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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
comap_id : @comap X X Z _ _ id = id | by { ext, simp only [continuous_id, id.def, function.comp.right_id, coe_comap] } | lemma | locally_constant.comap_id | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp [topological_space Z]
(f : X → Y) (g : Y → Z) (hf : continuous f) (hg : continuous g) :
@comap _ _ α _ _ f ∘ comap g = comap (g ∘ f) | by { ext, simp only [hf, hg, hg.comp hf, coe_comap] } | lemma | locally_constant.comap_comp | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_const (f : X → Y) (y : Y) (h : ∀ x, f x = y) :
(comap f : locally_constant Y Z → locally_constant X Z) =
λ g, ⟨λ x, g y, is_locally_constant.const _⟩ | begin
ext, rw coe_comap,
{ simp only [h, coe_mk, function.comp_app] },
{ rw show f = λ x, y, by ext; apply h,
exact continuous_const }
end | lemma | locally_constant.comap_const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous_const",
"is_locally_constant.const",
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc {X α β : Type*} [topological_space X] {g : α → β} (f : X → α) (h : locally_constant X β)
(cond : g ∘ f = h) (inj : function.injective g) : locally_constant X α | { to_fun := f,
is_locally_constant := is_locally_constant.desc _ g (by { rw cond, exact h.2 }) inj } | def | locally_constant.desc | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"is_locally_constant.desc",
"locally_constant",
"topological_space"
] | If a locally constant function factors through an injection, then it factors through a locally
constant function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_desc {X α β : Type*} [topological_space X] (f : X → α) (g : α → β)
(h : locally_constant X β) (cond : g ∘ f = h) (inj : function.injective g) :
⇑(desc f h cond inj) = f | rfl | lemma | locally_constant.coe_desc | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator (hU : is_clopen U) :
locally_constant X R | { to_fun := set.mul_indicator U f,
is_locally_constant :=
begin
rw is_locally_constant.iff_exists_open, rintros x,
obtain ⟨V, hV, hx, h'⟩ := (is_locally_constant.iff_exists_open _).1 f.is_locally_constant x,
by_cases x ∈ U,
{ refine ⟨U ∩ V, is_open.inter hU.1 hV, set.mem_inter h hx, _⟩, ri... | def | locally_constant.mul_indicator | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_clopen",
"is_clopen.compl",
"is_locally_constant",
"is_locally_constant.iff_exists_open",
"is_open.inter",
"locally_constant",
"set.mem_compl_iff",
"set.mem_inter",
"set.mem_inter_iff",
"set.mul_indicator",
"set.mul_indicator_of_mem"
] | Given a clopen set `U` and a locally constant function `f`, `locally_constant.mul_indicator`
returns the locally constant function that is `f` on `U` and `1` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_apply_eq_if (hU : is_clopen U) :
mul_indicator f hU a = if a ∈ U then f a else 1 | set.mul_indicator_apply U f a | theorem | locally_constant.mul_indicator_apply_eq_if | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_clopen",
"set.mul_indicator_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_of_mem (hU : is_clopen U) (h : a ∈ U) : f.mul_indicator hU a = f a | by{ rw mul_indicator_apply, apply set.mul_indicator_of_mem h, } | theorem | locally_constant.mul_indicator_of_mem | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_clopen",
"set.mul_indicator_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_of_not_mem (hU : is_clopen U) (h : a ∉ U) : f.mul_indicator hU a = 1 | by{ rw mul_indicator_apply, apply set.mul_indicator_of_not_mem h, } | theorem | locally_constant.mul_indicator_of_not_mem | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_clopen",
"set.mul_indicator_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_lipschitz_add [add_monoid β] : Prop | ( lipschitz_add : ∃ C, lipschitz_with C (λ p : β × β, p.1 + p.2) ) | class | has_lipschitz_add | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"add_monoid",
"lipschitz_with"
] | Class `has_lipschitz_add M` says that the addition `(+) : X × X → X` is Lipschitz jointly in
the two arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_lipschitz_mul [monoid β] : Prop | ( lipschitz_mul : ∃ C, lipschitz_with C (λ p : β × β, p.1 * p.2) ) | class | has_lipschitz_mul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"lipschitz_with",
"monoid"
] | Class `has_lipschitz_mul M` says that the multiplication `(*) : X × X → X` is Lipschitz jointly
in the two arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_lipschitz_mul.C [_i : has_lipschitz_mul β] : ℝ≥0 | classical.some _i.lipschitz_mul | def | has_lipschitz_mul.C | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"has_lipschitz_mul"
] | The Lipschitz constant of a monoid `β` satisfying `has_lipschitz_mul` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_with_lipschitz_const_mul_edist [_i : has_lipschitz_mul β] :
lipschitz_with (has_lipschitz_mul.C β) (λ p : β × β, p.1 * p.2) | classical.some_spec _i.lipschitz_mul | lemma | lipschitz_with_lipschitz_const_mul_edist | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"has_lipschitz_mul",
"has_lipschitz_mul.C",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with_lipschitz_const_mul :
∀ p q : β × β, dist (p.1 * p.2) (q.1 * q.2) ≤ (has_lipschitz_mul.C β) * dist p q | begin
rw ← lipschitz_with_iff_dist_le_mul,
exact lipschitz_with_lipschitz_const_mul_edist,
