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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to_GH_space_eq_to_GH_space_iff_isometry_equiv {X : Type u} [metric_space X] [compact_space X]
[nonempty X] {Y : Type v} [metric_space Y] [compact_space Y] [nonempty Y] :
to_GH_space X = to_GH_space Y ↔ nonempty (X ≃ᵢ Y) | ⟨begin
simp only [to_GH_space, quotient.eq],
rintro ⟨e⟩,
have I : ((nonempty_compacts.Kuratowski_embedding X) ≃ᵢ
(nonempty_compacts.Kuratowski_embedding Y))
= ((range (Kuratowski_embedding X)) ≃ᵢ (range (Kuratowski_embedding Y))),
by { dunfold nonempty_compacts.Kuratowski_embedding, ref... | lemma | Gromov_Hausdorff.to_GH_space_eq_to_GH_space_iff_isometry_equiv | 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",
"metric_space",
"nonempty_compacts.Kuratowski_embedding",
"quotient.eq"
] | Two nonempty compact spaces have the same image in `GH_space` if and only if they are
isometric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
GH_dist (X : Type u) (Y : Type v) [metric_space X] [nonempty X] [compact_space X]
[metric_space Y] [nonempty Y] [compact_space Y] : ℝ | dist (to_GH_space X) (to_GH_space Y) | def | Gromov_Hausdorff.GH_dist | 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"
] | The Gromov-Hausdorff distance between two nonempty compact metric spaces, equal by definition to
the distance of the equivalence classes of these spaces in the Gromov-Hausdorff space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_GH_dist (p q : GH_space) : dist p q = GH_dist p.rep (q.rep) | by rw [GH_dist, p.to_GH_space_rep, q.to_GH_space_rep] | lemma | Gromov_Hausdorff.dist_GH_dist | 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_dist_le_Hausdorff_dist {X : Type u} [metric_space X] [compact_space X] [nonempty X]
{Y : Type v} [metric_space Y] [compact_space Y] [nonempty Y]
{γ : Type w} [metric_space γ] {Φ : X → γ} {Ψ : Y → γ} (ha : isometry Φ) (hb : isometry Ψ) :
GH_dist X Y ≤ Hausdorff_dist (range Φ) (range Ψ) | begin
/- For the proof, we want to embed `γ` in `ℓ^∞(ℝ)`, to say that the Hausdorff distance is realized
in `ℓ^∞(ℝ)` and therefore bounded below by the Gromov-Hausdorff-distance. However, `γ` is not
separable in general. We restrict to the union of the images of `X` and `Y` in `γ`, which is
separable and theref... | theorem | Gromov_Hausdorff.GH_dist_le_Hausdorff_dist | 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",
"and_imp",
"cInf_le",
"compact_space",
"forall_exists_index",
"is_compact",
"is_compact_range",
"isometry",
"isometry_subtype_coe",
"lower_bounds",
"metric_space"
] | The Gromov-Hausdorff distance between two spaces is bounded by the Hausdorff distance
of isometric copies of the spaces, in any metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_optimal {X : Type u} [metric_space X] [compact_space X] [nonempty X]
{Y : Type v} [metric_space Y] [compact_space Y] [nonempty Y] :
Hausdorff_dist (range (optimal_GH_injl X Y)) (range (optimal_GH_injr X Y)) = GH_dist X Y | begin
inhabit X, inhabit Y,
/- we only need to check the inequality `≤`, as the other one follows from the previous lemma.
As the Gromov-Hausdorff distance is an infimum, we need to check that the Hausdorff distance
in the optimal coupling is smaller than the Hausdorff distance of any coupling.
First... | lemma | Gromov_Hausdorff.Hausdorff_dist_optimal | 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"
] | [
"bound",
"cinfi_le",
"compact_space",
"csupr_le",
"dist_comm",
"dist_eq_zero",
"dist_triangle",
"forall_const",
"is_compact.bounded",
"le_cInf",
"le_of_forall_le_of_dense",
"le_rfl",
"max_le_iff",
"metric_space",
"ring",
"set.nonempty.prod"
] | The optimal coupling constructed above realizes exactly the Gromov-Hausdorff distance,
essentially by design. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
GH_dist_eq_Hausdorff_dist (X : Type u) [metric_space X] [compact_space X] [nonempty X]
(Y : Type v) [metric_space Y] [compact_space Y] [nonempty Y] :
∃ Φ : X → ℓ_infty_ℝ, ∃ Ψ : Y → ℓ_infty_ℝ, isometry Φ ∧ isometry Ψ ∧
GH_dist X Y = Hausdorff_dist (range Φ) (range Ψ) | begin
let F := Kuratowski_embedding (optimal_GH_coupling X Y),
let Φ := F ∘ optimal_GH_injl X Y,
let Ψ := F ∘ optimal_GH_injr X Y,
refine ⟨Φ, Ψ, _, _, _⟩,
{ exact (Kuratowski_embedding.isometry _).comp (isometry_optimal_GH_injl X Y) },
{ exact (Kuratowski_embedding.isometry _).comp (isometry_optimal_GH_injr... | theorem | Gromov_Hausdorff.GH_dist_eq_Hausdorff_dist | 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"
] | The Gromov-Hausdorff distance can also be realized by a coupling in `ℓ^∞(ℝ)`, by embedding
the optimal coupling through its Kuratowski embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_space.nonempty_compacts.to_GH_space {X : Type u} [metric_space X]
(p : nonempty_compacts X) : Gromov_Hausdorff.GH_space | Gromov_Hausdorff.to_GH_space p | definition | topological_space.nonempty_compacts.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"
] | [
"Gromov_Hausdorff.GH_space",
"Gromov_Hausdorff.to_GH_space",
"metric_space"
] | In particular, nonempty compacts of a metric space map to `GH_space`. We register this
in the topological_space namespace to take advantage of the notation `p.to_GH_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
GH_dist_le_nonempty_compacts_dist (p q : nonempty_compacts X) :
dist p.to_GH_space q.to_GH_space ≤ dist p q | begin
have ha : isometry (coe : p → X) := isometry_subtype_coe,
have hb : isometry (coe : q → X) := isometry_subtype_coe,
have A : dist p q = Hausdorff_dist (p : set X) q := rfl,
have I : ↑p = range (coe : p → X) := subtype.range_coe_subtype.symm,
have J : ↑q = range (coe : q → X) := subtype.range_coe_subtype... | theorem | Gromov_Hausdorff.GH_dist_le_nonempty_compacts_dist | 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",
