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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dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ | by { rw [add_right_comm, dist_comm y₁], apply dist_triangle4 } | lemma | dist_triangle4_right | 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_triangle4"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
dist (f m) (f n) ≤ ∑ i in finset.Ico m n, dist (f i) (f (i + 1)) | begin
revert n,
apply nat.le_induction,
{ simp only [finset.sum_empty, finset.Ico_self, dist_self] },
{ assume n hn hrec,
calc dist (f m) (f (n+1)) ≤ dist (f m) (f n) + dist _ _ : dist_triangle _ _ _
... ≤ ∑ i in finset.Ico m n, _ + _ : add_le_add hrec le_rfl
... = ∑ i in finset.Ico m (n+1), _ :... | lemma | dist_le_Ico_sum_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_self",
"dist_triangle",
"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 |
dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) :
dist (f 0) (f n) ≤ ∑ i in finset.range n, dist (f i) (f (i + 1)) | nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_dist f (nat.zero_le n) | lemma | dist_le_range_sum_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_le_Ico_sum_dist",
"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 |
dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n)
{d : ℕ → ℝ} (hd : ∀ {k}, m ≤ k → k < n → dist (f k) (f (k + 1)) ≤ d k) :
dist (f m) (f n) ≤ ∑ i in finset.Ico m n, d i | le_trans (dist_le_Ico_sum_dist f hmn) $
finset.sum_le_sum $ λ k hk, hd (finset.mem_Ico.1 hk).1 (finset.mem_Ico.1 hk).2 | lemma | dist_le_Ico_sum_of_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_le_Ico_sum_dist",
"finset.Ico"
] | A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced
with an upper estimate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ)
{d : ℕ → ℝ} (hd : ∀ {k}, k < n → dist (f k) (f (k + 1)) ≤ d k) :
dist (f 0) (f n) ≤ ∑ i in finset.range n, d i | nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_of_dist_le (zero_le n) (λ _ _, hd) | lemma | dist_le_range_sum_of_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_le_Ico_sum_of_dist_le",
"finset.range",
"nat.Ico_zero_eq_range"
] | A version of `dist_le_range_sum_dist` with each intermediate distance replaced
with an upper estimate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
swap_dist : function.swap (@dist α _) = dist | by funext x y; exact dist_comm _ _ | theorem | swap_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_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y | abs_sub_le_iff.2
⟨sub_le_iff_le_add.2 (dist_triangle _ _ _),
sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩ | theorem | abs_dist_sub_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_triangle",
"dist_triangle_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_nonneg {x y : α} : 0 ≤ dist x y | pseudo_metric_space.dist_nonneg' dist dist_self dist_comm dist_triangle | theorem | dist_nonneg | 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",
"pseudo_metric_space.dist_nonneg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.tactic.positivity_dist : expr → tactic strictness | | `(dist %%a %%b) := nonnegative <$> mk_app ``dist_nonneg [a, b]
| _ := failed | def | tactic.positivity_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_nonneg"
] | Extension for the `positivity` tactic: distances are nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_dist {a b : α} : |dist a b| = dist a b | abs_of_nonneg dist_nonneg | theorem | abs_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"abs_of_nonneg",
"dist_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nndist (α : Type*) | (nndist : α → α → ℝ≥0) | class | has_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | A version of `has_dist` that takes value in `ℝ≥0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.to_has_nndist : has_nndist α | ⟨λ a b, ⟨dist a b, dist_nonneg⟩⟩ | instance | pseudo_metric_space.to_has_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"has_nndist"
] | Distance as a nonnegative real number. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nndist_edist (x y : α) : nndist x y = (edist x y).to_nnreal | by simp [nndist, edist_dist, real.to_nnreal, max_eq_left dist_nonneg, ennreal.of_real] | lemma | nndist_edist | 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",
"edist_dist",
"ennreal.of_real",
"real.to_nnreal"
] | Express `nndist` in terms of `edist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_nndist (x y : α) : edist x y = ↑(nndist x y) | by { simpa only [edist_dist, ennreal.of_real_eq_coe_nnreal dist_nonneg] } | lemma | edist_nndist | 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",
"edist_dist",
"ennreal.of_real_eq_coe_nnreal"
] | Express `edist` in terms of `nndist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y | (edist_nndist x y).symm | lemma | coe_nnreal_ennreal_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_nndist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_lt_coe {x y : α} {c : ℝ≥0} :
edist x y < c ↔ nndist x y < c | by rw [edist_nndist, ennreal.coe_lt_coe] | lemma | edist_lt_coe | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_nndist",
"ennreal.coe_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_coe {x y : α} {c : ℝ≥0} :
edist x y ≤ c ↔ nndist x y ≤ c | by rw [edist_nndist, ennreal.coe_le_coe] | lemma | edist_le_coe | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_nndist",
