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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|---|---|---|---|---|---|---|---|---|---|---|
diam_le {d : ℝ≥0∞} (h : ∀ (x ∈ s) (y ∈ s), edist x y ≤ d) : diam s ≤ d | diam_le_iff.2 h | lemma | emetric.diam_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | If the distance between any two points in a set is bounded by some constant, this constant
bounds the diameter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_subsingleton (hs : s.subsingleton) : diam s = 0 | nonpos_iff_eq_zero.1 $ diam_le $
λ x hx y hy, (hs hx hy).symm ▸ edist_self y ▸ le_rfl | lemma | emetric.diam_subsingleton | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"le_rfl"
] | The diameter of a subsingleton vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_Union_mem_option {ι : Type*} (o : option ι) (s : ι → set α) :
diam (⋃ i ∈ o, s i) = ⨆ i ∈ o, diam (s i) | by cases o; simp | lemma | emetric.diam_Union_mem_option | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_insert : diam (insert x s) = max (⨆ y ∈ s, edist x y) (diam s) | eq_of_forall_ge_iff $ λ d, by simp only [diam_le_iff, ball_insert_iff,
edist_self, edist_comm x, max_le_iff, supr_le_iff, zero_le, true_and,
forall_and_distrib, and_self, ← and_assoc] | lemma | emetric.diam_insert | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"eq_of_forall_ge_iff",
"forall_and_distrib",
"max_le_iff",
"supr_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_pair : diam ({x, y} : set α) = edist x y | by simp only [supr_singleton, diam_insert, diam_singleton, ennreal.max_zero_right] | lemma | emetric.diam_pair | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"ennreal.max_zero_right",
"supr_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_triple :
diam ({x, y, z} : set α) = max (max (edist x y) (edist x z)) (edist y z) | by simp only [diam_insert, supr_insert, supr_singleton, diam_singleton,
ennreal.max_zero_right, ennreal.sup_eq_max] | lemma | emetric.diam_triple | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"ennreal.max_zero_right",
"ennreal.sup_eq_max",
"supr_insert",
"supr_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_mono {s t : set α} (h : s ⊆ t) : diam s ≤ diam t | diam_le $ λ x hx y hy, edist_le_diam_of_mem (h hx) (h hy) | lemma | emetric.diam_mono | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | The diameter is monotonous with respect to inclusion | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_union {t : set α} (xs : x ∈ s) (yt : y ∈ t) :
diam (s ∪ t) ≤ diam s + edist x y + diam t | begin
have A : ∀a ∈ s, ∀b ∈ t, edist a b ≤ diam s + edist x y + diam t := λa ha b hb, calc
edist a b ≤ edist a x + edist x y + edist y b : edist_triangle4 _ _ _ _
... ≤ diam s + edist x y + diam t :
add_le_add (add_le_add (edist_le_diam_of_mem ha xs) le_rfl) (edist_le_diam_of_mem yt hb),
refine diam_l... | lemma | emetric.diam_union | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_triangle4",
"le_rfl"
] | The diameter of a union is controlled by the diameter of the sets, and the edistance
between two points in the sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_union' {t : set α} (h : (s ∩ t).nonempty) : diam (s ∪ t) ≤ diam s + diam t | let ⟨x, ⟨xs, xt⟩⟩ := h in by simpa using diam_union xs xt | lemma | emetric.diam_union' | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_closed_ball {r : ℝ≥0∞} : diam (closed_ball x r) ≤ 2 * r | diam_le $ λa ha b hb, calc
edist a b ≤ edist a x + edist b x : edist_triangle_right _ _ _
... ≤ r + r : add_le_add ha hb
... = 2 * r : (two_mul r).symm | lemma | emetric.diam_closed_ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_triangle_right",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_ball {r : ℝ≥0∞} : diam (ball x r) ≤ 2 * r | le_trans (diam_mono ball_subset_closed_ball) diam_closed_ball | lemma | emetric.diam_ball | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_pi_le_of_le {π : β → Type*} [fintype β] [∀ b, pseudo_emetric_space (π b)]
{s : Π (b : β), set (π b)} {c : ℝ≥0∞} (h : ∀ b, diam (s b) ≤ c) :
diam (set.pi univ s) ≤ c | begin
apply diam_le (λ x hx y hy, edist_pi_le_iff.mpr _),
rw [mem_univ_pi] at hx hy,
exact λ b, diam_le_iff.1 (h b) (x b) (hx b) (y b) (hy b),
end | lemma | emetric.diam_pi_le_of_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"fintype",
"pseudo_emetric_space",
"set.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emetric_space (α : Type u) extends pseudo_emetric_space α : Type u | (eq_of_edist_eq_zero : ∀ {x y : α}, edist x y = 0 → x = y) | class | emetric_space | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"pseudo_emetric_space"
] | We now define `emetric_space`, extending `pseudo_emetric_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_eq_zero {x y : γ} : edist x y = 0 ↔ x = y | iff.intro eq_of_edist_eq_zero (assume : x = y, this ▸ edist_self _) | theorem | edist_eq_zero | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | Characterize the equality of points by the vanishing of their extended distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_eq_edist {x y : γ} : 0 = edist x y ↔ x = y | iff.intro (assume h, eq_of_edist_eq_zero (h.symm))