"isometry_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_GH_space_lipschitz :
lipschitz_with 1 (nonempty_compacts.to_GH_space : nonempty_compacts X → GH_space) | lipschitz_with.mk_one GH_dist_le_nonempty_compacts_dist | lemma | Gromov_Hausdorff.to_GH_space_lipschitz | 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"
] | [
"lipschitz_with",
"lipschitz_with.mk_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_GH_space_continuous :
continuous (nonempty_compacts.to_GH_space : nonempty_compacts X → GH_space) | to_GH_space_lipschitz.continuous | lemma | Gromov_Hausdorff.to_GH_space_continuous | 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"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
GH_dist_le_of_approx_subsets {s : set X} (Φ : s → Y) {ε₁ ε₂ ε₃ : ℝ}
(hs : ∀ x : X, ∃ y ∈ s, dist x y ≤ ε₁) (hs' : ∀ x : Y, ∃ y : s, dist x (Φ y) ≤ ε₃)
(H : ∀ x y : s, |dist x y - dist (Φ x) (Φ y)| ≤ ε₂) :
GH_dist X Y ≤ ε₁ + ε₂ / 2 + ε₃ | begin
refine le_of_forall_pos_le_add (λ δ δ0, _),
rcases exists_mem_of_nonempty X with ⟨xX, _⟩,
rcases hs xX with ⟨xs, hxs, Dxs⟩,
have sne : s.nonempty := ⟨xs, hxs⟩,
letI : nonempty s := sne.to_subtype,
have : 0 ≤ ε₂ := le_trans (abs_nonneg _) (H ⟨xs, hxs⟩ ⟨xs, hxs⟩),
have : ∀ p q : s, |dist p q - dist (Φ... | theorem | Gromov_Hausdorff.GH_dist_le_of_approx_subsets | 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"
] | [
"abs_nonneg",
"dist_comm",
"dist_nonneg",
"dist_self",
"is_compact_range",
"isometry",
"metric_space",
"set.mem_image"
] | If there are subsets which are `ε₁`-dense and `ε₃`-dense in two spaces, and
isometric up to `ε₂`, then the Gromov-Hausdorff distance between the spaces is bounded by
`ε₁ + ε₂/2 + ε₃`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded {t : set GH_space} {C : ℝ} {u : ℕ → ℝ} {K : ℕ → ℕ}
(ulim : tendsto u at_top (𝓝 0))
(hdiam : ∀ p ∈ t, diam (univ : set (GH_space.rep p)) ≤ C)
(hcov : ∀ p ∈ t, ∀ n:ℕ, ∃ s : set (GH_space.rep p),
cardinal.mk s ≤ K n ∧ univ ⊆ ⋃x∈s, ball x (u n)) :
totally_bounded t | begin
/- Let `δ>0`, and `ε = δ/5`. For each `p`, we construct a finite subset `s p` of `p`, which
is `ε`-dense and has cardinality at most `K n`. Encoding the mutual distances of points in `s p`,
up to `ε`, we will get a map `F` associating to `p` finitely many data, and making it possible to
reconstruct `p` up... | lemma | Gromov_Hausdorff.totally_bounded | 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"
] | [
"abs_mul",
"abs_of_nonneg",
"bot_le",
"cardinal.mk",
"cardinal.mk_fin",
"cardinal.nat_cast_le",
"cardinal.nat_lt_aleph_0",
"dist_nonneg",
"equiv",
"equiv.apply_symm_apply",
"fin.cast",
"fin.coe_cast",
"fin.coe_mk",
"fin.ext_iff",
"fintype.card_eq",
"fintype.card_fin",
"int.floor_to_n... | Compactness criterion: a closed set of compact metric spaces is compact if the spaces have
a uniformly bounded diameter, and for all `ε` the number of balls of radius `ε` required
to cover the spaces is uniformly bounded. This is an equivalence, but we only prove the
interesting direction that these conditions imply co... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aux_gluing_struct (A : Type) [metric_space A] : Type 1 | (space : Type)
(metric : metric_space space)
(embed : A → space)
(isom : isometry embed) | structure | Gromov_Hausdorff.aux_gluing_struct | 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",
"metric_space"
] | Auxiliary structure used to glue metric spaces below, recording an isometric embedding
of a type `A` in another metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aux_gluing (n : ℕ) : aux_gluing_struct (X n) | nat.rec_on n default $ λ n Y,
{ space := glue_space Y.isom (isometry_optimal_GH_injl (X n) (X (n+1))),
metric := by apply_instance,
embed := (to_glue_r Y.isom (isometry_optimal_GH_injl (X n) (X (n+1))))
∘ (optimal_GH_injr (X n) (X (n+1))),
isom := (to_glue_r_isometry _ _).comp (isometry_... | def | Gromov_Hausdorff.aux_gluing | 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"
] | [] | Auxiliary sequence of metric spaces, containing copies of `X 0`, ..., `X n`, where each
`X i` is glued to `X (i+1)` in an optimal way. The space at step `n+1` is obtained from the space
at step `n` by adding `X (n+1)`, glued in an optimal way to the `X n` already sitting there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_space_fun : Type* | ((X ⊕ Y) × (X ⊕ Y)) → ℝ | def | Gromov_Hausdorff.prod_space_fun | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Cb : Type* | bounded_continuous_function ((X ⊕ Y) × (X ⊕ Y)) ℝ | def | Gromov_Hausdorff.Cb | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"bounded_continuous_function"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_var : ℝ≥0 | 2 * ⟨diam (univ : set X), diam_nonneg⟩ + 1 + 2 * ⟨diam (univ : set Y), diam_nonneg⟩ | def | Gromov_Hausdorff.max_var | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_max_var : 1 ≤ max_var X Y | calc
(1 : real) = 2 * 0 + 1 + 2 * 0 : by simp
... ≤ 2 * diam (univ : set X) + 1 + 2 * diam (univ : set Y) :
by apply_rules [add_le_add, mul_le_mul_of_nonneg_left, diam_nonneg]; norm_num | lemma | Gromov_Hausdorff.one_le_max_var | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"mul_le_mul_of_nonneg_left",
"real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates : set (prod_space_fun X Y) | {f | (((((∀ x y : X, f (sum.inl x, sum.inl y) = dist x y)
∧ (∀ x y : Y, f (sum.inr x, sum.inr y) = dist x y))
∧ (∀ x y, f (x, y) = f (y, x)))
∧ (∀ x y z, f (x, z) ≤ f (x, y) + f (y, z)))
∧ (∀ x, f (x, x) = 0))
∧ (∀ x y, f (x, y) ≤ max_var X Y) } | def | Gromov_Hausdorff.candidates | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | The set of functions on `X ⊕ Y` that are candidates distances to realize the
minimum of the Hausdorff distances between `X` and `Y` in a coupling | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
