ∑' (x : s), f x + ∑' (x : t \ s), f x :
tsum_union_disjoint disjoint_sdiff_self_right ennreal.summable ennreal.summable
... ≤ ∑' (x : s), f x + ∑' (x : t), f x :
add_le_add le_rfl (tsum_mono_subtype _ ... | lemma | ennreal.tsum_union_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"disjoint_sdiff_self_right",
"ennreal.summable",
"le_rfl",
"tsum_congr_subtype",
"tsum_union_disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_bUnion_le {ι : Type*} (f : α → ℝ≥0∞) (s : finset ι) (t : ι → set α) :
∑' (x : ⋃ (i ∈ s), t i), f x ≤ ∑ i in s, ∑' (x : t i), f x | begin
classical,
induction s using finset.induction_on with i s hi ihs h, { simp },
have : (⋃ (j ∈ insert i s), t j) = t i ∪ (⋃ (j ∈ s), t j), by simp,
rw tsum_congr_subtype f this,
calc ∑' (x : (t i ∪ (⋃ (j ∈ s), t j))), f x ≤
∑' (x : t i), f x + ∑' (x : ⋃ (j ∈ s), t j), f x : tsum_union_le _ _ _
... ≤ ∑... | lemma | ennreal.tsum_bUnion_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset",
"finset.induction_on",
"le_rfl",
"tsum_congr_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_Union_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 | begin
classical,
have : (⋃ i, t i) = (⋃ (i ∈ (finset.univ : finset ι)), t i), by simp,
rw tsum_congr_subtype f this,
exact tsum_bUnion_le _ _ _
end | lemma | ennreal.tsum_Union_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset",
"finset.univ",
"fintype",
"tsum_congr_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_add_tsum_ite {f : β → ℝ≥0∞} (b : β) : ∑' x, f x = f b + ∑' x, ite (x = b) 0 (f x) | tsum_eq_add_tsum_ite' b ennreal.summable | lemma | ennreal.tsum_eq_add_tsum_ite | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.summable",
"tsum_eq_add_tsum_ite",
"tsum_eq_add_tsum_ite'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_add_one_eq_top {f : ℕ → ℝ≥0∞} (hf : ∑' n, f n = ∞) (hf0 : f 0 ≠ ∞) :
∑' n, f (n + 1) = ∞ | begin
rw ← tsum_eq_tsum_of_has_sum_iff_has_sum (λ _, (not_mem_range_equiv 1).has_sum_iff),
swap, { apply_instance },
have h₁ : (∑' b : {n // n ∈ finset.range 1}, f b) + (∑' b : {n // n ∉ finset.range 1}, f b) =
∑' b, f b,
{ exact tsum_add_tsum_compl ennreal.summable ennreal.summable },
rw [finset.tsum_sub... | lemma | ennreal.tsum_add_one_eq_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"coe_not_mem_range_equiv",
"ennreal.add_eq_top",
"ennreal.summable",
"finset.range",
"finset.tsum_subtype",
"multiset.mem_range",
"not_mem_range_equiv",
"subtype.coe_mk",
"tsub_add_cancel_of_le",
"tsum_add_tsum_compl",
"tsum_congr",
"tsum_eq_tsum_of_has_sum_iff_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | begin
by_cases ε_infty : ε = ∞,
{ rw ε_infty,
by_contra maybe_infinite,
obtain ⟨j, hj⟩ := set.infinite.nonempty maybe_infinite,
exact tsum_ne_top (le_antisymm le_top (le_trans hj (le_tsum' (@ennreal.summable _ a) j))), },
have key := (nnreal.summable_coe.mpr
(summable_to_nnreal_of_tsum_... | lemma | ennreal.finite_const_le_of_tsum_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"Iio_mem_nhds",
"by_contra",
"ennreal.ne_top_of_tsum_ne_top",
"ennreal.summable",
"filter.mem_cofinite",
"filter.mem_map",
"finite",
"le_top",
"le_tsum'",
"set.infinite.nonempty"
] | A sum of extended nonnegative reals which is finite can have only finitely many terms
above any positive threshold. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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.to_finset.card ≤ c / ε | begin
by_cases ε = ∞,
{ have obs : {i : ι | ε ≤ a i} = ∅,
{ rw eq_empty_iff_forall_not_mem,
intros i hi,
have oops := (le_trans hi (le_tsum' (@ennreal.summable _ a) i)).trans tsum_le_c,
rw h at oops,
exact c_ne_top (le_antisymm le_top oops), },
simp only [obs, finite_empty, finite.to... | lemma | ennreal.finset_card_const_le_le_of_tsum_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"algebra_map.coe_zero",
"ennreal.div_le_div_right",
"ennreal.finite_const_le_of_tsum_ne_top",
"ennreal.le_div_iff_mul_le",
"ennreal.summable",
"exists_true_left",
"finite",
"finset.card_empty",
"le_top",
"le_tsum'",
"nsmul_eq_mul",
"sum_le_tsum",
"zero_le'"
] | Markov's inequality for `finset.card` and `tsum` in `ℝ≥0∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_to_real_iff {ι} {fi : filter ι} {f : ι → ℝ≥0∞} (hf : ∀ i, f i ≠ ∞) {x : ℝ≥0∞}
(hx : x ≠ ∞) :
fi.tendsto (λ n, (f n).to_real) (𝓝 x.to_real) ↔ fi.tendsto f (𝓝 x) | begin
refine ⟨λ h, _, λ h, tendsto.comp (ennreal.tendsto_to_real hx) h⟩,
have h_eq : f = (λ n, ennreal.of_real (f n).to_real),
