fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
MetricSpace.replaceUniformity_eq {γ} [U : UniformSpace γ] (m : MetricSpace γ)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m := by
ext; rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] | Mathlib/Topology/MetricSpace/Defs.lean | MetricSpace.replaceUniformity_eq | null |
MetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : MetricSpace γ)
(H : U = m.toPseudoMetricSpace.toUniformSpace.toTopologicalSpace) : MetricSpace γ :=
@MetricSpace.replaceUniformity γ (m.toUniformSpace.replaceTopology H) m rfl | abbrev | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] | Mathlib/Topology/MetricSpace/Defs.lean | MetricSpace.replaceTopology | Build a new metric space from an old one where the bundled topological structure is provably
(but typically non-definitionaly) equal to some given topological structure.
See Note [forgetful inheritance].
See Note [reducible non-instances]. |
MetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : MetricSpace γ)
(H : U = m.toPseudoMetricSpace.toUniformSpace.toTopologicalSpace) :
m.replaceTopology H = m := by
ext; rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] | Mathlib/Topology/MetricSpace/Defs.lean | MetricSpace.replaceTopology_eq | null |
MetricSpace.replaceBornology {α} [B : Bornology α] (m : MetricSpace α)
(H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) : MetricSpace α :=
{ PseudoMetricSpace.replaceBornology _ H, m with toBornology := B } | abbrev | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] | Mathlib/Topology/MetricSpace/Defs.lean | MetricSpace.replaceBornology | Build a new metric space from an old one where the bundled bornology structure is provably
(but typically non-definitionaly) equal to some given bornology structure.
See Note [forgetful inheritance].
See Note [reducible non-instances]. |
MetricSpace.replaceBornology_eq {α} [m : MetricSpace α] [B : Bornology α]
(H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
MetricSpace.replaceBornology _ H = m := by
ext
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] | Mathlib/Topology/MetricSpace/Defs.lean | MetricSpace.replaceBornology_eq | null |
@[simp] dist_ofMul (a b : X) : dist (ofMul a) (ofMul b) = dist a b := rfl
@[simp] theorem dist_ofAdd (a b : X) : dist (ofAdd a) (ofAdd b) = dist a b := rfl
@[simp] theorem dist_toMul (a b : Additive X) : dist a.toMul b.toMul = dist a b := rfl
@[simp] theorem dist_toAdd (a b : Multiplicative X) : dist a.toAdd b.toAdd = dist a b := rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] | Mathlib/Topology/MetricSpace/Defs.lean | dist_ofMul | null |
@[simp] dist_toDual (a b : X) : dist (toDual a) (toDual b) = dist a b := rfl
@[simp] theorem dist_ofDual (a b : Xᵒᵈ) : dist (ofDual a) (ofDual b) = dist a b := rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] | Mathlib/Topology/MetricSpace/Defs.lean | dist_toDual | null |
Dilation where
/-- The underlying function.
Do NOT use directly. Use the coercion instead. -/
toFun : α → β
edist_eq' : ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ x y : α, edist (toFun x) (toFun y) = r * edist x y
@[inherit_doc] infixl:25 " →ᵈ " => Dilation | structure | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | Dilation | A dilation is a map that uniformly scales the edistance between any two points. |
DilationClass (F : Type*) (α β : outParam Type*) [PseudoEMetricSpace α] [PseudoEMetricSpace β]
[FunLike F α β] : Prop where
edist_eq' : ∀ f : F, ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ x y : α, edist (f x) (f y) = r * edist x y | class | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | DilationClass | `DilationClass F α β r` states that `F` is a type of `r`-dilations.
