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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|---|---|---|---|---|---|---|---|---|---|---|
nhds_top_basis : (𝓝 ∞).has_basis (λ a, a < ∞) (λ a, Ioi a) | nhds_top_basis | lemma | ennreal.nhds_top_basis | 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"
] | [
"nhds_top_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_top_iff_nnreal {m : α → ℝ≥0∞} {f : filter α} :
tendsto m f (𝓝 ⊤) ↔ ∀ x : ℝ≥0, ∀ᶠ a in f, ↑x < m a | by simp only [nhds_top', tendsto_infi, tendsto_principal, mem_Ioi] | lemma | ennreal.tendsto_nhds_top_iff_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"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_top_iff_nat {m : α → ℝ≥0∞} {f : filter α} :
tendsto m f (𝓝 ⊤) ↔ ∀ n : ℕ, ∀ᶠ a in f, ↑n < m a | tendsto_nhds_top_iff_nnreal.trans ⟨λ h n, by simpa only [ennreal.coe_nat] using h n,
λ h x, let ⟨n, hn⟩ := exists_nat_gt x in
(h n).mono (λ y, lt_trans $ by rwa [← ennreal.coe_nat, coe_lt_coe])⟩ | lemma | ennreal.tendsto_nhds_top_iff_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.coe_nat",
"exists_nat_gt",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_top {m : α → ℝ≥0∞} {f : filter α}
(h : ∀ n : ℕ, ∀ᶠ a in f, ↑n < m a) : tendsto m f (𝓝 ⊤) | tendsto_nhds_top_iff_nat.2 h | lemma | ennreal.tendsto_nhds_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"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nat_nhds_top : tendsto (λ n : ℕ, ↑n) at_top (𝓝 ∞) | tendsto_nhds_top $ λ n, mem_at_top_sets.2
⟨n + 1, λ m hm, mem_set_of.2 $ nat.cast_lt.2 $ nat.lt_of_succ_le hm⟩ | lemma | ennreal.tendsto_nat_nhds_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_coe_nhds_top {f : α → ℝ≥0} {l : filter α} :
tendsto (λ x, (f x : ℝ≥0∞)) l (𝓝 ∞) ↔ tendsto f l at_top | by rw [tendsto_nhds_top_iff_nnreal, at_top_basis_Ioi.tendsto_right_iff];
[simp, apply_instance, apply_instance] | lemma | ennreal.tendsto_coe_nhds_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"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_of_real_at_top : tendsto ennreal.of_real at_top (𝓝 ∞) | tendsto_coe_nhds_top.2 tendsto_real_to_nnreal_at_top | lemma | ennreal.tendsto_of_real_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"
] | [
"ennreal.of_real",
"tendsto_real_to_nnreal_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_zero : 𝓝 (0 : ℝ≥0∞) = ⨅ a ≠ 0, 𝓟 (Iio a) | nhds_bot_order.trans $ by simp [bot_lt_iff_ne_bot, Iio] | lemma | ennreal.nhds_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"
] | [
"bot_lt_iff_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_zero_basis : (𝓝 (0 : ℝ≥0∞)).has_basis (λ a : ℝ≥0∞, 0 < a) (λ a, Iio a) | nhds_bot_basis | lemma | ennreal.nhds_zero_basis | 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"
] | [
"nhds_bot_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_zero_basis_Iic : (𝓝 (0 : ℝ≥0∞)).has_basis (λ a : ℝ≥0∞, 0 < a) Iic | nhds_bot_basis_Iic | lemma | ennreal.nhds_zero_basis_Iic | 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"
] | [
"nhds_bot_basis_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioi_coe_ne_bot {r : ℝ≥0} : (𝓝[>] (r : ℝ≥0∞)).ne_bot | nhds_within_Ioi_self_ne_bot' ⟨⊤, ennreal.coe_lt_top⟩ | lemma | ennreal.nhds_within_Ioi_coe_ne_bot | 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"
] | [
"nhds_within_Ioi_self_ne_bot'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Ioi_zero_ne_bot : (𝓝[>] (0 : ℝ≥0∞)).ne_bot | nhds_within_Ioi_coe_ne_bot | lemma | ennreal.nhds_within_Ioi_zero_ne_bot | 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 | |
Icc_mem_nhds (xt : x ≠ ⊤) (ε0 : ε ≠ 0) : Icc (x - ε) (x + ε) ∈ 𝓝 x | begin
rw _root_.mem_nhds_iff,
by_cases x0 : x = 0,
{ use Iio (x + ε),
have : Iio (x + ε) ⊆ Icc (x - ε) (x + ε), assume a, rw x0, simpa using le_of_lt,
use this, exact ⟨is_open_Iio, mem_Iio_self_add xt ε0⟩ },
{ use Ioo (x - ε) (x + ε), use Ioo_subset_Icc_self,
exact ⟨is_open_Ioo, mem_Ioo_self_sub_add... | lemma | ennreal.Icc_mem_nhds | 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"
] | [
"Icc_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_of_ne_top (xt : x ≠ ⊤) : 𝓝 x = ⨅ ε > 0, 𝓟 (Icc (x - ε) (x + ε)) | begin
refine le_antisymm _ _,
-- first direction
simp only [le_infi_iff, le_principal_iff], assume ε ε0, exact Icc_mem_nhds xt ε0.lt.ne',
-- second direction
rw nhds_generate_from, refine le_infi (assume s, le_infi $ assume hs, _),
rcases hs with ⟨xs, ⟨a, (rfl : s = Ioi a)|(rfl : s = Iio a)⟩⟩,
{ rcases ex... | lemma | ennreal.nhds_of_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"
] | [
"Icc_mem_nhds",
"add_tsub_cancel_of_le",
"exists_between",
"infi_le_of_le",
"le_infi",
"le_infi_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds {f : filter α} {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha : a ≠ ⊤) :
tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, (u x) ∈ Icc (a - ε) (a + ε) | by simp only [nhds_of_ne_top ha, tendsto_infi, tendsto_principal, mem_Icc] | theorem | ennreal.tendsto_nhds | 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",