candidates_b : set (Cb X Y) | {f : Cb X Y | (f : _ → ℝ) ∈ candidates X Y} | def | Gromov_Hausdorff.candidates_b | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | Version of the set of candidates in bounded_continuous_functions, to apply
Arzela-Ascoli | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
max_var_bound : dist x y ≤ max_var X Y | calc
dist x y ≤ diam (univ : set (X ⊕ Y)) :
dist_le_diam_of_mem bounded_of_compact_space (mem_univ _) (mem_univ _)
... = diam (range inl ∪ range inr : set (X ⊕ Y)) :
by rw [range_inl_union_range_inr]
... ≤ diam (range inl : set (X ⊕ Y)) + dist (inl default) (inr default) +
diam (range inr : set ... | lemma | Gromov_Hausdorff.max_var_bound | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_symm (fA : f ∈ candidates X Y) : f (x, y) = f (y, x) | fA.1.1.1.2 x y | lemma | Gromov_Hausdorff.candidates_symm | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_triangle (fA : f ∈ candidates X Y) : f (x, z) ≤ f (x, y) + f (y, z) | fA.1.1.2 x y z | lemma | Gromov_Hausdorff.candidates_triangle | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_refl (fA : f ∈ candidates X Y) : f (x, x) = 0 | fA.1.2 x | lemma | Gromov_Hausdorff.candidates_refl | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_nonneg (fA : f ∈ candidates X Y) : 0 ≤ f (x, y) | begin
have : 0 ≤ 2 * f (x, y) := calc
0 = f (x, x) : (candidates_refl fA).symm
... ≤ f (x, y) + f (y, x) : candidates_triangle fA
... = f (x, y) + f (x, y) : by rw [candidates_symm fA]
... = 2 * f (x, y) : by ring,
by linarith
end | lemma | Gromov_Hausdorff.candidates_nonneg | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_dist_inl (fA : f ∈ candidates X Y) (x y: X) :
f (inl x, inl y) = dist x y | fA.1.1.1.1.1 x y | lemma | Gromov_Hausdorff.candidates_dist_inl | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_dist_inr (fA : f ∈ candidates X Y) (x y : Y) :
f (inr x, inr y) = dist x y | fA.1.1.1.1.2 x y | lemma | Gromov_Hausdorff.candidates_dist_inr | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_le_max_var (fA : f ∈ candidates X Y) : f (x, y) ≤ max_var X Y | fA.2 x y | lemma | Gromov_Hausdorff.candidates_le_max_var | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_dist_bound (fA : f ∈ candidates X Y) :
∀ {x y : X ⊕ Y}, f (x, y) ≤ max_var X Y * dist x y | | (inl x) (inl y) := calc
f (inl x, inl y) = dist x y : candidates_dist_inl fA x y
... = dist (inl x) (inl y) : by { rw @sum.dist_eq X Y, refl }
... = 1 * dist (inl x) (inl y) : by simp
... ≤ max_var X Y * dist (inl x) (inl y) :
mul_le_mul_of_nonneg_right (one_le_max_var X Y) dist_nonneg
| (inl x)... | lemma | Gromov_Hausdorff.candidates_dist_bound | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"dist_nonneg",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"zero_le_one"
] | candidates are bounded by `max_var X Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
candidates_lipschitz_aux (fA : f ∈ candidates X Y) :
f (x, y) - f (z, t) ≤ 2 * max_var X Y * dist (x, y) (z, t) | calc
f (x, y) - f(z, t) ≤ f (x, t) + f (t, y) - f (z, t) : sub_le_sub_right (candidates_triangle fA) _
... ≤ (f (x, z) + f (z, t) + f(t, y)) - f (z, t) :
sub_le_sub_right (add_le_add_right (candidates_triangle fA) _ ) _
... = f (x, z) + f (t, y) : by simp [sub_eq_add_neg, add_assoc]
... ≤ max_var X Y * dist... | lemma | Gromov_Hausdorff.candidates_lipschitz_aux | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"dist_comm",
"mul_le_mul_of_nonneg_left",
"ring"
] | Technical lemma to prove that candidates are Lipschitz | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
candidates_lipschitz (fA : f ∈ candidates X Y) :
lipschitz_with (2 * max_var X Y) f | begin
apply lipschitz_with.of_dist_le_mul,
rintros ⟨x, y⟩ ⟨z, t⟩,
rw [real.dist_eq, abs_sub_le_iff],
use candidates_lipschitz_aux fA,
rw [dist_comm],
exact candidates_lipschitz_aux fA
end | lemma | Gromov_Hausdorff.candidates_lipschitz | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"abs_sub_le_iff",
"dist_comm",
"lipschitz_with",
"real.dist_eq"
] | Candidates are Lipschitz | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
candidates_b_of_candidates (f : prod_space_fun X Y) (fA : f ∈ candidates X Y) : Cb X Y | bounded_continuous_function.mk_of_compact ⟨f, (candidates_lipschitz fA).continuous⟩ | def | Gromov_Hausdorff.candidates_b_of_candidates | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"bounded_continuous_function.mk_of_compact"
] | candidates give rise to elements of bounded_continuous_functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
candidates_b_of_candidates_mem (f : prod_space_fun X Y) (fA : f ∈ candidates X Y) :
candidates_b_of_candidates f fA ∈ candidates_b X Y | fA | lemma | Gromov_Hausdorff.candidates_b_of_candidates_mem | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mem_candidates : (λp : (X ⊕ Y) × (X ⊕ Y), dist p.1 p.2) ∈ candidates X Y | begin
simp only [candidates, dist_comm, forall_const, and_true, add_comm, eq_self_iff_true,
and_self, sum.forall, set.mem_set_of_eq, dist_self],
repeat { split
<|> exact (λa y z, dist_triangle_left _ _ _)
<|> exact (λx y, by refl)
<|> exact (λx y, max_var_bound) }
end | lemma | Gromov_Hausdorff.dist_mem_candidates | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"dist_comm",
"dist_self",
"dist_triangle_left",
"forall_const"
] | The distance on `X ⊕ Y` is a candidate | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
candidates_b_dist (X : Type u) (Y : Type v) [metric_space X] [compact_space X] [inhabited X]
[metric_space Y] [compact_space Y] [inhabited Y] : Cb X Y | candidates_b_of_candidates _ dist_mem_candidates | def | Gromov_Hausdorff.candidates_b_dist | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"compact_space",
"metric_space"
