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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tendsto_dist_left_cocompact_at_top [proper_space α] (x : α) :
tendsto (dist x) (cocompact α) at_top | by simpa only [dist_comm] using tendsto_dist_right_cocompact_at_top x | lemma | tendsto_dist_left_cocompact_at_top | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_comm",
"proper_space",
"tendsto_dist_right_cocompact_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proper_space_of_compact_closed_ball_of_le
(R : ℝ) (h : ∀x:α, ∀r, R ≤ r → is_compact (closed_ball x r)) :
proper_space α | ⟨begin
assume x r,
by_cases hr : R ≤ r,
{ exact h x r hr },
{ have : closed_ball x r = closed_ball x R ∩ closed_ball x r,
{ symmetry,
apply inter_eq_self_of_subset_right,
exact closed_ball_subset_closed_ball (le_of_lt (not_le.1 hr)) },
rw this,
exact (h x R le_rfl).inter_right is_closed_... | lemma | proper_space_of_compact_closed_ball_of_le | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_compact",
"le_rfl",
"proper_space"
] | If all closed balls of large enough radius are compact, then the space is proper. Especially
useful when the lower bound for the radius is 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proper_of_compact [compact_space α] : proper_space α | ⟨assume x r, is_closed_ball.is_compact⟩ | instance | proper_of_compact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"compact_space",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_compact_of_proper [proper_space α] :
locally_compact_space α | locally_compact_space_of_has_basis (λ x, nhds_basis_closed_ball) $
λ x ε ε0, is_compact_closed_ball _ _ | instance | locally_compact_of_proper | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"locally_compact_space",
"locally_compact_space_of_has_basis",
"proper_space"
] | A proper space is locally compact | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_of_proper [proper_space α] : complete_space α | ⟨begin
intros f hf,
/- We want to show that the Cauchy filter `f` is converging. It suffices to find a closed
ball (therefore compact by properness) where it is nontrivial. -/
obtain ⟨t, t_fset, ht⟩ : ∃ t ∈ f, ∀ x y ∈ t, dist x y < 1 :=
(metric.cauchy_iff.1 hf).2 1 zero_lt_one,
rcases hf.1.nonempty_of_mem... | instance | complete_of_proper | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"complete_space",
"proper_space",
"zero_lt_one"
] | A proper space is complete | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_proper_space {α : Type*} {β : Type*} [pseudo_metric_space α] [pseudo_metric_space β]
[proper_space α] [proper_space β] :
proper_space (α × β) | { is_compact_closed_ball := begin
rintros ⟨x, y⟩ r,
rw ← closed_ball_prod_same x y,
apply (is_compact_closed_ball x r).prod (is_compact_closed_ball y r),
end } | instance | prod_proper_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closed_ball_prod_same",
"proper_space",
"pseudo_metric_space"
] | A binary product of proper spaces is proper. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_proper_space {π : β → Type*} [fintype β] [∀b, pseudo_metric_space (π b)]
[h : ∀b, proper_space (π b)] : proper_space (Πb, π b) | begin
refine proper_space_of_compact_closed_ball_of_le 0 (λx r hr, _),
rw closed_ball_pi _ hr,
apply is_compact_univ_pi (λb, _),
apply (h b).is_compact_closed_ball
end | instance | pi_proper_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closed_ball_pi",
"fintype",
"is_compact_univ_pi",
"proper_space",
"proper_space_of_compact_closed_ball_of_le",
"pseudo_metric_space"
] | A finite product of proper spaces is proper. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_pos_lt_subset_ball (hr : 0 < r) (hs : is_closed s) (h : s ⊆ ball x r) :
∃ r' ∈ Ioo 0 r, s ⊆ ball x r' | begin
unfreezingI { rcases eq_empty_or_nonempty s with rfl|hne },
{ exact ⟨r / 2, ⟨half_pos hr, half_lt_self hr⟩, empty_subset _⟩ },
have : is_compact s,
from is_compact_of_is_closed_subset (is_compact_closed_ball x r) hs
(subset.trans h ball_subset_closed_ball),
obtain ⟨y, hys, hy⟩ : ∃ y ∈ s, s ⊆ clo... | lemma | exists_pos_lt_subset_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_const",
"continuous_on",
"exists_between",
"is_closed",
"is_compact",
"is_compact_of_is_closed_subset"
] | If a nonempty ball in a proper space includes a closed set `s`, then there exists a nonempty
ball with the same center and a strictly smaller radius that includes `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_lt_subset_ball (hs : is_closed s) (h : s ⊆ ball x r) :
∃ r' < r, s ⊆ ball x r' | begin
cases le_or_lt r 0 with hr hr,
{ rw [ball_eq_empty.2 hr, subset_empty_iff] at h, unfreezingI { subst s },
exact (exists_lt r).imp (λ r' hr', ⟨hr', empty_subset _⟩) },
{ exact (exists_pos_lt_subset_ball hr hs h).imp (λ r' hr', ⟨hr'.fst.2, hr'.snd⟩) }
end | lemma | exists_lt_subset_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_pos_lt_subset_ball",