end | lemma | lipschitz_with_lipschitz_const_mul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"has_lipschitz_mul.C",
"lipschitz_with_iff_dist_le_mul",
"lipschitz_with_lipschitz_const_mul_edist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_lipschitz_mul.has_continuous_mul : has_continuous_mul β | ⟨ lipschitz_with_lipschitz_const_mul_edist.continuous ⟩ | instance | has_lipschitz_mul.has_continuous_mul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.has_lipschitz_mul (s : submonoid β) : has_lipschitz_mul s | { lipschitz_mul := ⟨has_lipschitz_mul.C β, begin
rintros ⟨x₁, x₂⟩ ⟨y₁, y₂⟩,
convert lipschitz_with_lipschitz_const_mul_edist ⟨(x₁:β), x₂⟩ ⟨y₁, y₂⟩ using 1
end⟩ } | instance | submonoid.has_lipschitz_mul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"has_lipschitz_mul",
"lipschitz_with_lipschitz_const_mul_edist",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_opposite.has_lipschitz_mul : has_lipschitz_mul βᵐᵒᵖ | { lipschitz_mul := ⟨has_lipschitz_mul.C β, λ ⟨x₁, x₂⟩ ⟨y₁, y₂⟩,
(lipschitz_with_lipschitz_const_mul_edist ⟨x₂.unop, x₁.unop⟩ ⟨y₂.unop, y₁.unop⟩).trans_eq
(congr_arg _ $ max_comm _ _)⟩ } | instance | mul_opposite.has_lipschitz_mul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"has_lipschitz_mul",
"lipschitz_with_lipschitz_const_mul_edist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.has_lipschitz_add : has_lipschitz_add ℝ | { lipschitz_add := ⟨2, begin
rw lipschitz_with_iff_dist_le_mul,
intros p q,
simp only [real.dist_eq, prod.dist_eq, prod.fst_sub, prod.snd_sub, nnreal.coe_one,
nnreal.coe_bit0],
convert le_trans (abs_add (p.1 - q.1) (p.2 - q.2)) _ using 2,
{ abel },
have := le_max_left (|p.1 - q.1|) (|p.2 -... | instance | real.has_lipschitz_add | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"abs_add",
"has_lipschitz_add",
"lipschitz_with_iff_dist_le_mul",
"nnreal.coe_bit0",
"nnreal.coe_one",
"prod.dist_eq",
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.has_lipschitz_add : has_lipschitz_add ℝ≥0 | { lipschitz_add := ⟨has_lipschitz_add.C ℝ, begin
rintros ⟨x₁, x₂⟩ ⟨y₁, y₂⟩,
convert lipschitz_with_lipschitz_const_add_edist ⟨(x₁:ℝ), x₂⟩ ⟨y₁, y₂⟩ using 1
end⟩ } | instance | nnreal.has_lipschitz_add | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"has_lipschitz_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_bounded_smul : Prop | ( dist_smul_pair' : ∀ x : α, ∀ y₁ y₂ : β, dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂ )
( dist_pair_smul' : ∀ x₁ x₂ : α, ∀ y : β, dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0 ) | class | has_bounded_smul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [] | Mixin typeclass on a scalar action of a metric space `α` on a metric space `β` both with
distinguished points `0`, requiring compatibility of the action in the sense that
`dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂` and
`dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_smul_pair (x : α) (y₁ y₂ : β) : dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂ | has_bounded_smul.dist_smul_pair' x y₁ y₂ | lemma | dist_smul_pair | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pair_smul (x₁ x₂ : α) (y : β) : dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0 | has_bounded_smul.dist_pair_smul' x₁ x₂ y | lemma | dist_pair_smul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_bounded_smul.has_continuous_smul : has_continuous_smul α β | { continuous_smul := begin
rw metric.continuous_iff,
rintros ⟨a, b⟩ ε hε,
have : 0 ≤ dist a 0 := dist_nonneg,
have : 0 ≤ dist b 0 := dist_nonneg,
let δ : ℝ := min 1 ((dist a 0 + dist b 0 + 2)⁻¹ * ε),
have hδ_pos : 0 < δ,
{ refine lt_min_iff.mpr ⟨by norm_num, mul_pos _ hε⟩,
rw inv_pos,
... | instance | has_bounded_smul.has_continuous_smul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"dist_comm",
"dist_nonneg",
"dist_pair_smul",
"dist_smul_pair",
"dist_triangle",
"has_continuous_smul",
"inv_mul_lt_iff",
"inv_pos",
"metric.continuous_iff"
] | The typeclass `has_bounded_smul` on a metric-space scalar action implies continuity of the
action. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.has_bounded_smul : has_bounded_smul ℝ ℝ | { dist_smul_pair' := λ x y₁ y₂, by simpa [real.dist_eq, mul_sub] using (abs_mul x (y₁ - y₂)).le,
dist_pair_smul' := λ x₁ x₂ y, by simpa [real.dist_eq, sub_mul] using (abs_mul (x₁ - x₂) y).le } | instance | real.has_bounded_smul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"abs_mul",
"has_bounded_smul",
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.has_bounded_smul : has_bounded_smul ℝ≥0 ℝ≥0 | { dist_smul_pair' := λ x y₁ y₂, by convert dist_smul_pair (x:ℝ) (y₁:ℝ) y₂ using 1,
dist_pair_smul' := λ x₁ x₂ y, by convert dist_pair_smul (x₁:ℝ) x₂ (y:ℝ) using 1 } | instance | nnreal.has_bounded_smul | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"dist_pair_smul",
"dist_smul_pair",
"has_bounded_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_bounded_smul.op [has_smul αᵐᵒᵖ β] [is_central_scalar α β] :
has_bounded_smul αᵐᵒᵖ β | { dist_smul_pair' := mul_opposite.rec $ λ x y₁ y₂,
by simpa only [op_smul_eq_smul] using dist_smul_pair x y₁ y₂,
dist_pair_smul' := mul_opposite.rec $ λ x₁, mul_opposite.rec $ λ x₂ y,
by simpa only [op_smul_eq_smul] using dist_pair_smul x₁ x₂ y } | instance | has_bounded_smul.op | topology.metric_space | src/topology/metric_space/algebra.lean | [
"topology.algebra.mul_action",
"topology.metric_space.lipschitz"
] | [
"dist_pair_smul",
"dist_smul_pair",
"has_bounded_smul",
"has_smul",
"is_central_scalar",