] | The distance on `X ⊕ Y` as a candidate | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
candidates_b_dist_mem_candidates_b : candidates_b_dist X Y ∈ candidates_b X Y | candidates_b_of_candidates_mem _ _ | lemma | Gromov_Hausdorff.candidates_b_dist_mem_candidates_b | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
candidates_b_nonempty : (candidates_b X Y).nonempty | ⟨_, candidates_b_dist_mem_candidates_b⟩ | lemma | Gromov_Hausdorff.candidates_b_nonempty | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_candidates_b : is_closed (candidates_b X Y) | begin
have I1 : ∀ x y, is_closed {f : Cb X Y | f (inl x, inl y) = dist x y} :=
λx y, is_closed_eq continuous_eval_const continuous_const,
have I2 : ∀ x y, is_closed {f : Cb X Y | f (inr x, inr y) = dist x y } :=
λx y, is_closed_eq continuous_eval_const continuous_const,
have I3 : ∀ x y, is_closed {f : Cb ... | lemma | Gromov_Hausdorff.closed_candidates_b | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"continuous_const",
"is_closed",
"is_closed.inter",
"is_closed_Inter",
"is_closed_eq",
"is_closed_le"
] | To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_candidates_b : is_compact (candidates_b X Y) | begin
refine arzela_ascoli₂ (Icc 0 (max_var X Y)) is_compact_Icc (candidates_b X Y)
closed_candidates_b _ _,
{ rintros f ⟨x1, x2⟩ hf,
simp only [set.mem_Icc],
exact ⟨candidates_nonneg hf, candidates_le_max_var hf⟩ },
{ refine equicontinuous_of_continuity_modulus (λt, 2 * max_var X Y * t) _ _ _,
{ ha... | lemma | Gromov_Hausdorff.is_compact_candidates_b | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"is_compact",
"set.mem_Icc"
] | Compactness of candidates (in bounded_continuous_functions) follows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
HD (f : Cb X Y) | max (⨆ x, ⨅ y, f (inl x, inr y)) (⨆ y, ⨅ x, f (inl x, inr y)) | def | Gromov_Hausdorff.HD | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | We will then choose the candidate minimizing the Hausdorff distance. Except that we are not
in a metric space setting, so we need to define our custom version of Hausdorff distance,
called HD, and prove its basic properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
HD_below_aux1 {f : Cb X Y} (C : ℝ) {x : X} :
bdd_below (range (λ (y : Y), f (inl x, inr y) + C)) | let ⟨cf, hcf⟩ := (real.bounded_iff_bdd_below_bdd_above.1 f.bounded_range).1 in
⟨cf + C, forall_range_iff.2 (λi, add_le_add_right ((λx, hcf (mem_range_self x)) _) _)⟩ | lemma | Gromov_Hausdorff.HD_below_aux1 | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HD_bound_aux1 (f : Cb X Y) (C : ℝ) :
bdd_above (range (λ (x : X), ⨅ y, f (inl x, inr y) + C)) | begin
rcases (real.bounded_iff_bdd_below_bdd_above.1 f.bounded_range).2 with ⟨Cf, hCf⟩,
refine ⟨Cf + C, forall_range_iff.2 (λx, _)⟩,
calc (⨅ y, f (inl x, inr y) + C) ≤ f (inl x, inr default) + C :
cinfi_le (HD_below_aux1 C) default
... ≤ Cf + C : add_le_add ((λx, hCf (mem_range_self x)) _) le_rfl
end | lemma | Gromov_Hausdorff.HD_bound_aux1 | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"bdd_above",
"cinfi_le",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HD_below_aux2 {f : Cb X Y} (C : ℝ) {y : Y} :
bdd_below (range (λ (x : X), f (inl x, inr y) + C)) | let ⟨cf, hcf⟩ := (real.bounded_iff_bdd_below_bdd_above.1 f.bounded_range).1 in
⟨cf + C, forall_range_iff.2 (λi, add_le_add_right ((λx, hcf (mem_range_self x)) _) _)⟩ | lemma | Gromov_Hausdorff.HD_below_aux2 | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HD_bound_aux2 (f : Cb X Y) (C : ℝ) :
bdd_above (range (λ (y : Y), ⨅ x, f (inl x, inr y) + C)) | begin
rcases (real.bounded_iff_bdd_below_bdd_above.1 f.bounded_range).2 with ⟨Cf, hCf⟩,
refine ⟨Cf + C, forall_range_iff.2 (λy, _)⟩,
calc (⨅ x, f (inl x, inr y) + C) ≤ f (inl default, inr y) + C :
cinfi_le (HD_below_aux2 C) default
... ≤ Cf + C : add_le_add ((λx, hCf (mem_range_self x)) _) le_rfl
end | lemma | Gromov_Hausdorff.HD_bound_aux2 | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"bdd_above",
"cinfi_le",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HD_candidates_b_dist_le :
HD (candidates_b_dist X Y) ≤ diam (univ : set X) + 1 + diam (univ : set Y) | begin
refine max_le (csupr_le (λx, _)) (csupr_le (λy, _)),
{ have A : (⨅ y, candidates_b_dist X Y (inl x, inr y)) ≤
candidates_b_dist X Y (inl x, inr default) :=
cinfi_le (by simpa using HD_below_aux1 0) default,
have B : dist (inl x) (inr default) ≤ diam (univ : set X) + 1 + diam (univ : set Y) := ... | lemma | Gromov_Hausdorff.HD_candidates_b_dist_le | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"cinfi_le",
"csupr_le",
"le_rfl"
] | Explicit bound on `HD (dist)`. This means that when looking for minimizers it will
be sufficient to look for functions with `HD(f)` bounded by this bound. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
HD_lipschitz_aux1 (f g : Cb X Y) :
(⨆ x, ⨅ y, f (inl x, inr y)) ≤ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g | begin
rcases (real.bounded_iff_bdd_below_bdd_above.1 g.bounded_range).1 with ⟨cg, hcg⟩,
have Hcg : ∀ x, cg ≤ g x := λx, hcg (mem_range_self x),
rcases (real.bounded_iff_bdd_below_bdd_above.1 f.bounded_range).1 with ⟨cf, hcf⟩,
have Hcf : ∀ x, cf ≤ f x := λx, hcf (mem_range_self x),
-- prove the inequality but... | lemma | Gromov_Hausdorff.HD_lipschitz_aux1 | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"bdd_below",
"cinfi_mono",
"continuous_at_const",
"csupr_mono",
"monotone.map_cinfi_of_continuous_at",
"monotone.map_csupr_of_continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HD_lipschitz_aux2 (f g : Cb X Y) :
(⨆ y, ⨅ x, f (inl x, inr y)) ≤ (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g | begin
rcases (real.bounded_iff_bdd_below_bdd_above.1 g.bounded_range).1 with ⟨cg, hcg⟩,
have Hcg : ∀ x, cg ≤ g x := λx, hcg (mem_range_self x),