"ennreal.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_lt_top {α : Type*} [pseudo_metric_space α] (x y : α) : edist x y < ⊤ | (edist_dist x y).symm ▸ ennreal.of_real_lt_top | lemma | edist_lt_top | 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",
"ennreal.of_real_lt_top",
"pseudo_metric_space"
] | In a pseudometric space, the extended distance is always finite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_ne_top (x y : α) : edist x y ≠ ⊤ | (edist_lt_top x y).ne | lemma | edist_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"
] | [
"edist_lt_top"
] | In a pseudometric space, the extended distance is always finite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nndist_self (a : α) : nndist a a = 0 | (nnreal.coe_eq_zero _).1 (dist_self a) | lemma | nndist_self | 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",
"nnreal.coe_eq_zero"
] | `nndist x x` vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_nndist (x y : α) : dist x y = ↑(nndist x y) | rfl | lemma | dist_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | Express `dist` in terms of `nndist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_nndist (x y : α) : ↑(nndist x y) = dist x y | (dist_nndist x y).symm | lemma | coe_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_nndist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_coe {x y : α} {c : ℝ≥0} :
dist x y < c ↔ nndist x y < c | iff.rfl | lemma | dist_lt_coe | 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_le_coe {x y : α} {c : ℝ≥0} :
dist x y ≤ c ↔ nndist x y ≤ c | iff.rfl | lemma | dist_le_coe | 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 | |
edist_lt_of_real {x y : α} {r : ℝ} : edist x y < ennreal.of_real r ↔ dist x y < r | by rw [edist_dist, ennreal.of_real_lt_of_real_iff_of_nonneg dist_nonneg] | lemma | edist_lt_of_real | 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",
"edist_dist",
"ennreal.of_real",
"ennreal.of_real_lt_of_real_iff_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_of_real {x y : α} {r : ℝ} (hr : 0 ≤ r) :
edist x y ≤ ennreal.of_real r ↔ dist x y ≤ r | by rw [edist_dist, ennreal.of_real_le_of_real_iff hr] | lemma | edist_le_of_real | 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",
"ennreal.of_real",
"ennreal.of_real_le_of_real_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_dist (x y : α) : nndist x y = real.to_nnreal (dist x y) | by rw [dist_nndist, real.to_nnreal_coe] | lemma | nndist_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_nndist",
"real.to_nnreal",
"real.to_nnreal_coe"
] | Express `nndist` in terms of `dist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nndist_comm (x y : α) : nndist x y = nndist y x | by simpa only [dist_nndist, nnreal.coe_eq] using dist_comm x y | theorem | nndist_comm | 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_nndist",
"nnreal.coe_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z | dist_triangle _ _ _ | theorem | nndist_triangle | 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"
] | Triangle inequality for the nonnegative distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y | dist_triangle_left _ _ _ | theorem | nndist_triangle_left | 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_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z | dist_triangle_right _ _ _ | theorem | nndist_triangle_right | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_edist (x y : α) : dist x y = (edist x y).to_real | by rw [edist_dist, ennreal.to_real_of_real (dist_nonneg)] | lemma | dist_edist | 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",
"edist_dist",
"ennreal.to_real_of_real"
] | Express `dist` in terms of `edist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball (x : α) (ε : ℝ) : set α | {y | dist y x < ε} | def | metric.ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | `ball x ε` is the set of all points `y` with `dist y x < ε` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ball : y ∈ ball x ε ↔ dist y x < ε | iff.rfl | theorem | metric.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball' : y ∈ ball x ε ↔ dist x y < ε | by rw [dist_comm, mem_ball] | theorem | metric.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"
] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε | dist_nonneg.trans_lt hy | theorem | metric.pos_of_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_self (h : 0 < ε) : x ∈ ball x ε | show dist x x < ε, by rw dist_self; assumption | theorem | metric.mem_ball_self | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_ball : (ball x ε).nonempty ↔ 0 < ε | ⟨λ ⟨x, hx⟩, pos_of_mem_ball hx, λ h, ⟨x, mem_ball_self h⟩⟩ | lemma | metric.nonempty_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 | |
ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 | by rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt] | lemma | metric.ball_eq_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 | |
ball_zero : ball x 0 = ∅ | by rw [ball_eq_empty] | lemma | metric.ball_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 | |
exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' | begin
simp only [mem_ball] at h ⊢,
exact ⟨(ε + dist x y) / 2, by linarith, by linarith⟩,
end | lemma | metric.exists_lt_mem_ball_of_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"
] | [] | If a point belongs to an open ball, then there is a strictly smaller radius whose ball also
contains it.