(assume : x = y, this ▸ (edist_self _).symm) | theorem | zero_eq_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_zero {x y : γ} : (edist x y ≤ 0) ↔ x = y | nonpos_iff_eq_zero.trans edist_eq_zero | theorem | edist_le_zero | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_pos {x y : γ} : 0 < edist x y ↔ x ≠ y | by simp [← not_le] | theorem | edist_pos | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_forall_edist_le {x y : γ} (h : ∀ε > 0, edist x y ≤ ε) : x = y | eq_of_edist_eq_zero (eq_of_le_of_forall_le_of_dense bot_le h) | theorem | eq_of_forall_edist_le | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"bot_le",
"eq_of_le_of_forall_le_of_dense"
] | Two points coincide if their distance is `< ε` for all positive ε | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_separated : separated_space γ | separated_def.2 $ λ x y h, eq_of_forall_edist_le $
λ ε ε0, le_of_lt (h _ (edist_mem_uniformity ε0)) | instance | to_separated | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"edist_mem_uniformity",
"eq_of_forall_edist_le",
"separated_space"
] | An emetric space is separated | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emetric.uniform_embedding_iff' [emetric_space β] {f : γ → β} :
uniform_embedding f ↔
(∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, edist a b < δ → edist (f a) (f b) < ε) ∧
(∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, edist (f a) (f b) < ε → edist a b < δ) | begin
simp only [uniform_embedding_iff_uniform_inducing,
uniformity_basis_edist.uniform_inducing_iff uniformity_basis_edist, exists_prop],
refl
end | theorem | emetric.uniform_embedding_iff' | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric_space",
"exists_prop",
"uniform_embedding",
"uniform_embedding_iff_uniform_inducing",
"uniformity_basis_edist"
] | A map between emetric spaces is a uniform embedding if and only if the edistance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y` and conversely. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emetric_space.of_t0_pseudo_emetric_space (α : Type*) [pseudo_emetric_space α] [t0_space α] :
emetric_space α | { eq_of_edist_eq_zero := λ x y hdist, (emetric.inseparable_iff.2 hdist).eq,
..‹pseudo_emetric_space α› } | def | emetric_space.of_t0_pseudo_emetric_space | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric_space",
"pseudo_emetric_space",
"t0_space"
] | If a `pseudo_emetric_space` is a T₀ space, then it is an `emetric_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emetric_space.replace_uniformity {γ} [U : uniform_space γ] (m : emetric_space γ)
(H : 𝓤[U] = 𝓤[pseudo_emetric_space.to_uniform_space]) :
emetric_space γ | { edist := @edist _ m.to_has_edist,
edist_self := edist_self,
eq_of_edist_eq_zero := @eq_of_edist_eq_zero _ _,
edist_comm := edist_comm,
edist_triangle := edist_triangle,
to_uniform_space := U,
uniformity_edist := H.trans (@pseudo_emetric_space.uniformity_edist γ _... | def | emetric_space.replace_uniformity | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric_space",
"uniform_space",
"uniformity_edist"
] | Auxiliary function to replace the uniformity on an emetric space with
a uniformity which is equal to the original one, but maybe not defeq.
This is useful if one wants to construct an emetric space with a
specified uniformity. See Note [forgetful inheritance] explaining why having definitionally
the right uniformity is... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emetric_space.induced {γ β} (f : γ → β) (hf : function.injective f)
(m : emetric_space β) : emetric_space γ | { edist := λ x y, edist (f x) (f y),
edist_self := λ x, edist_self _,
eq_of_edist_eq_zero := λ x y h, hf (edist_eq_zero.1 h),
edist_comm := λ x y, edist_comm _ _,
edist_triangle := λ x y z, edist_triangle _ _ _,
to_uniform_space := uniform_space.comap f m.to_uniform_spa... | def | emetric_space.induced | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric_space",
"uniform_space.comap",
"uniformity_edist"
] | The extended metric induced by an injective function taking values in a emetric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.emetric_space_max [emetric_space β] : emetric_space (γ × β) | { eq_of_edist_eq_zero := λ x y h, begin
cases max_le_iff.1 (le_of_eq h) with h₁ h₂,
have A : x.fst = y.fst := edist_le_zero.1 h₁,
have B : x.snd = y.snd := edist_le_zero.1 h₂,
exact prod.ext_iff.2 ⟨A, B⟩
end,
..prod.pseudo_emetric_space_max } | instance | prod.emetric_space_max | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric_space",
"prod.pseudo_emetric_space_max"
] | The product of two emetric spaces, with the max distance, is an extended
metric spaces. We make sure that the uniform structure thus constructed is the one
corresponding to the product of uniform spaces, to avoid diamond problems. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_edist :
𝓤 γ = ⨅ ε>0, 𝓟 {p:γ×γ | edist p.1 p.2 < ε} | pseudo_emetric_space.uniformity_edist | theorem | uniformity_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | Reformulation of the uniform structure in terms of the extended distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emetric_space_pi [∀b, emetric_space (π b)] : emetric_space (Πb, π b) | { eq_of_edist_eq_zero := assume f g eq0,
begin
have eq1 : sup univ (λ (b : β), edist (f b) (g b)) ≤ 0 := le_of_eq eq0,
simp only [finset.sup_le_iff] at eq1,
exact (funext $ assume b, edist_le_zero.1 $ eq1 b $ mem_univ b),
end,
..pseudo_emetric_space_pi } | instance | emetric_space_pi | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"emetric_space",
"finset.sup_le_iff",
"pseudo_emetric_space_pi"
] | The product of a finite number of emetric spaces, with the max distance, is still
an emetric space.