"is_closed"
] | If a ball in a proper space includes a closed set `s`, then there exists a ball with the same
center and a strictly smaller radius that includes `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.is_separable {s : set α} (hs : is_compact s) :
is_separable s | begin
haveI : compact_space s := is_compact_iff_compact_space.mp hs,
exact is_separable_of_separable_space_subtype s,
end | lemma | is_compact.is_separable | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"compact_space",
"is_compact",
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
second_countable_of_almost_dense_set
(H : ∀ε > (0 : ℝ), ∃ s : set α, s.countable ∧ (∀x, ∃y ∈ s, dist x y ≤ ε)) :
second_countable_topology α | begin
refine emetric.second_countable_of_almost_dense_set (λ ε ε0, _),
rcases ennreal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩,
choose s hsc y hys hyx using H ε' (by exact_mod_cast ε'0),
refine ⟨s, hsc, Union₂_eq_univ_iff.2 (λ x, ⟨y x, hys _, le_trans _ ε'ε.le⟩)⟩,
exact_mod_cast hyx x
end | lemma | metric.second_countable_of_almost_dense_set | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.second_countable_of_almost_dense_set"
] | A pseudometric space is second countable if, for every `ε > 0`, there is a countable set which
is `ε`-dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lebesgue_number_lemma_of_metric
{s : set α} {ι} {c : ι → set α} (hs : is_compact s)
(hc₁ : ∀ i, is_open (c i)) (hc₂ : s ⊆ ⋃ i, c i) :
∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i | let ⟨n, en, hn⟩ := lebesgue_number_lemma hs hc₁ hc₂,
⟨δ, δ0, hδ⟩ := mem_uniformity_dist.1 en in
⟨δ, δ0, assume x hx, let ⟨i, hi⟩ := hn x hx in
⟨i, assume y hy, hi (hδ (mem_ball'.mp hy))⟩⟩ | lemma | lebesgue_number_lemma_of_metric | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_compact",
"is_open",
"lebesgue_number_lemma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lebesgue_number_lemma_of_metric_sUnion
{s : set α} {c : set (set α)} (hs : is_compact s)
(hc₁ : ∀ t ∈ c, is_open t) (hc₂ : s ⊆ ⋃₀ c) :
∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t | by rw sUnion_eq_Union at hc₂;
simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂ | lemma | lebesgue_number_lemma_of_metric_sUnion | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_compact",
"is_open",
"lebesgue_number_lemma_of_metric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded (s : set α) : Prop | ∃C, ∀x y ∈ s, dist x y ≤ C | def | metric.bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | Boundedness of a subset of a pseudometric space. We formulate the definition to work
even in the empty space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_iff_is_bounded (s : set α) : bounded s ↔ is_bounded s | begin
change bounded s ↔ sᶜ ∈ (cobounded α).sets,
simp [pseudo_metric_space.cobounded_sets, metric.bounded],
end | lemma | metric.bounded_iff_is_bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric.bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_empty : bounded (∅ : set α) | ⟨0, by simp⟩ | lemma | metric.bounded_empty | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_of_unbounded (h : ¬ bounded s) : s.nonempty | begin
rw nonempty_iff_ne_empty,
rintro rfl,
exact h bounded_empty
end | lemma | metric.nonempty_of_unbounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_iff_mem_bounded : bounded s ↔ ∀ x ∈ s, bounded s | ⟨λ h _ _, h, λ H,
s.eq_empty_or_nonempty.elim
(λ hs, hs.symm ▸ bounded_empty)
(λ ⟨x, hx⟩, H x hx)⟩ | lemma | metric.bounded_iff_mem_bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded.mono (incl : s ⊆ t) : bounded t → bounded s | Exists.imp $ λ C hC x hx y hy, hC x (incl hx) y (incl hy) | lemma | metric.bounded.mono | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"Exists.imp"
] | Subsets of a bounded set are also bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_closed_ball : bounded (closed_ball x r) | ⟨r + r, λ y hy z hz, begin
simp only [mem_closed_ball] at *,
calc dist y z ≤ dist y x + dist z x : dist_triangle_right _ _ _
... ≤ r + r : add_le_add hy hz
end⟩ | lemma | metric.bounded_closed_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_triangle_right"
] | Closed balls are bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_ball : bounded (ball x r) | bounded_closed_ball.mono ball_subset_closed_ball | lemma | metric.bounded_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | Open balls are bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_sphere : bounded (sphere x r) | bounded_closed_ball.mono sphere_subset_closed_ball | lemma | metric.bounded_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | Spheres are bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_iff_subset_ball (c : α) : bounded s ↔ ∃r, s ⊆ closed_ball c r | begin
split; rintro ⟨C, hC⟩,
{ cases s.eq_empty_or_nonempty with h h,
{ subst s, exact ⟨0, by simp⟩ },
{ rcases h with ⟨x, hx⟩,