"mul_opposite.rec"
] | If a scalar is central, then its right action is bounded when its left action is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antilipschitz_with [pseudo_emetric_space α] [pseudo_emetric_space β] (K : ℝ≥0) (f : α → β) | ∀ x y, edist x y ≤ K * edist (f x) (f y) | def | antilipschitz_with | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"pseudo_emetric_space"
] | We say that `f : α → β` is `antilipschitz_with K` if for any two points `x`, `y` we have
`edist x y ≤ K * edist (f x) (f y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antilipschitz_with.edist_lt_top [pseudo_emetric_space α] [pseudo_metric_space β] {K : ℝ≥0}
{f : α → β} (h : antilipschitz_with K f) (x y : α) : edist x y < ⊤ | (h x y).trans_lt $ ennreal.mul_lt_top ennreal.coe_ne_top (edist_ne_top _ _) | lemma | antilipschitz_with.edist_lt_top | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"edist_ne_top",
"ennreal.coe_ne_top",
"ennreal.mul_lt_top",
"pseudo_emetric_space",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_with.edist_ne_top [pseudo_emetric_space α] [pseudo_metric_space β] {K : ℝ≥0}
{f : α → β} (h : antilipschitz_with K f) (x y : α) : edist x y ≠ ⊤ | (h.edist_lt_top x y).ne | lemma | antilipschitz_with.edist_ne_top | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"pseudo_emetric_space",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_with_iff_le_mul_nndist :
antilipschitz_with K f ↔ ∀ x y, nndist x y ≤ K * nndist (f x) (f y) | by { simp only [antilipschitz_with, edist_nndist], norm_cast } | lemma | antilipschitz_with_iff_le_mul_nndist | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"edist_nndist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_with_iff_le_mul_dist :
antilipschitz_with K f ↔ ∀ x y, dist x y ≤ K * dist (f x) (f y) | by { simp only [antilipschitz_with_iff_le_mul_nndist, dist_nndist], norm_cast } | lemma | antilipschitz_with_iff_le_mul_dist | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"antilipschitz_with_iff_le_mul_nndist",
"dist_nndist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_nndist (hf : antilipschitz_with K f) (x y : α) :
K⁻¹ * nndist x y ≤ nndist (f x) (f y) | by simpa only [div_eq_inv_mul] using nnreal.div_le_of_le_mul' (hf.le_mul_nndist x y) | lemma | antilipschitz_with.mul_le_nndist | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"div_eq_inv_mul",
"nnreal.div_le_of_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_dist (hf : antilipschitz_with K f) (x y : α) :
(K⁻¹ * dist x y : ℝ) ≤ dist (f x) (f y) | by exact_mod_cast hf.mul_le_nndist x y | lemma | antilipschitz_with.mul_le_dist | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
K (hf : antilipschitz_with K f) : ℝ≥0 | K | def | antilipschitz_with.K | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with"
] | Extract the constant from `hf : antilipschitz_with K f`. This is useful, e.g.,
if `K` is given by a long formula, and we want to reuse this value. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective {α : Type*} {β : Type*} [emetric_space α] [pseudo_emetric_space β]
{K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) : function.injective f | λ x y h, by simpa only [h, edist_self, mul_zero, edist_le_zero] using hf x y | lemma | antilipschitz_with.injective | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"edist_le_zero",
"emetric_space",
"mul_zero",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_edist (hf : antilipschitz_with K f) (x y : α) :
(K⁻¹ * edist x y : ℝ≥0∞) ≤ edist (f x) (f y) | begin
rw [mul_comm, ← div_eq_mul_inv],
exact ennreal.div_le_of_le_mul' (hf x y)
end | lemma | antilipschitz_with.mul_le_edist | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"div_eq_mul_inv",
"ennreal.div_le_of_le_mul'",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_preimage_le (hf : antilipschitz_with K f) (s : set β) : diam (f ⁻¹' s) ≤ K * diam s | diam_le $ λ x hx y hy, (hf x y).trans $ mul_le_mul_left' (edist_le_diam_of_mem hx hy) K | lemma | antilipschitz_with.ediam_preimage_le | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_ediam_image (hf : antilipschitz_with K f) (s : set α) : diam s ≤ K * diam (f '' s) | (diam_mono (subset_preimage_image _ _)).trans (hf.ediam_preimage_le (f '' s)) | lemma | antilipschitz_with.le_mul_ediam_image | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : antilipschitz_with 1 (id : α → α) | λ x y, by simp only [ennreal.coe_one, one_mul, id, le_refl] | lemma | antilipschitz_with.id | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"ennreal.coe_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {Kg : ℝ≥0} {g : β → γ} (hg : antilipschitz_with Kg g)
{Kf : ℝ≥0} {f : α → β} (hf : antilipschitz_with Kf f) :
antilipschitz_with (Kf * Kg) (g ∘ f) | λ x y,
calc edist x y ≤ Kf * edist (f x) (f y) : hf x y
... ≤ Kf * (Kg * edist (g (f x)) (g (f y))) : ennreal.mul_left_mono (hg _ _)
... = _ : by rw [ennreal.coe_mul, mul_assoc] | lemma | antilipschitz_with.comp | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"ennreal.coe_mul",
"ennreal.mul_left_mono",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict (hf : antilipschitz_with K f) (s : set α) :
antilipschitz_with K (s.restrict f) | λ x y, hf x y | lemma | antilipschitz_with.restrict | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cod_restrict (hf : antilipschitz_with K f) {s : set β} (hs : ∀ x, f x ∈ s) :