This construction would also work for infinite products, but it would not give rise
to the product topology. Hence, we only formalize it in the good situation of finitely many
spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_closure_of_compact {s : set γ} (hs : is_compact s) :
∃ t ⊆ s, (t.countable ∧ s = closure t) | begin
rcases subset_countable_closure_of_compact hs with ⟨t, hts, htc, hsub⟩,
exact ⟨t, hts, htc, subset.antisymm hsub (closure_minimal hts hs.is_closed)⟩
end | lemma | emetric.countable_closure_of_compact | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"closure",
"closure_minimal",
"is_compact"
] | A compact set in an emetric space is separable, i.e., it is the closure of a countable set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_eq_zero_iff : diam s = 0 ↔ s.subsingleton | ⟨λ h x hx y hy, edist_le_zero.1 $ h ▸ edist_le_diam_of_mem hx hy, diam_subsingleton⟩ | lemma | emetric.diam_eq_zero_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_pos_iff : 0 < diam s ↔ ∃ (x ∈ s) (y ∈ s), x ≠ y | by simp only [pos_iff_ne_zero, ne.def, diam_eq_zero_iff, set.subsingleton, not_forall] | lemma | emetric.diam_pos_iff | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"not_forall",
"set.subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.separation_quotient.edist_mk [pseudo_emetric_space X] (x y : X) :
@edist (uniform_space.separation_quotient X) _ (quot.mk _ x) (quot.mk _ y) = edist x y | rfl | theorem | uniform_space.separation_quotient.edist_mk | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"pseudo_emetric_space",
"uniform_space.separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_of_mul (a b : X) : edist (of_mul a) (of_mul b) = edist a b | rfl | lemma | edist_of_mul | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_of_add (a b : X) : edist (of_add a) (of_add b) = edist a b | rfl | lemma | edist_of_add | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_to_mul (a b : additive X) : edist (to_mul a) (to_mul b) = edist a b | rfl | lemma | edist_to_mul | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_to_add (a b : multiplicative X) : edist (to_add a) (to_add b) = edist a b | rfl | lemma | edist_to_add | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_to_dual (a b : X) : edist (to_dual a) (to_dual b) = edist a b | rfl | lemma | edist_to_dual | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_of_dual (a b : Xᵒᵈ) : edist (of_dual a) (of_dual b) = edist a b | rfl | lemma | edist_of_dual | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equicontinuous_at_iff_right {ι : Type*} [topological_space β] {F : ι → β → α} {x₀ : β} :
equicontinuous_at F x₀ ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 x₀, ∀ i, dist (F i x₀) (F i x) < ε | uniformity_basis_dist.equicontinuous_at_iff_right | lemma | metric.equicontinuous_at_iff_right | topology.metric_space | src/topology/metric_space/equicontinuity.lean | [
"topology.metric_space.basic",
"topology.uniform_space.equicontinuity"
] | [
"equicontinuous_at",
"topological_space"
] | Characterization of equicontinuity for families of functions taking values in a (pseudo) metric
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_at_iff {ι : Type*} [pseudo_metric_space β] {F : ι → β → α} {x₀ : β} :
equicontinuous_at F x₀ ↔ ∀ ε > 0, ∃ δ > 0, ∀ x, dist x x₀ < δ → ∀ i, dist (F i x₀) (F i x) < ε | nhds_basis_ball.equicontinuous_at_iff uniformity_basis_dist | lemma | metric.equicontinuous_at_iff | topology.metric_space | src/topology/metric_space/equicontinuity.lean | [
"topology.metric_space.basic",
"topology.uniform_space.equicontinuity"
] | [
"equicontinuous_at",
"pseudo_metric_space"
] | Characterization of equicontinuity for families of functions between (pseudo) metric spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_at_iff_pair {ι : Type*} [topological_space β] {F : ι → β → α}
{x₀ : β} :
equicontinuous_at F x₀ ↔ ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ (x x' ∈ U), ∀ i, dist (F i x) (F i x') < ε | begin
rw equicontinuous_at_iff_pair,
split; intros H,
{ intros ε hε,
refine exists_imp_exists (λ V, exists_imp_exists $ λ hV h, _) (H _ (dist_mem_uniformity hε)),
exact λ x hx x' hx', h _ hx _ hx' },
{ intros U hU,
rcases mem_uniformity_dist.mp hU with ⟨ε, hε, hεU⟩,
refine exists_imp_exists (λ V... | lemma | metric.equicontinuous_at_iff_pair | topology.metric_space | src/topology/metric_space/equicontinuity.lean | [
"topology.metric_space.basic",
"topology.uniform_space.equicontinuity"
] | [
"equicontinuous_at",
"equicontinuous_at_iff_pair",
"topological_space"
] | Reformulation of `equicontinuous_at_iff_pair` for families of functions taking values in a
(pseudo) metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous_iff_right {ι : Type*} [uniform_space β] {F : ι → β → α} :
uniform_equicontinuous F ↔
∀ ε > 0, ∀ᶠ (xy : β × β) in 𝓤 β, ∀ i, dist (F i xy.1) (F i xy.2) < ε | uniformity_basis_dist.uniform_equicontinuous_iff_right | lemma | metric.uniform_equicontinuous_iff_right | topology.metric_space | src/topology/metric_space/equicontinuity.lean | [
"topology.metric_space.basic",
"topology.uniform_space.equicontinuity"
] | [
"uniform_equicontinuous",
"uniform_space"
] | Characterization of uniform equicontinuity for families of functions taking values in a
(pseudo) metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous_iff {ι : Type*} [pseudo_metric_space β] {F : ι → β → α} :
uniform_equicontinuous F ↔
∀ ε > 0, ∃ δ > 0, ∀ x y, dist x y < δ → ∀ i, dist (F i x) (F i y) < ε | uniformity_basis_dist.uniform_equicontinuous_iff uniformity_basis_dist | lemma | metric.uniform_equicontinuous_iff | topology.metric_space | src/topology/metric_space/equicontinuity.lean | [
"topology.metric_space.basic",
"topology.uniform_space.equicontinuity"