by { ext1 n, rw ennreal.of_real_to_real (hf n), },
rw [h_eq, ← ennreal.of_real_to_real hx],
exact ennreal.tendsto_of_real h,
end | lemma | ennreal.tendsto_to_real_iff | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.of_real",
"ennreal.of_real_to_real",
"ennreal.tendsto_of_real",
"ennreal.tendsto_to_real",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_coe_ne_top_iff_summable_coe {f : α → ℝ≥0} :
∑' a, (f a : ℝ≥0∞) ≠ ∞ ↔ summable (λ a, (f a : ℝ)) | begin
rw nnreal.summable_coe,
exact tsum_coe_ne_top_iff_summable,
end | lemma | ennreal.tsum_coe_ne_top_iff_summable_coe | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"nnreal.summable_coe",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_coe_eq_top_iff_not_summable_coe {f : α → ℝ≥0} :
∑' a, (f a : ℝ≥0∞) = ∞ ↔ ¬ summable (λ a, (f a : ℝ)) | begin
rw [← @not_not (∑' a, ↑(f a) = ⊤)],
exact not_congr tsum_coe_ne_top_iff_summable_coe
end | lemma | ennreal.tsum_coe_eq_top_iff_not_summable_coe | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"not_not",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_to_real {f : α → ℝ≥0∞} (hsum : ∑' x, f x ≠ ∞) :
has_sum (λ x, (f x).to_real) (∑' x, (f x).to_real) | begin
lift f to α → ℝ≥0 using ennreal.ne_top_of_tsum_ne_top hsum,
simp only [coe_to_real, ← nnreal.coe_tsum, nnreal.has_sum_coe],
exact (tsum_coe_ne_top_iff_summable.1 hsum).has_sum
end | lemma | ennreal.has_sum_to_real | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.ne_top_of_tsum_ne_top",
"has_sum",
"lift",
"nnreal.coe_tsum",
"nnreal.has_sum_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_to_real {f : α → ℝ≥0∞} (hsum : ∑' x, f x ≠ ∞) :
summable (λ x, (f x).to_real) | (has_sum_to_real hsum).summable | lemma | ennreal.summable_to_real | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_to_nnreal_tsum {f : β → ℝ≥0} :
(∑' b, f b) = (∑' b, (f b : ℝ≥0∞)).to_nnreal | begin
by_cases h : summable f,
{ rw [← ennreal.coe_tsum h, ennreal.to_nnreal_coe] },
{ have A := tsum_eq_zero_of_not_summable h,
simp only [← ennreal.tsum_coe_ne_top_iff_summable, not_not] at h,
simp only [h, ennreal.top_to_nnreal, A] }
end | lemma | nnreal.tsum_eq_to_nnreal_tsum | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.coe_tsum",
"ennreal.to_nnreal_coe",
"ennreal.top_to_nnreal",
"ennreal.tsum_coe_ne_top_iff_summable",
"not_not",
"summable",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_le_has_sum_of_le {f g : β → ℝ≥0} {r : ℝ≥0}
(hgf : ∀b, g b ≤ f b) (hfr : has_sum f r) : ∃p≤r, has_sum g p | have ∑'b, (g b : ℝ≥0∞) ≤ r,
begin
refine has_sum_le (assume b, _) ennreal.summable.has_sum (ennreal.has_sum_coe.2 hfr),
exact ennreal.coe_le_coe.2 (hgf _)
end,
let ⟨p, eq, hpr⟩ := ennreal.le_coe_iff.1 this in
⟨p, hpr, ennreal.has_sum_coe.1 $ eq ▸ ennreal.summable.has_sum⟩ | lemma | nnreal.exists_le_has_sum_of_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"has_sum",
"has_sum_le"
] | Comparison test of convergence of `ℝ≥0`-valued series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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_has_sum_of_le hgf hfr in hp.summable | lemma | nnreal.summable_of_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"summable"
] | Comparison test of convergence of `ℝ≥0`-valued series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_iff_tendsto_nat {f : ℕ → ℝ≥0} {r : ℝ≥0} :
has_sum f r ↔ tendsto (λn:ℕ, ∑ i in finset.range n, f i) at_top (𝓝 r) | begin
rw [← ennreal.has_sum_coe, ennreal.has_sum_iff_tendsto_nat],
simp only [ennreal.coe_finset_sum.symm],
exact ennreal.tendsto_coe
end | lemma | nnreal.has_sum_iff_tendsto_nat | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.has_sum_coe",
"ennreal.has_sum_iff_tendsto_nat",
"ennreal.tendsto_coe",
"finset.range",
"has_sum"
] | A series of non-negative real numbers converges to `r` in the sense of `has_sum` if and only if
the sequence of partial sum converges to `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_summable_iff_tendsto_nat_at_top {f : ℕ → ℝ≥0} :
¬ summable f ↔ tendsto (λ n : ℕ, ∑ i in finset.range n, f i) at_top at_top | begin
split,
{ intros h,
refine ((tendsto_of_monotone _).resolve_right h).comp _,
exacts [finset.sum_mono_set _, tendsto_finset_range] },
{ rintros hnat ⟨r, hr⟩,
exact not_tendsto_nhds_of_tendsto_at_top hnat _ (has_sum_iff_tendsto_nat.1 hr) }
end | lemma | nnreal.not_summable_iff_tendsto_nat_at_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"not_tendsto_nhds_of_tendsto_at_top",
"summable",