rcases (real.bounded_iff_bdd_below_bdd_above.1 f.bounded_range).1 with ⟨cf, hcf⟩,
have Hcf : ∀ x, cf ≤ f x := λx, hcf (mem_range_self x),
-- prove the inequality but... | lemma | Gromov_Hausdorff.HD_lipschitz_aux2 | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"bdd_below",
"cinfi_mono",
"continuous_at_const",
"csupr_mono",
"monotone.map_cinfi_of_continuous_at",
"monotone.map_csupr_of_continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HD_lipschitz_aux3 (f g : Cb X Y) : HD f ≤ HD g + dist f g | max_le (le_trans (HD_lipschitz_aux1 f g) (add_le_add_right (le_max_left _ _) _))
(le_trans (HD_lipschitz_aux2 f g) (add_le_add_right (le_max_right _ _) _)) | lemma | Gromov_Hausdorff.HD_lipschitz_aux3 | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HD_continuous : continuous (HD : Cb X Y → ℝ) | lipschitz_with.continuous (lipschitz_with.of_le_add HD_lipschitz_aux3) | lemma | Gromov_Hausdorff.HD_continuous | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"continuous",
"lipschitz_with.continuous",
"lipschitz_with.of_le_add"
] | Conclude that HD, being Lipschitz, is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_minimizer : ∃ f ∈ candidates_b X Y, ∀ g ∈ candidates_b X Y, HD f ≤ HD g | is_compact_candidates_b.exists_forall_le candidates_b_nonempty HD_continuous.continuous_on | lemma | Gromov_Hausdorff.exists_minimizer | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
optimal_GH_dist : Cb X Y | classical.some (exists_minimizer X Y) | definition | Gromov_Hausdorff.optimal_GH_dist | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
optimal_GH_dist_mem_candidates_b : optimal_GH_dist X Y ∈ candidates_b X Y | by cases (classical.some_spec (exists_minimizer X Y)); assumption | lemma | Gromov_Hausdorff.optimal_GH_dist_mem_candidates_b | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HD_optimal_GH_dist_le (g : Cb X Y) (hg : g ∈ candidates_b X Y) :
HD (optimal_GH_dist X Y) ≤ HD g | let ⟨Z1, Z2⟩ := classical.some_spec (exists_minimizer X Y) in Z2 g hg | lemma | Gromov_Hausdorff.HD_optimal_GH_dist_le | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
premetric_optimal_GH_dist : pseudo_metric_space (X ⊕ Y) | { dist := λp q, optimal_GH_dist X Y (p, q),
dist_self := λx, candidates_refl (optimal_GH_dist_mem_candidates_b X Y),
dist_comm := λx y, candidates_symm (optimal_GH_dist_mem_candidates_b X Y),
dist_triangle := λx y z, candidates_triangle (optimal_GH_dist_mem_candidates_b X Y) } | def | Gromov_Hausdorff.premetric_optimal_GH_dist | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"pseudo_metric_space"
] | With the optimal candidate, construct a premetric space structure on `X ⊕ Y`, on which the
predistance is given by the candidate. Then, we will identify points at `0` predistance
to obtain a genuine metric space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
optimal_GH_coupling : Type* | @uniform_space.separation_quotient (X ⊕ Y) (premetric_optimal_GH_dist X Y).to_uniform_space | definition | Gromov_Hausdorff.optimal_GH_coupling | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"uniform_space.separation_quotient"
] | A metric space which realizes the optimal coupling between `X` and `Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
optimal_GH_injl (x : X) : optimal_GH_coupling X Y | quotient.mk' (inl x) | def | Gromov_Hausdorff.optimal_GH_injl | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"quotient.mk'"
] | Injection of `X` in the optimal coupling between `X` and `Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_optimal_GH_injl : isometry (optimal_GH_injl X Y) | isometry.of_dist_eq $ λ x y, candidates_dist_inl (optimal_GH_dist_mem_candidates_b X Y) _ _ | lemma | Gromov_Hausdorff.isometry_optimal_GH_injl | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"isometry"
] | The injection of `X` in the optimal coupling between `X` and `Y` is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
optimal_GH_injr (y : Y) : optimal_GH_coupling X Y | quotient.mk' (inr y) | def | Gromov_Hausdorff.optimal_GH_injr | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"quotient.mk'"
] | Injection of `Y` in the optimal coupling between `X` and `Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_optimal_GH_injr : isometry (optimal_GH_injr X Y) | isometry.of_dist_eq $ λ x y, candidates_dist_inr (optimal_GH_dist_mem_candidates_b X Y) _ _ | lemma | Gromov_Hausdorff.isometry_optimal_GH_injr | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"isometry"
] | The injection of `Y` in the optimal coupling between `X` and `Y` is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space_optimal_GH_coupling : compact_space (optimal_GH_coupling X Y) | ⟨begin
rw [← range_quotient_mk'],
exact is_compact_range (continuous_sum_dom.2 ⟨(isometry_optimal_GH_injl X Y).continuous,
(isometry_optimal_GH_injr X Y).continuous⟩)
end⟩ | instance | Gromov_Hausdorff.compact_space_optimal_GH_coupling | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"compact_space",
"continuous",
"is_compact_range"
] | The optimal coupling between two compact spaces `X` and `Y` is still a compact space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_optimal_le_HD {f} (h : f ∈ candidates_b X Y) :
Hausdorff_dist (range (optimal_GH_injl X Y)) (range (optimal_GH_injr X Y)) ≤ HD f | begin
refine le_trans (le_of_forall_le_of_dense (λ r hr, _)) (HD_optimal_GH_dist_le X Y f h),
have A : ∀ x ∈ range (optimal_GH_injl X Y), ∃ y ∈ range (optimal_GH_injr X Y), dist x y ≤ r,
{ rintro _ ⟨z, rfl⟩,
have I1 : (⨆ x, ⨅ y, optimal_GH_dist X Y (inl x, inr y)) < r :=
lt_of_le_of_lt (le_max_left _ _)... | lemma | Gromov_Hausdorff.Hausdorff_dist_optimal_le_HD | topology.metric_space | src/topology/metric_space/gromov_hausdorff_realized.lean | [
"topology.metric_space.gluing",
"topology.metric_space.hausdorff_distance",
"topology.continuous_function.bounded"