] | [
"pseudo_metric_space",
"uniform_equicontinuous"
] | Characterization of uniform equicontinuity for families of functions between
(pseudo) metric spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_at_of_continuity_modulus {ι : Type*} [topological_space β] {x₀ : β}
(b : β → ℝ)
(b_lim : tendsto b (𝓝 x₀) (𝓝 0))
(F : ι → β → α)
(H : ∀ᶠ x in 𝓝 x₀, ∀ i, dist (F i x₀) (F i x) ≤ b x) :
equicontinuous_at F x₀ | begin
rw metric.equicontinuous_at_iff_right,
intros ε ε0,
filter_upwards [b_lim (Iio_mem_nhds ε0), H] using λ x hx₁ hx₂ i, (hx₂ i).trans_lt hx₁
end | lemma | metric.equicontinuous_at_of_continuity_modulus | topology.metric_space | src/topology/metric_space/equicontinuity.lean | [
"topology.metric_space.basic",
"topology.uniform_space.equicontinuity"
] | [
"Iio_mem_nhds",
"equicontinuous_at",
"metric.equicontinuous_at_iff_right",
"topological_space"
] | For a family of functions to a (pseudo) metric spaces, a convenient way to prove
equicontinuity at a point is to show that all of the functions share a common *local* continuity
modulus. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous_of_continuity_modulus {ι : Type*} [pseudo_metric_space β] (b : ℝ → ℝ)
(b_lim : tendsto b (𝓝 0) (𝓝 0))
(F : ι → β → α)
(H : ∀ (x y : β) i, dist (F i x) (F i y) ≤ b (dist x y)) :
uniform_equicontinuous F | begin
rw metric.uniform_equicontinuous_iff,
intros ε ε0,
rcases tendsto_nhds_nhds.1 b_lim ε ε0 with ⟨δ, δ0, hδ⟩,
refine ⟨δ, δ0, λ x y hxy i, _⟩,
calc
dist (F i x) (F i y) ≤ b (dist x y) : H x y i
... ≤ |b (dist x y)| : le_abs_self _
... = dist (b (dist x y)) 0 : by simp [real.dist_eq]
... < ε ... | lemma | metric.uniform_equicontinuous_of_continuity_modulus | topology.metric_space | src/topology/metric_space/equicontinuity.lean | [
"topology.metric_space.basic",
"topology.uniform_space.equicontinuity"
] | [
"abs_dist",
"le_abs_self",
"metric.uniform_equicontinuous_iff",
"pseudo_metric_space",
"real.dist_eq",
"tsub_zero",
"uniform_equicontinuous"
] | For a family of functions between (pseudo) metric spaces, a convenient way to prove
uniform equicontinuity is to show that all of the functions share a common *global* continuity
modulus. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_of_continuity_modulus {ι : Type*} [pseudo_metric_space β] (b : ℝ → ℝ)
(b_lim : tendsto b (𝓝 0) (𝓝 0))
(F : ι → β → α)
(H : ∀ (x y : β) i, dist (F i x) (F i y) ≤ b (dist x y)) :
equicontinuous F | (uniform_equicontinuous_of_continuity_modulus b b_lim F H).equicontinuous | lemma | metric.equicontinuous_of_continuity_modulus | topology.metric_space | src/topology/metric_space/equicontinuity.lean | [
"topology.metric_space.basic",
"topology.uniform_space.equicontinuity"
] | [
"equicontinuous",
"pseudo_metric_space"
] | For a family of functions between (pseudo) metric spaces, a convenient way to prove
equicontinuity is to show that all of the functions share a common *global* continuity modulus. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glue_dist (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : X ⊕ Y → X ⊕ Y → ℝ | | (inl x) (inl y) := dist x y
| (inr x) (inr y) := dist x y
| (inl x) (inr y) := (⨅ p, dist x (Φ p) + dist y (Ψ p)) + ε
| (inr x) (inl y) := (⨅ p, dist y (Φ p) + dist x (Ψ p)) + ε | def | metric.glue_dist | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [] | Define a predistance on `X ⊕ Y`, for which `Φ p` and `Ψ p` are at distance `ε` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glue_dist_self (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : ∀ x, glue_dist Φ Ψ ε x x = 0 | | (inl x) := dist_self _
| (inr x) := dist_self _ | lemma | metric.glue_dist_self | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_dist_glued_points [nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (p : Z) :
glue_dist Φ Ψ ε (inl (Φ p)) (inr (Ψ p)) = ε | begin
have : (⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q)) = 0,
{ have A : ∀ q, 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) :=
λq, by rw ← add_zero (0 : ℝ); exact add_le_add dist_nonneg dist_nonneg,
refine le_antisymm _ (le_cinfi A),
have : 0 = dist (Φ p) (Φ p) + dist (Ψ p) (Ψ p), by simp,
rw this,
e... | lemma | metric.glue_dist_glued_points | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"cinfi_le",
"dist_nonneg",
"le_cinfi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_dist_comm (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :
∀ x y, glue_dist Φ Ψ ε x y = glue_dist Φ Ψ ε y x | | (inl x) (inl y) := dist_comm _ _
| (inr x) (inr y) := dist_comm _ _
| (inl x) (inr y) := rfl
| (inr x) (inl y) := rfl | lemma | metric.glue_dist_comm | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_dist_triangle (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)
(H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) :
∀ x y z, glue_dist Φ Ψ ε x z ≤ glue_dist Φ Ψ ε x y + glue_dist Φ Ψ ε y z | | (inl x) (inl y) (inl z) := dist_triangle _ _ _
| (inr x) (inr y) (inr z) := dist_triangle _ _ _
| (inr x) (inl y) (inl z) := begin
have B : ∀ a b, bdd_below (range (λ (p : Z), dist a (Φ p) + dist b (Ψ p))) :=
λa b, ⟨0, forall_range_iff.2 (λp, add_nonneg dist_nonneg dist_nonneg)⟩,
unfold glue_dist,
h... | lemma | metric.glue_dist_triangle | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"bdd_below",
"cinfi_mono",
"continuous_at_const",
"continuous_at_id",
"dist_comm",
"dist_nonneg",
"dist_triangle",
"dist_triangle4",
"dist_triangle_left",
"exists_lt_of_cinfi_lt",
"infi",
"le_abs_self",
"le_rfl",
"monotone.map_cinfi_of_continuous_at",
"neg_le_abs_self",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_eq_of_dist_eq_zero (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε) :
∀ p q : X ⊕ Y, glue_dist Φ Ψ ε p q = 0 → p = q | | (inl x) (inl y) h := by rw eq_of_dist_eq_zero h