You should extend this typeclass when you extend `Dilation`. |
funLike : FunLike (α →ᵈ β) α β where
coe := toFun
coe_injective' f g h := by cases f; cases g; congr | instance | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | funLike | null |
toDilationClass : DilationClass (α →ᵈ β) α β where
edist_eq' f := edist_eq' f
@[simp] | instance | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | toDilationClass | null |
toFun_eq_coe {f : α →ᵈ β} : f.toFun = (f : α → β) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | toFun_eq_coe | null |
coe_mk (f : α → β) (h) : ⇑(⟨f, h⟩ : α →ᵈ β) = f :=
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | coe_mk | null |
protected congr_fun {f g : α →ᵈ β} (h : f = g) (x : α) : f x = g x :=
DFunLike.congr_fun h x | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | congr_fun | null |
protected congr_arg (f : α →ᵈ β) {x y : α} (h : x = y) : f x = f y :=
DFunLike.congr_arg f h
@[ext] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | congr_arg | null |
ext {f g : α →ᵈ β} (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext f g h
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ext | null |
mk_coe (f : α →ᵈ β) (h) : Dilation.mk f h = f :=
ext fun _ => rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mk_coe | null |
@[simps -fullyApplied]
protected copy (f : α →ᵈ β) (f' : α → β) (h : f' = ⇑f) : α →ᵈ β where
toFun := f'
edist_eq' := h.symm ▸ f.edist_eq' | def | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | copy | Copy of a `Dilation` with a new `toFun` equal to the old one. Useful to fix definitional
equalities. |
copy_eq_self (f : α →ᵈ β) {f' : α → β} (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h
variable [FunLike F α β]
open Classical in | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | copy_eq_self | null |
ratio [DilationClass F α β] (f : F) : ℝ≥0 :=
if ∀ x y : α, edist x y = 0 ∨ edist x y = ⊤ then 1 else (DilationClass.edist_eq' f).choose | def | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio | The ratio of a dilation `f`. If the ratio is undefined (i.e., the distance between any two
points in `α` is either zero or infinity), then we choose one as the ratio. |
ratio_of_trivial [DilationClass F α β] (f : F)
(h : ∀ x y : α, edist x y = 0 ∨ edist x y = ∞) : ratio f = 1 :=
if_pos h
@[nontriviality] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_of_trivial | null |
ratio_of_subsingleton [Subsingleton α] [DilationClass F α β] (f : F) : ratio f = 1 :=
if_pos fun x y ↦ by simp [Subsingleton.elim x y] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_of_subsingleton | null |
ratio_ne_zero [DilationClass F α β] (f : F) : ratio f ≠ 0 := by
rw [ratio]; split_ifs
· exact one_ne_zero
exact (DilationClass.edist_eq' f).choose_spec.1 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_ne_zero | null |
ratio_pos [DilationClass F α β] (f : F) : 0 < ratio f :=
(ratio_ne_zero f).bot_lt
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_pos | null |
edist_eq [DilationClass F α β] (f : F) (x y : α) :
edist (f x) (f y) = ratio f * edist x y := by
rw [ratio]; split_ifs with key
· rcases DilationClass.edist_eq' f with ⟨r, hne, hr⟩
replace hr := hr x y
rcases key x y with h | h
· simp only [hr, h, mul_zero]
· simp [hr, h, hne]
exact (DilationClass.edist_eq' f).choose_spec.2 x y
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | edist_eq | null |
nndist_eq {α β F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β]
[DilationClass F α β] (f : F) (x y : α) :
nndist (f x) (f y) = ratio f * nndist x y := by
simp only [← ENNReal.coe_inj, ← edist_nndist, ENNReal.coe_mul, edist_eq]
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | nndist_eq | null |
dist_eq {α β F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β]
[DilationClass F α β] (f : F) (x y : α) :
dist (f x) (f y) = ratio f * dist x y := by
simp only [dist_nndist, nndist_eq, NNReal.coe_mul] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | dist_eq | null |
ratio_unique [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (h₀ : edist x y ≠ 0)
(htop : edist x y ≠ ⊤) (hr : edist (f x) (f y) = r * edist x y) : r = ratio f := by
simpa only [hr, ENNReal.mul_left_inj h₀ htop, ENNReal.coe_inj] using edist_eq f x y | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_unique | The `ratio` is equal to the distance ratio for any two points with nonzero finite distance.