"tendsto_nhds"
] | Characterization of neighborhoods for `ℝ≥0∞` numbers. See also `tendsto_order`
for a version with strict inequalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_nhds_zero {f : filter α} {u : α → ℝ≥0∞} :
tendsto u f (𝓝 0) ↔ ∀ ε > 0, ∀ᶠ x in f, u x ≤ ε | begin
rw ennreal.tendsto_nhds zero_ne_top,
simp only [true_and, zero_tsub, zero_le, zero_add, set.mem_Icc],
end | lemma | ennreal.tendsto_nhds_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.tendsto_nhds",
"filter",
"set.mem_Icc",
"zero_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top [nonempty β] [semilattice_sup β] {f : β → ℝ≥0∞} {a : ℝ≥0∞}
(ha : a ≠ ⊤) : tendsto f at_top (𝓝 a) ↔ ∀ε>0, ∃N, ∀n≥N, (f n) ∈ Icc (a - ε) (a + ε) | by simp only [ennreal.tendsto_nhds ha, mem_at_top_sets, mem_set_of_eq, filter.eventually] | lemma | ennreal.tendsto_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"
] | [
"ennreal.tendsto_nhds",
"filter.eventually",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_zero [hβ : nonempty β] [semilattice_sup β] {f : β → ℝ≥0∞} :
filter.at_top.tendsto f (𝓝 0) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, f n ≤ ε | begin
rw ennreal.tendsto_at_top zero_ne_top,
{ simp_rw [set.mem_Icc, zero_add, zero_tsub, zero_le _, true_and], },
{ exact hβ, },
end | lemma | ennreal.tendsto_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.tendsto_at_top",
"semilattice_sup",
"set.mem_Icc",
"zero_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_sub {a b : ℝ≥0∞} (h : a ≠ ∞ ∨ b ≠ ∞) :
tendsto (λ p : ℝ≥0∞ × ℝ≥0∞, p.1 - p.2) (𝓝 (a, b)) (𝓝 (a - b)) | begin
cases a; cases b,
{ simp only [eq_self_iff_true, not_true, ne.def, none_eq_top, or_self] at h, contradiction },
{ simp only [some_eq_coe, with_top.top_sub_coe, none_eq_top],
apply tendsto_nhds_top_iff_nnreal.2 (λ n, _),
rw [nhds_prod_eq, eventually_prod_iff],
refine ⟨λ z, ((n + (b + 1)) : ℝ≥0∞) ... | lemma | ennreal.tendsto_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"
] | [
"Iio_mem_nhds",
"Ioi_mem_nhds",
"continuity",
"continuous.tendsto",
"ennreal.add_lt_add",
"ennreal.coe_sub",
"ennreal.lt_add_right",
"lt_tsub_iff_right",
"nhds_prod_eq",
"one_ne_zero",
"tsub_eq_zero_iff_le",
"with_top.sub_top",
"with_top.top_sub_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.sub {f : filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hma : tendsto ma f (𝓝 a)) (hmb : tendsto mb f (𝓝 b)) (h : a ≠ ∞ ∨ b ≠ ∞) :
tendsto (λ a, ma a - mb a) f (𝓝 (a - b)) | show tendsto ((λ p : ℝ≥0∞ × ℝ≥0∞, p.1 - p.2) ∘ (λa, (ma a, mb a))) f (𝓝 (a - b)), from
tendsto.comp (ennreal.tendsto_sub h) (hma.prod_mk_nhds hmb) | lemma | ennreal.tendsto.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.tendsto_sub",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_mul (ha : a ≠ 0 ∨ b ≠ ⊤) (hb : b ≠ 0 ∨ a ≠ ⊤) :
tendsto (λp:ℝ≥0∞×ℝ≥0∞, p.1 * p.2) (𝓝 (a, b)) (𝓝 (a * b)) | have ht : ∀b:ℝ≥0∞, b ≠ 0 → tendsto (λp:ℝ≥0∞×ℝ≥0∞, p.1 * p.2) (𝓝 ((⊤:ℝ≥0∞), b)) (𝓝 ⊤),
begin
refine assume b hb, tendsto_nhds_top_iff_nnreal.2 $ assume n, _,
rcases lt_iff_exists_nnreal_btwn.1 (pos_iff_ne_zero.2 hb) with ⟨ε, hε, hεb⟩,
have : ∀ᶠ c : ℝ≥0∞ × ℝ≥0∞ in 𝓝 (∞, b), ↑n / ↑ε < c.1 ∧ ↑ε < c.2,
from (lt... | lemma | ennreal.tendsto_mul | 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.div_mul_cancel",
"lt_mem_nhds",
"mul_comm",
"mul_lt_mul",
"nhds_swap",
"tendsto_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.mul {f : filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hma : tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ ⊤) (hmb : tendsto mb f (𝓝 b)) (hb : b ≠ 0 ∨ a ≠ ⊤) :
tendsto (λa, ma a * mb a) f (𝓝 (a * b)) | show tendsto ((λp:ℝ≥0∞×ℝ≥0∞, p.1 * p.2) ∘ (λa, (ma a, mb a))) f (𝓝 (a * b)), from
tendsto.comp (ennreal.tendsto_mul ha hb) (hma.prod_mk_nhds hmb) | lemma | ennreal.tendsto.mul | 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.tendsto_mul",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_on.ennreal_mul [topological_space α] {f g : α → ℝ≥0∞} {s : set α}
(hf : continuous_on f s) (hg : continuous_on g s) (h₁ : ∀ x ∈ s, f x ≠ 0 ∨ g x ≠ ∞)
(h₂ : ∀ x ∈ s, g x ≠ 0 ∨ f x ≠ ∞) :
continuous_on (λ x, f x * g x) s | λ x hx, ennreal.tendsto.mul (hf x hx) (h₁ x hx) (hg x hx) (h₂ x hx) | lemma | continuous_on.ennreal_mul | 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_on",
"ennreal.tendsto.mul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous.ennreal_mul [topological_space α] {f g : α → ℝ≥0∞} (hf : continuous f)
(hg : continuous g) (h₁ : ∀ x, f x ≠ 0 ∨ g x ≠ ∞) (h₂ : ∀ x, g x ≠ 0 ∨ f x ≠ ∞) :
continuous (λ x, f x * g x) | continuous_iff_continuous_at.2 $
λ x, ennreal.tendsto.mul hf.continuous_at (h₁ x) hg.continuous_at (h₂ x) | lemma | continuous.ennreal_mul | 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",
