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feat: port Topology.MetricSpace.Holder (#4381)
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/- | ||
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Yury G. Kudryashov | ||
! This file was ported from Lean 3 source module topology.metric_space.holder | ||
! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8 | ||
! Please do not edit these lines, except to modify the commit id | ||
! if you have ported upstream changes. | ||
-/ | ||
import Mathlib.Topology.MetricSpace.Lipschitz | ||
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity | ||
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/-! | ||
# Hölder continuous functions | ||
In this file we define Hölder continuity on a set and on the whole space. We also prove some basic | ||
properties of Hölder continuous functions. | ||
## Main definitions | ||
* `HolderOnWith`: `f : X → Y` is said to be *Hölder continuous* with constant `C : ℝ≥0` and | ||
exponent `r : ℝ≥0` on a set `s`, if `edist (f x) (f y) ≤ C * edist x y ^ r` for all `x y ∈ s`; | ||
* `HolderWith`: `f : X → Y` is said to be *Hölder continuous* with constant `C : ℝ≥0` and exponent | ||
`r : ℝ≥0`, if `edist (f x) (f y) ≤ C * edist x y ^ r` for all `x y : X`. | ||
## Implementation notes | ||
We use the type `ℝ≥0` (a.k.a. `NNReal`) for `C` because this type has coercion both to `ℝ` and | ||
`ℝ≥0∞`, so it can be easily used both in inequalities about `dist` and `edist`. We also use `ℝ≥0` | ||
for `r` to ensure that `d ^ r` is monotone in `d`. It might be a good idea to use | ||
`ℝ>0` for `r` but we don't have this type in `mathlib` (yet). | ||
## Tags | ||
Hölder continuity, Lipschitz continuity | ||
-/ | ||
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variable {X Y Z : Type _} | ||
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open Filter Set | ||
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open NNReal ENNReal Topology | ||
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section Emetric | ||
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variable [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [PseudoEMetricSpace Z] | ||
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/-- A function `f : X → Y` between two `PseudoEMetricSpace`s is Hölder continuous with constant | ||
`C : ℝ≥0` and exponent `r : ℝ≥0`, if `edist (f x) (f y) ≤ C * edist x y ^ r` for all `x y : X`. -/ | ||
def HolderWith (C r : ℝ≥0) (f : X → Y) : Prop := | ||
∀ x y, edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) | ||
#align holder_with HolderWith | ||
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/-- A function `f : X → Y` between two `PseudoEMetricSpace`s is Hölder continuous with constant | ||
`C : ℝ≥0` and exponent `r : ℝ≥0` on a set `s : set X`, if `edist (f x) (f y) ≤ C * edist x y ^ r` | ||
for all `x y ∈ s`. -/ | ||
def HolderOnWith (C r : ℝ≥0) (f : X → Y) (s : Set X) : Prop := | ||
∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) | ||
#align holder_on_with HolderOnWith | ||
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@[simp] | ||
theorem holderOnWith_empty (C r : ℝ≥0) (f : X → Y) : HolderOnWith C r f ∅ := fun _ hx => hx.elim | ||
#align holder_on_with_empty holderOnWith_empty | ||
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@[simp] | ||
theorem holderOnWith_singleton (C r : ℝ≥0) (f : X → Y) (x : X) : HolderOnWith C r f {x} := by | ||
rintro a (rfl : a = x) b (rfl : b = a) | ||
rw [edist_self] | ||
exact zero_le _ | ||
#align holder_on_with_singleton holderOnWith_singleton | ||
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theorem Set.Subsingleton.holderOnWith {s : Set X} (hs : s.Subsingleton) (C r : ℝ≥0) (f : X → Y) : | ||
HolderOnWith C r f s := | ||
hs.induction_on (holderOnWith_empty C r f) (holderOnWith_singleton C r f) | ||
#align set.subsingleton.holder_on_with Set.Subsingleton.holderOnWith | ||
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theorem holderOnWith_univ {C r : ℝ≥0} {f : X → Y} : HolderOnWith C r f univ ↔ HolderWith C r f := by | ||
simp only [HolderOnWith, HolderWith, mem_univ, true_imp_iff] | ||
#align holder_on_with_univ holderOnWith_univ | ||
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@[simp] | ||
theorem holderOnWith_one {C : ℝ≥0} {f : X → Y} {s : Set X} : | ||
HolderOnWith C 1 f s ↔ LipschitzOnWith C f s := by | ||
simp only [HolderOnWith, LipschitzOnWith, NNReal.coe_one, ENNReal.rpow_one] | ||
#align holder_on_with_one holderOnWith_one | ||
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alias holderOnWith_one ↔ _ LipschitzOnWith.holderOnWith | ||
#align lipschitz_on_with.holder_on_with LipschitzOnWith.holderOnWith | ||
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@[simp] | ||
theorem holderWith_one {C : ℝ≥0} {f : X → Y} : HolderWith C 1 f ↔ LipschitzWith C f := | ||
holderOnWith_univ.symm.trans <| holderOnWith_one.trans lipschitz_on_univ | ||
#align holder_with_one holderWith_one | ||
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alias holderWith_one ↔ _ LipschitzWith.holderWith | ||
#align lipschitz_with.holder_with LipschitzWith.holderWith | ||
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theorem holderWith_id : HolderWith 1 1 (id : X → X) := | ||