exact ⟨C + dist x c, λ y hy, calc
dist y c ≤ dist y x + dist x c : dist_triangle _ _ _
... ≤ C + dist x c : add_le_add_right (hC y hy x hx) _⟩ } },
{ ex... | lemma | metric.bounded_iff_subset_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_triangle"
] | Given a point, a bounded subset is included in some ball around this point | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded.subset_ball (h : bounded s) (c : α) : ∃ r, s ⊆ closed_ball c r | (bounded_iff_subset_ball c).1 h | lemma | metric.bounded.subset_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded.subset_ball_lt (h : bounded s) (a : ℝ) (c : α) : ∃ r, a < r ∧ s ⊆ closed_ball c r | begin
rcases h.subset_ball c with ⟨r, hr⟩,
refine ⟨max r (a+1), lt_of_lt_of_le (by linarith) (le_max_right _ _), _⟩,
exact subset.trans hr (closed_ball_subset_closed_ball (le_max_left _ _))
end | lemma | metric.bounded.subset_ball_lt | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_closure_of_bounded (h : bounded s) : bounded (closure s) | let ⟨C, h⟩ := h in
⟨C, λ a ha b hb, (is_closed_le' C).closure_subset $ map_mem_closure₂ continuous_dist ha hb h⟩ | lemma | metric.bounded_closure_of_bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closure",
"continuous_dist",
"is_closed_le'",
"map_mem_closure₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_closure_iff : bounded (closure s) ↔ bounded s | ⟨λ h, h.mono subset_closure, λ h, h.closure⟩ | lemma | metric.bounded_closure_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded.union (hs : bounded s) (ht : bounded t) : bounded (s ∪ t) | begin
refine bounded_iff_mem_bounded.2 (λ x _, _),
rw bounded_iff_subset_ball x at hs ht ⊢,
rcases hs with ⟨Cs, hCs⟩, rcases ht with ⟨Ct, hCt⟩,
exact ⟨max Cs Ct, union_subset
(subset.trans hCs $ closed_ball_subset_closed_ball $ le_max_left _ _)
(subset.trans hCt $ closed_ball_subset_closed_ball $ le_max... | lemma | metric.bounded.union | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | The union of two bounded sets is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_union : bounded (s ∪ t) ↔ bounded s ∧ bounded t | ⟨λ h, ⟨h.mono (by simp), h.mono (by simp)⟩, λ h, h.1.union h.2⟩ | lemma | metric.bounded_union | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | The union of two sets is bounded iff each of the sets is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_bUnion {I : set β} {s : β → set α} (H : I.finite) :
bounded (⋃i∈I, s i) ↔ ∀i ∈ I, bounded (s i) | finite.induction_on H (by simp) $ λ x I _ _ IH,
by simp [or_imp_distrib, forall_and_distrib, IH] | lemma | metric.bounded_bUnion | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"forall_and_distrib",
"or_imp_distrib"
] | A finite union of bounded sets is bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded.prod [pseudo_metric_space β] {s : set α} {t : set β}
(hs : bounded s) (ht : bounded t) : bounded (s ×ˢ t) | begin
refine bounded_iff_mem_bounded.mpr (λ x hx, _),
rcases hs.subset_ball x.1 with ⟨rs, hrs⟩,
rcases ht.subset_ball x.2 with ⟨rt, hrt⟩,
suffices : s ×ˢ t ⊆ closed_ball x (max rs rt),
from bounded_closed_ball.mono this,
rw [← @prod.mk.eta _ _ x, ← closed_ball_prod_same],
exact prod_mono (hrs.trans $ cl... | lemma | metric.bounded.prod | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closed_ball_prod_same",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.totally_bounded.bounded {s : set α} (h : totally_bounded s) : bounded s | -- We cover the totally bounded set by finitely many balls of radius 1,
-- and then argue that a finite union of bounded sets is bounded
let ⟨t, fint, subs⟩ := (totally_bounded_iff.mp h) 1 zero_lt_one in
bounded.mono subs $ (bounded_bUnion fint).2 $ λ i hi, bounded_ball | lemma | totally_bounded.bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded",
"zero_lt_one"
] | A totally bounded set is bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_compact.bounded {s : set α} (h : is_compact s) : bounded s | -- A compact set is totally bounded, thus bounded
h.totally_bounded.bounded | lemma | is_compact.bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_compact"
] | A compact set is bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_of_finite {s : set α} (h : s.finite) : bounded s | h.is_compact.bounded | lemma | metric.bounded_of_finite | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | A finite set is bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_singleton {x : α} : bounded ({x} : set α) | bounded_of_finite $ finite_singleton _ | lemma | metric.bounded_singleton | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | A singleton is bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_range_iff {f : β → α} : bounded (range f) ↔ ∃C, ∀x y, dist (f x) (f y) ≤ C | exists_congr $ λ C, ⟨
λ H x y, H _ ⟨x, rfl⟩ _ ⟨y, rfl⟩,