| (inl x) (inr y) h := begin
have : 0 ≤ (⨅ p, dist x (Φ p) + dist y (Ψ p)) :=
le_cinfi (λp, by simpa using add_le_add (@dist_nonneg _ _ x _) (@dist_nonneg _ _ y _)),
have : 0 + ε ≤ glue_dist Φ Ψ ε (inl x) (inr y) := add_le_add this (le_refl ε),
exfalso... | lemma | metric.glue_eq_of_dist_eq_zero | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_nonneg",
"eq_of_dist_eq_zero",
"le_cinfi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_metric_approx (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε)
(H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) : metric_space (X ⊕ Y) | { dist := glue_dist Φ Ψ ε,
dist_self := glue_dist_self Φ Ψ ε,
dist_comm := glue_dist_comm Φ Ψ ε,
dist_triangle := glue_dist_triangle Φ Ψ ε H,
eq_of_dist_eq_zero := glue_eq_of_dist_eq_zero Φ Ψ ε ε0 } | def | metric.glue_metric_approx | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"eq_of_dist_eq_zero",
"metric_space"
] | Given two maps `Φ` and `Ψ` intro metric spaces `X` and `Y` such that the distances between
`Φ p` and `Φ q`, and between `Ψ p` and `Ψ q`, coincide up to `2 ε` where `ε > 0`, one can almost
glue the two spaces `X` and `Y` along the images of `Φ` and `Ψ`, so that `Φ p` and `Ψ p` are
at distance `ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum.dist : X ⊕ Y → X ⊕ Y → ℝ | | (inl a) (inl a') := dist a a'
| (inr b) (inr b') := dist b b'
| (inl a) (inr b) := dist a (nonempty.some ⟨a⟩) + 1 + dist (nonempty.some ⟨b⟩) b
| (inr b) (inl a) := dist b (nonempty.some ⟨b⟩) + 1 + dist (nonempty.some ⟨a⟩) a | def | metric.sum.dist | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"nonempty.some"
] | Distance on a disjoint union. There are many (noncanonical) ways to put a distance compatible
with each factor.
If the two spaces are bounded, one can say for instance that each point in the first is at distance
`diam X + diam Y + 1` of each point in the second.
Instead, we choose a construction that works for unbounde... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum.dist_eq_glue_dist {p q : X ⊕ Y} (x : X) (y : Y) :
sum.dist p q = glue_dist (λ _ : unit, nonempty.some ⟨x⟩) (λ _ : unit, nonempty.some ⟨y⟩) 1 p q | by cases p; cases q; refl <|> simp [sum.dist, glue_dist, dist_comm, add_comm, add_left_comm] | lemma | metric.sum.dist_eq_glue_dist | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_comm",
"nonempty.some"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum.dist_comm (x y : X ⊕ Y) : sum.dist x y = sum.dist y x | by cases x; cases y; simp only [sum.dist, dist_comm, add_comm, add_left_comm] | lemma | metric.sum.dist_comm | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum.one_dist_le {x : X} {y : Y} : 1 ≤ sum.dist (inl x) (inr y) | le_trans (le_add_of_nonneg_right dist_nonneg) $
add_le_add_right (le_add_of_nonneg_left dist_nonneg) _ | lemma | metric.sum.one_dist_le | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum.one_dist_le' {x : X} {y : Y} : 1 ≤ sum.dist (inr y) (inl x) | by rw sum.dist_comm; exact sum.one_dist_le | lemma | metric.sum.one_dist_le' | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum.mem_uniformity (s : set ((X ⊕ Y) × (X ⊕ Y))) :
s ∈ 𝓤 (X ⊕ Y) ↔ ∃ ε > 0, ∀ a b, sum.dist a b < ε → (a, b) ∈ s | begin
split,
{ rintro ⟨hsX, hsY⟩,
rcases mem_uniformity_dist.1 hsX with ⟨εX, εX0, hX⟩,
rcases mem_uniformity_dist.1 hsY with ⟨εY, εY0, hY⟩,
refine ⟨min (min εX εY) 1, lt_min (lt_min εX0 εY0) zero_lt_one, _⟩,
rintro (a|a) (b|b) h,
{ exact hX (lt_of_lt_of_le h (le_trans (min_le_left _ _) (min_le_l... | lemma | metric.sum.mem_uniformity | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"filter.mem_map",
"filter.mem_sets",
"not_le_of_lt",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric_space_sum : metric_space (X ⊕ Y) | { dist := sum.dist,
dist_self := λx, by cases x; simp only [sum.dist, dist_self],
dist_comm := sum.dist_comm,
dist_triangle := λ p q r,
begin
cases p; cases q; cases r,
{ exact dist_triangle _ _ _ },
{ simp only [dist, sum.dist_eq_glue_dist p r],
exact glue... | def | metric.metric_space_sum | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_comm",
"dist_eq_zero",
"dist_self",
"dist_triangle",
"eq_of_dist_eq_zero",
"imp_self",
"metric_space",
"sum.uniform_space",
"uniformity_dist_of_mem_uniformity",
"zero_lt_one"
] | The distance on the disjoint union indeed defines a metric space. All the distance properties
follow from our choice of the distance. The harder work is to show that the uniform structure
defined by the distance coincides with the disjoint union uniform structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum.dist_eq {x y : X ⊕ Y} : dist x y = sum.dist x y | rfl | lemma | metric.sum.dist_eq | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_inl : isometry (sum.inl : X → (X ⊕ Y)) | isometry.of_dist_eq $ λ x y, rfl | lemma | metric.isometry_inl | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | The left injection of a space in a disjoint union is an isometry | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_inr : isometry (sum.inr : Y → (X ⊕ Y)) | isometry.of_dist_eq $ λ x y, rfl | lemma | metric.isometry_inr | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | The right injection of a space in a disjoint union is an isometry | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist : (Σ i, E i) → (Σ i, E i) → ℝ | | ⟨i, x⟩ ⟨j, y⟩ :=
if h : i = j then by { have : E j = E i, by rw h, exact has_dist.dist x (cast this y) }
else has_dist.dist x (nonempty.some ⟨x⟩) + 1 + has_dist.dist (nonempty.some ⟨y⟩) y | def | metric.sigma.dist | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"nonempty.some"
] | Distance on a disjoint union. There are many (noncanonical) ways to put a distance compatible
with each factor.