"tendsto_of_monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_iff_not_tendsto_nat_at_top {f : ℕ → ℝ≥0} :
summable f ↔ ¬ tendsto (λ n : ℕ, ∑ i in finset.range n, f i) at_top at_top | by rw [← not_iff_not, not_not, not_summable_iff_tendsto_nat_at_top] | lemma | nnreal.summable_iff_not_tendsto_nat_at_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"not_iff_not",
"not_not",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_sum_range_le {f : ℕ → ℝ≥0} {c : ℝ≥0}
(h : ∀ n, ∑ i in finset.range n, f i ≤ c) : summable f | begin
apply summable_iff_not_tendsto_nat_at_top.2 (λ H, _),
rcases exists_lt_of_tendsto_at_top H 0 c with ⟨n, -, hn⟩,
exact lt_irrefl _ (hn.trans_le (h n)),
end | lemma | nnreal.summable_of_sum_range_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"summable",
"summable_of_sum_range_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_le_of_sum_range_le {f : ℕ → ℝ≥0} {c : ℝ≥0}
(h : ∀ n, ∑ i in finset.range n, f i ≤ c) : ∑' n, f n ≤ c | tsum_le_of_sum_range_le (summable_of_sum_range_le h) h | lemma | nnreal.tsum_le_of_sum_range_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"summable_of_sum_range_le",
"tsum_le_of_sum_range_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | tsum_le_tsum_of_inj i hi (λ c hc, zero_le _) (λ b, le_rfl) (summable_comp_injective hf hi) hf | lemma | nnreal.tsum_comp_le_tsum_of_inj | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"le_rfl",
"summable",
"tsum_comp_le_tsum_of_inj",
"tsum_le_tsum_of_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_sigma {β : Π x : α, Type*} {f : (Σ x, β x) → ℝ≥0} :
summable f ↔ (∀ x, summable (λ y, f ⟨x, y⟩)) ∧ summable (λ x, ∑' y, f ⟨x, y⟩) | begin
split,
{ simp only [← nnreal.summable_coe, nnreal.coe_tsum],
exact λ 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 }
end | lemma | nnreal.summable_sigma | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.coe_tsum",
"ennreal.tsum_coe_ne_top_iff_summable",
"ennreal.tsum_sigma'",
"nnreal.coe_tsum",
"nnreal.summable_coe",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_summable {f : α → ℝ≥0} (hf : summable f) (s : set α) :
summable (s.indicator f) | begin
refine nnreal.summable_of_le (λ a, le_trans (le_of_eq (s.indicator_apply f a)) _) hf,
split_ifs,
exact le_refl (f a),
exact zero_le_coe,
end | lemma | nnreal.indicator_summable | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"nnreal.summable_of_le",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | λ h', let ⟨a, ha, hap⟩ := h in
hap (trans (set.indicator_apply_eq_self.mpr (absurd ha)).symm
(((tsum_eq_zero_iff (indicator_summable hf s)).1 h') a)) | lemma | nnreal.tsum_indicator_ne_zero | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"summable",
"tsum_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_sum_nat_add (f : ℕ → ℝ≥0) : tendsto (λ i, ∑' k, f (k + i)) at_top (𝓝 0) | begin
rw ← tendsto_coe,
convert tendsto_sum_nat_add (λ i, (f i : ℝ)),
norm_cast,
end | lemma | nnreal.tendsto_sum_nat_add | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_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 : has_sum f sf) (hg : has_sum g sg) : sf < sg | begin
have A : ∀ (a : α), (f a : ℝ) ≤ g a := λ a, nnreal.coe_le_coe.2 (h a),
have : (sf : ℝ) < sg :=
has_sum_lt A (nnreal.coe_lt_coe.2 hi) (has_sum_coe.2 hf) (has_sum_coe.2 hg),
exact nnreal.coe_lt_coe.1 this
end | lemma | nnreal.has_sum_lt | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"has_sum",
"has_sum_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_strict_mono
{f g : α → ℝ≥0} {sf sg : ℝ≥0} (hf : has_sum f sf) (hg : has_sum g sg) (h : f < g) : sf < sg | let ⟨hle, i, hi⟩ := pi.lt_def.mp h in has_sum_lt hle hi hf hg | lemma | nnreal.has_sum_strict_mono | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"has_sum",
"has_sum_lt",
"has_sum_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | has_sum_lt h hi (summable_of_le h hg).has_sum hg.has_sum | lemma | nnreal.tsum_lt_tsum | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"has_sum",
"has_sum_lt",
"summable",
"tsum_lt_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 in tsum_lt_tsum hle hi hg | lemma | nnreal.tsum_strict_mono | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"summable",
"tsum_lt_tsum",
"tsum_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 (λ a, zero_le _) hi hg } | lemma | nnreal.tsum_pos | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"summable",
"tsum_lt_tsum",
"tsum_pos",