See also `exists_lt_subset_ball`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball_eq_ball (ε : ℝ) (x : α) :
uniform_space.ball x {p | dist p.2 p.1 < ε} = metric.ball x ε | rfl | lemma | metric.ball_eq_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric.ball",
"uniform_space.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_eq_ball' (ε : ℝ) (x : α) :
uniform_space.ball x {p | dist p.1 p.2 < ε} = metric.ball x ε | by { ext, simp [dist_comm, uniform_space.ball] } | lemma | metric.ball_eq_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_comm",
"metric.ball",
"uniform_space.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_ball_nat (x : α) : (⋃ n : ℕ, ball x n) = univ | Union_eq_univ_iff.2 $ λ y, exists_nat_gt (dist y x) | lemma | metric.Union_ball_nat | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_nat_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_ball_nat_succ (x : α) : (⋃ n : ℕ, ball x (n + 1)) = univ | Union_eq_univ_iff.2 $ λ y, (exists_nat_gt (dist y x)).imp $ λ n hn,
hn.trans (lt_add_one _) | lemma | metric.Union_ball_nat_succ | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_nat_gt",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball (x : α) (ε : ℝ) | {y | dist y x ≤ ε} | def | metric.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"
] | [] | `closed_ball x ε` is the set of all points `y` with `dist y x ≤ ε` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closed_ball : y ∈ closed_ball x ε ↔ dist y x ≤ ε | iff.rfl | theorem | metric.mem_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball' : y ∈ closed_ball x ε ↔ dist x y ≤ ε | by rw [dist_comm, mem_closed_ball] | theorem | metric.mem_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_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sphere (x : α) (ε : ℝ) | {y | dist y x = ε} | def | metric.sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | `sphere x ε` is the set of all points `y` with `dist y x = ε` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_sphere : y ∈ sphere x ε ↔ dist y x = ε | iff.rfl | theorem | metric.mem_sphere | 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 | |
mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε | by rw [dist_comm, mem_sphere] | theorem | metric.mem_sphere' | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x | by { contrapose! hε, symmetry, simpa [hε] using h } | theorem | metric.ne_of_mem_sphere | 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 | |
nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε | dist_nonneg.trans_eq hy | theorem | metric.nonneg_of_mem_sphere | 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 | |
sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ | set.eq_empty_iff_forall_not_mem.mpr $ λ y hy, (nonneg_of_mem_sphere hy).not_lt hε | theorem | metric.sphere_eq_empty_of_neg | 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 | |
sphere_eq_empty_of_subsingleton [subsingleton α] (hε : ε ≠ 0) :
sphere x ε = ∅ | set.eq_empty_iff_forall_not_mem.mpr $ λ y hy, ne_of_mem_sphere hy hε (subsingleton.elim _ _) | theorem | metric.sphere_eq_empty_of_subsingleton | 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 | |
sphere_is_empty_of_subsingleton [subsingleton α] (hε : ε ≠ 0) :
is_empty (sphere x ε) | by simp only [sphere_eq_empty_of_subsingleton hε, set.has_emptyc.emptyc.is_empty α] | theorem | metric.sphere_is_empty_of_subsingleton | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball_self (h : 0 ≤ ε) : x ∈ closed_ball x ε | show dist x x ≤ ε, by rw dist_self; assumption | theorem | metric.mem_closed_ball_self | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_closed_ball : (closed_ball x ε).nonempty ↔ 0 ≤ ε | ⟨λ ⟨x, hx⟩, dist_nonneg.trans hx, λ h, ⟨x, mem_closed_ball_self h⟩⟩ | lemma | metric.nonempty_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_eq_empty : closed_ball x ε = ∅ ↔ ε < 0 | by rw [← not_nonempty_iff_eq_empty, nonempty_closed_ball, not_le] | lemma | metric.closed_ball_eq_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 | |