] | [
"dist_comm",
"dist_nonneg",
"exists_lt_of_cInf_lt",
"le_cSup",
"le_of_forall_le_of_dense"
] | For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that HD of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH (s : set X) : ℝ≥0∞ | by { borelize X, exact ⨆ (d : ℝ≥0) (hd : @hausdorff_measure X _ _ ⟨rfl⟩ d s = ∞), d } | def | dimH | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [] | Hausdorff dimension of a set in an (e)metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_def (s : set X) : dimH s = ⨆ (d : ℝ≥0) (hd : μH[d] s = ∞), d | by { borelize X, rw dimH } | lemma | dimH_def | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | Unfold the definition of `dimH` using `[measurable_space X] [borel_space X]` from the
environment. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hausdorff_measure_of_lt_dimH {s : set X} {d : ℝ≥0} (h : ↑d < dimH s) : μH[d] s = ∞ | begin
simp only [dimH_def, lt_supr_iff] at h,
rcases h with ⟨d', hsd', hdd'⟩,
rw [ennreal.coe_lt_coe, ← nnreal.coe_lt_coe] at hdd',
exact top_unique (hsd' ▸ hausdorff_measure_mono hdd'.le _)
end | lemma | hausdorff_measure_of_lt_dimH | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_def",
"ennreal.coe_lt_coe",
"lt_supr_iff",
"nnreal.coe_lt_coe",
"top_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_le {s : set X} {d : ℝ≥0∞} (H : ∀ d' : ℝ≥0, μH[d'] s = ∞ → ↑d' ≤ d) : dimH s ≤ d | (dimH_def s).trans_le $ supr₂_le H | lemma | dimH_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_def",
"supr₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_le_of_hausdorff_measure_ne_top {s : set X} {d : ℝ≥0} (h : μH[d] s ≠ ∞) :
dimH s ≤ d | le_of_not_lt $ mt hausdorff_measure_of_lt_dimH h | lemma | dimH_le_of_hausdorff_measure_ne_top | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"hausdorff_measure_of_lt_dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_dimH_of_hausdorff_measure_eq_top {s : set X} {d : ℝ≥0} (h : μH[d] s = ∞) :
↑d ≤ dimH s | by { rw dimH_def, exact le_supr₂ d h } | lemma | le_dimH_of_hausdorff_measure_eq_top | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_def",
"le_supr₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hausdorff_measure_of_dimH_lt {s : set X} {d : ℝ≥0}
(h : dimH s < d) : μH[d] s = 0 | begin
rw dimH_def at h,
rcases ennreal.lt_iff_exists_nnreal_btwn.1 h with ⟨d', hsd', hd'd⟩,
rw [ennreal.coe_lt_coe, ← nnreal.coe_lt_coe] at hd'd,
exact (hausdorff_measure_zero_or_top hd'd s).resolve_right (λ h, hsd'.not_le $ le_supr₂ d' h)
end | lemma | hausdorff_measure_of_dimH_lt | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_def",
"ennreal.coe_lt_coe",
"le_supr₂",
"nnreal.coe_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_zero_of_dimH_lt {μ : measure X} {d : ℝ≥0}
(h : μ ≪ μH[d]) {s : set X} (hd : dimH s < d) :
μ s = 0 | h $ hausdorff_measure_of_dimH_lt hd | lemma | measure_zero_of_dimH_lt | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"hausdorff_measure_of_dimH_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_dimH_of_hausdorff_measure_ne_zero {s : set X} {d : ℝ≥0} (h : μH[d] s ≠ 0) :
↑d ≤ dimH s | le_of_not_lt $ mt hausdorff_measure_of_dimH_lt h | lemma | le_dimH_of_hausdorff_measure_ne_zero | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"hausdorff_measure_of_dimH_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_of_hausdorff_measure_ne_zero_ne_top {d : ℝ≥0} {s : set X} (h : μH[d] s ≠ 0)
(h' : μH[d] s ≠ ∞) : dimH s = d | le_antisymm (dimH_le_of_hausdorff_measure_ne_top h') (le_dimH_of_hausdorff_measure_ne_zero h) | lemma | dimH_of_hausdorff_measure_ne_zero_ne_top | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_le_of_hausdorff_measure_ne_top",
"le_dimH_of_hausdorff_measure_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_mono {s t : set X} (h : s ⊆ t) : dimH s ≤ dimH t | begin
borelize X,
exact dimH_le (λ d hd, le_dimH_of_hausdorff_measure_eq_top $
top_unique $ hd ▸ measure_mono h)
end | lemma | dimH_mono | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_le",
"le_dimH_of_hausdorff_measure_eq_top",
"top_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_subsingleton {s : set X} (h : s.subsingleton) : dimH s = 0 | begin
borelize X,
apply le_antisymm _ (zero_le _),
refine dimH_le_of_hausdorff_measure_ne_top _,
exact ((hausdorff_measure_le_one_of_subsingleton h le_rfl).trans_lt ennreal.one_lt_top).ne,
end | lemma | dimH_subsingleton | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_le_of_hausdorff_measure_ne_top",
"ennreal.one_lt_top",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_empty : dimH (∅ : set X) = 0 | subsingleton_empty.dimH_zero | lemma | dimH_empty | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_singleton (x : X) : dimH ({x} : set X) = 0 | subsingleton_singleton.dimH_zero | lemma | dimH_singleton | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_Union [encodable ι] (s : ι → set X) :
dimH (⋃ i, s i) = ⨆ i, dimH (s i) | begin
borelize X,
refine le_antisymm (dimH_le $ λ d hd, _) (supr_le $ λ i, dimH_mono $ subset_Union _ _),
contrapose! hd,
have : ∀ i, μH[d] (s i) = 0,
from λ i, hausdorff_measure_of_dimH_lt ((le_supr (λ i, dimH (s i)) i).trans_lt hd),
rw measure_Union_null this,
exact ennreal.zero_ne_top
end | lemma | dimH_Union | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_le",
"dimH_mono",
"encodable",
"ennreal.zero_ne_top",
"hausdorff_measure_of_dimH_lt",
"le_supr",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_bUnion {s : set ι} (hs : s.countable) (t : ι → set X) :
dimH (⋃ i ∈ s, t i) = ⨆ i ∈ s, dimH (t i) | begin
haveI := hs.to_encodable,
rw [bUnion_eq_Union, dimH_Union, ← supr_subtype'']
end | lemma | dimH_bUnion | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_Union",