antilipschitz_with K (s.cod_restrict f hs) | λ x y, hf x y | lemma | antilipschitz_with.cod_restrict | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_right_inv_on' {s : set α} (hf : antilipschitz_with K (s.restrict f))
{g : β → α} {t : set β} (g_maps : maps_to g t s) (g_inv : right_inv_on g f t) :
lipschitz_with K (t.restrict g) | λ x y, by simpa only [restrict_apply, g_inv x.mem, g_inv y.mem, subtype.edist_eq, subtype.coe_mk]
using hf ⟨g x, g_maps x.mem⟩ ⟨g y, g_maps y.mem⟩ | lemma | antilipschitz_with.to_right_inv_on' | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"lipschitz_with",
"subtype.coe_mk",
"subtype.edist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_right_inv_on (hf : antilipschitz_with K f) {g : β → α} {t : set β}
(h : right_inv_on g f t) :
lipschitz_with K (t.restrict g) | (hf.restrict univ).to_right_inv_on' (maps_to_univ g t) h | lemma | antilipschitz_with.to_right_inv_on | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_right_inverse (hf : antilipschitz_with K f) {g : β → α} (hg : function.right_inverse g f) :
lipschitz_with K g | begin
intros x y,
have := hf (g x) (g y),
rwa [hg x, hg y] at this
end | lemma | antilipschitz_with.to_right_inverse | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_uniformity_le (hf : antilipschitz_with K f) :
(𝓤 β).comap (prod.map f f) ≤ 𝓤 α | begin
refine ((uniformity_basis_edist.comap _).le_basis_iff uniformity_basis_edist).2 (λ ε h₀, _),
refine ⟨K⁻¹ * ε, ennreal.mul_pos (ennreal.inv_ne_zero.2 ennreal.coe_ne_top) h₀.ne', _⟩,
refine λ x hx, (hf x.1 x.2).trans_lt _,
rw [mul_comm, ← div_eq_mul_inv] at hx,
rw mul_comm,
exact ennreal.mul_lt_of_lt_di... | lemma | antilipschitz_with.comap_uniformity_le | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"div_eq_mul_inv",
"ennreal.coe_ne_top",
"ennreal.mul_lt_of_lt_div",
"ennreal.mul_pos",
"mul_comm",
"uniformity_basis_edist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
uniform_inducing f | ⟨le_antisymm hf.comap_uniformity_le hfc.le_comap⟩ | lemma | antilipschitz_with.uniform_inducing | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"uniform_continuous",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding {α : Type*} {β : Type*} [emetric_space α] [pseudo_emetric_space β]
{K : ℝ≥0} {f : α → β} (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
uniform_embedding f | ⟨hf.uniform_inducing hfc, hf.injective⟩ | lemma | antilipschitz_with.uniform_embedding | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"emetric_space",
"pseudo_emetric_space",
"uniform_continuous",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_range [complete_space α] (hf : antilipschitz_with K f)
(hfc : uniform_continuous f) : is_complete (range f) | (hf.uniform_inducing hfc).is_complete_range | lemma | antilipschitz_with.is_complete_range | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"complete_space",
"is_complete",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_range {α β : Type*} [pseudo_emetric_space α] [emetric_space β] [complete_space α]
{f : α → β} {K : ℝ≥0} (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
is_closed (range f) | (hf.is_complete_range hfc).is_closed | lemma | antilipschitz_with.is_closed_range | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"complete_space",
"emetric_space",
"is_closed",
"pseudo_emetric_space",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding {α : Type*} {β : Type*} [emetric_space α] [emetric_space β] {K : ℝ≥0}
{f : α → β} [complete_space α] (hf : antilipschitz_with K f) (hfc : uniform_continuous f) :
closed_embedding f | { closed_range := hf.is_closed_range hfc,
.. (hf.uniform_embedding hfc).embedding } | lemma | antilipschitz_with.closed_embedding | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"closed_embedding",
"complete_space",
"embedding",
"emetric_space",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_coe (s : set α) : antilipschitz_with 1 (coe : s → α) | antilipschitz_with.id.restrict s | lemma | antilipschitz_with.subtype_coe | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subsingleton [subsingleton α] {K : ℝ≥0} : antilipschitz_with K f | λ x y, by simp only [subsingleton.elim x y, edist_self, zero_le] | lemma | antilipschitz_with.of_subsingleton | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton {α β} [emetric_space α] [pseudo_emetric_space β] {f : α → β}
(h : antilipschitz_with 0 f) : subsingleton α | ⟨λ x y, edist_le_zero.1 $ (h x y).trans_eq $ zero_mul _⟩ | lemma | antilipschitz_with.subsingleton | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"emetric_space",
"pseudo_emetric_space",
"zero_mul"
] | If `f : α → β` is `0`-antilipschitz, then `α` is a `subsingleton`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_preimage (hf : antilipschitz_with K f)
{s : set β} (hs : bounded s) :
bounded (f ⁻¹' s) | exists.intro (K * diam s) $ λ x hx y hy,
calc dist x y ≤ K * dist (f x) (f y) : hf.le_mul_dist x y
... ≤ K * diam s : mul_le_mul_of_nonneg_left (dist_le_diam_of_mem hs hx hy) K.2 | lemma | antilipschitz_with.bounded_preimage | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cobounded (hf : antilipschitz_with K f) : tendsto f (cobounded α) (cobounded β) | compl_surjective.forall.2 $ λ s (hs : is_bounded s), metric.is_bounded_iff.2 $
hf.bounded_preimage $ metric.is_bounded_iff.1 hs | lemma | antilipschitz_with.tendsto_cobounded | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proper_space {α : Type*} [metric_space α] {K : ℝ≥0} {f : α → β} [proper_space α]
(hK : antilipschitz_with K f) (f_cont : continuous f) (hf : function.surjective f) :