closed_ball_eq_sphere_of_nonpos (hε : ε ≤ 0) : closed_ball x ε = sphere x ε | set.ext $ λ _, (hε.trans dist_nonneg).le_iff_eq | theorem | metric.closed_ball_eq_sphere_of_nonpos | 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",
"set.ext"
] | Closed balls and spheres coincide when the radius is non-positive | 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 | metric.ball_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sphere_subset_closed_ball : sphere x ε ⊆ closed_ball x ε | λ y, le_of_eq | theorem | metric.sphere_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_disjoint_ball (h : δ + ε ≤ dist x y) : disjoint (closed_ball x δ) (ball y ε) | set.disjoint_left.mpr $
λ a ha1 ha2, (h.trans $ dist_triangle_left _ _ _).not_lt $ add_lt_add_of_le_of_lt ha1 ha2 | lemma | metric.closed_ball_disjoint_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"disjoint",
"dist_triangle_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_disjoint_closed_ball (h : δ + ε ≤ dist x y) : disjoint (ball x δ) (closed_ball y ε) | (closed_ball_disjoint_ball $ by rwa [add_comm, dist_comm]).symm | lemma | metric.ball_disjoint_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"
] | [
"disjoint",
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_disjoint_ball (h : δ + ε ≤ dist x y) : disjoint (ball x δ) (ball y ε) | (closed_ball_disjoint_ball h).mono_left ball_subset_closed_ball | lemma | metric.ball_disjoint_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_disjoint_closed_ball (h : δ + ε < dist x y) :
disjoint (closed_ball x δ) (closed_ball y ε) | set.disjoint_left.mpr $
λ a ha1 ha2, h.not_le $ (dist_triangle_left _ _ _).trans $ add_le_add ha1 ha2 | lemma | metric.closed_ball_disjoint_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"
] | [
"disjoint",
"dist_triangle_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sphere_disjoint_ball : disjoint (sphere x ε) (ball x ε) | set.disjoint_left.mpr $ λ y hy₁ hy₂, absurd hy₁ $ ne_of_lt hy₂ | theorem | metric.sphere_disjoint_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_union_sphere : ball x ε ∪ sphere x ε = closed_ball x ε | set.ext $ λ y, (@le_iff_lt_or_eq ℝ _ _ _).symm | theorem | metric.ball_union_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sphere_union_ball : sphere x ε ∪ ball x ε = closed_ball x ε | by rw [union_comm, ball_union_sphere] | theorem | metric.sphere_union_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 | |
closed_ball_diff_sphere : closed_ball x ε \ sphere x ε = ball x ε | by rw [← ball_union_sphere, set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot] | theorem | metric.closed_ball_diff_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"set.union_diff_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_diff_ball : closed_ball x ε \ ball x ε = sphere x ε | by rw [← ball_union_sphere, set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot] | theorem | metric.closed_ball_diff_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"set.union_diff_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε | by rw [mem_ball', mem_ball] | theorem | metric.mem_ball_comm | 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 | |
mem_closed_ball_comm : x ∈ closed_ball y ε ↔ y ∈ closed_ball x ε | by rw [mem_closed_ball', mem_closed_ball] | theorem | metric.mem_closed_ball_comm | 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 | |
mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε | by rw [mem_sphere', mem_sphere] | theorem | metric.mem_sphere_comm | 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 | |
ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ | λ y (yx : _ < ε₁), lt_of_lt_of_le yx h | theorem | metric.ball_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 | |
closed_ball_eq_bInter_ball : closed_ball x ε = ⋂ δ > ε, ball x δ | by ext y; rw [mem_closed_ball, ← forall_lt_iff_le', mem_Inter₂]; refl | lemma | metric.closed_ball_eq_bInter_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"forall_lt_iff_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ | λ z hz, calc