`dist` and `nndist` versions below |
ratio_unique_of_nndist_ne_zero {α β F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
[FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (hxy : nndist x y ≠ 0)
(hr : nndist (f x) (f y) = r * nndist x y) : r = ratio f :=
ratio_unique (by rwa [edist_nndist, ENNReal.coe_ne_zero]) (edist_ne_top x y)
(by rw [edist_nndist, edist_nndist, hr, ENNReal.coe_mul]) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_unique_of_nndist_ne_zero | The `ratio` is equal to the distance ratio for any two points
with nonzero finite distance; `nndist` version |
ratio_unique_of_dist_ne_zero {α β} {F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
[FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (hxy : dist x y ≠ 0)
(hr : dist (f x) (f y) = r * dist x y) : r = ratio f :=
ratio_unique_of_nndist_ne_zero (NNReal.coe_ne_zero.1 hxy) <|
NNReal.eq <| by rw [coe_nndist, hr, NNReal.coe_mul, coe_nndist] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_unique_of_dist_ne_zero | The `ratio` is equal to the distance ratio for any two points
with nonzero finite distance; `dist` version |
mkOfNNDistEq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β)
(h : ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ x y : α, nndist (f x) (f y) = r * nndist x y) : α →ᵈ β where
toFun := f
edist_eq' := by
rcases h with ⟨r, hne, h⟩
refine ⟨r, hne, fun x y => ?_⟩
rw [edist_nndist, edist_nndist, ← ENNReal.coe_mul, h x y]
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mkOfNNDistEq | Alternative `Dilation` constructor when the distance hypothesis is over `nndist` |
coe_mkOfNNDistEq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h) :
⇑(mkOfNNDistEq f h : α →ᵈ β) = f :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | coe_mkOfNNDistEq | null |
mk_coe_of_nndist_eq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α →ᵈ β)
(h) : Dilation.mkOfNNDistEq f h = f :=
ext fun _ => rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mk_coe_of_nndist_eq | null |
mkOfDistEq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β)
(h : ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ x y : α, dist (f x) (f y) = r * dist x y) : α →ᵈ β :=
mkOfNNDistEq f <|
h.imp fun r hr =>
⟨hr.1, fun x y => NNReal.eq <| by rw [coe_nndist, hr.2, NNReal.coe_mul, coe_nndist]⟩
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mkOfDistEq | Alternative `Dilation` constructor when the distance hypothesis is over `dist` |
coe_mkOfDistEq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h) :
⇑(mkOfDistEq f h : α →ᵈ β) = f :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | coe_mkOfDistEq | null |
mk_coe_of_dist_eq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α →ᵈ β) (h) :
Dilation.mkOfDistEq f h = f :=
ext fun _ => rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mk_coe_of_dist_eq | null |
@[simps]
_root_.Isometry.toDilation (f : α → β) (hf : Isometry f) : α →ᵈ β where
toFun := f
edist_eq' := ⟨1, one_ne_zero, by simpa using hf⟩
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | _root_.Isometry.toDilation | Every isometry is a dilation of ratio `1`. |
_root_.Isometry.toDilation_ratio {f : α → β} {hf : Isometry f} : ratio hf.toDilation = 1 := by
by_cases h : ∀ x y : α, edist x y = 0 ∨ edist x y = ⊤
· exact ratio_of_trivial hf.toDilation h
· push_neg at h
obtain ⟨x, y, h₁, h₂⟩ := h
exact ratio_unique h₁ h₂ (by simp [hf x y]) |>.symm | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | _root_.Isometry.toDilation_ratio | null |
lipschitz : LipschitzWith (ratio f) (f : α → β) := fun x y => (edist_eq f x y).le | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | lipschitz | null |
antilipschitz : AntilipschitzWith (ratio f)⁻¹ (f : α → β) := fun x y => by
have hr : ratio f ≠ 0 := ratio_ne_zero f
exact mod_cast
(ENNReal.mul_le_iff_le_inv (ENNReal.coe_ne_zero.2 hr) ENNReal.coe_ne_top).1 (edist_eq f x y).ge | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | antilipschitz | null |
protected injective {α : Type*} [EMetricSpace α] [FunLike F α β] [DilationClass F α β]
(f : F) :
Injective f :=
(antilipschitz f).injective | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | injective | A dilation from an emetric space is injective |