proper_space β | begin
apply proper_space_of_compact_closed_ball_of_le 0 (λx₀ r hr, _),
let K := f ⁻¹' (closed_ball x₀ r),
have A : is_closed K := is_closed_ball.preimage f_cont,
have B : bounded K := hK.bounded_preimage bounded_closed_ball,
have : is_compact K := is_compact_iff_is_closed_bounded.2 ⟨A, B⟩,
convert this.imag... | lemma | antilipschitz_with.proper_space | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"continuous",
"is_closed",
"is_compact",
"metric_space",
"proper_space",
"proper_space_of_compact_closed_ball_of_le"
] | The image of a proper space under an expanding onto map is proper. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_of_image2_left (f : α → β → γ) {K₁ : ℝ≥0}
(hf : ∀ b, antilipschitz_with K₁ (λ a, f a b))
{s : set α} {t : set β} (hst : bounded (set.image2 f s t)) :
bounded s ∨ bounded t | begin
contrapose! hst,
obtain ⟨b, hb⟩ : t.nonempty := nonempty_of_unbounded hst.2,
have : ¬bounded (set.image2 f s {b}),
{ intro h,
apply hst.1,
rw set.image2_singleton_right at h,
replace h := (hf b).bounded_preimage h,
refine h.mono (subset_preimage_image _ _) },
exact mt (bounded.mono (imag... | lemma | antilipschitz_with.bounded_of_image2_left | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"set.image2",
"set.image2_singleton_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_of_image2_right {f : α → β → γ} {K₂ : ℝ≥0}
(hf : ∀ a, antilipschitz_with K₂ (f a))
{s : set α} {t : set β} (hst : bounded (set.image2 f s t)) :
bounded s ∨ bounded t | or.symm $ bounded_of_image2_left (flip f) hf $ image2_swap f s t ▸ hst | lemma | antilipschitz_with.bounded_of_image2_right | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"set.image2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.to_right_inverse [pseudo_emetric_space α] [pseudo_emetric_space β] {K : ℝ≥0}
{f : α → β} (hf : lipschitz_with K f) {g : β → α} (hg : function.right_inverse g f) :
antilipschitz_with K g | λ x y, by simpa only [hg _] using hf (g x) (g y) | lemma | lipschitz_with.to_right_inverse | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"antilipschitz_with",
"lipschitz_with",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.proper_space [pseudo_metric_space α] [metric_space β] [proper_space β]
{K : ℝ≥0} {f : α ≃ₜ β} (hK : lipschitz_with K f) :
proper_space α | (hK.to_right_inverse f.right_inv).proper_space f.symm.continuous f.symm.surjective | theorem | lipschitz_with.proper_space | topology.metric_space | src/topology/metric_space/antilipschitz.lean | [
"topology.metric_space.lipschitz",
"topology.uniform_space.complete_separated"
] | [
"lipschitz_with",
"metric_space",
"proper_space",
"pseudo_metric_space"
] | The preimage of a proper space under a Lipschitz homeomorphism is proper. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
baire_space (α : Type*) [topological_space α] : Prop | (baire_property : ∀ f : ℕ → set α, (∀ n, is_open (f n)) → (∀ n, dense (f n)) → dense (⋂n, f n)) | class | baire_space | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"is_open",
"topological_space"
] | The property `baire_space α` means that the topological space `α` has the Baire property:
any countable intersection of open dense subsets is dense.
Formulated here when the source space is ℕ (and subsumed below by `dense_Inter_of_open` working
with any encodable source space). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
baire_category_theorem_emetric_complete [pseudo_emetric_space α] [complete_space α] :
baire_space α | begin
refine ⟨λ f ho hd, _⟩,
let B : ℕ → ℝ≥0∞ := λn, 1/2^n,
have Bpos : ∀n, 0 < B n,
{ intro n,
simp only [B, one_div, one_mul, ennreal.inv_pos],
exact pow_ne_top two_ne_top },
/- Translate the density assumption into two functions `center` and `radius` associating
to any n, x, δ, δpos a center and ... | instance | baire_category_theorem_emetric_complete | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"baire_space",
"cauchy_seq",
"cauchy_seq_of_edist_le_geometric_two",
"cauchy_seq_tendsto_of_complete",
"closure",
"complete_space",
"ennreal.add_halves",
"ennreal.half_pos",
"ennreal.inv_pos",
"exists_prop",
"filter.eventually_ge_at_top",
"le_rfl",
"mem_closure_iff_nhds_basis",
"nat.le_ind... | Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete pseudo_emetric spaces are Baire. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
baire_category_theorem_locally_compact [topological_space α] [t2_space α]
[locally_compact_space α] :
baire_space α | begin
constructor,
intros f ho hd,
/- To prove that an intersection of open dense subsets is dense, prove that its intersection
with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,... | instance | baire_category_theorem_locally_compact | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"baire_space",
"exists_positive_compacts_subset",
"interior",
"interior_subset",
"is_compact",
"is_compact.is_closed",
"is_compact.nonempty_Inter_of_sequence_nonempty_compact_closed",
"is_open_interior",
"locally_compact_space",
"t2_space",
"topological_space"
] | The second theorem states that locally compact spaces are Baire. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_Inter_of_open_nat {f : ℕ → set α} (ho : ∀ n, is_open (f n)) (hd : ∀ n, dense (f n)) :
dense (⋂ n, f n) | baire_space.baire_property f ho hd | theorem | dense_Inter_of_open_nat | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"is_open"