dist z y ≤ dist z x + dist x y : dist_triangle _ _ _
... < ε₁ + dist x y : add_lt_add_right hz _
... ≤ ε₂ : h | lemma | metric.ball_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_subset_closed_ball (h : ε₁ ≤ ε₂) :
closed_ball x ε₁ ⊆ closed_ball x ε₂ | λ y (yx : _ ≤ ε₁), le_trans yx h | theorem | metric.closed_ball_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_subset_closed_ball' (h : ε₁ + dist x y ≤ ε₂) :
closed_ball x ε₁ ⊆ closed_ball y ε₂ | λ z hz, calc
dist z y ≤ dist z x + dist x y : dist_triangle _ _ _
... ≤ ε₁ + dist x y : add_le_add_right hz _
... ≤ ε₂ : h | lemma | metric.closed_ball_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_subset_ball (h : ε₁ < ε₂) :
closed_ball x ε₁ ⊆ ball x ε₂ | λ y (yh : dist y x ≤ ε₁), lt_of_le_of_lt yh h | theorem | metric.closed_ball_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 | |
closed_ball_subset_ball' (h : ε₁ + dist x y < ε₂) :
closed_ball x ε₁ ⊆ ball y ε₂ | λ z hz, calc
dist z y ≤ dist z x + dist x y : dist_triangle _ _ _
... ≤ ε₁ + dist x y : add_le_add_right hz _
... < ε₂ : h | lemma | metric.closed_ball_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_add_of_nonempty_closed_ball_inter_closed_ball
(h : (closed_ball x ε₁ ∩ closed_ball y ε₂).nonempty) :
dist x y ≤ ε₁ + ε₂ | let ⟨z, hz⟩ := h in calc
dist x y ≤ dist z x + dist z y : dist_triangle_left _ _ _
... ≤ ε₁ + ε₂ : add_le_add hz.1 hz.2 | lemma | metric.dist_le_add_of_nonempty_closed_ball_inter_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_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_add_of_nonempty_closed_ball_inter_ball (h : (closed_ball x ε₁ ∩ ball y ε₂).nonempty) :
dist x y < ε₁ + ε₂ | let ⟨z, hz⟩ := h in calc
dist x y ≤ dist z x + dist z y : dist_triangle_left _ _ _
... < ε₁ + ε₂ : add_lt_add_of_le_of_lt hz.1 hz.2 | lemma | metric.dist_lt_add_of_nonempty_closed_ball_inter_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_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_add_of_nonempty_ball_inter_closed_ball (h : (ball x ε₁ ∩ closed_ball y ε₂).nonempty) :
dist x y < ε₁ + ε₂ | begin
rw inter_comm at h,
rw [add_comm, dist_comm],
exact dist_lt_add_of_nonempty_closed_ball_inter_ball h
end | lemma | metric.dist_lt_add_of_nonempty_ball_inter_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_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).nonempty) :
dist x y < ε₁ + ε₂ | dist_lt_add_of_nonempty_closed_ball_inter_ball $
h.mono (inter_subset_inter ball_subset_closed_ball subset.rfl) | lemma | metric.dist_lt_add_of_nonempty_ball_inter_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 | |
Union_closed_ball_nat (x : α) : (⋃ n : ℕ, closed_ball x n) = univ | Union_eq_univ_iff.2 $ λ y, exists_nat_ge (dist y x) | lemma | metric.Union_closed_ball_nat | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_nat_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_inter_closed_ball_nat (s : set α) (x : α) :
(⋃ (n : ℕ), s ∩ closed_ball x n) = s | by rw [← inter_Union, Union_closed_ball_nat, inter_univ] | lemma | metric.Union_inter_closed_ball_nat | 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 | |
ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ | λ z zx, by rw ← add_sub_cancel'_right ε₁ ε₂; exact
lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h) | theorem | metric.ball_subset | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε | ball_subset $ by rw sub_self_div_two; exact le_of_lt h | theorem | metric.ball_half_subset | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"sub_self_div_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε | ⟨_, sub_pos.2 h, ball_subset $ by rw sub_sub_self⟩ | theorem | metric.exists_ball_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 | |
forall_of_forall_mem_closed_ball (p : α → Prop) (x : α)
(H : ∃ᶠ (R : ℝ) in at_top, ∀ y ∈ closed_ball x R, p y) (y : α) :
p y | begin
obtain ⟨R, hR, h⟩ : ∃ (R : ℝ) (H : dist y x ≤ R), ∀ (z : α), z ∈ closed_ball x R → p z :=
frequently_iff.1 H (Ici_mem_at_top (dist y x)),
exact h _ hR
