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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edist_congr_right {x y z : α} (h : edist x y = 0) : edist x z = edist y z | begin
apply le_antisymm,
{ rw [←zero_add (edist y z), ←h],
apply edist_triangle, },
{ rw edist_comm at h,
rw [←zero_add (edist x z), ←h],
apply edist_triangle, },
end | lemma | edist_congr_right | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_congr_left {x y z : α} (h : edist x y = 0) : edist z x = edist z y | by { rw [edist_comm z x, edist_comm z y], apply edist_congr_right h, } | lemma | edist_congr_left | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_congr_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_triangle4 (x y z t : α) :
edist x t ≤ edist x y + edist y z + edist z t | calc
edist x t ≤ edist x z + edist z t : edist_triangle x z t
... ≤ (edist x y + edist y z) + edist z t : add_le_add_right (edist_triangle x y z) _ | lemma | edist_triangle4 | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_Ico_sum_edist (f : ℕ → α) {m n} (h : m ≤ n) :
edist (f m) (f n) ≤ ∑ i in finset.Ico m n, edist (f i) (f (i + 1)) | begin
revert n,
refine nat.le_induction _ _,
{ simp only [finset.sum_empty, finset.Ico_self, edist_self],
-- TODO: Why doesn't Lean close this goal automatically? `exact le_rfl` fails too.
exact le_refl (0:ℝ≥0∞) },
{ assume n hn hrec,
calc edist (f m) (f (n+1)) ≤ edist (f m) (f n) + edist (f n) (f (... | lemma | edist_le_Ico_sum_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"finset.Ico",
"finset.Ico_self",
"le_rfl",
"nat.Ico_succ_right_eq_insert_Ico",
"nat.le_induction"
] | The triangle (polygon) inequality for sequences of points; `finset.Ico` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_le_range_sum_edist (f : ℕ → α) (n : ℕ) :
edist (f 0) (f n) ≤ ∑ i in finset.range n, edist (f i) (f (i + 1)) | nat.Ico_zero_eq_range ▸ edist_le_Ico_sum_edist f (nat.zero_le n) | lemma | edist_le_range_sum_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_le_Ico_sum_edist",
"finset.range",
"nat.Ico_zero_eq_range"
] | The triangle (polygon) inequality for sequences of points; `finset.range` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_le_Ico_sum_of_edist_le {f : ℕ → α} {m n} (hmn : m ≤ n)
{d : ℕ → ℝ≥0∞} (hd : ∀ {k}, m ≤ k → k < n → edist (f k) (f (k + 1)) ≤ d k) :
edist (f m) (f n) ≤ ∑ i in finset.Ico m n, d i | le_trans (edist_le_Ico_sum_edist f hmn) $
finset.sum_le_sum $ λ k hk, hd (finset.mem_Ico.1 hk).1 (finset.mem_Ico.1 hk).2 | lemma | edist_le_Ico_sum_of_edist_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_le_Ico_sum_edist",
"finset.Ico"
] | A version of `edist_le_Ico_sum_edist` with each intermediate distance replaced
with an upper estimate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_le_range_sum_of_edist_le {f : ℕ → α} (n : ℕ)
{d : ℕ → ℝ≥0∞} (hd : ∀ {k}, k < n → edist (f k) (f (k + 1)) ≤ d k) :
edist (f 0) (f n) ≤ ∑ i in finset.range n, d i | nat.Ico_zero_eq_range ▸ edist_le_Ico_sum_of_edist_le (zero_le n) (λ _ _, hd) | lemma | edist_le_range_sum_of_edist_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_le_Ico_sum_of_edist_le",
"finset.range",
"nat.Ico_zero_eq_range"
] | A version of `edist_le_range_sum_edist` with each intermediate distance replaced
with an upper estimate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_pseudoedist :
𝓤 α = ⨅ ε>0, 𝓟 {p:α×α | edist p.1 p.2 < ε} | pseudo_emetric_space.uniformity_edist | theorem | uniformity_pseudoedist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | Reformulation of the uniform structure in terms of the extended distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space_edist : ‹pseudo_emetric_space α›.to_uniform_space =
uniform_space_of_edist edist edist_self edist_comm edist_triangle | uniform_space_eq uniformity_pseudoedist | theorem | uniform_space_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"uniform_space_eq",
"uniform_space_of_edist",
"uniformity_pseudoedist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_edist :
(𝓤 α).has_basis (λ ε : ℝ≥0∞, 0 < ε) (λ ε, {p:α×α | edist p.1 p.2 < ε}) | (@uniform_space_edist α _).symm ▸ uniform_space.has_basis_of_fun ⟨1, one_pos⟩ _ _ _ _ _ | theorem | uniformity_basis_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"uniform_space.has_basis_of_fun",
"uniform_space_edist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_uniformity_edist {s : set (α×α)} :
s ∈ 𝓤 α ↔ (∃ε>0, ∀{a b:α}, edist a b < ε → (a, b) ∈ s) | uniformity_basis_edist.mem_uniformity_iff | theorem | mem_uniformity_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | Characterization of the elements of the uniformity in terms of the extended distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emetric.mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ≥0∞}
(hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x (hx : p x), f x ≤ ε) :
(𝓤 α).has_basis p (λ x, {p:α×α | edist p.1 p.2 < f x}) | begin
refine ⟨λ s, uniformity_basis_edist.mem_iff.trans _⟩,
split,
{ rintros ⟨ε, ε₀, hε⟩,
rcases hf ε ε₀ with ⟨i, hi, H⟩,
exact ⟨i, hi, λ x hx, hε $ lt_of_lt_of_le hx H⟩ },
{ exact λ ⟨i, hi, H⟩, ⟨f i, hf₀ i hi, H⟩ }