] | Definition of a Baire space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_sInter_of_open {S : set (set α)} (ho : ∀s∈S, is_open s) (hS : S.countable)
(hd : ∀s∈S, dense s) : dense (⋂₀S) | begin
cases S.eq_empty_or_nonempty with h h,
{ simp [h] },
{ rcases hS.exists_eq_range h with ⟨f, hf⟩,
have F : ∀n, f n ∈ S := λn, by rw hf; exact mem_range_self _,
rw [hf, sInter_range],
exact dense_Inter_of_open_nat (λn, ho _ (F n)) (λn, hd _ (F n)) }
end | theorem | dense_sInter_of_open | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_Inter_of_open_nat",
"is_open"
] | Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_bInter_of_open {S : set β} {f : β → set α} (ho : ∀s∈S, is_open (f s))
(hS : S.countable) (hd : ∀s∈S, dense (f s)) : dense (⋂s∈S, f s) | begin
rw ← sInter_image,
apply dense_sInter_of_open,
{ rwa ball_image_iff },
{ exact hS.image _ },
{ rwa ball_image_iff }
end | theorem | dense_bInter_of_open | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_sInter_of_open",
"is_open"
] | Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_Inter_of_open [encodable β] {f : β → set α} (ho : ∀s, is_open (f s))
(hd : ∀s, dense (f s)) : dense (⋂s, f s) | begin
rw ← sInter_range,
apply dense_sInter_of_open,
{ rwa forall_range_iff },
{ exact countable_range _ },
{ rwa forall_range_iff }
end | theorem | dense_Inter_of_open | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_sInter_of_open",
"encodable",
"is_open"
] | Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_residual {s : set α} :
s ∈ residual α ↔ ∃ t ⊆ s, is_Gδ t ∧ dense t | begin
split,
{ rw mem_residual_iff,
rintros ⟨S, hSo, hSd, Sct, Ss⟩,
refine ⟨_, Ss, ⟨_, λ t ht, hSo _ ht, Sct, rfl⟩, _⟩,
exact dense_sInter_of_open hSo Sct hSd, },
rintros ⟨t, ts, ho, hd⟩,
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts,
end | lemma | mem_residual | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_sInter_of_open",
"is_Gδ",
"mem_residual_iff",
"residual",
"residual_of_dense_Gδ"
] | A set is residual (comeagre) if and only if it includes a dense `Gδ` set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_residual {p : α → Prop} :
(∀ᶠ x in residual α, p x) ↔ ∃ (t : set α), is_Gδ t ∧ dense t ∧ ∀ (x : α), x ∈ t → p x | begin
-- this can probably be improved...
convert @mem_residual _ _ _ p,
simp_rw [exists_prop, and_comm ((_ : set α) ⊆ p), and_assoc],
refl,
end | lemma | eventually_residual | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"exists_prop",
"is_Gδ",
"mem_residual",
"residual"
] | A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_of_mem_residual {s : set α} (hs : s ∈ residual α) : dense s | let ⟨t, hts, _, hd⟩ := mem_residual.1 hs in hd.mono hts | lemma | dense_of_mem_residual | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"residual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_sInter_of_Gδ {S : set (set α)} (ho : ∀s∈S, is_Gδ s) (hS : S.countable)
(hd : ∀s∈S, dense s) : dense (⋂₀S) | dense_of_mem_residual ((countable_sInter_mem hS).mpr
(λ s hs, residual_of_dense_Gδ (ho _ hs) (hd _ hs))) | theorem | dense_sInter_of_Gδ | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"countable_sInter_mem",
"dense",
"dense_of_mem_residual",
"is_Gδ",
"residual_of_dense_Gδ"
] | Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with ⋂₀. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_Inter_of_Gδ [encodable β] {f : β → set α} (ho : ∀s, is_Gδ (f s))
(hd : ∀s, dense (f s)) : dense (⋂s, f s) | begin
rw ← sInter_range,
exact dense_sInter_of_Gδ (forall_range_iff.2 ‹_›) (countable_range _) (forall_range_iff.2 ‹_›)
end | theorem | dense_Inter_of_Gδ | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_sInter_of_Gδ",
"encodable",
"is_Gδ"
] | Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is an encodable type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_bInter_of_Gδ {S : set β} {f : Π x ∈ S, set α} (ho : ∀s∈S, is_Gδ (f s ‹_›))
(hS : S.countable) (hd : ∀s∈S, dense (f s ‹_›)) : dense (⋂s∈S, f s ‹_›) | begin
rw bInter_eq_Inter,
haveI := hS.to_encodable,
exact dense_Inter_of_Gδ (λ s, ho s s.2) (λ s, hd s s.2)
end | theorem | dense_bInter_of_Gδ | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_Inter_of_Gδ",
"is_Gδ"
] | Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable set in any type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense.inter_of_Gδ {s t : set α} (hs : is_Gδ s) (ht : is_Gδ t) (hsc : dense s)
(htc : dense t) :
dense (s ∩ t) | begin
rw [inter_eq_Inter],
apply dense_Inter_of_Gδ; simp [bool.forall_bool, *]
end | theorem | dense.inter_of_Gδ | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"bool.forall_bool",
"dense",
"dense_Inter_of_Gδ",
"is_Gδ"
] | Baire theorem: the intersection of two dense Gδ sets is dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Gδ.dense_Union_interior_of_closed [encodable ι] {s : set α} (hs : is_Gδ s)
(hd : dense s) {f : ι → set α} (hc : ∀ i, is_closed (f i)) (hU : s ⊆ ⋃ i, f i) :
dense (⋃ i, interior (f i)) | begin
let g := λ i, (frontier (f i))ᶜ,
have hgo : ∀ i, is_open (g i), from λ i, is_closed_frontier.is_open_compl,
have hgd : dense (⋂ i, g i),
{ refine dense_Inter_of_open hgo (λ i x, _),
rw [closure_compl, interior_frontier (hc _)],
exact id },
refine (hd.inter_of_Gδ hs (is_Gδ_Inter_of_open $ λ i, hg... | lemma | is_Gδ.dense_Union_interior_of_closed | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"closure_compl",
"dense",
"dense_Inter_of_open",
"encodable",
"frontier",
"interior",
"interior_frontier",
"is_Gδ",
"is_Gδ_Inter_of_open",