"tsum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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) | begin
refine tsum_eq_add_tsum_ite' i (nnreal.summable_of_le (λ i', _) hf),
rw [function.update_apply],
split_ifs; simp only [zero_le', le_rfl]
end | lemma | nnreal.tsum_eq_add_tsum_ite | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"le_rfl",
"nnreal.summable_of_le",
"summable",
"tsum_eq_add_tsum_ite",
"tsum_eq_add_tsum_ite'",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_to_nnreal_eq {f : α → ℝ≥0∞} (hf : ∀ a, f a ≠ ∞) :
(∑' a, f a).to_nnreal = ∑' a, (f a).to_nnreal | (congr_arg ennreal.to_nnreal (tsum_congr $ λ x, (coe_to_nnreal (hf x)).symm)).trans
nnreal.tsum_eq_to_nnreal_tsum.symm | lemma | ennreal.tsum_to_nnreal_eq | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_nnreal",
"tsum_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_to_real_eq {f : α → ℝ≥0∞} (hf : ∀ a, f a ≠ ∞) :
(∑' a, f a).to_real = ∑' a, (f a).to_real | by simp only [ennreal.to_real, tsum_to_nnreal_eq hf, nnreal.coe_tsum] | lemma | ennreal.tsum_to_real_eq | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_real",
"nnreal.coe_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_sum_nat_add (f : ℕ → ℝ≥0∞) (hf : ∑' i, f i ≠ ∞) :
tendsto (λ i, ∑' k, f (k + i)) at_top (𝓝 0) | begin
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
end | lemma | ennreal.tendsto_sum_nat_add | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.coe_tsum",
"ennreal.coe_zero",
"ennreal.ne_top_of_tsum_ne_top",
"lift",
"nnreal.summable_nat_add",
"nnreal.tendsto_sum_nat_add",
"summable",
"tendsto_sum_nat_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_le_of_sum_range_le {f : ℕ → ℝ≥0∞} {c : ℝ≥0∞}
(h : ∀ n, ∑ i in finset.range n, f i ≤ c) : ∑' n, f n ≤ c | tsum_le_of_sum_range_le ennreal.summable h | lemma | ennreal.tsum_le_of_sum_range_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.summable",
"finset.range",
"tsum_le_of_sum_range_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_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 : has_sum f sf) (hg : has_sum g sg) : sf < sg | begin
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 λ 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 usin... | lemma | ennreal.has_sum_lt | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.ne_top_of_tsum_ne_top",
"has_sum",
"has_sum_lt",
"le_top",
"lift",
"nnreal.has_sum_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | has_sum_lt h hi hfi ennreal.summable.has_sum ennreal.summable.has_sum | lemma | ennreal.tsum_lt_tsum | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"has_sum_lt",
"tsum",
"tsum_lt_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | begin
lift f to α → ℝ≥0 using hn,
rw nnreal.summable_coe at hf,
simpa only [(∘), ← nnreal.coe_tsum] using nnreal.tsum_comp_le_tsum_of_inj hf hi
end | lemma | tsum_comp_le_tsum_of_inj | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"lift",
"nnreal.coe_tsum",
"nnreal.summable_coe",
"nnreal.tsum_comp_le_tsum_of_inj",
"summable",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | begin
lift f to β → ℝ≥0 using λ 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 (λ b, nnreal.coe_le_coe.1 (hgf b)) hf
end | lemma | summable_of_nonneg_of_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"lift",
"nnreal.summable_coe",
"nnreal.summable_of_le",
"summable"
] | Comparison test of convergence of series of non-negative real numbers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable.to_nnreal {f : α → ℝ} (hf : summable f) :
summable (λ n, (f n).to_nnreal) | begin
apply nnreal.summable_coe.1,
refine summable_of_nonneg_of_le (λ n, nnreal.coe_nonneg _) (λ n, _) hf.abs,
simp only [le_abs_self, real.coe_to_nnreal', max_le_iff, abs_nonneg, and_self]
end | lemma | summable.to_nnreal | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"abs_nonneg",
"le_abs_self",
"max_le_iff",
"nnreal.coe_nonneg",
"real.coe_to_nnreal'",
"summable",
"summable_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_iff_tendsto_nat_of_nonneg {f : ℕ → ℝ} (hf : ∀i, 0 ≤ f i) (r : ℝ) :
has_sum f r ↔ tendsto (λ n : ℕ, ∑ i in finset.range n, f i) at_top (𝓝 r) | begin
lift f to ℕ → ℝ≥0 using hf,
simp only [has_sum, ← nnreal.coe_sum, nnreal.tendsto_coe'],
exact exists_congr (λ hr, nnreal.has_sum_iff_tendsto_nat)
end | lemma | has_sum_iff_tendsto_nat_of_nonneg | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"has_sum",
"lift",
"nnreal.coe_sum",
"nnreal.has_sum_iff_tendsto_nat",
"nnreal.tendsto_coe'"