"ennreal.tendsto.mul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.const_mul {f : filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hm : tendsto m f (𝓝 b)) (hb : b ≠ 0 ∨ a ≠ ⊤) : tendsto (λb, a * m b) f (𝓝 (a * b)) | by_cases
(assume : a = 0, by simp [this, tendsto_const_nhds])
(assume ha : a ≠ 0, ennreal.tendsto.mul tendsto_const_nhds (or.inl ha) hm hb) | lemma | ennreal.tendsto.const_mul | 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.tendsto.mul",
"filter",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.mul_const {f : filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hm : tendsto m f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ ⊤) : tendsto (λx, m x * b) f (𝓝 (a * b)) | by simpa only [mul_comm] using ennreal.tendsto.const_mul hm ha | lemma | ennreal.tendsto.mul_const | 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.tendsto.const_mul",
"filter",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_finset_prod_of_ne_top {ι : Type*} {f : ι → α → ℝ≥0∞} {x : filter α} {a : ι → ℝ≥0∞}
(s : finset ι) (h : ∀ i ∈ s, tendsto (f i) x (𝓝 (a i))) (h' : ∀ i ∈ s, a i ≠ ∞):
tendsto (λ b, ∏ c in s, f c b) x (𝓝 (∏ c in s, a c)) | begin
induction s using finset.induction with a s has IH, { simp [tendsto_const_nhds] },
simp only [finset.prod_insert has],
apply tendsto.mul (h _ (finset.mem_insert_self _ _)),
{ right,
exact (prod_lt_top (λ i hi, h' _ (finset.mem_insert_of_mem hi))).ne },
{ exact IH (λ i hi, h _ (finset.mem_insert_of_m... | lemma | ennreal.tendsto_finset_prod_of_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"
] | [
"filter",
"finset",
"finset.induction",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.prod_insert",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const_mul {a b : ℝ≥0∞} (h : a ≠ ⊤ ∨ b ≠ 0) :
continuous_at ((*) a) b | tendsto.const_mul tendsto_id h.symm | lemma | ennreal.continuous_at_const_mul | 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_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_mul_const {a b : ℝ≥0∞} (h : a ≠ ⊤ ∨ b ≠ 0) :
continuous_at (λ x, x * a) b | tendsto.mul_const tendsto_id h.symm | lemma | ennreal.continuous_at_mul_const | 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_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_const_mul {a : ℝ≥0∞} (ha : a ≠ ⊤) : continuous ((*) a) | continuous_iff_continuous_at.2 $ λ x, ennreal.continuous_at_const_mul (or.inl ha) | lemma | ennreal.continuous_const_mul | 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",
"ennreal.continuous_at_const_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_mul_const {a : ℝ≥0∞} (ha : a ≠ ⊤) : continuous (λ x, x * a) | continuous_iff_continuous_at.2 $ λ x, ennreal.continuous_at_mul_const (or.inl ha) | lemma | ennreal.continuous_mul_const | 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",
"ennreal.continuous_at_mul_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_div_const (c : ℝ≥0∞) (c_ne_zero : c ≠ 0) :
continuous (λ (x : ℝ≥0∞), x / c) | begin
simp_rw [div_eq_mul_inv, continuous_iff_continuous_at],
intro x,
exact ennreal.continuous_at_mul_const (or.intro_left _ (inv_ne_top.mpr c_ne_zero)),
end | lemma | ennreal.continuous_div_const | 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_iff_continuous_at",
"div_eq_mul_inv",
"ennreal.continuous_at_mul_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_pow (n : ℕ) : continuous (λ a : ℝ≥0∞, a ^ n) | begin
induction n with n IH,
{ simp [continuous_const] },
simp_rw [nat.succ_eq_add_one, pow_add, pow_one, continuous_iff_continuous_at],
assume x,
refine ennreal.tendsto.mul (IH.tendsto _) _ tendsto_id _;
by_cases H : x = 0,
{ simp only [H, zero_ne_top, ne.def, or_true, not_false_iff]},
{ exact or.inl (... | lemma | ennreal.continuous_pow | 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_const",
"continuous_iff_continuous_at",
"continuous_pow",
"ennreal.tendsto.mul",
"pow_add",
"pow_eq_zero",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_sub :
continuous_on (λ p : ℝ≥0∞ × ℝ≥0∞, p.fst - p.snd) { p : ℝ≥0∞ × ℝ≥0∞ | p ≠ ⟨∞, ∞⟩ } | begin
rw continuous_on,
rintros ⟨x, y⟩ hp,
simp only [ne.def, set.mem_set_of_eq, prod.mk.inj_iff] at hp,
refine tendsto_nhds_within_of_tendsto_nhds (tendsto_sub (not_and_distrib.mp hp)),
end | lemma | ennreal.continuous_on_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"
] | [
"continuous_on",
"prod.mk.inj_iff",
"tendsto_nhds_within_of_tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sub_left {a : ℝ≥0∞} (a_ne_top : a ≠ ⊤) :
continuous (λ x, a - x) | begin
rw (show (λ x, a - x) = (λ p : ℝ≥0∞ × ℝ≥0∞, p.fst - p.snd) ∘ (λ x, ⟨a, x⟩), by refl),
apply continuous_on.comp_continuous continuous_on_sub (continuous.prod.mk a),
intro x,
simp only [a_ne_top, ne.def, mem_set_of_eq, prod.mk.inj_iff, false_and, not_false_iff],
end | lemma | ennreal.continuous_sub_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"
] | [
"continuous",
"continuous.prod.mk",
"continuous_on.comp_continuous",
"prod.mk.inj_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_nnreal_sub {a : ℝ≥0} :