"is_closed",
"is_open",
"self_diff_frontier"
] | If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Gδ.dense_bUnion_interior_of_closed {t : set ι} {s : set α} (hs : is_Gδ s)
(hd : dense s) (ht : t.countable) {f : ι → set α} (hc : ∀ i ∈ t, is_closed (f i))
(hU : s ⊆ ⋃ i ∈ t, f i) :
dense (⋃ i ∈ t, interior (f i)) | begin
haveI := ht.to_encodable,
simp only [bUnion_eq_Union, set_coe.forall'] at *,
exact hs.dense_Union_interior_of_closed hd hc hU
end | lemma | is_Gδ.dense_bUnion_interior_of_closed | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"interior",
"is_Gδ",
"is_closed",
"set_coe.forall'"
] | If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with a union over a countable set in any type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Gδ.dense_sUnion_interior_of_closed {T : set (set α)} {s : set α} (hs : is_Gδ s)
(hd : dense s) (hc : T.countable) (hc' : ∀ t ∈ T, is_closed t) (hU : s ⊆ ⋃₀ T) :
dense (⋃ t ∈ T, interior t) | hs.dense_bUnion_interior_of_closed hd hc hc' $ by rwa [← sUnion_eq_bUnion] | lemma | is_Gδ.dense_sUnion_interior_of_closed | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"interior",
"is_Gδ",
"is_closed"
] | If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_bUnion_interior_of_closed {S : set β} {f : β → set α} (hc : ∀s∈S, is_closed (f s))
(hS : S.countable) (hU : (⋃s∈S, f s) = univ) : dense (⋃s∈S, interior (f s)) | is_Gδ_univ.dense_bUnion_interior_of_closed dense_univ hS hc hU.ge | theorem | dense_bUnion_interior_of_closed | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_univ",
"interior",
"is_closed"
] | Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with an index set which is a countable set in any type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_sUnion_interior_of_closed {S : set (set α)} (hc : ∀s∈S, is_closed s)
(hS : S.countable) (hU : (⋃₀ S) = univ) : dense (⋃s∈S, interior s) | is_Gδ_univ.dense_sUnion_interior_of_closed dense_univ hS hc hU.ge | theorem | dense_sUnion_interior_of_closed | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_univ",
"interior",
"is_closed"
] | Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with `⋃₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_Union_interior_of_closed [encodable β] {f : β → set α} (hc : ∀s, is_closed (f s))
(hU : (⋃s, f s) = univ) : dense (⋃s, interior (f s)) | is_Gδ_univ.dense_Union_interior_of_closed dense_univ hc hU.ge | theorem | dense_Union_interior_of_closed | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense",
"dense_univ",
"encodable",
"interior",
"is_closed"
] | Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with an index set which is an encodable type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_interior_of_Union_of_closed [nonempty α] [encodable β] {f : β → set α}
(hc : ∀s, is_closed (f s)) (hU : (⋃s, f s) = univ) :
∃s, (interior $ f s).nonempty | by simpa using (dense_Union_interior_of_closed hc hU).nonempty | theorem | nonempty_interior_of_Union_of_closed | topology.metric_space | src/topology/metric_space/baire.lean | [
"analysis.specific_limits.basic",
"topology.G_delta",
"topology.sets.compacts"
] | [
"dense_Union_interior_of_closed",
"encodable",
"interior",
"is_closed"
] | One of the most useful consequences of Baire theorem: if a countable union of closed sets
covers the space, then one of the sets has nonempty interior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space_of_dist
(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) : uniform_space α | uniform_space.of_fun dist dist_self dist_comm dist_triangle $ λ ε ε0,
⟨ε / 2, half_pos ε0, λ x hx y hy, add_halves ε ▸ add_lt_add hx hy⟩ | def | uniform_space_of_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"add_halves",
"dist_comm",
"dist_self",
"dist_triangle",
"half_pos",
"uniform_space",
"uniform_space.of_fun"
] | Construct a uniform structure from a distance function and metric space axioms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_iff_aux {α : Type*} (dist : α → α → ℝ)
(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)
(s : set α) (a : α) :
(∃ c, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ c) ↔ (∃ r, ∀ ⦃x⦄, x ∈ s → dist x a ≤ r) | begin
split; rintro ⟨C, hC⟩,
{ rcases s.eq_empty_or_nonempty with rfl | ⟨x, hx⟩,
{ exact ⟨0, by simp⟩ },
{ exact ⟨C + dist x a, λ y hy,
(dist_triangle y x a).trans (add_le_add_right (hC hy hx) _)⟩ } },
{ exact ⟨C + C, λ x hx y hy,
(dist_triangle x a y).trans (add_le_add (hC hx) (by... | lemma | bounded_iff_aux | 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_triangle"
] | This is an internal lemma used to construct a bornology from a metric in `bornology.of_dist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bornology.of_dist {α : Type*} (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) :
bornology α | bornology.of_bounded
{ s : set α | ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C }
⟨0, λ x hx y, hx.elim⟩
(λ s ⟨c, hc⟩ t h, ⟨c, λ x hx y hy, hc (h hx) (h hy)⟩)
(λ s hs t ht,
begin
rcases s.eq_empty_or_nonempty with rfl | ⟨z, hz⟩,
{ exact (empty_union t).symm ▸ ht },
{ simp only [λ u, bou... | def | bornology.of_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"bornology",
"bornology.of_bounded",
"bounded_iff_aux",
"dist_comm",
"dist_self",
"dist_triangle"
] | Construct a bornology from a distance function and metric space axioms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_dist (α : Type*) | (dist : α → α → ℝ) | class | has_dist | 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 function (given an ambient metric space on `α`), which returns