end | theorem | emetric.mk_uniformity_basis | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | Given `f : β → ℝ≥0∞`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
For specific bases see `uniformity_basis_edist`, `uniformity_basis_edist'`,
`uniformity_basis_edist_nnreal`, and `uniformity_basis_edist_inv_nat`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emetric.mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ≥0∞}
(hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x (hx : p x), f x ≤ ε) :
(𝓤 α).has_basis p (λ x, {p:α×α | edist p.1 p.2 ≤ f x}) | begin
refine ⟨λ s, uniformity_basis_edist.mem_iff.trans _⟩,
split,
{ rintros ⟨ε, ε₀, hε⟩,
rcases exists_between ε₀ with ⟨ε', hε'⟩,
rcases hf ε' hε'.1 with ⟨i, hi, H⟩,
exact ⟨i, hi, λ x hx, hε $ lt_of_le_of_lt (le_trans hx H) hε'.2⟩ },
{ exact λ ⟨i, hi, H⟩, ⟨f i, hf₀ i hi, λ x hx, H (le_of_lt hx)⟩ }
... | theorem | emetric.mk_uniformity_basis_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"exists_between"
] | Given `f : β → ℝ≥0∞`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then closed `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
For specific bases see `uniformity_basis_edist_le` and `uniformity_basis_edist_le'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_basis_edist_le :
(𝓤 α).has_basis (λ ε : ℝ≥0∞, 0 < ε) (λ ε, {p:α×α | edist p.1 p.2 ≤ ε}) | emetric.mk_uniformity_basis_le (λ _, id) (λ ε ε₀, ⟨ε, ε₀, le_refl ε⟩) | theorem | uniformity_basis_edist_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric.mk_uniformity_basis_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_edist' (ε' : ℝ≥0∞) (hε' : 0 < ε') :
(𝓤 α).has_basis (λ ε : ℝ≥0∞, ε ∈ Ioo 0 ε') (λ ε, {p:α×α | edist p.1 p.2 < ε}) | emetric.mk_uniformity_basis (λ _, and.left)
(λ ε ε₀, let ⟨δ, hδ⟩ := exists_between hε' in
⟨min ε δ, ⟨lt_min ε₀ hδ.1, lt_of_le_of_lt (min_le_right _ _) hδ.2⟩, min_le_left _ _⟩) | theorem | uniformity_basis_edist' | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric.mk_uniformity_basis",
"exists_between"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_edist_le' (ε' : ℝ≥0∞) (hε' : 0 < ε') :
(𝓤 α).has_basis (λ ε : ℝ≥0∞, ε ∈ Ioo 0 ε') (λ ε, {p:α×α | edist p.1 p.2 ≤ ε}) | emetric.mk_uniformity_basis_le (λ _, and.left)
(λ ε ε₀, let ⟨δ, hδ⟩ := exists_between hε' in
⟨min ε δ, ⟨lt_min ε₀ hδ.1, lt_of_le_of_lt (min_le_right _ _) hδ.2⟩, min_le_left _ _⟩) | theorem | uniformity_basis_edist_le' | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric.mk_uniformity_basis_le",
"exists_between"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_edist_nnreal :
(𝓤 α).has_basis (λ ε : ℝ≥0, 0 < ε) (λ ε, {p:α×α | edist p.1 p.2 < ε}) | emetric.mk_uniformity_basis (λ _, ennreal.coe_pos.2)
(λ ε ε₀, let ⟨δ, hδ⟩ := ennreal.lt_iff_exists_nnreal_btwn.1 ε₀ in
⟨δ, ennreal.coe_pos.1 hδ.1, le_of_lt hδ.2⟩) | theorem | uniformity_basis_edist_nnreal | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric.mk_uniformity_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_edist_nnreal_le :
(𝓤 α).has_basis (λ ε : ℝ≥0, 0 < ε) (λ ε, {p:α×α | edist p.1 p.2 ≤ ε}) | emetric.mk_uniformity_basis_le (λ _, ennreal.coe_pos.2)
(λ ε ε₀, let ⟨δ, hδ⟩ := ennreal.lt_iff_exists_nnreal_btwn.1 ε₀ in
⟨δ, ennreal.coe_pos.1 hδ.1, le_of_lt hδ.2⟩) | theorem | uniformity_basis_edist_nnreal_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric.mk_uniformity_basis_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_edist_inv_nat :
(𝓤 α).has_basis (λ _, true) (λ n:ℕ, {p:α×α | edist p.1 p.2 < (↑n)⁻¹}) | emetric.mk_uniformity_basis
(λ n _, ennreal.inv_pos.2 $ ennreal.nat_ne_top n)
(λ ε ε₀, let ⟨n, hn⟩ := ennreal.exists_inv_nat_lt (ne_of_gt ε₀) in ⟨n, trivial, le_of_lt hn⟩) | theorem | uniformity_basis_edist_inv_nat | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric.mk_uniformity_basis",
"ennreal.exists_inv_nat_lt",
"ennreal.nat_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_edist_inv_two_pow :
(𝓤 α).has_basis (λ _, true) (λ n:ℕ, {p:α×α | edist p.1 p.2 < 2⁻¹ ^ n}) | emetric.mk_uniformity_basis
(λ n _, ennreal.pow_pos (ennreal.inv_pos.2 ennreal.two_ne_top) _)
(λ ε ε₀, let ⟨n, hn⟩ := ennreal.exists_inv_two_pow_lt (ne_of_gt ε₀) in ⟨n, trivial, le_of_lt hn⟩) | theorem | uniformity_basis_edist_inv_two_pow | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric.mk_uniformity_basis",
"ennreal.exists_inv_two_pow_lt",
"ennreal.pow_pos",
"ennreal.two_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_mem_uniformity {ε:ℝ≥0∞} (ε0 : 0 < ε) :
{p:α×α | edist p.1 p.2 < ε} ∈ 𝓤 α | mem_uniformity_edist.2 ⟨ε, ε0, λ a b, id⟩ | theorem | edist_mem_uniformity | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | Fixed size neighborhoods of the diagonal belong to the uniform structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_on_iff [pseudo_emetric_space β] {f : α → β} {s : set α} :
uniform_continuous_on f s ↔ ∀ ε > 0, ∃ δ > 0,
∀ {a b ∈ s}, edist a b < δ → edist (f a) (f b) < ε | uniformity_basis_edist.uniform_continuous_on_iff uniformity_basis_edist | theorem | emetric.uniform_continuous_on_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"pseudo_emetric_space",
"uniform_continuous_on",
"uniformity_basis_edist"
] | ε-δ characterization of uniform continuity on a set for pseudoemetric spaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_iff [pseudo_emetric_space β] {f : α → β} :