continuous (λ (x : ℝ≥0∞), (a : ℝ≥0∞) - x) | continuous_sub_left coe_ne_top | lemma | ennreal.continuous_nnreal_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"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_sub_left (a : ℝ≥0∞) :
continuous_on (λ x, a - x) {x : ℝ≥0∞ | x ≠ ∞} | begin
rw (show (λ x, a - x) = (λ p : ℝ≥0∞ × ℝ≥0∞, p.fst - p.snd) ∘ (λ x, ⟨a, x⟩), by refl),
apply continuous_on.comp continuous_on_sub (continuous.continuous_on (continuous.prod.mk a)),
rintros _ h (_|_),
exact h none_eq_top,
end | lemma | ennreal.continuous_on_sub_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"
] | [
"continuous.continuous_on",
"continuous.prod.mk",
"continuous_on",
"continuous_on.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sub_right (a : ℝ≥0∞) :
continuous (λ x : ℝ≥0∞, x - a) | begin
by_cases a_infty : a = ∞,
{ simp [a_infty, continuous_const], },
{ rw (show (λ x, x - a) = (λ p : ℝ≥0∞ × ℝ≥0∞, p.fst - p.snd) ∘ (λ x, ⟨x, a⟩), by refl),
apply continuous_on.comp_continuous
continuous_on_sub (continuous_id'.prod_mk continuous_const),
intro x,
simp only [a_infty, ne.def, mem... | lemma | ennreal.continuous_sub_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"
] | [
"continuous",
"continuous_const",
"continuous_on.comp_continuous",
"prod.mk.inj_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.pow {f : filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} {n : ℕ}
(hm : tendsto m f (𝓝 a)) :
tendsto (λ x, (m x) ^ n) f (𝓝 (a ^ n)) | ((continuous_pow n).tendsto a).comp hm | lemma | ennreal.tendsto.pow | 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_pow",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_lt_one_mul_le {x y : ℝ≥0∞} (h : ∀ a < 1, a * x ≤ y) : x ≤ y | begin
have : tendsto (* x) (𝓝[<] 1) (𝓝 (1 * x)) :=
(ennreal.continuous_at_mul_const (or.inr one_ne_zero)).mono_left inf_le_left,
rw one_mul at this,
haveI : (𝓝[<] (1 : ℝ≥0∞)).ne_bot := nhds_within_Iio_self_ne_bot' ⟨0, zero_lt_one⟩,
exact le_of_tendsto this (eventually_nhds_within_iff.2 $ eventually_of_fo... | lemma | ennreal.le_of_forall_lt_one_mul_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.continuous_at_mul_const",
"inf_le_left",
"le_of_forall_lt_one_mul_le",
"le_of_tendsto",
"nhds_within_Iio_self_ne_bot'",
"one_mul",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_mul_left' {ι} {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
(h : a = ⊤ → (⨅ i, f i) = 0 → ∃ i, f i = 0) (h0 : a = 0 → nonempty ι) :
(⨅ i, a * f i) = a * ⨅ i, f i | begin
by_cases H : a = ⊤ ∧ (⨅ i, f i) = 0,
{ rcases h H.1 H.2 with ⟨i, hi⟩,
rw [H.2, mul_zero, ← bot_eq_zero, infi_eq_bot],
exact λ b hb, ⟨i, by rwa [hi, mul_zero, ← bot_eq_zero]⟩ },
{ rw not_and_distrib at H,
casesI is_empty_or_nonempty ι,
{ rw [infi_of_empty, infi_of_empty, mul_top, if_neg],
... | lemma | ennreal.infi_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.continuous_at_const_mul",
"infi_eq_bot",
"infi_of_empty",
"is_empty_or_nonempty",
"mul_zero",
"not_and_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_mul_left {ι} [nonempty ι] {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
(h : a = ⊤ → (⨅ i, f i) = 0 → ∃ i, f i = 0) :
(⨅ i, a * f i) = a * ⨅ i, f i | infi_mul_left' h (λ _, ‹nonempty ι›) | lemma | ennreal.infi_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_mul_right' {ι} {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
(h : a = ⊤ → (⨅ i, f i) = 0 → ∃ i, f i = 0) (h0 : a = 0 → nonempty ι) :
(⨅ i, f i * a) = (⨅ i, f i) * a | by simpa only [mul_comm a] using infi_mul_left' h h0 | lemma | ennreal.infi_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"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_mul_right {ι} [nonempty ι] {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
(h : a = ⊤ → (⨅ i, f i) = 0 → ∃ i, f i = 0) :
(⨅ i, f i * a) = (⨅ i, f i) * a | infi_mul_right' h (λ _, ‹nonempty ι›) | lemma | ennreal.infi_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_map_infi {ι : Sort*} {x : ι → ℝ≥0∞} :
(infi x)⁻¹ = (⨆ i, (x i)⁻¹) | order_iso.inv_ennreal.map_infi x | lemma | ennreal.inv_map_infi | 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"
] | [
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_map_supr {ι : Sort*} {x : ι → ℝ≥0∞} :
(supr x)⁻¹ = (⨅ i, (x i)⁻¹) | order_iso.inv_ennreal.map_supr x | lemma | ennreal.inv_map_supr | 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"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_limsup {ι : Sort*} {x : ι → ℝ≥0∞} {l : filter ι} :
(limsup x l)⁻¹ = liminf (λ i, (x i)⁻¹) l | by simp only [limsup_eq_infi_supr, inv_map_infi, inv_map_supr, liminf_eq_supr_infi] | lemma | ennreal.inv_limsup | 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 | |
inv_liminf {ι : Sort*} {x : ι → ℝ≥0∞} {l : filter ι} :
(liminf x l)⁻¹ = limsup (λ i, (x i)⁻¹) l | by simp only [limsup_eq_infi_supr, inv_map_infi, inv_map_supr, liminf_eq_supr_infi] | lemma | ennreal.inv_liminf | 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 | |