We choose a construction that works for unbounded spaces, but requires basepoints,
chosen arbitrarily.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the dista... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_dist : has_dist (Σ i, E i) | ⟨sigma.dist⟩ | def | metric.sigma.has_dist | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"has_dist"
] | A `has_dist` instance on the disjoint union `Σ i, E i`.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the distance from `a` to `b` is the sum of the distances of `a` and `b` to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an ar... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_same (i : ι) (x : E i) (y : E i) :
dist (⟨i, x⟩ : Σ j, E j) ⟨i, y⟩ = dist x y | by simp [has_dist.dist, sigma.dist] | lemma | metric.sigma.dist_same | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_ne {i j : ι} (h : i ≠ j) (x : E i) (y : E j) :
dist (⟨i, x⟩ : Σ k, E k) ⟨j, y⟩ = dist x (nonempty.some ⟨x⟩) + 1 + dist (nonempty.some ⟨y⟩) y | by simp [has_dist.dist, sigma.dist, h] | lemma | metric.sigma.dist_ne | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"nonempty.some"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_dist_of_ne {i j : ι} (h : i ≠ j) (x : E i) (y : E j) :
1 ≤ dist (⟨i, x⟩ : Σ k, E k) ⟨j, y⟩ | begin
rw sigma.dist_ne h x y,
linarith [@dist_nonneg _ _ x (nonempty.some ⟨x⟩), @dist_nonneg _ _ (nonempty.some ⟨y⟩) y]
end | lemma | metric.sigma.one_le_dist_of_ne | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_nonneg",
"nonempty.some"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_eq_of_dist_lt_one (x y : Σ i, E i) (h : dist x y < 1) :
x.1 = y.1 | begin
cases x, cases y,
contrapose! h,
apply one_le_dist_of_ne h,
end | lemma | metric.sigma.fst_eq_of_dist_lt_one | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_triangle (x y z : Σ i, E i) :
dist x z ≤ dist x y + dist y z | begin
rcases x with ⟨i, x⟩, rcases y with ⟨j, y⟩, rcases z with ⟨k, z⟩,
rcases eq_or_ne i k with rfl|hik,
{ rcases eq_or_ne i j with rfl|hij,
{ simpa using dist_triangle x y z },
{ simp only [hij, hij.symm, sigma.dist_same, sigma.dist_ne, ne.def, not_false_iff],
calc dist x z ≤ dist x (nonempty.some... | lemma | metric.sigma.dist_triangle | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_nonneg",
"dist_triangle",
"eq_or_ne",
"le_rfl",
"nonempty.some",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff (s : set (Σ i, E i)) :
is_open s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s | begin
split,
{ rintros hs ⟨i, x⟩ hx,
obtain ⟨ε, εpos, hε⟩ : ∃ (ε : ℝ) (H : ε > 0), ball x ε ⊆ sigma.mk i ⁻¹' s :=
metric.is_open_iff.1 (is_open_sigma_iff.1 hs i) x hx,
refine ⟨min ε 1, lt_min εpos zero_lt_one, _⟩,
rintros ⟨j, y⟩ hy,
rcases eq_or_ne i j with rfl|hij,
{ simp only [sigma.dist... | lemma | metric.sigma.is_open_iff | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"eq_or_ne",
"is_open",
"lt_min_iff",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric_space : metric_space (Σ i, E i) | begin
refine metric_space.of_dist_topology sigma.dist _ _ sigma.dist_triangle
sigma.is_open_iff _,
{ rintros ⟨i, x⟩, simp [sigma.dist] },
{ rintros ⟨i, x⟩ ⟨j, y⟩,
rcases eq_or_ne i j with rfl|h,
{ simp [sigma.dist, dist_comm] },
{ simp only [sigma.dist, dist_comm, h, h.symm, not_false_iff, dif_neg... | def | metric.sigma.metric_space | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_comm",
"eq_or_ne",
"metric_space",
"metric_space.of_dist_topology",
"zero_lt_one"
] | A metric space structure on the disjoint union `Σ i, E i`.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the distance from `a` to `b` is the sum of the distances of `a` and `b` to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_mk (i : ι) : isometry (sigma.mk i : E i → Σ k, E k) | isometry.of_dist_eq (λ x y, by simp) | lemma | metric.sigma.isometry_mk | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | The injection of a space in a disjoint union is an isometry | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space [∀ i, complete_space (E i)] : complete_space (Σ i, E i) | begin
set s : ι → set (Σ i, E i) := λ i, (sigma.fst ⁻¹' {i}),
set U := {p : (Σ k, E k) × (Σ k, E k) | dist p.1 p.2 < 1},
have hc : ∀ i, is_complete (s i),
{ intro i,
simp only [s, ← range_sigma_mk],
exact (isometry_mk i).uniform_inducing.is_complete_range },
have hd : ∀ i j (x ∈ s i) (y ∈ s j), (x, y)... | lemma | metric.sigma.complete_space | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"complete_space",
"complete_space_of_is_complete_univ",
"is_complete",
"is_complete_Union_separated",
"uniform_inducing.is_complete_range",
"zero_lt_one"
] | A disjoint union of complete metric spaces is complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glue_premetric (hΦ : isometry Φ) (hΨ : isometry Ψ) : pseudo_metric_space (X ⊕ Y) | { dist := glue_dist Φ Ψ 0,
dist_self := glue_dist_self Φ Ψ 0,
dist_comm := glue_dist_comm Φ Ψ 0,
dist_triangle := glue_dist_triangle Φ Ψ 0 $ λp q, by rw [hΦ.dist_eq, hΨ.dist_eq]; simp } | def | metric.glue_premetric | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"isometry",
"pseudo_metric_space"
] | Given two isometric embeddings `Φ : Z → X` and `Ψ : Z → Y`, we define a pseudo metric space