] | A series of non-negative real numbers converges to `r` in the sense of `has_sum` if and only if
the sequence of partial sum converges to `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ennreal.of_real_tsum_of_nonneg {f : α → ℝ} (hf_nonneg : ∀ n, 0 ≤ f n) (hf : summable f) :
ennreal.of_real (∑' n, f n) = ∑' n, ennreal.of_real (f n) | by simp_rw [ennreal.of_real, ennreal.tsum_coe_eq
(nnreal.has_sum_real_to_nnreal_of_nonneg hf_nonneg hf)] | lemma | ennreal.of_real_tsum_of_nonneg | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.of_real",
"ennreal.tsum_coe_eq",
"nnreal.has_sum_real_to_nnreal_of_nonneg",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_summable_iff_tendsto_nat_at_top_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
¬ summable f ↔ tendsto (λ n : ℕ, ∑ i in finset.range n, f i) at_top at_top | begin
lift f to ℕ → ℝ≥0 using hf,
exact_mod_cast nnreal.not_summable_iff_tendsto_nat_at_top
end | lemma | not_summable_iff_tendsto_nat_at_top_of_nonneg | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"lift",
"nnreal.not_summable_iff_tendsto_nat_at_top",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_iff_not_tendsto_nat_at_top_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
summable f ↔ ¬ tendsto (λ n : ℕ, ∑ i in finset.range n, f i) at_top at_top | by rw [← not_iff_not, not_not, not_summable_iff_tendsto_nat_at_top_of_nonneg hf] | lemma | summable_iff_not_tendsto_nat_at_top_of_nonneg | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"not_iff_not",
"not_not",
"not_summable_iff_tendsto_nat_at_top_of_nonneg",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_sigma_of_nonneg {β : Π x : α, Type*} {f : (Σ x, β x) → ℝ} (hf : ∀ x, 0 ≤ f x) :
summable f ↔ (∀ x, summable (λ y, f ⟨x, y⟩)) ∧ summable (λ x, ∑' y, f ⟨x, y⟩) | by { lift f to (Σ x, β x) → ℝ≥0 using hf, exact_mod_cast nnreal.summable_sigma } | lemma | summable_sigma_of_nonneg | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"lift",
"nnreal.summable_sigma",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_sum_le {ι : Type*} {f : ι → ℝ} {c : ℝ} (hf : 0 ≤ f)
(h : ∀ u : finset ι, ∑ x in u, f x ≤ c) :
summable f | ⟨ ⨆ u : finset ι, ∑ x in u, f x,
tendsto_at_top_csupr (finset.sum_mono_set_of_nonneg hf) ⟨c, λ y ⟨u, hu⟩, hu ▸ h u⟩ ⟩ | lemma | summable_of_sum_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset",
"summable",
"tendsto_at_top_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
(h : ∀ n, ∑ i in finset.range n, f i ≤ c) : summable f | begin
apply (summable_iff_not_tendsto_nat_at_top_of_nonneg hf).2 (λ H, _),
rcases exists_lt_of_tendsto_at_top H 0 c with ⟨n, -, hn⟩,
exact lt_irrefl _ (hn.trans_le (h n)),
end | lemma | summable_of_sum_range_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"summable",
"summable_iff_not_tendsto_nat_at_top_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.tsum_le_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
(h : ∀ n, ∑ i in finset.range n, f i ≤ c) : ∑' n, f n ≤ c | tsum_le_of_sum_range_le (summable_of_sum_range_le hf h) h | lemma | real.tsum_le_of_sum_range_le | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"finset.range",
"summable_of_sum_range_le",
"tsum_le_of_sum_range_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_lt_tsum_of_nonneg {i : ℕ} {f g : ℕ → ℝ}
(h0 : ∀ (b : ℕ), 0 ≤ f b) (h : ∀ (b : ℕ), f b ≤ g b) (hi : f i < g i) (hg : summable g) :
∑' n, f n < ∑' n, g n | tsum_lt_tsum h hi (summable_of_nonneg_of_le h0 h hg) hg | lemma | tsum_lt_tsum_of_nonneg | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"summable",
"summable_of_nonneg_of_le",
"tsum_lt_tsum"
] | If a sequence `f` with non-negative terms is dominated by a sequence `g` with summable
series and at least one term of `f` is strictly smaller than the corresponding term in `g`,
then the series of `f` is strictly smaller than the series of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_ne_top_of_mem_ball {a : β} {r : ℝ≥0∞} (x y : ball a r) : edist x.1 y.1 ≠ ⊤ | lt_top_iff_ne_top.1 $
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 add_lt_add x.2 y.2
... ≤ ⊤ : le_top | lemma | edist_ne_top_of_mem_ball | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"edist_triangle_left",
"le_top"
] | In an emetric ball, the distance between points is everywhere finite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space_emetric_ball (a : β) (r : ℝ≥0∞) : metric_space (ball a r) | emetric_space.to_metric_space edist_ne_top_of_mem_ball | def | metric_space_emetric_ball | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"edist_ne_top_of_mem_ball",