uniform_continuous f ↔ ∀ ε > 0, ∃ δ > 0,
∀{a b:α}, edist a b < δ → edist (f a) (f b) < ε | uniformity_basis_edist.uniform_continuous_iff uniformity_basis_edist | theorem | emetric.uniform_continuous_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"pseudo_emetric_space",
"uniform_continuous",
"uniform_continuous_iff",
"uniformity_basis_edist"
] | ε-δ characterization of uniform continuity on pseudoemetric spaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_iff [pseudo_emetric_space β] {f : α → β} :
uniform_embedding f ↔ function.injective f ∧ uniform_continuous f ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ | begin
simp only [uniformity_basis_edist.uniform_embedding_iff uniformity_basis_edist, exists_prop],
refl
end | theorem | emetric.uniform_embedding_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"exists_prop",
"pseudo_emetric_space",
"uniform_continuous",
"uniform_embedding",
"uniformity_basis_edist"
] | ε-δ characterization of uniform embeddings on pseudoemetric spaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
controlled_of_uniform_embedding [pseudo_emetric_space β] {f : α → β} :
uniform_embedding f →
(∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, edist a b < δ → edist (f a) (f b) < ε) ∧
(∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ) | λ h, ⟨uniform_continuous_iff.1 (uniform_embedding_iff.1 h).2.1, (uniform_embedding_iff.1 h).2.2⟩ | theorem | emetric.controlled_of_uniform_embedding | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"pseudo_emetric_space",
"uniform_embedding"
] | If a map between pseudoemetric spaces is a uniform embedding then the edistance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_iff {f : filter α} :
cauchy f ↔ f ≠ ⊥ ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x y ∈ t, edist x y < ε | by rw ← ne_bot_iff; exact uniformity_basis_edist.cauchy_iff | lemma | emetric.cauchy_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"cauchy",
"cauchy_iff",
"filter"
] | ε-δ characterization of Cauchy sequences on pseudoemetric spaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_of_convergent_controlled_sequences (B : ℕ → ℝ≥0∞) (hB : ∀n, 0 < B n)
(H : ∀u : ℕ → α, (∀N n m : ℕ, N ≤ n → N ≤ m → edist (u n) (u m) < B N) →
∃x, tendsto u at_top (𝓝 x)) :
complete_space α | uniform_space.complete_of_convergent_controlled_sequences
(λ n, {p:α×α | edist p.1 p.2 < B n}) (λ n, edist_mem_uniformity $ hB n) H | theorem | emetric.complete_of_convergent_controlled_sequences | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"complete_space",
"edist_mem_uniformity",
"uniform_space.complete_of_convergent_controlled_sequences"
] | A very useful criterion to show that a space is complete is to show that all sequences
which satisfy a bound of the form `edist (u n) (u m) < B N` for all `n m ≥ N` are
converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to
`0`, which makes it possible to use arguments of converging... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_of_cauchy_seq_tendsto :
(∀ u : ℕ → α, cauchy_seq u → ∃a, tendsto u at_top (𝓝 a)) → complete_space α | uniform_space.complete_of_cauchy_seq_tendsto | theorem | emetric.complete_of_cauchy_seq_tendsto | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"cauchy_seq",
"complete_space",
"uniform_space.complete_of_cauchy_seq_tendsto"
] | A sequentially complete pseudoemetric space is complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on_iff {ι : Type*} [topological_space β]
{F : ι → β → α} {f : β → α} {p : filter ι} {s : set β} :
tendsto_locally_uniformly_on F f p s ↔
∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, edist (f y) (F n y) < ε | begin
refine ⟨λ H ε hε, H _ (edist_mem_uniformity hε), λ H u hu x hx, _⟩,
rcases mem_uniformity_edist.1 hu with ⟨ε, εpos, hε⟩,
rcases H ε εpos x hx with ⟨t, ht, Ht⟩,
exact ⟨t, ht, Ht.mono (λ n hs x hx, hε (hs x hx))⟩
end | lemma | emetric.tendsto_locally_uniformly_on_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_mem_uniformity",
"filter",
"tendsto_locally_uniformly_on",
"topological_space"
] | Expressing locally uniform convergence on a set using `edist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_iff {ι : Type*}
{F : ι → β → α} {f : β → α} {p : filter ι} {s : set β} :
tendsto_uniformly_on F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, edist (f x) (F n x) < ε | begin
refine ⟨λ H ε hε, H _ (edist_mem_uniformity hε), λ H u hu, _⟩,
rcases mem_uniformity_edist.1 hu with ⟨ε, εpos, hε⟩,
exact (H ε εpos).mono (λ n hs x hx, hε (hs x hx))
end | lemma | emetric.tendsto_uniformly_on_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_mem_uniformity",
"filter",
"tendsto_uniformly_on"
] | Expressing uniform convergence on a set using `edist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_iff {ι : Type*} [topological_space β]
{F : ι → β → α} {f : β → α} {p : filter ι} :
tendsto_locally_uniformly F f p ↔
∀ ε > 0, ∀ (x : β), ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, edist (f y) (F n y) < ε | by simp only [← tendsto_locally_uniformly_on_univ, tendsto_locally_uniformly_on_iff,