protected id (α) [PseudoEMetricSpace α] : α →ᵈ α where
toFun := id
edist_eq' := ⟨1, one_ne_zero, fun x y => by simp only [id, ENNReal.coe_one, one_mul]⟩ | def | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | id | The identity is a dilation |
@[simp]
protected coe_id : ⇑(Dilation.id α) = id :=
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | coe_id | null |
ratio_id : ratio (Dilation.id α) = 1 := by
by_cases h : ∀ x y : α, edist x y = 0 ∨ edist x y = ∞
· rw [ratio, if_pos h]
· push_neg at h
rcases h with ⟨x, y, hne⟩
refine (ratio_unique hne.1 hne.2 ?_).symm
simp | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_id | null |
comp (g : β →ᵈ γ) (f : α →ᵈ β) : α →ᵈ γ where
toFun := g ∘ f
edist_eq' := ⟨ratio g * ratio f, mul_ne_zero (ratio_ne_zero g) (ratio_ne_zero f),
fun x y => by simp_rw [Function.comp, edist_eq, ENNReal.coe_mul, mul_assoc]⟩ | def | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | comp | The composition of dilations is a dilation |
comp_assoc {δ : Type*} [PseudoEMetricSpace δ] (f : α →ᵈ β) (g : β →ᵈ γ)
(h : γ →ᵈ δ) : (h.comp g).comp f = h.comp (g.comp f) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | comp_assoc | null |
coe_comp (g : β →ᵈ γ) (f : α →ᵈ β) : (g.comp f : α → γ) = g ∘ f :=
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | coe_comp | null |
comp_apply (g : β →ᵈ γ) (f : α →ᵈ β) (x : α) : (g.comp f : α → γ) x = g (f x) :=
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | comp_apply | null |
ratio_comp' {g : β →ᵈ γ} {f : α →ᵈ β}
(hne : ∃ x y : α, edist x y ≠ 0 ∧ edist x y ≠ ⊤) : ratio (g.comp f) = ratio g * ratio f := by
rcases hne with ⟨x, y, hα⟩
have hgf := (edist_eq (g.comp f) x y).symm
simp_rw [coe_comp, Function.comp, edist_eq, ← mul_assoc, ENNReal.mul_left_inj hα.1 hα.2]
at hgf
rwa [← ENNReal.coe_inj, ENNReal.coe_mul]
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_comp' | Ratio of the composition `g.comp f` of two dilations is the product of their ratios. We assume
that there exist two points in `α` at extended distance neither `0` nor `∞` because otherwise
`Dilation.ratio (g.comp f) = Dilation.ratio f = 1` while `Dilation.ratio g` can be any number. This
version works for most general spaces, see also `Dilation.ratio_comp` for a version assuming that
`α` is a nontrivial metric space. |
comp_id (f : α →ᵈ β) : f.comp (Dilation.id α) = f :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | comp_id | null |
id_comp (f : α →ᵈ β) : (Dilation.id β).comp f = f :=
ext fun _ => rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | id_comp | null |
one_def : (1 : α →ᵈ α) = Dilation.id α :=
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | one_def | null |
mul_def (f g : α →ᵈ α) : f * g = f.comp g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mul_def | null |
coe_one : ⇑(1 : α →ᵈ α) = id :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | coe_one | null |
coe_mul (f g : α →ᵈ α) : ⇑(f * g) = f ∘ g :=
rfl
@[simp] theorem ratio_one : ratio (1 : α →ᵈ α) = 1 := ratio_id
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | coe_mul | null |
ratio_mul (f g : α →ᵈ α) : ratio (f * g) = ratio f * ratio g := by
by_cases h : ∀ x y : α, edist x y = 0 ∨ edist x y = ∞
· simp [ratio_of_trivial, h]
push_neg at h
exact ratio_comp' h | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_mul | null |
@[simps]
ratioHom : (α →ᵈ α) →* ℝ≥0 := ⟨⟨ratio, ratio_one⟩, ratio_mul⟩
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratioHom | `Dilation.ratio` as a monoid homomorphism from `α →ᵈ α` to `ℝ≥0`. |
ratio_pow (f : α →ᵈ α) (n : ℕ) : ratio (f ^ n) = ratio f ^ n :=
ratioHom.map_pow _ _
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_pow | null |
cancel_right {g₁ g₂ : β →ᵈ γ} {f : α →ᵈ β} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h => Dilation.ext <| hf.forall.2 (Dilation.ext_iff.1 h), fun h => h ▸ rfl⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | cancel_right | null |
cancel_left {g : β →ᵈ γ} {f₁ f₂ : α →ᵈ β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h => Dilation.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | cancel_left | null |