structure on `X ⊕ Y` by declaring that `Φ x` and `Ψ x` are at distance `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glue_space (hΦ : isometry Φ) (hΨ : isometry Ψ) : Type* | @uniform_space.separation_quotient _ (glue_premetric hΦ hΨ).to_uniform_space | def | metric.glue_space | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry",
"uniform_space.separation_quotient"
] | Given two isometric embeddings `Φ : Z → X` and `Ψ : Z → Y`, we define a
space `glue_space hΦ hΨ` by identifying in `X ⊕ Y` the points `Φ x` and `Ψ x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_glue_l (hΦ : isometry Φ) (hΨ : isometry Ψ) (x : X) : glue_space hΦ hΨ | quotient.mk' (inl x) | def | metric.to_glue_l | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry",
"quotient.mk'"
] | The canonical map from `X` to the space obtained by gluing isometric subsets in `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_glue_r (hΦ : isometry Φ) (hΨ : isometry Ψ) (y : Y) : glue_space hΦ hΨ | quotient.mk' (inr y) | def | metric.to_glue_r | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry",
"quotient.mk'"
] | The canonical map from `Y` to the space obtained by gluing isometric subsets in `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_left (hΦ : isometry Φ) (hΨ : isometry Ψ) [inhabited X] :
inhabited (glue_space hΦ hΨ) | ⟨to_glue_l _ _ default⟩ | instance | metric.inhabited_left | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited_right (hΦ : isometry Φ) (hΨ : isometry Ψ) [inhabited Y] :
inhabited (glue_space hΦ hΨ) | ⟨to_glue_r _ _ default⟩ | instance | metric.inhabited_right | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_glue_commute (hΦ : isometry Φ) (hΨ : isometry Ψ) :
(to_glue_l hΦ hΨ) ∘ Φ = (to_glue_r hΦ hΨ) ∘ Ψ | begin
letI i : pseudo_metric_space (X ⊕ Y) := glue_premetric hΦ hΨ,
letI := i.to_uniform_space,
funext,
simp only [comp, to_glue_l, to_glue_r],
refine uniform_space.separation_quotient.mk_eq_mk.2 (metric.inseparable_iff.2 _),
exact glue_dist_glued_points Φ Ψ 0 x
end | lemma | metric.to_glue_commute | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_glue_l_isometry (hΦ : isometry Φ) (hΨ : isometry Ψ) : isometry (to_glue_l hΦ hΨ) | isometry.of_dist_eq $ λ_ _, rfl | lemma | metric.to_glue_l_isometry | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_glue_r_isometry (hΦ : isometry Φ) (hΨ : isometry Ψ) : isometry (to_glue_r hΦ hΨ) | isometry.of_dist_eq $ λ_ _, rfl | lemma | metric.to_glue_r_isometry | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inductive_limit_dist (f : Π n, X n → X (n+1)) (x y : Σ n, X n) : ℝ | dist (le_rec_on (le_max_left x.1 y.1) f x.2 : X (max x.1 y.1))
(le_rec_on (le_max_right x.1 y.1) f y.2 : X (max x.1 y.1)) | def | metric.inductive_limit_dist | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [] | Predistance on the disjoint union `Σ n, X n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inductive_limit_dist_eq_dist (I : ∀ n, isometry (f n))
(x y : Σ n, X n) (m : ℕ) : ∀ hx : x.1 ≤ m, ∀ hy : y.1 ≤ m,
inductive_limit_dist f x y = dist (le_rec_on hx f x.2 : X m) (le_rec_on hy f y.2 : X m) | begin
induction m with m hm,
{ assume hx hy,
have A : max x.1 y.1 = 0, { rw [nonpos_iff_eq_zero.1 hx, nonpos_iff_eq_zero.1 hy], simp },
unfold inductive_limit_dist,
congr; simp only [A] },
{ assume hx hy,
by_cases h : max x.1 y.1 = m.succ,
{ unfold inductive_limit_dist,
congr; simp only ... | lemma | metric.inductive_limit_dist_eq_dist | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | The predistance on the disjoint union `Σ n, X n` can be computed in any `X k` for large
enough `k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inductive_premetric (I : ∀ n, isometry (f n)) :
pseudo_metric_space (Σ n, X n) | { dist := inductive_limit_dist f,
dist_self := λx, by simp [dist, inductive_limit_dist],
dist_comm := λx y, begin
let m := max x.1 y.1,
have hx : x.1 ≤ m := le_max_left _ _,
have hy : y.1 ≤ m := le_max_right _ _,
unfold dist,
rw [inductive_limit_dist_eq_dist I x y m hx hy, induc... | def | metric.inductive_premetric | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"isometry",
"pseudo_metric_space"
] | Premetric space structure on `Σ n, X n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inductive_limit (I : ∀ n, isometry (f n)) : Type* | @uniform_space.separation_quotient _ (inductive_premetric I).to_uniform_space | def | metric.inductive_limit | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry",
"uniform_space.separation_quotient"
] | The type giving the inductive limit in a metric space context. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_inductive_limit (I : ∀ n, isometry (f n)) (n : ℕ) (x : X n) : metric.inductive_limit I | quotient.mk' (sigma.mk n x) | def | metric.to_inductive_limit | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry",
"metric.inductive_limit",
"quotient.mk'"
] | Mapping each `X n` to the inductive limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_inductive_limit_isometry (I : ∀ n, isometry (f n)) (n : ℕ) :
isometry (to_inductive_limit I n) | isometry.of_dist_eq $ λ x y,
begin
change inductive_limit_dist f ⟨n, x⟩ ⟨n, y⟩ = dist x y,
rw [inductive_limit_dist_eq_dist I ⟨n, x⟩ ⟨n, y⟩ n (le_refl n) (le_refl n),
le_rec_on_self, le_rec_on_self]