by rintro H _ ⟨x, rfl⟩ _ ⟨y, rfl⟩; exact H x y⟩ | lemma | metric.bounded_range_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | Characterization of the boundedness of the range of a function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_range_of_tendsto_cofinite_uniformity {f : β → α}
(hf : tendsto (prod.map f f) (cofinite ×ᶠ cofinite) (𝓤 α)) :
bounded (range f) | begin
rcases (has_basis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one
with ⟨s, hsf, hs1⟩,
rw [← image_univ, ← union_compl_self s, image_union, bounded_union],
use [(hsf.image f).bounded, 1],
rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩,
exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩)
end | lemma | metric.bounded_range_of_tendsto_cofinite_uniformity | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_range_of_cauchy_map_cofinite {f : β → α} (hf : cauchy (map f cofinite)) :
bounded (range f) | bounded_range_of_tendsto_cofinite_uniformity $ (cauchy_map_iff.1 hf).2 | lemma | metric.bounded_range_of_cauchy_map_cofinite | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"cauchy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.cauchy_seq.bounded_range {f : ℕ → α} (hf : cauchy_seq f) : bounded (range f) | bounded_range_of_cauchy_map_cofinite $ by rwa nat.cofinite_eq_at_top | lemma | cauchy_seq.bounded_range | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"cauchy_seq",
"nat.cofinite_eq_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : tendsto f cofinite (𝓝 a)) :
bounded (range f) | bounded_range_of_tendsto_cofinite_uniformity $
(hf.prod_map hf).mono_right $ nhds_prod_eq.symm.trans_le (nhds_le_uniformity a) | lemma | metric.bounded_range_of_tendsto_cofinite | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_of_compact_space [compact_space α] : bounded s | is_compact_univ.bounded.mono (subset_univ _) | lemma | metric.bounded_of_compact_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"compact_space"
] | In a compact space, all sets are bounded | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : tendsto u at_top (𝓝 x)) :
bounded (range u) | hu.cauchy_seq.bounded_range | lemma | metric.bounded_range_of_tendsto | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at
[topological_space β] {k s : set β} {f : β → α}
(hk : is_compact k) (hf : ∀ x ∈ k, continuous_within_at f s x) :
∃ t, k ⊆ t ∧ is_open t ∧ bounded (f '' (t ∩ s)) | begin
apply hk.induction_on,
{ exact ⟨∅, subset.refl _, is_open_empty,
by simp only [image_empty, bounded_empty, empty_inter]⟩ },
{ rintros s s' hss' ⟨t, s't, t_open, t_bounded⟩,
exact ⟨t, hss'.trans s't, t_open, t_bounded⟩ },
{ rintros s s' ⟨t, st, t_open, t_bounded⟩ ⟨t', s't', t'_open, t'_bounded⟩,
... | lemma | metric.exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_within_at",
"is_compact",
"is_open",
"is_open_empty",
"nhds_within_le_nhds",
"topological_space",
"zero_lt_one"
] | If a function is continuous within a set `s` at every point of a compact set `k`, then it is
bounded on some open neighborhood of `k` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at
[topological_space β] {k : set β} {f : β → α}
(hk : is_compact k) (hf : ∀ x ∈ k, continuous_at f x) :
∃ t, k ⊆ t ∧ is_open t ∧ bounded (f '' t) | begin
simp_rw ← continuous_within_at_univ at hf,
simpa only [inter_univ] using
exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at hk hf,
end | lemma | metric.exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_at",
"continuous_within_at_univ",
"is_compact",
"is_open",
"topological_space"
] | If a function is continuous at every point of a compact set `k`, then it is bounded on
some open neighborhood of `k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on
[topological_space β] {k s : set β} {f : β → α}
(hk : is_compact k) (hks : k ⊆ s) (hf : continuous_on f s) :
∃ t, k ⊆ t ∧ is_open t ∧ bounded (f '' (t ∩ s)) | exists_is_open_bounded_image_inter_of_is_compact_of_forall_continuous_within_at hk
(λ x hx, hf x (hks hx)) | lemma | metric.exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_on",
"is_compact",
"is_open",
"topological_space"
] | If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
some open neighborhood of `k` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_open_bounded_image_of_is_compact_of_continuous_on
[topological_space β] {k s : set β} {f : β → α}
(hk : is_compact k) (hs : is_open s) (hks : k ⊆ s) (hf : continuous_on f s) :
∃ t, k ⊆ t ∧ is_open t ∧ bounded (f '' t) | exists_is_open_bounded_image_of_is_compact_of_forall_continuous_at hk
(λ x hx, hf.continuous_at (hs.mem_nhds (hks hx))) | lemma | metric.exists_is_open_bounded_image_of_is_compact_of_continuous_on | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_on",
"is_compact",
"is_open",
"topological_space"
] | If a function is continuous on a neighborhood of a compact set `k`, then it is bounded on
some open neighborhood of `k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_of_is_closed_bounded [proper_space α] (hc : is_closed s) (hb : bounded s) :