mem_univ, forall_const, exists_prop, nhds_within_univ] | lemma | emetric.tendsto_locally_uniformly_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"exists_prop",
"filter",
"forall_const",
"nhds_within_univ",
"tendsto_locally_uniformly",
"tendsto_locally_uniformly_on_univ",
"topological_space"
] | Expressing locally uniform convergence using `edist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_iff {ι : Type*}
{F : ι → β → α} {f : β → α} {p : filter ι} :
tendsto_uniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, edist (f x) (F n x) < ε | by simp only [← tendsto_uniformly_on_univ, tendsto_uniformly_on_iff, mem_univ, forall_const] | lemma | emetric.tendsto_uniformly_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"filter",
"forall_const",
"tendsto_uniformly",
"tendsto_uniformly_on_univ"
] | Expressing uniform convergence using `edist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_emetric_space.replace_uniformity {α} [U : uniform_space α] (m : pseudo_emetric_space α)
(H : 𝓤[U] = 𝓤[pseudo_emetric_space.to_uniform_space]) :
pseudo_emetric_space α | { edist := @edist _ m.to_has_edist,
edist_self := edist_self,
edist_comm := edist_comm,
edist_triangle := edist_triangle,
to_uniform_space := U,
uniformity_edist := H.trans (@pseudo_emetric_space.uniformity_edist α _) } | def | pseudo_emetric_space.replace_uniformity | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"pseudo_emetric_space",
"uniform_space",
"uniformity_edist"
] | Auxiliary function to replace the uniformity on a pseudoemetric space with
a uniformity which is equal to the original one, but maybe not defeq.
This is useful if one wants to construct a pseudoemetric space with a
specified uniformity. See Note [forgetful inheritance] explaining why having definitionally
the right uni... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_emetric_space.induced {α β} (f : α → β)
(m : pseudo_emetric_space β) : pseudo_emetric_space α | { edist := λ x y, edist (f x) (f y),
edist_self := λ x, edist_self _,
edist_comm := λ x y, edist_comm _ _,
edist_triangle := λ x y z, edist_triangle _ _ _,
to_uniform_space := uniform_space.comap f m.to_uniform_space,
uniformity_edist := (uniformity_basis_edist.comap... | def | pseudo_emetric_space.induced | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"pseudo_emetric_space",
"uniform_space.comap",
"uniformity_edist"
] | The extended pseudometric induced by a function taking values in a pseudoemetric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype.edist_eq {p : α → Prop} (x y : subtype p) : edist x y = edist (x : α) y | rfl | theorem | subtype.edist_eq | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | The extended psuedodistance on a subset of a pseudoemetric space is the restriction of
the original pseudodistance, by definition | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_unop (x y : αᵐᵒᵖ) : edist (unop x) (unop y) = edist x y | rfl | theorem | mul_opposite.edist_unop | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_op (x y : α) : edist (op x) (op y) = edist x y | rfl | theorem | mul_opposite.edist_op | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.edist_eq (x y : ulift α) : edist x y = edist x.down y.down | rfl | lemma | ulift.edist_eq | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.edist_up_up (x y : α) : edist (ulift.up x) (ulift.up y) = edist x y | rfl | lemma | ulift.edist_up_up | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.pseudo_emetric_space_max [pseudo_emetric_space β] : pseudo_emetric_space (α × β) | { edist := λ x y, edist x.1 y.1 ⊔ edist x.2 y.2,
edist_self := λ x, by simp,
edist_comm := λ x y, by simp [edist_comm],
edist_triangle := λ x y z, max_le
(le_trans (edist_triangle _ _ _) (add_le_add (le_max_left _ _) (le_max_left _ _)))
(le_trans (edist_triangle _ _ _) (add_le_add (le_max_right _ _) (le_m... | instance | prod.pseudo_emetric_space_max | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"infi_inf_eq",
"max_lt_iff",
"pseudo_emetric_space",
"uniformity_edist"
] | The product of two pseudoemetric spaces, with the max distance, is an extended
pseudometric spaces. We make sure that the uniform structure thus constructed is the one
corresponding to the product of uniform spaces, to avoid diamond problems. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.edist_eq [pseudo_emetric_space β] (x y : α × β) :
edist x y = max (edist x.1 y.1) (edist x.2 y.2) | rfl | lemma | prod.edist_eq | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_emetric_space_pi [∀b, pseudo_emetric_space (π b)] :
pseudo_emetric_space (Πb, π b) | { edist := λ f g, finset.sup univ (λb, edist (f b) (g b)),
edist_self := assume f, bot_unique $ finset.sup_le $ by simp,
edist_comm := assume f g, by unfold edist; congr; funext a; exact edist_comm _ _,
edist_triangle := assume f g h,
begin
simp only [finset.sup_le_iff],
assume b hb,
exact l... | instance | pseudo_emetric_space_pi | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"Pi.uniform_space",
"Pi.uniformity",
"bot_unique",
"finset.sup",
"finset.sup_le_iff",
"gt_iff_lt",
"infi_comm",
"pseudo_emetric_space",
"set.ext_iff",
"uniformity_edist"
] | The product of a finite number of pseudoemetric spaces, with the max distance, is still
a pseudoemetric space.