tendsto_inv_iff {f : filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} :
tendsto (λ x, (m x)⁻¹) f (𝓝 a⁻¹) ↔ tendsto m f (𝓝 a) | ⟨λ h, by simpa only [inv_inv] using tendsto.inv h, tendsto.inv⟩ | lemma | ennreal.tendsto_inv_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"
] | [
"filter",
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.div {f : filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hma : tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) (hmb : tendsto mb f (𝓝 b)) (hb : b ≠ ⊤ ∨ a ≠ ⊤) :
tendsto (λa, ma a / mb a) f (𝓝 (a / b)) | by { apply tendsto.mul hma _ (ennreal.tendsto_inv_iff.2 hmb) _; simp [ha, hb] } | lemma | ennreal.tendsto.div | 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 | |
tendsto.const_div {f : filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hm : tendsto m f (𝓝 b)) (hb : b ≠ ⊤ ∨ a ≠ ⊤) : tendsto (λb, a / m b) f (𝓝 (a / b)) | by { apply tendsto.const_mul (ennreal.tendsto_inv_iff.2 hm), simp [hb] } | lemma | ennreal.tendsto.const_div | 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 | |
tendsto.div_const {f : filter α} {m : α → ℝ≥0∞} {a b : ℝ≥0∞}
(hm : tendsto m f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ 0) : tendsto (λx, m x / b) f (𝓝 (a / b)) | by { apply tendsto.mul_const hm, simp [ha] } | lemma | ennreal.tendsto.div_const | 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 | |
tendsto_inv_nat_nhds_zero : tendsto (λ n : ℕ, (n : ℝ≥0∞)⁻¹) at_top (𝓝 0) | ennreal.inv_top ▸ ennreal.tendsto_inv_iff.2 tendsto_nat_nhds_top | lemma | ennreal.tendsto_inv_nat_nhds_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.inv_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_add {ι : Sort*} {s : ι → ℝ≥0∞} [h : nonempty ι] : supr s + a = ⨆b, s b + a | monotone.map_supr_of_continuous_at' (continuous_at_id.add continuous_at_const) $
monotone_id.add monotone_const | lemma | ennreal.supr_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"
] | [
"continuous_at_const",
"monotone.map_supr_of_continuous_at'",
"monotone_const",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsupr_add' {ι : Sort*} {p : ι → Prop} (h : ∃ i, p i) {f : ι → ℝ≥0∞} :
(⨆ i (hi : p i), f i) + a = ⨆ i (hi : p i), f i + a | by { haveI : nonempty {i // p i} := nonempty_subtype.2 h, simp only [supr_subtype', supr_add] } | lemma | ennreal.bsupr_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"
] | [
"supr_subtype'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_bsupr' {ι : Sort*} {p : ι → Prop} (h : ∃ i, p i) {f : ι → ℝ≥0∞} :
a + (⨆ i (hi : p i), f i) = ⨆ i (hi : p i), a + f i | by simp only [add_comm a, bsupr_add' h] | lemma | ennreal.add_bsupr' | 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 | |
bsupr_add {ι} {s : set ι} (hs : s.nonempty) {f : ι → ℝ≥0∞} :
(⨆ i ∈ s, f i) + a = ⨆ i ∈ s, f i + a | bsupr_add' hs | lemma | ennreal.bsupr_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_bsupr {ι} {s : set ι} (hs : s.nonempty) {f : ι → ℝ≥0∞} :
a + (⨆ i ∈ s, f i) = ⨆ i ∈ s, a + f i | add_bsupr' hs | lemma | ennreal.add_bsupr | 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 | |
Sup_add {s : set ℝ≥0∞} (hs : s.nonempty) : Sup s + a = ⨆b∈s, b + a | by rw [Sup_eq_supr, bsupr_add hs] | lemma | ennreal.Sup_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"
] | [
"Sup_eq_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_supr {ι : Sort*} {s : ι → ℝ≥0∞} [nonempty ι] : a + supr s = ⨆b, a + s b | by rw [add_comm, supr_add]; simp [add_comm] | lemma | ennreal.add_supr | 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"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_add_supr_le {ι ι' : Sort*} [nonempty ι] [nonempty ι']
{f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i j, f i + g j ≤ a) :
supr f + supr g ≤ a | by simpa only [add_supr, supr_add] using supr₂_le h | lemma | ennreal.supr_add_supr_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"
] | [
"supr",
"supr₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsupr_add_bsupr_le' {ι ι'} {p : ι → Prop} {q : ι' → Prop} (hp : ∃ i, p i) (hq : ∃ j, q j)
{f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i (hi : p i) j (hj : q j), f i + g j ≤ a) :
(⨆ i (hi : p i), f i) + (⨆ j (hj : q j), g j) ≤ a | by { simp_rw [bsupr_add' hp, add_bsupr' hq], exact supr₂_le (λ i hi, supr₂_le (h i hi)) } | lemma | ennreal.bsupr_add_bsupr_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"
] | [
"supr₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsupr_add_bsupr_le {ι ι'} {s : set ι} {t : set ι'} (hs : s.nonempty) (ht : t.nonempty)
{f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ (i ∈ s) (j ∈ t), f i + g j ≤ a) :
(⨆ i ∈ s, f i) + (⨆ j ∈ t, g j) ≤ a | bsupr_add_bsupr_le' hs ht h | lemma | ennreal.bsupr_add_bsupr_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_add_supr {ι : Sort*} {f g : ι → ℝ≥0∞} (h : ∀i j, ∃k, f i + g j ≤ f k + g k) :
supr f + supr g = (⨆ a, f a + g a) | begin
casesI is_empty_or_nonempty ι,