is_compact s | begin
unfreezingI { rcases eq_empty_or_nonempty s with (rfl|⟨x, hx⟩) },
{ exact is_compact_empty },
{ rcases hb.subset_ball x with ⟨r, hr⟩,
exact is_compact_of_is_closed_subset (is_compact_closed_ball x r) hc hr }
end | lemma | metric.is_compact_of_is_closed_bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_closed",
"is_compact",
"is_compact_empty",
"is_compact_of_is_closed_subset",
"proper_space"
] | The **Heine–Borel theorem**: In a proper space, a closed bounded set is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded.is_compact_closure [proper_space α] (h : bounded s) :
is_compact (closure s) | is_compact_of_is_closed_bounded is_closed_closure h.closure | lemma | metric.bounded.is_compact_closure | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closure",
"is_closed_closure",
"is_compact",
"proper_space"
] | The **Heine–Borel theorem**: In a proper space, the closure of a bounded set is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_iff_is_closed_bounded [t2_space α] [proper_space α] :
is_compact s ↔ is_closed s ∧ bounded s | ⟨λ h, ⟨h.is_closed, h.bounded⟩, λ h, is_compact_of_is_closed_bounded h.1 h.2⟩ | lemma | metric.is_compact_iff_is_closed_bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_closed",
"is_compact",
"proper_space",
"t2_space"
] | The **Heine–Borel theorem**:
In a proper Hausdorff space, a set is compact if and only if it is closed and bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space_iff_bounded_univ [proper_space α] : compact_space α ↔ bounded (univ : set α) | ⟨@bounded_of_compact_space α _ _, λ hb, ⟨is_compact_of_is_closed_bounded is_closed_univ hb⟩⟩ | lemma | metric.compact_space_iff_bounded_univ | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"compact_space",
"is_closed_univ",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_Icc (a b : α) : bounded (Icc a b) | (totally_bounded_Icc a b).bounded | lemma | metric.bounded_Icc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_Ico (a b : α) : bounded (Ico a b) | (totally_bounded_Ico a b).bounded | lemma | metric.bounded_Ico | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_Ioc (a b : α) : bounded (Ioc a b) | (totally_bounded_Ioc a b).bounded | lemma | metric.bounded_Ioc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_Ioo (a b : α) : bounded (Ioo a b) | (totally_bounded_Ioo a b).bounded | lemma | metric.bounded_Ioo | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_of_bdd_above_of_bdd_below {s : set α} (h₁ : bdd_above s) (h₂ : bdd_below s) :
bounded s | let ⟨u, hu⟩ := h₁, ⟨l, hl⟩ := h₂ in
bounded.mono (λ x hx, mem_Icc.mpr ⟨hl hx, hu hx⟩) (bounded_Icc l u) | lemma | metric.bounded_of_bdd_above_of_bdd_below | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"bdd_above",
"bdd_below"
] | In a pseudo metric space with a conditionally complete linear order such that the order and the
metric structure give the same topology, any order-bounded set is metric-bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam (s : set α) : ℝ | ennreal.to_real (emetric.diam s) | def | metric.diam | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.diam",
"ennreal.to_real"
] | The diameter of a set in a metric space. To get controllable behavior even when the diameter
should be infinite, we express it in terms of the emetric.diameter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_nonneg : 0 ≤ diam s | ennreal.to_real_nonneg | lemma | metric.diam_nonneg | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"ennreal.to_real_nonneg"
] | The diameter of a set is always nonnegative | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_subsingleton (hs : s.subsingleton) : diam s = 0 | by simp only [diam, emetric.diam_subsingleton hs, ennreal.zero_to_real] | lemma | metric.diam_subsingleton | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.diam_subsingleton",
"ennreal.zero_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_empty : diam (∅ : set α) = 0 | diam_subsingleton subsingleton_empty | lemma | metric.diam_empty | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | The empty set has zero diameter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_singleton : diam ({x} : set α) = 0 | diam_subsingleton subsingleton_singleton | lemma | metric.diam_singleton | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | A singleton has zero diameter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_one [has_one α] : diam (1 : set α) = 0 | diam_singleton | lemma | metric.diam_one | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_pair : diam ({x, y} : set α) = dist x y | by simp only [diam, emetric.diam_pair, dist_edist] | lemma | metric.diam_pair | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_edist",
"emetric.diam_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_triple :
metric.diam ({x, y, z} : set α) = max (max (dist x y) (dist x z)) (dist y z) | begin