This construction would also work for infinite products, but it would not give rise
to the product topology. Hence, we only formalize it in the good situation of finitely many
spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_pi_def [Π b, pseudo_emetric_space (π b)] (f g : Π b, π b) :
edist f g = finset.sup univ (λb, edist (f b) (g b)) | rfl | lemma | edist_pi_def | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"finset.sup",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_pi_edist [Π b, pseudo_emetric_space (π b)] (f g : Π b, π b) (b : β) :
edist (f b) (g b) ≤ edist f g | finset.le_sup (finset.mem_univ b) | lemma | edist_le_pi_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"finset.le_sup",
"finset.mem_univ",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_pi_le_iff [Π b, pseudo_emetric_space (π b)] {f g : Π b, π b} {d : ℝ≥0∞} :
edist f g ≤ d ↔ ∀ b, edist (f b) (g b) ≤ d | finset.sup_le_iff.trans $ by simp only [finset.mem_univ, forall_const] | lemma | edist_pi_le_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"finset.mem_univ",
"forall_const",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_pi_const_le (a b : α) : edist (λ _ : β, a) (λ _, b) ≤ edist a b | edist_pi_le_iff.2 $ λ _, le_rfl | lemma | edist_pi_const_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_pi_const [nonempty β] (a b : α) : edist (λ x : β, a) (λ _, b) = edist a b | finset.sup_const univ_nonempty (edist a b) | lemma | edist_pi_const | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"finset.sup_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball (x : α) (ε : ℝ≥0∞) : set α | {y | edist y x < ε} | def | emetric.ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | `emetric.ball x ε` is the set of all points `y` with `edist y x < ε` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ball : y ∈ ball x ε ↔ edist y x < ε | iff.rfl | theorem | emetric.mem_ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball' : y ∈ ball x ε ↔ edist x y < ε | by rw [edist_comm, mem_ball] | theorem | emetric.mem_ball' | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball (x : α) (ε : ℝ≥0∞) | {y | edist y x ≤ ε} | def | emetric.closed_ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | `emetric.closed_ball x ε` is the set of all points `y` with `edist y x ≤ ε` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closed_ball : y ∈ closed_ball x ε ↔ edist y x ≤ ε | iff.rfl | theorem | emetric.mem_closed_ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball' : y ∈ closed_ball x ε ↔ edist x y ≤ ε | by rw [edist_comm, mem_closed_ball] | theorem | emetric.mem_closed_ball' | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_top (x : α) : closed_ball x ∞ = univ | eq_univ_of_forall $ λ y, le_top | theorem | emetric.closed_ball_top | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_subset_closed_ball : ball x ε ⊆ closed_ball x ε | assume y hy, le_of_lt hy | theorem | emetric.ball_subset_closed_ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε | lt_of_le_of_lt (zero_le _) hy | theorem | emetric.pos_of_mem_ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_self (h : 0 < ε) : x ∈ ball x ε | show edist x x < ε, by rw edist_self; assumption | theorem | emetric.mem_ball_self | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball_self : x ∈ closed_ball x ε | show edist x x ≤ ε, by rw edist_self; exact bot_le | theorem | emetric.mem_closed_ball_self | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"bot_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_disjoint (h : ε₁ + ε₂ ≤ edist x y) : disjoint (ball x ε₁) (ball y ε₂) | set.disjoint_left.mpr $ λ z h₁ h₂,
(edist_triangle_left x y z).not_lt $ (ennreal.add_lt_add h₁ h₂).trans_le h | theorem | emetric.ball_disjoint | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"disjoint",
"edist_triangle_left",
"ennreal.add_lt_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_subset (h : edist x y + ε₁ ≤ ε₂) (h' : edist x y ≠ ∞) : ball x ε₁ ⊆ ball y ε₂ | λ z zx, calc
edist z y ≤ edist z x + edist x y : edist_triangle _ _ _
... = edist x y + edist z x : add_comm _ _
... < edist x y + ε₁ : ennreal.add_lt_add_left h' zx
... ≤ ε₂ : h | theorem | emetric.ball_subset | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"ennreal.add_lt_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε | begin
have : 0 < ε - edist y x := by simpa using h,
refine ⟨ε - edist y x, this, ball_subset _ (ne_top_of_lt h)⟩,
exact (add_tsub_cancel_of_le (mem_ball.mp h).le).le
end | theorem | emetric.exists_ball_subset_ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"add_tsub_cancel_of_le",
"ne_top_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_eq_empty_iff : ball x ε = ∅ ↔ ε = 0 | eq_empty_iff_forall_not_mem.trans