"emetric_space.to_metric_space",
"metric_space"
] | Each ball in an extended metric space gives us a metric space, as the edist
is everywhere finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_eq_nhds_emetric_ball (a x : β) (r : ℝ≥0∞) (h : x ∈ ball a r) :
𝓝 x = map (coe : ball a r → β) (𝓝 ⟨x, h⟩) | (map_nhds_subtype_coe_eq _ $ is_open.mem_nhds emetric.is_open_ball h).symm | lemma | nhds_eq_nhds_emetric_ball | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"emetric.is_open_ball",
"is_open.mem_nhds",
"map_nhds_subtype_coe_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_edist_tendsto_0 {l : filter β} {f : β → α} {y : α} :
tendsto f l (𝓝 y) ↔ tendsto (λ 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] | lemma | tendsto_iff_edist_tendsto_0 | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"emetric.mem_ball",
"ennreal.not_lt_zero",
"filter",
"forall_const",
"forall_prop_of_false",
"tendsto_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emetric.cauchy_seq_iff_le_tendsto_0 [nonempty β] [semilattice_sup β] {s : β → α} :
cauchy_seq s ↔ (∃ (b: β → ℝ≥0∞), (∀ n m N : β, N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N)
∧ (tendsto b at_top (𝓝 0))) | ⟨begin
assume hs,
rw emetric.cauchy_seq_iff at hs,
/- `s` is Cauchy sequence. The sequence `b` will be constructed by taking
the supremum of the distances between `s n` and `s m` for `n m ≥ N`-/
let b := λN, Sup ((λ(p : β × β), edist (s p.1) (s p.2))''{p | p.1 ≥ N ∧ p.2 ≥ N}),
--Prove that it bounds the dis... | lemma | emetric.cauchy_seq_iff_le_tendsto_0 | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"Sup_le",
"and_imp",
"cauchy_seq",
"emetric.cauchy_seq_iff",
"ennreal.not_lt_zero",
"exists_between",
"exists_imp_distrib",
"le_Sup",
"semilattice_sup",
"set.mem_image"
] | Yet another metric characterization of Cauchy sequences on integers. This one is often the
most efficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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 | begin
rcases eq_or_ne C 0 with (rfl|C0),
{ simp only [zero_mul, add_zero] at h,
exact continuous_of_const (λ x y, le_antisymm (h _ _) (h _ _)) },
{ refine continuous_iff_continuous_at.2 (λ x, _),
by_cases hx : f x = ∞,
{ have : f =ᶠ[𝓝 x] (λ _, ∞),
{ filter_upwards [emetric.ball_mem_nhds x ennre... | lemma | continuous_of_le_add_edist | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous",
"continuous_of_const",
"emetric.ball_mem_nhds",
"emetric.closed_ball_mem_nhds",
"ennreal.coe_lt_top",
"ennreal.mul_div_cancel'",
"ennreal.mul_ne_top",
"ennreal.tendsto_nhds",
"eq_or_ne",
"mul_le_mul_left'",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_edist : continuous (λp:α×α, edist p.1 p.2) | begin
apply continuous_of_le_add_edist 2 (by norm_num),
rintros ⟨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 [edist_comm]; cc
... ≤ edist x' y' + (edist (x, y) (x', y') + edist (x, y) (x', y')) :
... | theorem | continuous_edist | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous",
"continuous_of_le_add_edist",
"edist_triangle4",
"mul_comm",
"mul_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.edist [topological_space β] {f g : β → α}
(hf : continuous f) (hg : continuous g) : continuous (λb, edist (f b) (g b)) | continuous_edist.comp (hf.prod_mk hg : _) | theorem | continuous.edist | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 (λx, edist (f x) (g x)) x (𝓝 (edist a b)) | (continuous_edist.tendsto (a, b)).comp (hf.prod_mk_nhds hg) | theorem | filter.tendsto.edist | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_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 ≠ ∞) :
cauchy_seq f | begin
lift d to (ℕ → nnreal) using (λ i, ennreal.ne_top_of_tsum_ne_top hd i),
rw ennreal.tsum_coe_ne_top_iff_summable at hd,
exact cauchy_seq_of_edist_le_of_summable d hf hd
end | lemma | cauchy_seq_of_edist_le_of_tsum_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"cauchy_seq",
"cauchy_seq_of_edist_le_of_summable",
"ennreal.ne_top_of_tsum_ne_top",
"ennreal.tsum_coe_ne_top_iff_summable",
"lift",
"nnreal",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emetric.is_closed_ball {a : α} {r : ℝ≥0∞} : is_closed (closed_ball a r) | is_closed_le (continuous_id.edist continuous_const) continuous_const | lemma | emetric.is_closed_ball | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous_const",
"is_closed",