simp only [metric.diam, emetric.diam_triple, dist_edist],
rw [ennreal.to_real_max, ennreal.to_real_max];
apply_rules [ne_of_lt, edist_lt_top, max_lt]
end | lemma | metric.diam_triple | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_edist",
"edist_lt_top",
"emetric.diam_triple",
"ennreal.to_real_max",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ (x ∈ s) (y ∈ s), dist x y ≤ C) :
emetric.diam s ≤ ennreal.of_real C | emetric.diam_le $
λ x hx y hy, (edist_dist x y).symm ▸ ennreal.of_real_le_of_real (h x hx y hy) | lemma | metric.ediam_le_of_forall_dist_le | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_dist",
"emetric.diam",
"emetric.diam_le",
"ennreal.of_real",
"ennreal.of_real_le_of_real"
] | If the distance between any two points in a set is bounded by some constant `C`,
then `ennreal.of_real C` bounds the emetric diameter of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ (x ∈ s) (y ∈ s), dist x y ≤ C) :
diam s ≤ C | ennreal.to_real_le_of_le_of_real h₀ (ediam_le_of_forall_dist_le h) | lemma | metric.diam_le_of_forall_dist_le | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"ennreal.to_real_le_of_le_of_real"
] | If the distance between any two points in a set is bounded by some non-negative constant,
this constant bounds the diameter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_le_of_forall_dist_le_of_nonempty (hs : s.nonempty) {C : ℝ}
(h : ∀ (x ∈ s) (y ∈ s), dist x y ≤ C) : diam s ≤ C | have h₀ : 0 ≤ C, from let ⟨x, hx⟩ := hs in le_trans dist_nonneg (h x hx x hx),
diam_le_of_forall_dist_le h₀ h | lemma | metric.diam_le_of_forall_dist_le_of_nonempty | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_nonneg"
] | If the distance between any two points in a nonempty set is bounded by some constant,
this constant bounds the diameter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_diam_of_mem' (h : emetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
dist x y ≤ diam s | begin
rw [diam, dist_edist],
rw ennreal.to_real_le_to_real (edist_ne_top _ _) h,
exact emetric.edist_le_diam_of_mem hx hy
end | lemma | metric.dist_le_diam_of_mem' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_edist",
"edist_ne_top",
"emetric.diam",
"emetric.edist_le_diam_of_mem",
"ennreal.to_real_le_to_real"
] | The distance between two points in a set is controlled by the diameter of the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_iff_ediam_ne_top : bounded s ↔ emetric.diam s ≠ ⊤ | iff.intro
(λ ⟨C, hC⟩, ne_top_of_le_ne_top ennreal.of_real_ne_top $ ediam_le_of_forall_dist_le hC)
(λ h, ⟨diam s, λ x hx y hy, dist_le_diam_of_mem' h hx hy⟩) | lemma | metric.bounded_iff_ediam_ne_top | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.diam",
"ennreal.of_real_ne_top",
"ne_top_of_le_ne_top"
] | Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded.ediam_ne_top (h : bounded s) : emetric.diam s ≠ ⊤ | bounded_iff_ediam_ne_top.1 h | lemma | metric.bounded.ediam_ne_top | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_univ_eq_top_iff_noncompact [proper_space α] :
emetric.diam (univ : set α) = ∞ ↔ noncompact_space α | by rw [← not_compact_space_iff, compact_space_iff_bounded_univ, bounded_iff_ediam_ne_top, not_not] | lemma | metric.ediam_univ_eq_top_iff_noncompact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.diam",
"noncompact_space",
"not_compact_space_iff",
"not_not",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_univ_of_noncompact [proper_space α] [noncompact_space α] :
emetric.diam (univ : set α) = ∞ | ediam_univ_eq_top_iff_noncompact.mpr ‹_› | lemma | metric.ediam_univ_of_noncompact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.diam",
"noncompact_space",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_univ_of_noncompact [proper_space α] [noncompact_space α] :
diam (univ : set α) = 0 | by simp [diam] | lemma | metric.diam_univ_of_noncompact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"noncompact_space",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_diam_of_mem (h : bounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s | dist_le_diam_of_mem' h.ediam_ne_top hx hy | lemma | metric.dist_le_diam_of_mem | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | The distance between two points in a set is controlled by the diameter of the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ediam_of_unbounded (h : ¬(bounded s)) : emetric.diam s = ∞ | by rwa [bounded_iff_ediam_ne_top, not_not] at h | lemma | metric.ediam_of_unbounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.diam",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_eq_zero_of_unbounded (h : ¬(bounded s)) : diam s = 0 | by rw [diam, ediam_of_unbounded h, ennreal.top_to_real] | lemma | metric.diam_eq_zero_of_unbounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"ennreal.top_to_real"
] | An unbounded set has zero diameter. If you would prefer to get the value ∞, use `emetric.diam`.