"supr_subtype''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_sUnion {S : set (set X)} (hS : S.countable) : dimH (⋃₀ S) = ⨆ s ∈ S, dimH s | by rw [sUnion_eq_bUnion, dimH_bUnion hS] | lemma | dimH_sUnion | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_union (s t : set X) : dimH (s ∪ t) = max (dimH s) (dimH t) | by rw [union_eq_Union, dimH_Union, supr_bool_eq, cond, cond, ennreal.sup_eq_max] | lemma | dimH_union | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_Union",
"ennreal.sup_eq_max",
"supr_bool_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_countable {s : set X} (hs : s.countable) : dimH s = 0 | bUnion_of_singleton s ▸ by simp only [dimH_bUnion hs, dimH_singleton, ennreal.supr_zero_eq_zero] | lemma | dimH_countable | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_bUnion",
"dimH_singleton",
"ennreal.supr_zero_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_finite {s : set X} (hs : s.finite) : dimH s = 0 | hs.countable.dimH_zero | lemma | dimH_finite | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_coe_finset (s : finset X) : dimH (s : set X) = 0 | s.finite_to_set.dimH_zero | lemma | dimH_coe_finset | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_nhds_within_lt_dimH_of_lt_dimH {s : set X} {r : ℝ≥0∞} (h : r < dimH s) :
∃ x ∈ s, ∀ t ∈ 𝓝[s] x, r < dimH t | begin
contrapose! h, choose! t htx htr using h,
rcases countable_cover_nhds_within htx with ⟨S, hSs, hSc, hSU⟩,
calc dimH s ≤ dimH (⋃ x ∈ S, t x) : dimH_mono hSU
... = ⨆ x ∈ S, dimH (t x) : dimH_bUnion hSc _
... ≤ r : supr₂_le (λ x hx, htr x $ hSs hx)
end | lemma | exists_mem_nhds_within_lt_dimH_of_lt_dimH | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_bUnion",
"dimH_mono",
"supr₂_le"
] | If `r` is less than the Hausdorff dimension of a set `s` in an (extended) metric space with
second countable topology, then there exists a point `x ∈ s` such that every neighborhood
`t` of `x` within `s` has Hausdorff dimension greater than `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bsupr_limsup_dimH (s : set X) : (⨆ x ∈ s, limsup dimH (𝓝[s] x).small_sets) = dimH s | begin
refine le_antisymm (supr₂_le $ λ x hx, _) _,
{ refine Limsup_le_of_le (by apply_auto_param) (eventually_map.2 _),
exact eventually_small_sets.2 ⟨s, self_mem_nhds_within, λ t, dimH_mono⟩ },
{ refine le_of_forall_ge_of_dense (λ r hr, _),
rcases exists_mem_nhds_within_lt_dimH_of_lt_dimH hr with ⟨x, hxs... | lemma | bsupr_limsup_dimH | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"exists_mem_nhds_within_lt_dimH_of_lt_dimH",
"le_Inf",
"le_of_forall_ge_of_dense",
"le_supr₂_of_le",
"self_mem_nhds_within",
"supr₂_le"
] | In an (extended) metric space with second countable topology, the Hausdorff dimension
of a set `s` is the supremum over `x ∈ s` of the limit superiors of `dimH t` along
`(𝓝[s] x).small_sets`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
supr_limsup_dimH (s : set X) : (⨆ x, limsup dimH (𝓝[s] x).small_sets) = dimH s | begin
refine le_antisymm (supr_le $ λ x, _) _,
{ refine Limsup_le_of_le (by apply_auto_param) (eventually_map.2 _),
exact eventually_small_sets.2 ⟨s, self_mem_nhds_within, λ t, dimH_mono⟩ },
{ rw ← bsupr_limsup_dimH, exact supr₂_le_supr _ _ }
end | lemma | supr_limsup_dimH | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"bsupr_limsup_dimH",
"dimH",
"self_mem_nhds_within",
"supr_le",
"supr₂_le_supr"
] | In an (extended) metric space with second countable topology, the Hausdorff dimension
of a set `s` is the supremum over all `x` of the limit superiors of `dimH t` along
`(𝓝[s] x).small_sets`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
holder_on_with.dimH_image_le (h : holder_on_with C r f s) (hr : 0 < r) :
dimH (f '' s) ≤ dimH s / r | begin
borelize [X, Y],
refine dimH_le (λ d hd, _),
have := h.hausdorff_measure_image_le hr d.coe_nonneg,
rw [hd, ennreal.coe_rpow_of_nonneg _ d.coe_nonneg, top_le_iff] at this,
have Hrd : μH[(r * d : ℝ≥0)] s = ⊤,
{ contrapose this, exact ennreal.mul_ne_top ennreal.coe_ne_top this },
rw [ennreal.le_div_iff... | lemma | holder_on_with.dimH_image_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_le",
"ennreal.coe_mul",
"ennreal.coe_ne_top",
"ennreal.coe_rpow_of_nonneg",
"ennreal.le_div_iff_mul_le",
"ennreal.mul_ne_top",
"holder_on_with",
"le_dimH_of_hausdorff_measure_eq_top",
"mul_comm",
"top_le_iff"
] | If `f` is a Hölder continuous map with exponent `r > 0`, then `dimH (f '' s) ≤ dimH s / r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_image_le (h : holder_with C r f) (hr : 0 < r) (s : set X) :
dimH (f '' s) ≤ dimH s / r | (h.holder_on_with s).dimH_image_le hr | lemma | holder_with.dimH_image_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"holder_with"
] | If `f : X → Y` is Hölder continuous with a positive exponent `r`, then the Hausdorff dimension
of the image of a set `s` is at most `dimH s / r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_range_le (h : holder_with C r f) (hr : 0 < r) :
dimH (range f) ≤ dimH (univ : set X) / r | @image_univ _ _ f ▸ h.dimH_image_le hr univ | lemma | holder_with.dimH_range_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"holder_with"
] | If `f` is a Hölder continuous map with exponent `r > 0`, then the Hausdorff dimension of its
range is at most the Hausdorff dimension of its domain divided by `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_image_le_of_locally_holder_on [second_countable_topology X] {r : ℝ≥0} {f : X → Y}
(hr : 0 < r) {s : set X} (hf : ∀ x ∈ s, ∃ (C : ℝ≥0) (t ∈ 𝓝[s] x), holder_on_with C r f t) :
dimH (f '' s) ≤ dimH s / r | begin
choose! C t htn hC using hf,
rcases countable_cover_nhds_within htn with ⟨u, hus, huc, huU⟩,
replace huU := inter_eq_self_of_subset_left huU, rw inter_Union₂ at huU,
rw [← huU, image_Union₂, dimH_bUnion huc, dimH_bUnion huc], simp only [ennreal.supr_div],