⟨λh, le_bot_iff.1 (le_of_not_gt (λ ε0, h _ (mem_ball_self ε0))),
λε0 y h, not_lt_of_le (le_of_eq ε0) (pos_of_mem_ball h)⟩ | theorem | emetric.ball_eq_empty_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"not_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_connected_set_of_closed_ball_subset (x : α) (s : set α) :
ord_connected {r | closed_ball x r ⊆ s} | ⟨λ r₁ hr₁ r₂ hr₂ r hr, (closed_ball_subset_closed_ball hr.2).trans hr₂⟩ | lemma | emetric.ord_connected_set_of_closed_ball_subset | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_connected_set_of_ball_subset (x : α) (s : set α) :
ord_connected {r | ball x r ⊆ s} | ⟨λ r₁ hr₁ r₂ hr₂ r hr, (ball_subset_ball hr.2).trans hr₂⟩ | lemma | emetric.ord_connected_set_of_ball_subset | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_lt_top_setoid : setoid α | { r := λ x y, edist x y < ⊤,
iseqv := ⟨λ x, by { rw edist_self, exact ennreal.coe_lt_top },
λ x y h, by rwa edist_comm,
λ x y z hxy hyz, lt_of_le_of_lt (edist_triangle x y z) (ennreal.add_lt_top.2 ⟨hxy, hyz⟩)⟩ } | def | emetric.edist_lt_top_setoid | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"ennreal.coe_lt_top"
] | Relation “two points are at a finite edistance” is an equivalence relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball_zero : ball x 0 = ∅ | by rw [emetric.ball_eq_empty_iff] | lemma | emetric.ball_zero | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric.ball_eq_empty_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_eball : (𝓝 x).has_basis (λ ε:ℝ≥0∞, 0 < ε) (ball x) | nhds_basis_uniformity uniformity_basis_edist | theorem | emetric.nhds_basis_eball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"nhds_basis_uniformity",
"uniformity_basis_edist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_basis_eball : (𝓝[s] x).has_basis (λ ε : ℝ≥0∞, 0 < ε) (λ ε, ball x ε ∩ s) | nhds_within_has_basis nhds_basis_eball s | lemma | emetric.nhds_within_basis_eball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"nhds_within_has_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_closed_eball : (𝓝 x).has_basis (λ ε:ℝ≥0∞, 0 < ε) (closed_ball x) | nhds_basis_uniformity uniformity_basis_edist_le | theorem | emetric.nhds_basis_closed_eball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"nhds_basis_uniformity",
"uniformity_basis_edist_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_basis_closed_eball :
(𝓝[s] x).has_basis (λ ε : ℝ≥0∞, 0 < ε) (λ ε, closed_ball x ε ∩ s) | nhds_within_has_basis nhds_basis_closed_eball s | lemma | emetric.nhds_within_basis_closed_eball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"nhds_within_has_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq : 𝓝 x = (⨅ε>0, 𝓟 (ball x ε)) | nhds_basis_eball.eq_binfi | theorem | emetric.nhds_eq | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ε>0, ball x ε ⊆ s | nhds_basis_eball.mem_iff | theorem | emetric.mem_nhds_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_iff : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s | nhds_within_basis_eball.mem_iff | lemma | emetric.mem_nhds_within_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_nhds_within {t : set β} {a b} :
tendsto f (𝓝[s] a) (𝓝[t] b) ↔
∀ ε > 0, ∃ δ > 0, ∀ ⦃x⦄, x ∈ s → edist x a < δ → f x ∈ t ∧ edist (f x) b < ε | (nhds_within_basis_eball.tendsto_iff nhds_within_basis_eball).trans $
forall₂_congr $ λ ε hε, exists₂_congr $ λ δ hδ,
forall_congr $ λ x, by simp; itauto | lemma | emetric.tendsto_nhds_within_nhds_within | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"exists₂_congr",
"forall₂_congr",
"itauto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_nhds {a b} :
tendsto f (𝓝[s] a) (𝓝 b) ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, x ∈ s → edist x a < δ → edist (f x) b < ε | by { rw [← nhds_within_univ b, tendsto_nhds_within_nhds_within], simp only [mem_univ, true_and] } | lemma | emetric.tendsto_nhds_within_nhds | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_nhds {a b} :
tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x⦄, edist x a < δ → edist (f x) b < ε | nhds_basis_eball.tendsto_iff nhds_basis_eball | lemma | emetric.tendsto_nhds_nhds | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff : is_open s ↔ ∀x∈s, ∃ε>0, ball x ε ⊆ s | by simp [is_open_iff_nhds, mem_nhds_iff] | theorem | emetric.is_open_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"is_open",
"is_open_iff_nhds",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_ball_top : is_closed (ball x ⊤) | is_open_compl_iff.1 $ is_open_iff.2 $ λ y hy, ⟨⊤, ennreal.coe_lt_top,