{ simp only [supr_of_empty, bot_eq_zero, zero_add] },
{ refine le_antisymm _ (supr_le $ λ a, add_le_add (le_supr _ _) (le_supr _ _)),
refine supr_add_supr_le (λ i j, _),
rcases h i j with ⟨k, hk⟩,
exact le_supr_of_le k hk }
end | lemma | ennreal.supr_add_supr | 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_empty_or_nonempty",
"le_supr",
"le_supr_of_le",
"supr",
"supr_le",
"supr_of_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_add_supr_of_monotone {ι : Sort*} [semilattice_sup ι]
{f g : ι → ℝ≥0∞} (hf : monotone f) (hg : monotone g) :
supr f + supr g = (⨆ a, f a + g a) | supr_add_supr $ assume i j, ⟨i ⊔ j, add_le_add (hf $ le_sup_left) (hg $ le_sup_right)⟩ | lemma | ennreal.supr_add_supr_of_monotone | 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_sup_left",
"le_sup_right",
"monotone",
"semilattice_sup",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sum_supr_nat {α} {ι} [semilattice_sup ι] {s : finset α} {f : α → ι → ℝ≥0∞}
(hf : ∀a, monotone (f a)) :
∑ a in s, supr (f a) = (⨆ n, ∑ a in s, f a n) | begin
refine finset.induction_on s _ _,
{ simp, },
{ assume a s has ih,
simp only [finset.sum_insert has],
rw [ih, supr_add_supr_of_monotone (hf a)],
assume i j h,
exact (finset.sum_le_sum $ assume a ha, hf a h) }
end | lemma | ennreal.finset_sum_supr_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"
] | [
"finset",
"finset.induction_on",
"ih",
"monotone",
"semilattice_sup",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_supr {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : a * supr f = ⨆i, a * f i | begin
by_cases hf : ∀ i, f i = 0,
{ obtain rfl : f = (λ _, 0), from funext hf,
simp only [supr_zero_eq_zero, mul_zero] },
{ refine (monotone_id.const_mul' _).map_supr_of_continuous_at _ (mul_zero a),
refine ennreal.tendsto.const_mul tendsto_id (or.inl _),
exact mt supr_eq_zero.1 hf }
end | lemma | ennreal.mul_supr | 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.tendsto.const_mul",
"mul_zero",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_Sup {s : set ℝ≥0∞} {a : ℝ≥0∞} : a * Sup s = ⨆i∈s, a * i | by simp only [Sup_eq_supr, mul_supr] | lemma | ennreal.mul_Sup | 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_eq_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_mul {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : supr f * a = ⨆i, f i * a | by rw [mul_comm, mul_supr]; congr; funext; rw [mul_comm] | lemma | ennreal.supr_mul | 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"
] | [
"mul_comm",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_supr {ι : Sort*} {R} [has_smul R ℝ≥0∞] [is_scalar_tower R ℝ≥0∞ ℝ≥0∞]
(f : ι → ℝ≥0∞) (c : R) :
c • (⨆ i, f i) = ⨆ i, c • f i | by simp only [←smul_one_mul c (f _), ←smul_one_mul c (supr _), ennreal.mul_supr] | lemma | ennreal.smul_supr | 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.mul_supr",
"has_smul",
"is_scalar_tower",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_Sup {R} [has_smul R ℝ≥0∞] [is_scalar_tower R ℝ≥0∞ ℝ≥0∞]
(s : set ℝ≥0∞) (c : R) :
c • Sup s = ⨆ i ∈ s, c • i | by simp_rw [←smul_one_mul c (Sup _), ennreal.mul_Sup, smul_one_mul] | lemma | ennreal.smul_Sup | 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.mul_Sup",
"has_smul",
"is_scalar_tower",
"smul_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_div {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : supr f / a = ⨆i, f i / a | supr_mul | lemma | ennreal.supr_div | 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"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_coe_sub : ∀{b:ℝ≥0∞}, tendsto (λb:ℝ≥0∞, ↑r - b) (𝓝 b) (𝓝 (↑r - b)) | begin
refine forall_ennreal.2 ⟨λ a, _, _⟩,
{ simp [@nhds_coe a, tendsto_map'_iff, (∘), tendsto_coe, ← with_top.coe_sub],
exact tendsto_const_nhds.sub tendsto_id },
simp,
exact (tendsto.congr' (mem_of_superset (lt_mem_nhds $ @coe_lt_top r) $
by simp [le_of_lt] {contextual := tt})) tendsto_const_nhds
end | lemma | ennreal.tendsto_coe_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"
] | [
"lt_mem_nhds",
"tendsto_const_nhds",
"with_top.coe_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_supr {ι : Sort*} [nonempty ι] {b : ι → ℝ≥0∞} (hr : a < ⊤) :
a - (⨆i, b i) = (⨅i, a - b i) | let ⟨r, eq, _⟩ := lt_iff_exists_coe.mp hr in
have Inf ((λb, ↑r - b) '' range b) = ↑r - (⨆i, b i),
from is_glb.Inf_eq $ is_lub_supr.is_glb_of_tendsto
(assume x _ y _, tsub_le_tsub (le_refl (r : ℝ≥0∞)))
(range_nonempty _)
(ennreal.tendsto_coe_sub.comp (tendsto_id'.2 inf_le_left)),
by rw [eq, ←this]; simp [I... | lemma | ennreal.sub_supr | 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"
] | [
"Inf_image",
"inf_le_left",
"infi_range",
"is_glb.Inf_eq",
"le_rfl",
"tsub_le_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_countable_dense_no_zero_top :
∃ (s : set ℝ≥0∞), s.countable ∧ dense s ∧ 0 ∉ s ∧ ∞ ∉ s | begin
obtain ⟨s, s_count, s_dense, hs⟩ : ∃ s : set ℝ≥0∞, s.countable ∧ dense s ∧