end | lemma | metric.to_inductive_limit_isometry | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"isometry"
] | The map `to_inductive_limit n` mapping `X n` to the inductive limit is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_inductive_limit_commute (I : ∀ n, isometry (f n)) (n : ℕ) :
(to_inductive_limit I n.succ) ∘ (f n) = to_inductive_limit I n | begin
letI := inductive_premetric I,
funext,
simp only [comp, to_inductive_limit],
refine uniform_space.separation_quotient.mk_eq_mk.2 (metric.inseparable_iff.2 _),
show inductive_limit_dist f ⟨n.succ, f n x⟩ ⟨n, x⟩ = 0,
{ rw [inductive_limit_dist_eq_dist I ⟨n.succ, f n x⟩ ⟨n, x⟩ n.succ,
le_rec_on_s... | lemma | metric.to_inductive_limit_commute | topology.metric_space | src/topology/metric_space/gluing.lean | [
"topology.metric_space.isometry"
] | [
"dist_self",
"isometry",
"le_rfl"
] | The maps `to_inductive_limit n` are compatible with the maps `f n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_rel : nonempty_compacts ℓ_infty_ℝ → nonempty_compacts ℓ_infty_ℝ → Prop | λ x y, nonempty (x ≃ᵢ y) | def | Gromov_Hausdorff.isometry_rel | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [] | Equivalence relation identifying two nonempty compact sets which are isometric | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equivalence_isometry_rel : equivalence isometry_rel | ⟨λ x, ⟨isometry_equiv.refl _⟩, λ x y ⟨e⟩, ⟨e.symm⟩, λ x y z ⟨e⟩ ⟨f⟩, ⟨e.trans f⟩⟩ | lemma | Gromov_Hausdorff.is_equivalence_isometry_rel | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [] | This is indeed an equivalence relation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_rel.setoid : setoid (nonempty_compacts ℓ_infty_ℝ) | setoid.mk isometry_rel is_equivalence_isometry_rel | instance | Gromov_Hausdorff.isometry_rel.setoid | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [] | setoid instance identifying two isometric nonempty compact subspaces of ℓ^∞(ℝ) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
GH_space : Type | quotient (isometry_rel.setoid) | definition | Gromov_Hausdorff.GH_space | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [] | The Gromov-Hausdorff space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_GH_space (X : Type u) [metric_space X] [compact_space X] [nonempty X] : GH_space | ⟦nonempty_compacts.Kuratowski_embedding X⟧ | definition | Gromov_Hausdorff.to_GH_space | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [
"compact_space",
"metric_space"
] | Map any nonempty compact type to `GH_space` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
GH_space.rep (p : GH_space) : Type | (quotient.out p : nonempty_compacts ℓ_infty_ℝ) | def | Gromov_Hausdorff.GH_space.rep | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [
"quotient.out"
] | A metric space representative of any abstract point in `GH_space` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_GH_space_iff {X : Type u} [metric_space X] [compact_space X] [nonempty X]
{p : nonempty_compacts ℓ_infty_ℝ} :
⟦p⟧ = to_GH_space X ↔ ∃ Ψ : X → ℓ_infty_ℝ, isometry Ψ ∧ range Ψ = p | begin
simp only [to_GH_space, quotient.eq],
refine ⟨λ h, _, _⟩,
{ rcases setoid.symm h with ⟨e⟩,
have f := (Kuratowski_embedding.isometry X).isometry_equiv_on_range.trans e,
use [λ x, f x, isometry_subtype_coe.comp f.isometry],
rw [range_comp, f.range_eq_univ, set.image_univ, subtype.range_coe],
r... | lemma | Gromov_Hausdorff.eq_to_GH_space_iff | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [
"Kuratowski_embedding",
"Kuratowski_embedding.isometry",
"compact_space",
"isometry",
"metric_space",
"nonempty_compacts.Kuratowski_embedding",
"quotient.eq",
"set.image_univ",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_GH_space {p : nonempty_compacts ℓ_infty_ℝ} : ⟦p⟧ = to_GH_space p | eq_to_GH_space_iff.2 ⟨λ x, x, isometry_subtype_coe, subtype.range_coe⟩ | lemma | Gromov_Hausdorff.eq_to_GH_space | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [
"isometry_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rep_GH_space_metric_space {p : GH_space} : metric_space p.rep | by apply_instance | instance | Gromov_Hausdorff.rep_GH_space_metric_space | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [
"metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rep_GH_space_compact_space {p : GH_space} : compact_space p.rep | by apply_instance | instance | Gromov_Hausdorff.rep_GH_space_compact_space | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rep_GH_space_nonempty {p : GH_space} : nonempty p.rep | by apply_instance | instance | Gromov_Hausdorff.rep_GH_space_nonempty | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
GH_space.to_GH_space_rep (p : GH_space) : to_GH_space p.rep = p | begin
change to_GH_space (quot.out p : nonempty_compacts ℓ_infty_ℝ) = p,
rw ← eq_to_GH_space,
exact quot.out_eq p
end | lemma | Gromov_Hausdorff.GH_space.to_GH_space_rep | topology.metric_space | src/topology/metric_space/gromov_hausdorff.lean | [
"set_theory.cardinal.basic",
"topology.metric_space.closeds",
"topology.metric_space.completion",
"topology.metric_space.gromov_hausdorff_realized",
"topology.metric_space.kuratowski"
] | [
"quot.out",
"quot.out_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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