end | lemma | metric.forall_of_forall_mem_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"
] | [] | If a property holds for all points in closed balls of arbitrarily large radii, then it holds for
all points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forall_of_forall_mem_ball (p : α → Prop) (x : α)
(H : ∃ᶠ (R : ℝ) in at_top, ∀ y ∈ ball x R, p y) (y : α) :
p y | begin
obtain ⟨R, hR, h⟩ : ∃ (R : ℝ) (H : dist y x < R), ∀ (z : α), z ∈ ball x R → p z :=
frequently_iff.1 H (Ioi_mem_at_top (dist y x)),
exact h _ hR
end | lemma | metric.forall_of_forall_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"
] | [] | If a property holds for all points in balls of arbitrarily large radii, then it holds for all
points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_iff {s : set α} :
is_bounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C | by rw [is_bounded_def, ← filter.mem_sets, (@pseudo_metric_space.cobounded_sets α _).out,
mem_set_of_eq, compl_compl] | theorem | metric.is_bounded_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"compl_compl",
"filter.mem_sets"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_iff_eventually {s : set α} :
is_bounded s ↔ ∀ᶠ C in at_top, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C | is_bounded_iff.trans ⟨λ ⟨C, h⟩, eventually_at_top.2 ⟨C, λ C' hC' x hx y hy, (h hx hy).trans hC'⟩,
eventually.exists⟩ | theorem | metric.is_bounded_iff_eventually | 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 | |
is_bounded_iff_exists_ge {s : set α} (c : ℝ) :
is_bounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C | ⟨λ h, ((eventually_ge_at_top c).and (is_bounded_iff_eventually.1 h)).exists,
λ h, is_bounded_iff.2 $ h.imp $ λ _, and.right⟩ | theorem | metric.is_bounded_iff_exists_ge | 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 | |
is_bounded_iff_nndist {s : set α} :
is_bounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C | by simp only [is_bounded_iff_exists_ge 0, nnreal.exists, ← nnreal.coe_le_coe, ← dist_nndist,
nnreal.coe_mk, exists_prop] | theorem | metric.is_bounded_iff_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_nndist",
"exists_prop",
"nnreal.coe_le_coe",
"nnreal.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_uniform_space_eq : ‹pseudo_metric_space α›.to_uniform_space =
uniform_space_of_dist dist dist_self dist_comm dist_triangle | uniform_space_eq pseudo_metric_space.uniformity_dist | theorem | metric.to_uniform_space_eq | 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",
"uniform_space_eq",
"uniform_space_of_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_dist :
(𝓤 α).has_basis (λ ε : ℝ, 0 < ε) (λ ε, {p:α×α | dist p.1 p.2 < ε}) | begin
rw [to_uniform_space_eq],
exact uniform_space.has_basis_of_fun (exists_gt _) _ _ _ _ _
end | theorem | metric.uniformity_basis_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"uniform_space.has_basis_of_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ}
(hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ i (hi : p i), f i ≤ ε) :
(𝓤 α).has_basis p (λ i, {p:α×α | dist p.1 p.2 < f i}) | begin
refine ⟨λ s, uniformity_basis_dist.mem_iff.trans _⟩,
split,
{ rintros ⟨ε, ε₀, hε⟩,
obtain ⟨i, hi, H⟩ : ∃ i (hi : p i), f i ≤ ε, from hf ε₀,
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 | metric.mk_uniformity_basis | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | Given `f : β → ℝ`, 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_dist`, `uniformity_basis_dist_inv_nat_succ`,
and `uniformity_basis_dist_inv_nat_pos`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_basis_dist_rat :
(𝓤 α).has_basis (λ r : ℚ, 0 < r) (λ r, {p : α × α | dist p.1 p.2 < r}) | metric.mk_uniformity_basis (λ _, rat.cast_pos.2) $ λ ε hε,
let ⟨r, hr0, hrε⟩ := exists_rat_btwn hε in ⟨r, rat.cast_pos.1 hr0, hrε.le⟩ | theorem | metric.uniformity_basis_dist_rat | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_rat_btwn",
"metric.mk_uniformity_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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