(∀ x, is_bot x → x ∉ s) ∧ (∀ x, is_top x → x ∉ s) := exists_countable_dense_no_bot_top ℝ≥0∞,
exact ⟨s, s_count, s_dense, λ h, hs.1 0 (by simp) h, λ h, hs.2 ∞ (by simp) h⟩,
end | lemma | ennreal.exists_countable_dense_no_zero_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"
] | [
"dense",
"exists_countable_dense_no_bot_top",
"is_bot",
"is_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lt_add_of_lt_add {x y z : ℝ≥0∞} (h : x < y + z) (hy : y ≠ 0) (hz : z ≠ 0) :
∃ y' z', y' < y ∧ z' < z ∧ x < y' + z' | begin
haveI : ne_bot (𝓝[<] y) := nhds_within_Iio_self_ne_bot' ⟨0, pos_iff_ne_zero.2 hy⟩,
haveI : ne_bot (𝓝[<] z) := nhds_within_Iio_self_ne_bot' ⟨0, pos_iff_ne_zero.2 hz⟩,
have A : tendsto (λ (p : ℝ≥0∞ × ℝ≥0∞), p.1 + p.2) ((𝓝[<] y).prod (𝓝[<] z)) (𝓝 (y + z)),
{ apply tendsto.mono_left _ (filter.prod_mono n... | lemma | ennreal.exists_lt_add_of_lt_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"
] | [
"filter.prod_mem_prod",
"filter.prod_mono",
"nhds_prod_eq",
"nhds_within_Iio_self_ne_bot'",
"nhds_within_le_nhds",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_frequently_lt_of_liminf_ne_top
{ι : Type*} {l : filter ι} {x : ι → ℝ} (hx : liminf (λ n, ((x n).nnabs : ℝ≥0∞)) l ≠ ∞) :
∃ R, ∃ᶠ n in l, x n < R | begin
by_contra h,
simp_rw [not_exists, not_frequently, not_lt] at h,
refine hx (ennreal.eq_top_of_forall_nnreal_le $ λ r, le_Liminf_of_le (by is_bounded_default) _),
simp only [eventually_map, ennreal.coe_le_coe],
filter_upwards [h r] with i hi using hi.trans (le_abs_self (x i))
end | lemma | ennreal.exists_frequently_lt_of_liminf_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"
] | [
"by_contra",
"ennreal.coe_le_coe",
"ennreal.eq_top_of_forall_nnreal_le",
"filter",
"le_abs_self",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_frequently_lt_of_liminf_ne_top'
{ι : Type*} {l : filter ι} {x : ι → ℝ} (hx : liminf (λ n, ((x n).nnabs : ℝ≥0∞)) l ≠ ∞) :
∃ R, ∃ᶠ n in l, R < x n | begin
by_contra h,
simp_rw [not_exists, not_frequently, not_lt] at h,
refine hx (ennreal.eq_top_of_forall_nnreal_le $ λ r, le_Liminf_of_le (by is_bounded_default) _),
simp only [eventually_map, ennreal.coe_le_coe],
filter_upwards [h (-r)] with i hi using (le_neg.1 hi).trans (neg_le_abs_self _),
end | lemma | ennreal.exists_frequently_lt_of_liminf_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"
] | [
"by_contra",
"ennreal.coe_le_coe",
"ennreal.eq_top_of_forall_nnreal_le",
"filter",
"neg_le_abs_self",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_upcrossings_of_not_bounded_under
{ι : Type*} {l : filter ι} {x : ι → ℝ}
(hf : liminf (λ i, ((x i).nnabs : ℝ≥0∞)) l ≠ ∞)
(hbdd : ¬ is_bounded_under (≤) l (λ i, |x i|)) :
∃ a b : ℚ, a < b ∧ (∃ᶠ i in l, x i < a) ∧ (∃ᶠ i in l, ↑b < x i) | begin
rw [is_bounded_under_le_abs, not_and_distrib] at hbdd,
obtain hbdd | hbdd := hbdd,
{ obtain ⟨R, hR⟩ := exists_frequently_lt_of_liminf_ne_top hf,
obtain ⟨q, hq⟩ := exists_rat_gt R,
refine ⟨q, q + 1, (lt_add_iff_pos_right _).2 zero_lt_one, _, _⟩,
{ refine λ hcon, hR _,
filter_upwards [hcon] ... | lemma | ennreal.exists_upcrossings_of_not_bounded_under | 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"
] | [
"exists_prop",
"exists_rat_gt",
"exists_rat_lt",
"filter",
"ge_iff_le",
"not_and_distrib",
"not_exists",
"not_forall",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_coe {f : α → ℝ≥0} {r : ℝ≥0} :
has_sum (λa, (f a : ℝ≥0∞)) ↑r ↔ has_sum f r | have (λs:finset α, ∑ a in s, ↑(f a)) = (coe : ℝ≥0 → ℝ≥0∞) ∘ (λs:finset α, ∑ a in s, f a),
from funext $ assume s, ennreal.coe_finset_sum.symm,
by unfold has_sum; rw [this, tendsto_coe] | lemma | ennreal.has_sum_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"
] | [
"finset",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_coe_eq {f : α → ℝ≥0} (h : has_sum f r) : ∑'a, (f a : ℝ≥0∞) = r | (ennreal.has_sum_coe.2 h).tsum_eq | lemma | ennreal.tsum_coe_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"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_tsum {f : α → ℝ≥0} : summable f → ↑(tsum f) = ∑'a, (f a : ℝ≥0∞) | | ⟨r, hr⟩ := by rw [hr.tsum_eq, ennreal.tsum_coe_eq hr] | lemma | ennreal.coe_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.tsum_coe_eq",
"summable",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum : has_sum f (⨆s:finset α, ∑ a in s, f a) | tendsto_at_top_supr $ λ s t, finset.sum_le_sum_of_subset | lemma | ennreal.has_sum | 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",
"has_sum",
"tendsto_at_top_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable : summable f | ⟨_, ennreal.has_sum⟩ | lemma | ennreal.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"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_coe_ne_top_iff_summable {f : β → ℝ≥0} :
∑' b, (f b:ℝ≥0∞) ≠ ∞ ↔ summable f | begin
refine ⟨λ h, _, λ h, ennreal.coe_tsum h ▸ ennreal.coe_ne_top⟩,
lift (∑' b, (f b:ℝ≥0∞)) to ℝ≥0 using h with a ha,
refine ⟨a, ennreal.has_sum_coe.1 _⟩,
rw ha,
exact ennreal.summable.has_sum