a nonnegative real number `dist x y` given `x y : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.dist_nonneg' {α} {x y : 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): 0 ≤ dist x y | have 2 * dist x y ≥ 0,
from calc 2 * dist x y = dist x y + dist y x : by rw [dist_comm x y, two_mul]
... ≥ 0 : by rw ← dist_self x; apply dist_triangle,
nonneg_of_mul_nonneg_right this zero_lt_two | theorem | pseudo_metric_space.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",
"nonneg_of_mul_nonneg_right",
"two_mul",
"zero_lt_two"
] | This is an internal lemma used inside the default of `pseudo_metric_space.edist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.edist_dist_tac : tactic unit | tactic.intros >> `[exact (ennreal.of_real_eq_coe_nnreal _).symm <|> control_laws_tac] | def | pseudo_metric_space.edist_dist_tac | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"ennreal.of_real_eq_coe_nnreal"
] | This tactic is used to populate `pseudo_metric_space.edist_dist` when the default `edist` is
used. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space (α : Type u) extends has_dist α : Type u | (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)
(edist : α → α → ℝ≥0∞ := λ x y,
@coe (ℝ≥0) _ _ ⟨dist x y, pseudo_metric_space.dist_nonneg' _ ‹_› ‹_› ‹_›⟩)
(edist_dist : ∀ x y : α,
edist x y = ennreal.of_real (dist x y) . ... | class | pseudo_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"
] | [
"bornology",
"bornology.of_dist",
"dist_comm",
"dist_self",
"dist_triangle",
"edist_dist",
"ennreal.of_real",
"has_dist",
"pseudo_metric_space.dist_nonneg'",
"pseudo_metric_space.edist_dist_tac",
"uniform_space",
"uniform_space_of_dist"
] | Pseudo metric and Metric spaces
A pseudo metric space is endowed with a distance for which the requirement `d(x,y)=0 → x = y` might
not hold. A metric space is a pseudo metric space such that `d(x,y)=0 → x = y`.
Each pseudo metric space induces a canonical `uniform_space` and hence a canonical
`topological_space` This... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.ext {α : Type*} {m m' : pseudo_metric_space α}
(h : m.to_has_dist = m'.to_has_dist) : m = m' | begin
unfreezingI { rcases m, rcases m' },
dsimp at h,
unfreezingI { subst h },
congr,
{ ext x y : 2,
dsimp at m_edist_dist m'_edist_dist,
simp [m_edist_dist, m'_edist_dist] },
{ dsimp at m_uniformity_dist m'_uniformity_dist,
rw ← m'_uniformity_dist at m_uniformity_dist,
exact uniform_space_... | lemma | pseudo_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"
] | [
"pseudo_metric_space",
"uniform_space_eq"
] | Two pseudo metric space structures with the same distance function coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.to_has_edist : has_edist α | ⟨pseudo_metric_space.edist⟩ | instance | pseudo_metric_space.to_has_edist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"has_edist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_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) :
... | { dist := dist,
dist_self := dist_self,
dist_comm := dist_comm,
dist_triangle := dist_triangle,
to_uniform_space :=
{ is_open_uniformity := λ s, (H s).trans $ forall₂_congr $ λ x _,
((uniform_space.has_basis_of_fun (exists_gt (0 : ℝ))
dist _ _ _ _).comap (prod.mk x)).mem_iff.symm.trans mem_comap... | def | pseudo_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"
] | [
"bornology.of_dist",
"dist_comm",
"dist_self",
"dist_triangle",
"forall₂_congr",
"is_open",
"is_open_uniformity",
"pseudo_metric_space",
"topological_space",
"uniform_space.has_basis_of_fun",
"uniform_space_of_dist"
] | Construct a pseudo-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 |
dist_self (x : α) : dist x x = 0 | pseudo_metric_space.dist_self x | theorem | dist_self | 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_comm (x y : α) : dist x y = dist y x | pseudo_metric_space.dist_comm x y | theorem | dist_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 | |
edist_dist (x y : α) : edist x y = ennreal.of_real (dist x y) | pseudo_metric_space.edist_dist x y | theorem | edist_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z | pseudo_metric_space.dist_triangle x y z | theorem | dist_triangle | 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_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y | by rw dist_comm z; apply dist_triangle | theorem | dist_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_comm",
"dist_triangle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z | by rw dist_comm y; apply dist_triangle | theorem | dist_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_comm",
"dist_triangle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_triangle4 (x y z w : α) :
dist x w ≤ dist x y + dist y z + dist z w | calc dist x w ≤ dist x z + dist z w : dist_triangle x z w
... ≤ (dist x y + dist y z) + dist z w : add_le_add_right (dist_triangle x y z) _ | lemma | dist_triangle4 | 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_triangle4_left (x₁ y₁ x₂ y₂ : α) :
dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) | by { rw [add_left_comm, dist_comm x₁, ← add_assoc], apply dist_triangle4 } | lemma | dist_triangle4_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_comm",
"dist_triangle4"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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