isUniformInducing : IsUniformInducing (f : α → β) :=
(antilipschitz f).isUniformInducing (lipschitz f).uniformContinuous | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | isUniformInducing | A dilation from a metric space is a uniform inducing map |
tendsto_nhds_iff {ι : Type*} {g : ι → α} {a : Filter ι} {b : α} :
Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto ((f : α → β) ∘ g) a (𝓝 (f b)) :=
(Dilation.isUniformInducing f).isInducing.tendsto_nhds_iff | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | tendsto_nhds_iff | null |
toContinuous : Continuous (f : α → β) :=
(lipschitz f).continuous | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | toContinuous | A dilation is continuous. |
ediam_image (s : Set α) : EMetric.diam ((f : α → β) '' s) = ratio f * EMetric.diam s := by
refine ((lipschitz f).ediam_image_le s).antisymm ?_
apply ENNReal.mul_le_of_le_div'
rw [div_eq_mul_inv, mul_comm, ← ENNReal.coe_inv]
exacts [(antilipschitz f).le_mul_ediam_image s, ratio_ne_zero f] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ediam_image | Dilations scale the diameter by `ratio f` in pseudoemetric spaces. |
ediam_range : EMetric.diam (range (f : α → β)) = ratio f * EMetric.diam (univ : Set α) := by
rw [← image_univ]; exact ediam_image f univ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ediam_range | A dilation scales the diameter of the range by `ratio f`. |
mapsTo_emetric_ball (x : α) (r : ℝ≥0∞) :
MapsTo (f : α → β) (EMetric.ball x r) (EMetric.ball (f x) (ratio f * r)) :=
fun y hy => (edist_eq f y x).trans_lt <|
(ENNReal.mul_lt_mul_left (ENNReal.coe_ne_zero.2 <| ratio_ne_zero f) ENNReal.coe_ne_top).2 hy | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mapsTo_emetric_ball | A dilation maps balls to balls and scales the radius by `ratio f`. |
mapsTo_emetric_closedBall (x : α) (r' : ℝ≥0∞) :
MapsTo (f : α → β) (EMetric.closedBall x r') (EMetric.closedBall (f x) (ratio f * r')) :=
fun y hy => (edist_eq f y x).trans_le <| mul_le_mul_left' hy _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mapsTo_emetric_closedBall | A dilation maps closed balls to closed balls and scales the radius by `ratio f`. |
comp_continuousOn_iff {γ} [TopologicalSpace γ] {g : γ → α} {s : Set γ} :
ContinuousOn ((f : α → β) ∘ g) s ↔ ContinuousOn g s :=
(Dilation.isUniformInducing f).isInducing.continuousOn_iff.symm | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | comp_continuousOn_iff | null |
comp_continuous_iff {γ} [TopologicalSpace γ] {g : γ → α} :
Continuous ((f : α → β) ∘ g) ↔ Continuous g :=
(Dilation.isUniformInducing f).isInducing.continuous_iff.symm | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | comp_continuous_iff | null |
isUniformEmbedding [PseudoEMetricSpace β] [DilationClass F α β] (f : F) :
IsUniformEmbedding f :=
(antilipschitz f).isUniformEmbedding (lipschitz f).uniformContinuous | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | isUniformEmbedding | A dilation from a metric space is a uniform embedding |
isEmbedding [PseudoEMetricSpace β] [DilationClass F α β] (f : F) :
IsEmbedding (f : α → β) :=
(Dilation.isUniformEmbedding f).isEmbedding | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | isEmbedding | A dilation from a metric space is an embedding |
isClosedEmbedding [CompleteSpace α] [EMetricSpace β] [DilationClass F α β] (f : F) :
IsClosedEmbedding f :=
(antilipschitz f).isClosedEmbedding (lipschitz f).uniformContinuous | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | isClosedEmbedding | A dilation from a complete emetric space is a closed embedding |
@[simp]
ratio_comp [MetricSpace α] [Nontrivial α] [PseudoEMetricSpace β]
[PseudoEMetricSpace γ] {g : β →ᵈ γ} {f : α →ᵈ β} : ratio (g.comp f) = ratio g * ratio f :=
ratio_comp' <|
let ⟨x, y, hne⟩ := exists_pair_ne α; ⟨x, y, mt edist_eq_zero.1 hne, edist_ne_top _ _⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | ratio_comp | Ratio of the composition `g.comp f` of two dilations is the product of their ratios. We assume
that the domain `α` of `f` is a nontrivial metric space, otherwise
`Dilation.ratio f = Dilation.ratio (g.comp f) = 1` but `Dilation.ratio g` may have any value.