end | lemma | ennreal.tsum_coe_ne_top_iff_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"
] | [
"ennreal.coe_tsum",
"lift",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_supr_sum : ∑'a, f a = (⨆s:finset α, ∑ a in s, f a) | ennreal.has_sum.tsum_eq | lemma | ennreal.tsum_eq_supr_sum | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_supr_sum' {ι : Type*} (s : ι → finset α) (hs : ∀ t, ∃ i, t ⊆ s i) :
∑' a, f a = ⨆ i, ∑ a in s i, f a | begin
rw [ennreal.tsum_eq_supr_sum],
symmetry,
change (⨆i:ι, (λ t : finset α, ∑ a in t, f a) (s i)) = ⨆s:finset α, ∑ a in s, f a,
exact (finset.sum_mono_set f).supr_comp_eq hs
end | lemma | ennreal.tsum_eq_supr_sum' | 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_sum",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_sigma {β : α → Type*} (f : Πa, β a → ℝ≥0∞) :
∑'p:Σa, β a, f p.1 p.2 = ∑'a b, f a b | tsum_sigma' (assume b, ennreal.summable) ennreal.summable | lemma | ennreal.tsum_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.summable",
"tsum_sigma",
"tsum_sigma'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_sigma' {β : α → Type*} (f : (Σ a, β a) → ℝ≥0∞) :
∑'p:(Σa, β a), f p = ∑'a b, f ⟨a, b⟩ | tsum_sigma' (assume b, ennreal.summable) ennreal.summable | lemma | ennreal.tsum_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.summable",
"tsum_sigma'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_prod {f : α → β → ℝ≥0∞} : ∑' p : α × β, f p.1 p.2 = ∑' a b, f a b | tsum_prod' ennreal.summable $ λ _, ennreal.summable | lemma | ennreal.tsum_prod | 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_prod",
"tsum_prod'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_prod' {f : α × β → ℝ≥0∞} : ∑' p : α × β, f p = ∑' a b, f (a, b) | tsum_prod' ennreal.summable $ λ _, ennreal.summable | lemma | ennreal.tsum_prod' | 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_prod'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_comm {f : α → β → ℝ≥0∞} : ∑'a, ∑'b, f a b = ∑'b, ∑'a, f a b | tsum_comm' ennreal.summable (λ _, ennreal.summable) (λ _, ennreal.summable) | lemma | ennreal.tsum_comm | 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_comm",
"tsum_comm'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_add : ∑'a, (f a + g a) = (∑'a, f a) + (∑'a, g a) | tsum_add ennreal.summable ennreal.summable | lemma | ennreal.tsum_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.summable",
"tsum_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_le_tsum (h : ∀a, f a ≤ g a) : ∑'a, f a ≤ ∑'a, g a | tsum_le_tsum h ennreal.summable ennreal.summable | lemma | ennreal.tsum_le_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.summable",
"tsum_le_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le_tsum {f : α → ℝ≥0∞} (s : finset α) : ∑ x in s, f x ≤ ∑' x, f x | sum_le_tsum s (λ x hx, zero_le _) ennreal.summable | lemma | ennreal.sum_le_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.summable",
"finset",
"sum_le_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_supr_nat' {f : ℕ → ℝ≥0∞} {N : ℕ → ℕ} (hN : tendsto N at_top at_top) :
∑'i:ℕ, f i = (⨆i:ℕ, ∑ a in finset.range (N i), f a) | ennreal.tsum_eq_supr_sum' _ $ λ t,
let ⟨n, hn⟩ := t.exists_nat_subset_range,
⟨k, _, hk⟩ := exists_le_of_tendsto_at_top hN 0 n in
⟨k, finset.subset.trans hn (finset.range_mono hk)⟩ | lemma | ennreal.tsum_eq_supr_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_sum'",
"finset.range",
"finset.range_mono",
"finset.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_supr_nat {f : ℕ → ℝ≥0∞} :
∑'i:ℕ, f i = (⨆i:ℕ, ∑ a in finset.range i, f a) | ennreal.tsum_eq_supr_sum' _ finset.exists_nat_subset_range | lemma | ennreal.tsum_eq_supr_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_sum'",
"finset.exists_nat_subset_range",
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_liminf_sum_nat {f : ℕ → ℝ≥0∞} :
∑' i, f i = liminf (λ n, ∑ i in finset.range n, f i) at_top | begin
rw [ennreal.tsum_eq_supr_nat, filter.liminf_eq_supr_infi_of_nat],
congr,
refine funext (λ n, le_antisymm _ _),
{ refine le_infi₂ (λ i hi, finset.sum_le_sum_of_subset_of_nonneg _ (λ _ _ _, zero_le _)),
simpa only [finset.range_subset, add_le_add_iff_right] using hi, },
{ refine le_trans (infi_le _ n)... | lemma | ennreal.tsum_eq_liminf_sum_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",
"filter.liminf_eq_supr_infi_of_nat",
"finset.range",
"finset.range_subset",
"infi_le",
"le_infi₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsum (a : α) : f a ≤ ∑'a, f a | le_tsum' ennreal.summable a | lemma | ennreal.le_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.summable",
"le_tsum",
"le_tsum'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_zero : ∑' i, f i = 0 ↔ ∀ i, f i = 0 | ⟨λ h i, nonpos_iff_eq_zero.1 $ h ▸ ennreal.le_tsum i, λ h, by simp [h]⟩ | lemma | ennreal.tsum_eq_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.le_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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