(ball_disjoint $ by { rw top_add, exact le_of_not_lt hy }).subset_compl_right⟩ | theorem | emetric.is_closed_ball_top | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"ennreal.coe_lt_top",
"is_closed",
"top_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_mem_nhds (x : α) {ε : ℝ≥0∞} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x | is_open_ball.mem_nhds (mem_ball_self ε0) | theorem | emetric.ball_mem_nhds | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_mem_nhds (x : α) {ε : ℝ≥0∞} (ε0 : 0 < ε) : closed_ball x ε ∈ 𝓝 x | mem_of_superset (ball_mem_nhds x ε0) ball_subset_closed_ball | theorem | emetric.closed_ball_mem_nhds | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_prod_same [pseudo_emetric_space β] (x : α) (y : β) (r : ℝ≥0∞) :
ball x r ×ˢ ball y r = ball (x, y) r | ext $ λ z, max_lt_iff.symm | theorem | emetric.ball_prod_same | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"ball_prod_same",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_prod_same [pseudo_emetric_space β] (x : α) (y : β) (r : ℝ≥0∞) :
closed_ball x r ×ˢ closed_ball y r = closed_ball (x, y) r | ext $ λ z, max_le_iff.symm | theorem | emetric.closed_ball_prod_same | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"closed_ball_prod_same",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff :
x ∈ closure s ↔ ∀ε>0, ∃y ∈ s, edist x y < ε | (mem_closure_iff_nhds_basis nhds_basis_eball).trans $
by simp only [mem_ball, edist_comm x] | theorem | emetric.mem_closure_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"closure",
"mem_closure_iff",
"mem_closure_iff_nhds_basis"
] | ε-characterization of the closure in pseudoemetric spaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_nhds {f : filter β} {u : β → α} {a : α} :
tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, edist (u x) a < ε | nhds_basis_eball.tendsto_right_iff | theorem | emetric.tendsto_nhds | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"filter",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top [nonempty β] [semilattice_sup β] {u : β → α} {a : α} :
tendsto u at_top (𝓝 a) ↔ ∀ε>0, ∃N, ∀n≥N, edist (u n) a < ε | (at_top_basis.tendsto_iff nhds_basis_eball).trans $
by simp only [exists_prop, true_and, mem_Ici, mem_ball] | theorem | emetric.tendsto_at_top | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"exists_prop",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_iff : inseparable x y ↔ edist x y = 0 | by simp [inseparable_iff_mem_closure, mem_closure_iff, edist_comm, forall_lt_iff_le'] | theorem | emetric.inseparable_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"forall_lt_iff_le'",
"inseparable",
"inseparable_iff_mem_closure",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_iff [nonempty β] [semilattice_sup β] {u : β → α} :
cauchy_seq u ↔ ∀ε>0, ∃N, ∀m n≥N, edist (u m) (u n) < ε | uniformity_basis_edist.cauchy_seq_iff | theorem | emetric.cauchy_seq_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"cauchy_seq",
"cauchy_seq_iff",
"semilattice_sup"
] | In a pseudoemetric space, Cauchy sequences are characterized by the fact that, eventually,
the pseudoedistance between its elements is arbitrarily small | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_iff' [nonempty β] [semilattice_sup β] {u : β → α} :
cauchy_seq u ↔ ∀ε>(0 : ℝ≥0∞), ∃N, ∀n≥N, edist (u n) (u N) < ε | uniformity_basis_edist.cauchy_seq_iff' | theorem | emetric.cauchy_seq_iff' | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"cauchy_seq",
"cauchy_seq_iff'",
"semilattice_sup"
] | A variation around the emetric characterization of Cauchy sequences | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_iff_nnreal [nonempty β] [semilattice_sup β] {u : β → α} :
cauchy_seq u ↔ ∀ ε : ℝ≥0, 0 < ε → ∃ N, ∀ n, N ≤ n → edist (u n) (u N) < ε | uniformity_basis_edist_nnreal.cauchy_seq_iff' | theorem | emetric.cauchy_seq_iff_nnreal | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"cauchy_seq",
"semilattice_sup"
] | A variation of the emetric characterization of Cauchy sequences that deals with
`ℝ≥0` upper bounds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded_iff {s : set α} :
totally_bounded s ↔ ∀ ε > 0, ∃t : set α, t.finite ∧ s ⊆ ⋃y∈t, ball y ε | ⟨λ H ε ε0, H _ (edist_mem_uniformity ε0),
λ H r ru, let ⟨ε, ε0, hε⟩ := mem_uniformity_edist.1 ru,
⟨t, ft, h⟩ := H ε ε0 in
⟨t, ft, h.trans $ Union₂_mono $ λ y yt z, hε⟩⟩ | theorem | emetric.totally_bounded_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_mem_uniformity",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_iff' {s : set α} :
totally_bounded s ↔ ∀ ε > 0, ∃t⊆s, set.finite t ∧ s ⊆ ⋃y∈t, ball y ε | ⟨λ H ε ε0, (totally_bounded_iff_subset.1 H) _ (edist_mem_uniformity ε0),
λ H r ru, let ⟨ε, ε0, hε⟩ := mem_uniformity_edist.1 ru,