See also `Dilation.ratio_comp'` for a version that works for more general spaces. |
diam_image (s : Set α) : Metric.diam ((f : α → β) '' s) = ratio f * Metric.diam s := by
simp [Metric.diam, ediam_image, ENNReal.toReal_mul] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | diam_image | A dilation scales the diameter by `ratio f` in pseudometric spaces. |
diam_range : Metric.diam (range (f : α → β)) = ratio f * Metric.diam (univ : Set α) := by
rw [← image_univ, diam_image] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | diam_range | null |
mapsTo_ball (x : α) (r' : ℝ) :
MapsTo (f : α → β) (Metric.ball x r') (Metric.ball (f x) (ratio f * r')) :=
fun y hy => (dist_eq f y x).trans_lt <| by gcongr; exacts [ratio_pos _, hy] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mapsTo_ball | A dilation maps balls to balls and scales the radius by `ratio f`. |
mapsTo_sphere (x : α) (r' : ℝ) :
MapsTo (f : α → β) (Metric.sphere x r') (Metric.sphere (f x) (ratio f * r')) :=
fun y hy => Metric.mem_sphere.mp hy ▸ dist_eq f y x | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mapsTo_sphere | A dilation maps spheres to spheres and scales the radius by `ratio f`. |
mapsTo_closedBall (x : α) (r' : ℝ) :
MapsTo (f : α → β) (Metric.closedBall x r') (Metric.closedBall (f x) (ratio f * r')) :=
fun y hy => (dist_eq f y x).trans_le <| mul_le_mul_of_nonneg_left hy (NNReal.coe_nonneg _) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | mapsTo_closedBall | A dilation maps closed balls to closed balls and scales the radius by `ratio f`. |
tendsto_cobounded : Filter.Tendsto f (cobounded α) (cobounded β) :=
(Dilation.antilipschitz f).tendsto_cobounded
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | tendsto_cobounded | null |
comap_cobounded : Filter.comap f (cobounded β) = cobounded α :=
le_antisymm (lipschitz f).comap_cobounded_le (tendsto_cobounded f).le_comap | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Data.FunLike.Basic"
] | Mathlib/Topology/MetricSpace/Dilation.lean | comap_cobounded | null |
DilationEquivClass [EquivLike F X Y] : Prop where
edist_eq' : ∀ f : F, ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ x y : X, edist (f x) (f y) = r * edist x y | class | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | DilationEquivClass | Typeclass saying that `F` is a type of bundled equivalences such that all `e : F` are
dilations. |
DilationEquiv (X Y : Type*) [PseudoEMetricSpace X] [PseudoEMetricSpace Y]
extends X ≃ Y, Dilation X Y
@[inherit_doc] infixl:25 " ≃ᵈ " => DilationEquiv | structure | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | DilationEquiv | Type of equivalences `X ≃ Y` such that `∀ x y, edist (f x) (f y) = r * edist x y` for some
`r : ℝ≥0`, `r ≠ 0`. |
@[simp] coe_toEquiv (e : X ≃ᵈ Y) : ⇑e.toEquiv = e := rfl
@[ext] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | coe_toEquiv | null |
protected ext {e e' : X ≃ᵈ Y} (h : ∀ x, e x = e' x) : e = e' :=
DFunLike.ext _ _ h | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | ext | null |
symm (e : X ≃ᵈ Y) : Y ≃ᵈ X where
toEquiv := e.1.symm
edist_eq' := by
refine ⟨(ratio e)⁻¹, inv_ne_zero <| ratio_ne_zero e, e.surjective.forall₂.2 fun x y ↦ ?_⟩
simp_rw [Equiv.toFun_as_coe, Equiv.symm_apply_apply, coe_toEquiv, edist_eq]
rw [← mul_assoc, ← ENNReal.coe_mul, inv_mul_cancel₀ (ratio_ne_zero e),
ENNReal.coe_one, one_mul]
@[simp] theorem symm_symm (e : X ≃ᵈ Y) : e.symm.symm = e := rfl | def | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | symm | Inverse `DilationEquiv`. |
symm_bijective : Function.Bijective (DilationEquiv.symm : (X ≃ᵈ Y) → Y ≃ᵈ X) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩
@[simp] theorem apply_symm_apply (e : X ≃ᵈ Y) (x : Y) : e (e.symm x) = x := e.right_inv x
@[simp] theorem symm_apply_apply (e : X ≃ᵈ Y) (x : X) : e.symm (e x) = x := e.left_inv x | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | symm_bijective | null |
Simps.symm_apply (e : X ≃ᵈ Y) : Y → X := e.symm
initialize_simps_projections DilationEquiv (toFun → apply, invFun → symm_apply) | def | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | Simps.symm_apply | See Note [custom simps projection]. |
ratio_toDilation (e : X ≃ᵈ Y) : ratio e.toDilation = ratio e := rfl | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | ratio_toDilation | null |
@[simps! -fullyApplied apply]
refl (X : Type*) [PseudoEMetricSpace X] : X ≃ᵈ X where
toEquiv := .refl X
edist_eq' := ⟨1, one_ne_zero, fun _ _ ↦ by simp⟩
@[simp] theorem refl_symm : (refl X).symm = refl X := rfl
@[simp] theorem ratio_refl : ratio (refl X) = 1 := Dilation.ratio_id | def | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | refl | Identity map as a `DilationEquiv`. |
@[simps! -fullyApplied apply]
trans (e₁ : X ≃ᵈ Y) (e₂ : Y ≃ᵈ Z) : X ≃ᵈ Z where
toEquiv := e₁.1.trans e₂.1
__ := e₂.toDilation.comp e₁.toDilation
@[simp] theorem refl_trans (e : X ≃ᵈ Y) : (refl X).trans e = e := rfl
@[simp] theorem trans_refl (e : X ≃ᵈ Y) : e.trans (refl Y) = e := rfl
@[simp] theorem symm_trans_self (e : X ≃ᵈ Y) : e.symm.trans e = refl Y :=
DilationEquiv.ext e.apply_symm_apply
@[simp] theorem self_trans_symm (e : X ≃ᵈ Y) : e.trans e.symm = refl X :=
DilationEquiv.ext e.symm_apply_apply | def | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | trans | Composition of `DilationEquiv`s. |
protected surjective (e : X ≃ᵈ Y) : Surjective e := e.1.surjective | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | surjective | null |
protected bijective (e : X ≃ᵈ Y) : Bijective e := e.1.bijective | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | bijective | null |
protected injective (e : X ≃ᵈ Y) : Injective e := e.1.injective
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | injective | null |
ratio_trans (e : X ≃ᵈ Y) (e' : Y ≃ᵈ Z) : ratio (e.trans e') = ratio e * ratio e' := by
by_cases hX : ∀ x y : X, edist x y = 0 ∨ edist x y = ∞
· have hY : ∀ x y : Y, edist x y = 0 ∨ edist x y = ∞ := e.surjective.forall₂.2 fun x y ↦ by
refine (hX x y).imp (fun h ↦ ?_) fun h ↦ ?_ <;> simp [*, Dilation.ratio_ne_zero]
simp [Dilation.ratio_of_trivial, *]
push_neg at hX
exact (Dilation.ratio_comp' (g := e'.toDilation) (f := e.toDilation) hX).trans (mul_comm _ _)
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | ratio_trans | null |
ratio_symm (e : X ≃ᵈ Y) : ratio e.symm = (ratio e)⁻¹ :=
eq_inv_of_mul_eq_one_left <| by rw [← ratio_trans, symm_trans_self, ratio_refl] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | ratio_symm | null |
mul_def (e e' : X ≃ᵈ X) : e * e' = e'.trans e := rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | mul_def | null |
one_def : (1 : X ≃ᵈ X) = refl X := rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | one_def | null |
inv_def (e : X ≃ᵈ X) : e⁻¹ = e.symm := rfl
@[simp] theorem coe_mul (e e' : X ≃ᵈ X) : ⇑(e * e') = e ∘ e' := rfl
@[simp] theorem coe_one : ⇑(1 : X ≃ᵈ X) = id := rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | inv_def | null |
coe_inv (e : X ≃ᵈ X) : ⇑(e⁻¹) = e.symm := rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | coe_inv | null |
noncomputable ratioHom : (X ≃ᵈ X) →* ℝ≥0 where
toFun := Dilation.ratio
map_one' := ratio_refl
map_mul' _ _ := (ratio_trans _ _).trans (mul_comm _ _)
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | ratioHom | `Dilation.ratio` as a monoid homomorphism. |
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