exact supr₂_mono (λ x hx, ((hC x (hus hx)).mono ... | lemma | dimH_image_le_of_locally_holder_on | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_bUnion",
"ennreal.supr_div",
"holder_on_with",
"supr₂_mono"
] | If `s` is a set in a space `X` with second countable topology and `f : X → Y` is Hölder
continuous in a neighborhood within `s` of every point `x ∈ s` with the same positive exponent `r`
but possibly different coefficients, then the Hausdorff dimension of the image `f '' s` is at most
the Hausdorff dimension of `s` div... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_range_le_of_locally_holder_on [second_countable_topology X] {r : ℝ≥0} {f : X → Y}
(hr : 0 < r) (hf : ∀ x : X, ∃ (C : ℝ≥0) (s ∈ 𝓝 x), holder_on_with C r f s) :
dimH (range f) ≤ dimH (univ : set X) / r | begin
rw ← image_univ,
refine dimH_image_le_of_locally_holder_on hr (λ x _, _),
simpa only [exists_prop, nhds_within_univ] using hf x
end | lemma | dimH_range_le_of_locally_holder_on | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_image_le_of_locally_holder_on",
"exists_prop",
"holder_on_with",
"nhds_within_univ"
] | If `f : X → Y` is Hölder continuous in a neighborhood of every point `x : X` with the same
positive exponent `r` but possibly different coefficients, then the Hausdorff dimension of the range
of `f` is at most the Hausdorff dimension of `X` divided by `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with.dimH_image_le (h : lipschitz_on_with K f s) : dimH (f '' s) ≤ dimH s | by simpa using h.holder_on_with.dimH_image_le zero_lt_one | lemma | lipschitz_on_with.dimH_image_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"lipschitz_on_with",
"zero_lt_one"
] | If `f : X → Y` is Lipschitz continuous on `s`, then `dimH (f '' s) ≤ dimH s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_image_le (h : lipschitz_with K f) (s : set X) : dimH (f '' s) ≤ dimH s | (h.lipschitz_on_with s).dimH_image_le | lemma | lipschitz_with.dimH_image_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"lipschitz_with"
] | If `f` is a Lipschitz continuous map, then `dimH (f '' s) ≤ dimH s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_range_le (h : lipschitz_with K f) : dimH (range f) ≤ dimH (univ : set X) | @image_univ _ _ f ▸ h.dimH_image_le univ | lemma | lipschitz_with.dimH_range_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"lipschitz_with"
] | If `f` is a Lipschitz continuous map, then the Hausdorff dimension of its range is at most the
Hausdorff dimension of its domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_image_le_of_locally_lipschitz_on [second_countable_topology X] {f : X → Y}
{s : set X} (hf : ∀ x ∈ s, ∃ (C : ℝ≥0) (t ∈ 𝓝[s] x), lipschitz_on_with C f t) :
dimH (f '' s) ≤ dimH s | begin
have : ∀ x ∈ s, ∃ (C : ℝ≥0) (t ∈ 𝓝[s] x), holder_on_with C 1 f t,
by simpa only [holder_on_with_one] using hf,
simpa only [ennreal.coe_one, div_one]
using dimH_image_le_of_locally_holder_on zero_lt_one this
end | lemma | dimH_image_le_of_locally_lipschitz_on | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_image_le_of_locally_holder_on",
"div_one",
"ennreal.coe_one",
"holder_on_with",
"holder_on_with_one",
"lipschitz_on_with",
"zero_lt_one"
] | If `s` is a set in an extended metric space `X` with second countable topology and `f : X → Y`
is Lipschitz in a neighborhood within `s` of every point `x ∈ s`, then the Hausdorff dimension of
the image `f '' s` is at most the Hausdorff dimension of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_range_le_of_locally_lipschitz_on [second_countable_topology X] {f : X → Y}
(hf : ∀ x : X, ∃ (C : ℝ≥0) (s ∈ 𝓝 x), lipschitz_on_with C f s) :
dimH (range f) ≤ dimH (univ : set X) | begin
rw ← image_univ,
refine dimH_image_le_of_locally_lipschitz_on (λ x _, _),
simpa only [exists_prop, nhds_within_univ] using hf x
end | lemma | dimH_range_le_of_locally_lipschitz_on | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_image_le_of_locally_lipschitz_on",
"exists_prop",
"lipschitz_on_with",
"nhds_within_univ"
] | If `f : X → Y` is Lipschitz in a neighborhood of each point `x : X`, then the Hausdorff
dimension of `range f` is at most the Hausdorff dimension of `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimH_preimage_le (hf : antilipschitz_with K f) (s : set Y) :
dimH (f ⁻¹' s) ≤ dimH s | begin
borelize [X, Y],
refine dimH_le (λ d hd, le_dimH_of_hausdorff_measure_eq_top _),
have := hf.hausdorff_measure_preimage_le d.coe_nonneg s,
rw [hd, top_le_iff] at this,
contrapose! this,
exact ennreal.mul_ne_top (by simp) this
end | lemma | antilipschitz_with.dimH_preimage_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"antilipschitz_with",
"dimH",
"dimH_le",
"ennreal.mul_ne_top",
"le_dimH_of_hausdorff_measure_eq_top",
"top_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_dimH_image (hf : antilipschitz_with K f) (s : set X) :
dimH s ≤ dimH (f '' s) | calc dimH s ≤ dimH (f ⁻¹' (f '' s)) : dimH_mono (subset_preimage_image _ _)
... ≤ dimH (f '' s) : hf.dimH_preimage_le _ | lemma | antilipschitz_with.le_dimH_image | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"antilipschitz_with",
"dimH",
"dimH_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry.dimH_image (hf : isometry f) (s : set X) : dimH (f '' s) = dimH s | le_antisymm (hf.lipschitz.dimH_image_le _) (hf.antilipschitz.le_dimH_image _) | lemma | isometry.dimH_image | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_image (e : X ≃ᵢ Y) (s : set X) : dimH (e '' s) = dimH s | e.isometry.dimH_image s | lemma | isometry_equiv.dimH_image | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_preimage (e : X ≃ᵢ Y) (s : set Y) : dimH (e ⁻¹' s) = dimH s | by rw [← e.image_symm, e.symm.dimH_image] | lemma | isometry_equiv.dimH_preimage | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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