This lemma makes it possible to avoid side conditions in some situations | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_mono {s t : set α} (h : s ⊆ t) (ht : bounded t) : diam s ≤ diam t | begin
unfold diam,
rw ennreal.to_real_le_to_real (bounded.mono h ht).ediam_ne_top ht.ediam_ne_top,
exact emetric.diam_mono h
end | lemma | metric.diam_mono | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.diam_mono",
"ennreal.to_real_le_to_real"
] | If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_union {t : set α} (xs : x ∈ s) (yt : y ∈ t) :
diam (s ∪ t) ≤ diam s + dist x y + diam t | begin
by_cases H : bounded (s ∪ t),
{ have hs : bounded s, from H.mono (subset_union_left _ _),
have ht : bounded t, from H.mono (subset_union_right _ _),
rw [bounded_iff_ediam_ne_top] at H hs ht,
rw [dist_edist, diam, diam, diam, ← ennreal.to_real_add, ← ennreal.to_real_add,
ennreal.to_real_le_to... | lemma | metric.diam_union | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_edist",
"dist_nonneg",
"edist_ne_top",
"emetric.diam_union",
"ennreal.to_real_add",
"ennreal.to_real_le_to_real"
] | The diameter of a union is controlled by the sum of the diameters, and the distance between
any two points in each of the sets. This lemma is true without any side condition, since it is
obviously true if `s ∪ t` is unbounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_union' {t : set α} (h : (s ∩ t).nonempty) : diam (s ∪ t) ≤ diam s + diam t | begin
rcases h with ⟨x, ⟨xs, xt⟩⟩,
simpa using diam_union xs xt
end | lemma | metric.diam_union' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | If two sets intersect, the diameter of the union is bounded by the sum of the diameters. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_le_of_subset_closed_ball {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closed_ball x r) :
diam s ≤ 2 * r | diam_le_of_forall_dist_le (mul_nonneg zero_le_two hr) $ λa ha b hb, calc
dist a b ≤ dist a x + dist b x : dist_triangle_right _ _ _
... ≤ r + r : add_le_add (h ha) (h hb)
... = 2 * r : by simp [mul_two, mul_comm] | lemma | metric.diam_le_of_subset_closed_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_triangle_right",
"mul_comm",
"mul_two",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_closed_ball {r : ℝ} (h : 0 ≤ r) : diam (closed_ball x r) ≤ 2 * r | diam_le_of_subset_closed_ball h subset.rfl | lemma | metric.diam_closed_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | The diameter of a closed ball of radius `r` is at most `2 r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r | diam_le_of_subset_closed_ball h ball_subset_closed_ball | lemma | metric.diam_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | The diameter of a ball of radius `r` is at most `2 r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_complete.nonempty_Inter_of_nonempty_bInter {s : ℕ → set α} (h0 : is_complete (s 0))
(hs : ∀ n, is_closed (s n)) (h's : ∀ n, bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).nonempty)
(h' : tendsto (λ n, diam (s n)) at_top (𝓝 0)) :
(⋂ n, s n).nonempty | begin
let u := λ N, (h N).some,
have I : ∀ n N, n ≤ N → u N ∈ s n,
{ assume n N hn,
apply mem_of_subset_of_mem _ ((h N).some_spec),
assume x hx,
simp only [mem_Inter] at hx,
exact hx n hn },
have : ∀ n, u n ∈ s 0 := λ n, I 0 n (zero_le _),
have : cauchy_seq u,
{ apply cauchy_seq_of_le_tendst... | lemma | is_complete.nonempty_Inter_of_nonempty_bInter | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"cauchy_seq",
"cauchy_seq_of_le_tendsto_0",
"cauchy_seq_tendsto_of_is_complete",
"is_closed",
"is_complete"
] | If a family of complete sets with diameter tending to `0` is such that each finite intersection
is nonempty, then the total intersection is also nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_Inter_of_nonempty_bInter [complete_space α] {s : ℕ → set α}
(hs : ∀ n, is_closed (s n)) (h's : ∀ n, bounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).nonempty)
(h' : tendsto (λ n, diam (s n)) at_top (𝓝 0)) :
(⋂ n, s n).nonempty | (hs 0).is_complete.nonempty_Inter_of_nonempty_bInter hs h's h h' | lemma | metric.nonempty_Inter_of_nonempty_bInter | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"complete_space",
"is_closed",
"is_complete.nonempty_Inter_of_nonempty_bInter"
] | In a complete space, if a family of closed sets with diameter tending to `0` is such that each
finite intersection is nonempty, then the total intersection is also nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_local_min_mem_ball [proper_space α] [topological_space β]
[conditionally_complete_linear_order β] [order_topology β]
{f : α → β} {a z : α} {r : ℝ} (hf : continuous_on f (closed_ball a r))
(hz : z ∈ closed_ball a r) (hf1 : ∀ z' ∈ sphere a r, f z < f z') :
∃ z ∈ ball a r, is_local_min f z | begin
simp_rw [← closed_ball_diff_ball] at hf1,
exact (is_compact_closed_ball a r).exists_local_min_mem_open ball_subset_closed_ball hf hz hf1
is_open_ball,
end | lemma | metric.exists_local_min_mem_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"conditionally_complete_linear_order",
"continuous_on",
"is_local_min",
"order_topology",
"proper_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positivity_diam : expr → tactic strictness | | `(metric.diam %%s) := nonnegative <$> mk_app ``metric.diam_nonneg [s]
| e := pp e >>= fail ∘ format.bracket "The expression " " is not of the form `metric.diam s`" | def | tactic.positivity_diam | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric.diam_nonneg"