"is_closed_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emetric.diam_closure (s : set α) : diam (closure s) = diam s | begin
refine le_antisymm (diam_le $ λ x hx y hy, _) (diam_mono subset_closure),
have : edist x y ∈ closure (Iic (diam s)),
from map_mem_closure₂ continuous_edist hx hy (λ x hx y hy, edist_le_diam_of_mem hx hy),
rwa closure_Iic at this
end | lemma | emetric.diam_closure | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"closure",
"closure_Iic",
"continuous_edist",
"map_mem_closure₂",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.diam_closure {α : Type*} [pseudo_metric_space α] (s : set α) :
metric.diam (closure s) = diam s | by simp only [metric.diam, emetric.diam_closure] | lemma | metric.diam_closure | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"closure",
"emetric.diam_closure",
"metric.diam",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_of_lipschitz_on_with {α β} [pseudo_emetric_space α] [pseudo_emetric_space β]
(K : ℝ≥0) (s : set α) :
is_closed {f : α → β | lipschitz_on_with K f s} | begin
simp only [lipschitz_on_with, set_of_forall],
refine is_closed_bInter (λ x hx, is_closed_bInter $ λ y hy, is_closed_le _ _),
exacts [continuous.edist (continuous_apply x) (continuous_apply y), continuous_const]
end | lemma | is_closed_set_of_lipschitz_on_with | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous.edist",
"continuous_apply",
"continuous_const",
"is_closed",
"is_closed_bInter",
"is_closed_le",
"lipschitz_on_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_of_lipschitz_with {α β} [pseudo_emetric_space α] [pseudo_emetric_space β]
(K : ℝ≥0) :
is_closed {f : α → β | lipschitz_with K f} | by simp only [← lipschitz_on_univ, is_closed_set_of_lipschitz_on_with] | lemma | is_closed_set_of_lipschitz_with | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"is_closed",
"is_closed_set_of_lipschitz_on_with",
"lipschitz_on_univ",
"lipschitz_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_eq {s : set ℝ} (h : bounded s) :
emetric.diam s = ennreal.of_real (Sup s - Inf s) | begin
rcases eq_empty_or_nonempty s with rfl|hne, { simp },
refine le_antisymm (metric.ediam_le_of_forall_dist_le $ λ x hx y hy, _) _,
{ have := real.subset_Icc_Inf_Sup_of_bounded h,
exact real.dist_le_of_mem_Icc (this hx) (this hy) },
{ apply ennreal.of_real_le_of_le_to_real,
rw [← metric.diam, ← metri... | lemma | real.ediam_eq | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"cInf_mem_closure",
"cSup_mem_closure",
"closure",
"emetric.diam",
"ennreal.of_real",
"ennreal.of_real_le_of_le_to_real",
"le_abs_self",
"metric.diam",
"metric.diam_closure",
"metric.ediam_le_of_forall_dist_le",
"real.dist_le_of_mem_Icc",
"real.subset_Icc_Inf_Sup_of_bounded"
] | For a bounded set `s : set ℝ`, its `emetric.diam` is equal to `Sup s - Inf s` reinterpreted as
`ℝ≥0∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_eq {s : set ℝ} (h : bounded s) : metric.diam s = Sup s - Inf s | begin
rw [metric.diam, real.ediam_eq h, ennreal.to_real_of_real],
rw real.bounded_iff_bdd_below_bdd_above at h,
exact sub_nonneg.2 (real.Inf_le_Sup s h.1 h.2)
end | lemma | real.diam_eq | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_real_of_real",
"metric.diam",
"real.Inf_le_Sup",
"real.bounded_iff_bdd_below_bdd_above",
"real.ediam_eq"
] | For a bounded set `s : set ℝ`, its `metric.diam` is equal to `Sup s - Inf s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ediam_Ioo (a b : ℝ) :
emetric.diam (Ioo a b) = ennreal.of_real (b - a) | begin
rcases le_or_lt b a with h|h,
{ simp [h] },
{ rw [real.ediam_eq (bounded_Ioo _ _), cSup_Ioo h, cInf_Ioo h] },
end | lemma | real.ediam_Ioo | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"cInf_Ioo",
"cSup_Ioo",
"emetric.diam",
"ennreal.of_real",
"real.ediam_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_Icc (a b : ℝ) :
emetric.diam (Icc a b) = ennreal.of_real (b - a) | begin
rcases le_or_lt a b with h|h,
{ rw [real.ediam_eq (bounded_Icc _ _), cSup_Icc h, cInf_Icc h] },
{ simp [h, h.le] }
end | lemma | real.ediam_Icc | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"cInf_Icc",
"cSup_Icc",
"emetric.diam",
"ennreal.of_real",
"real.ediam_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_Ico (a b : ℝ) :
emetric.diam (Ico a b) = ennreal.of_real (b - a) | le_antisymm (ediam_Icc a b ▸ diam_mono Ico_subset_Icc_self)
(ediam_Ioo a b ▸ diam_mono Ioo_subset_Ico_self) | lemma | real.ediam_Ico | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"emetric.diam",