⟨t, _, ft, h⟩ := H ε ε0 in
⟨t, ft, h.trans $ Union₂_mono $ λ y yt z, hε⟩⟩ | theorem | emetric.totally_bounded_iff' | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_mem_uniformity",
"set.finite",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_countable_closure_of_almost_dense_set (s : set α)
(hs : ∀ ε > 0, ∃ t : set α, t.countable ∧ s ⊆ ⋃ x ∈ t, closed_ball x ε) :
∃ t ⊆ s, (t.countable ∧ s ⊆ closure t) | begin
rcases s.eq_empty_or_nonempty with rfl|⟨x₀, hx₀⟩,
{ exact ⟨∅, empty_subset _, countable_empty, empty_subset _⟩ },
choose! T hTc hsT using (λ n : ℕ, hs n⁻¹ (by simp)),
have : ∀ r x, ∃ y ∈ s, closed_ball x r ∩ s ⊆ closed_ball y (r * 2),
{ intros r x,
rcases (closed_ball x r ∩ s).eq_empty_or_nonempty w... | lemma | emetric.subset_countable_closure_of_almost_dense_set | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"closure",
"edist_triangle_right",
"ennreal.exists_inv_nat_lt",
"ennreal.half_pos",
"ennreal.mul_lt_of_lt_div",
"mul_two"
] | For a set `s` in a pseudo emetric space, if for every `ε > 0` there exists a countable
set that is `ε`-dense in `s`, then there exists a countable subset `t ⊆ s` that is dense in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_countable_closure_of_compact {s : set α} (hs : is_compact s) :
∃ t ⊆ s, (t.countable ∧ s ⊆ closure t) | begin
refine subset_countable_closure_of_almost_dense_set s (λ ε hε, _),
rcases totally_bounded_iff'.1 hs.totally_bounded ε hε with ⟨t, hts, htf, hst⟩,
exact ⟨t, htf.countable,
subset.trans hst $ Union₂_mono $ λ _ _, ball_subset_closed_ball⟩
end | lemma | emetric.subset_countable_closure_of_compact | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"closure",
"is_compact"
] | A compact set in a pseudo emetric space is separable, i.e., it is a subset of the closure of a
countable set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_countable_of_sigma_compact [sigma_compact_space α] :
second_countable_topology α | begin
suffices : separable_space α, by exactI uniform_space.second_countable_of_separable α,
choose T hTsub hTc hsubT
using λ n, subset_countable_closure_of_compact (is_compact_compact_covering α n),
refine ⟨⟨⋃ n, T n, countable_Union hTc, λ x, _⟩⟩,
rcases Union_eq_univ_iff.1 (Union_compact_covering α) x w... | lemma | emetric.second_countable_of_sigma_compact | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"Union_compact_covering",
"closure_mono",
"is_compact_compact_covering",
"sigma_compact_space",
"uniform_space.second_countable_of_separable"
] | A sigma compact pseudo emetric space has second countable topology. This is not an instance
to avoid a loop with `sigma_compact_space_of_locally_compact_second_countable`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_countable_of_almost_dense_set
(hs : ∀ ε > 0, ∃ t : set α, t.countable ∧ (⋃ x ∈ t, closed_ball x ε) = univ) :
second_countable_topology α | begin
suffices : separable_space α, by exactI uniform_space.second_countable_of_separable α,
rcases subset_countable_closure_of_almost_dense_set (univ : set α) (λ ε ε0, _)
with ⟨t, -, htc, ht⟩,
{ exact ⟨⟨t, htc, λ x, ht (mem_univ x)⟩⟩ },
{ rcases hs ε ε0 with ⟨t, htc, ht⟩,
exact ⟨t, htc, univ_subset_iff... | lemma | emetric.second_countable_of_almost_dense_set | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"uniform_space.second_countable_of_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam (s : set α) | ⨆ (x ∈ s) (y ∈ s), edist x y | def | emetric.diam | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | The diameter of a set in a pseudoemetric space, named `emetric.diam` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_le_iff {d : ℝ≥0∞} :
diam s ≤ d ↔ ∀ (x ∈ s) (y ∈ s), edist x y ≤ d | by simp only [diam, supr_le_iff] | lemma | emetric.diam_le_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"supr_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_image_le_iff {d : ℝ≥0∞} {f : β → α} {s : set β} :
diam (f '' s) ≤ d ↔ ∀ (x ∈ s) (y ∈ s), edist (f x) (f y) ≤ d | by simp only [diam_le_iff, ball_image_iff] | lemma | emetric.diam_image_le_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_of_diam_le {d} (hx : x ∈ s) (hy : y ∈ s) (hd : diam s ≤ d) : edist x y ≤ d | diam_le_iff.1 hd x hx y hy | lemma | emetric.edist_le_of_diam_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_diam_of_mem (hx : x ∈ s) (hy : y ∈ s) : edist x y ≤ diam s | edist_le_of_diam_le hx hy le_rfl | lemma | emetric.edist_le_diam_of_mem | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"le_rfl"
] | If two points belong to some set, their edistance is bounded by the diameter of the set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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