] | Extension for the `positivity` tactic: the diameter of a set is always nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_dist_right_at_top_le_cocompact (x : α) : comap (λ y, dist y x) at_top ≤ cocompact α | begin
refine filter.has_basis_cocompact.ge_iff.2 (λ s hs, mem_comap.2 _),
rcases hs.bounded.subset_ball x with ⟨r, hr⟩,
exact ⟨Ioi r, Ioi_mem_at_top r, λ y hy hys, (mem_closed_ball.1 $ hr hys).not_lt hy⟩
end | lemma | comap_dist_right_at_top_le_cocompact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_dist_left_at_top_le_cocompact (x : α) : comap (dist x) at_top ≤ cocompact α | by simpa only [dist_comm _ x] using comap_dist_right_at_top_le_cocompact x | lemma | comap_dist_left_at_top_le_cocompact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"comap_dist_right_at_top_le_cocompact",
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_dist_right_at_top_eq_cocompact [proper_space α] (x : α) :
comap (λ y, dist y x) at_top = cocompact α | (comap_dist_right_at_top_le_cocompact x).antisymm $ (tendsto_dist_right_cocompact_at_top x).le_comap | lemma | comap_dist_right_at_top_eq_cocompact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"comap_dist_right_at_top_le_cocompact",
"proper_space",
"tendsto_dist_right_cocompact_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_dist_left_at_top_eq_cocompact [proper_space α] (x : α) :
comap (dist x) at_top = cocompact α | (comap_dist_left_at_top_le_cocompact x).antisymm $ (tendsto_dist_left_cocompact_at_top x).le_comap | lemma | comap_dist_left_at_top_eq_cocompact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"comap_dist_left_at_top_le_cocompact",
"proper_space",
"tendsto_dist_left_cocompact_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cocompact_of_tendsto_dist_comp_at_top {f : β → α} {l : filter β} (x : α)
(h : tendsto (λ y, dist (f y) x) l at_top) : tendsto f l (cocompact α) | by { refine tendsto.mono_right _ (comap_dist_right_at_top_le_cocompact x), rwa tendsto_comap_iff } | lemma | tendsto_cocompact_of_tendsto_dist_comp_at_top | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"comap_dist_right_at_top_le_cocompact",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric_space (α : Type u) extends pseudo_metric_space α : Type u | (eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y) | class | metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eq_of_dist_eq_zero",
"pseudo_metric_space"
] | We now define `metric_space`, extending `pseudo_metric_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space.ext {α : Type*} {m m' : metric_space α}
(h : m.to_has_dist = m'.to_has_dist) : m = m' | begin
have h' : m.to_pseudo_metric_space = m'.to_pseudo_metric_space := pseudo_metric_space.ext h,
unfreezingI { rcases m, rcases m' },
dsimp at h',
unfreezingI { subst h' },
end | lemma | metric_space.ext | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric_space",
"pseudo_metric_space.ext"
] | Two metric space structures with the same distance coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric_space.of_dist_topology {α : Type u} [topological_space α] (dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
(H : ∀ s : set α, is_open s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s)
(eq_of_d... | { eq_of_dist_eq_zero := eq_of_dist_eq_zero,
..pseudo_metric_space.of_dist_topology dist dist_self dist_comm dist_triangle H } | def | metric_space.of_dist_topology | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"eq_of_dist_eq_zero",
"is_open",
"metric_space",
"pseudo_metric_space.of_dist_topology",
"topological_space"
] | Construct a metric space structure whose underlying topological space structure
(definitionally) agrees which a pre-existing topology which is compatible with a given distance
function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_dist_eq_zero {x y : γ} : dist x y = 0 → x = y | metric_space.eq_of_dist_eq_zero | theorem | eq_of_dist_eq_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_zero {x y : γ} : dist x y = 0 ↔ x = y | iff.intro eq_of_dist_eq_zero (assume : x = y, this ▸ dist_self _) | theorem | dist_eq_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_self",
"eq_of_dist_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_dist {x y : γ} : 0 = dist x y ↔ x = y | by rw [eq_comm, dist_eq_zero] | theorem | zero_eq_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_ne_zero {x y : γ} : dist x y ≠ 0 ↔ x ≠ y | by simpa only [not_iff_not] using dist_eq_zero | theorem | dist_ne_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_eq_zero",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_zero {x y : γ} : dist x y ≤ 0 ↔ x = y | by simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y | theorem | dist_le_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_eq_zero",
"dist_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y | by simpa only [not_le] using not_congr dist_le_zero | theorem | dist_pos | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_le_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x = y | eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h) | theorem | eq_of_forall_dist_le | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_nonneg",
"eq_of_dist_eq_zero",
"eq_of_le_of_forall_le_of_dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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