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_Ioc (a b : ℝ) :
emetric.diam (Ioc a b) = ennreal.of_real (b - a) | le_antisymm (ediam_Icc a b ▸ diam_mono Ioc_subset_Icc_self)
(ediam_Ioo a b ▸ diam_mono Ioo_subset_Ioc_self) | lemma | real.ediam_Ioc | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"emetric.diam",
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_Icc {a b : ℝ} (h : a ≤ b) : metric.diam (Icc a b) = b - a | by simp [metric.diam, ennreal.to_real_of_real, sub_nonneg.2 h] | lemma | real.diam_Icc | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_real_of_real",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_Ico {a b : ℝ} (h : a ≤ b) : metric.diam (Ico a b) = b - a | by simp [metric.diam, ennreal.to_real_of_real, sub_nonneg.2 h] | lemma | real.diam_Ico | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_real_of_real",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_Ioc {a b : ℝ} (h : a ≤ b) : metric.diam (Ioc a b) = b - a | by simp [metric.diam, ennreal.to_real_of_real, sub_nonneg.2 h] | lemma | real.diam_Ioc | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_real_of_real",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_Ioo {a b : ℝ} (h : a ≤ b) : metric.diam (Ioo a b) = b - a | by simp [metric.diam, ennreal.to_real_of_real, sub_nonneg.2 h] | lemma | real.diam_Ioo | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_real_of_real",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 at_top (𝓝 a)) (n : ℕ) :
edist (f n) a ≤ ∑' m, d (n + m) | begin
refine le_of_tendsto (tendsto_const_nhds.edist ha)
(mem_at_top_sets.2 ⟨n, λ m hnm, _⟩),
refine le_trans (edist_le_Ico_sum_of_edist_le hnm (λ k _ _, hf k)) _,
rw [finset.sum_Ico_eq_sum_range],
exact sum_le_tsum _ (λ _ _, zero_le _) ennreal.summable
end | lemma | edist_le_tsum_of_edist_le_of_tendsto | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"edist_le_Ico_sum_of_edist_le",
"ennreal.summable",
"le_of_tendsto",
"sum_le_tsum"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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 at_top (𝓝 a)) :
edist (f 0) a ≤ ∑' m, d m | by simpa using edist_le_tsum_of_edist_le_of_tendsto d hf ha 0 | lemma | edist_le_tsum_of_edist_le_of_tendsto₀ | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"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 0` to the limit is bounded by `∑'_{k=0}^∞ d k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding_coe : embedding (coe : ℝ → ereal) | ⟨⟨begin
refine le_antisymm _ _,
{ rw [@order_topology.topology_eq_generate_intervals ereal _,
← coinduced_le_iff_le_induced],
refine le_generate_from (assume s ha, _),
rcases ha with ⟨a, rfl | rfl⟩,
show is_open {b : ℝ | a < ↑b},
{ induction a using ereal.rec,
{ simp only [is_open_univ, ... | lemma | ereal.embedding_coe | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"coinduced_le_iff_le_induced",
"embedding",
"ereal",
"ereal.coe_eq_coe_iff",
"ereal.coe_lt_coe_iff",
"ereal.rec",
"imp_self",
"is_open",
"is_open_Iio",
"is_open_Ioi",
"is_open_empty",
"is_open_univ",
"le_generate_from",
"not_lt_bot",
"not_top_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_coe : open_embedding (coe : ℝ → ereal) | ⟨embedding_coe,
begin
convert @is_open_Ioo ereal _ _ _ ⊥ ⊤,
ext x,
induction x using ereal.rec,
{ simp only [left_mem_Ioo, mem_range, coe_ne_bot, exists_false, not_false_iff] },
{ simp only [mem_range_self, mem_Ioo, bot_lt_coe, coe_lt_top, and_self] },
{ simp only [mem_range, right_mem_Ioo, exists_false, co... | lemma | ereal.open_embedding_coe | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"ereal.rec",
"exists_false",
"is_open_Ioo",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_coe {α : Type*} {f : filter α} {m : α → ℝ} {a : ℝ} :
tendsto (λ a, (m a : ereal)) f (𝓝 ↑a) ↔ tendsto m f (𝓝 a) | embedding_coe.tendsto_nhds_iff.symm | lemma | ereal.tendsto_coe | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_coe_real_ereal : continuous (coe : ℝ → ereal) | embedding_coe.continuous | lemma | continuous_coe_real_ereal | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous",
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe_iff {f : α → ℝ} :
continuous (λa, (f a : ereal)) ↔ continuous f | embedding_coe.continuous_iff.symm | lemma | ereal.continuous_coe_iff | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous",
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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