LipschitzWith.id.holderWith | ||
#align holder_with_id holderWith_id | ||
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protected theorem HolderWith.holderOnWith {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f) | ||
(s : Set X) : HolderOnWith C r f s := fun x _ y _ => h x y | ||
#align holder_with.holder_on_with HolderWith.holderOnWith | ||
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namespace HolderOnWith | ||
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variable {C r : ℝ≥0} {f : X → Y} {s t : Set X} | ||
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theorem edist_le (h : HolderOnWith C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : | ||
edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) := | ||
h x hx y hy | ||
#align holder_on_with.edist_le HolderOnWith.edist_le | ||
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theorem edist_le_of_le (h : HolderOnWith C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) {d : ℝ≥0∞} | ||
(hd : edist x y ≤ d) : edist (f x) (f y) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := | ||
(h.edist_le hx hy).trans (mul_le_mul_left' (ENNReal.rpow_le_rpow hd r.coe_nonneg) _) | ||
#align holder_on_with.edist_le_of_le HolderOnWith.edist_le_of_le | ||
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theorem comp {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t) {Cf rf : ℝ≥0} | ||
{f : X → Y} (hf : HolderOnWith Cf rf f s) (hst : MapsTo f s t) : | ||
HolderOnWith (Cg * NNReal.rpow Cf rg) (rg * rf) (g ∘ f) s := by | ||
intro x hx y hy | ||
rw [ENNReal.coe_mul, mul_comm rg, NNReal.coe_mul, ENNReal.rpow_mul, mul_assoc, NNReal.rpow_eq_pow, | ||
← ENNReal.coe_rpow_of_nonneg _ rg.coe_nonneg, ← ENNReal.mul_rpow_of_nonneg _ _ rg.coe_nonneg] | ||
exact hg.edist_le_of_le (hst hx) (hst hy) (hf.edist_le hx hy) | ||
#align holder_on_with.comp HolderOnWith.comp | ||
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theorem comp_holderWith {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t) | ||
{Cf rf : ℝ≥0} {f : X → Y} (hf : HolderWith Cf rf f) (ht : ∀ x, f x ∈ t) : | ||
HolderWith (Cg * NNReal.rpow Cf rg) (rg * rf) (g ∘ f) := | ||
holderOnWith_univ.mp <| hg.comp (hf.holderOnWith univ) fun x _ => ht x | ||
#align holder_on_with.comp_holder_with HolderOnWith.comp_holderWith | ||
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/-- A Hölder continuous function is uniformly continuous -/ | ||
protected theorem uniformContinuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) : | ||
UniformContinuousOn f s := by | ||
refine' EMetric.uniformContinuousOn_iff.2 fun ε εpos => _ | ||
have : Tendsto (fun d : ℝ≥0∞ => (C : ℝ≥0∞) * d ^ (r : ℝ)) (𝓝 0) (𝓝 0) := | ||
ENNReal.tendsto_const_mul_rpow_nhds_zero_of_pos ENNReal.coe_ne_top h0 | ||
rcases ENNReal.nhds_zero_basis.mem_iff.1 (this (gt_mem_nhds εpos)) with ⟨δ, δ0, H⟩ | ||
exact ⟨δ, δ0, fun hx y hy h => (hf.edist_le hx hy).trans_lt (H h)⟩ | ||
#align holder_on_with.uniform_continuous_on HolderOnWith.uniformContinuousOn | ||
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protected theorem continuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) : ContinuousOn f s := | ||
(hf.uniformContinuousOn h0).continuousOn | ||
#align holder_on_with.continuous_on HolderOnWith.continuousOn | ||
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protected theorem mono (hf : HolderOnWith C r f s) (ht : t ⊆ s) : HolderOnWith C r f t := | ||
fun _ hx _ hy => hf.edist_le (ht hx) (ht hy) | ||
#align holder_on_with.mono HolderOnWith.mono | ||
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theorem ediam_image_le_of_le (hf : HolderOnWith C r f s) {d : ℝ≥0∞} (hd : EMetric.diam s ≤ d) : | ||
EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := | ||
EMetric.diam_image_le_iff.2 fun _ hx _ hy => | ||
hf.edist_le_of_le hx hy <| (EMetric.edist_le_diam_of_mem hx hy).trans hd | ||
#align holder_on_with.ediam_image_le_of_le HolderOnWith.ediam_image_le_of_le | ||
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theorem ediam_image_le (hf : HolderOnWith C r f s) : | ||
EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * EMetric.diam s ^ (r : ℝ) := | ||
hf.ediam_image_le_of_le le_rfl | ||
#align holder_on_with.ediam_image_le HolderOnWith.ediam_image_le | ||
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theorem ediam_image_le_of_subset (hf : HolderOnWith C r f s) (ht : t ⊆ s) : | ||
EMetric.diam (f '' t) ≤ (C : ℝ≥0∞) * EMetric.diam t ^ (r : ℝ) := | ||
(hf.mono ht).ediam_image_le | ||
#align holder_on_with.ediam_image_le_of_subset HolderOnWith.ediam_image_le_of_subset | ||
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theorem ediam_image_le_of_subset_of_le (hf : HolderOnWith C r f s) (ht : t ⊆ s) {d : ℝ≥0∞} | ||
(hd : EMetric.diam t ≤ d) : EMetric.diam (f '' t) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := | ||
(hf.mono ht).ediam_image_le_of_le hd | ||
#align holder_on_with.ediam_image_le_of_subset_of_le HolderOnWith.ediam_image_le_of_subset_of_le | ||
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theorem ediam_image_inter_le_of_le (hf : HolderOnWith C r f s) {d : ℝ≥0∞} | ||
(hd : EMetric.diam t ≤ d) : EMetric.diam (f '' (t ∩ s)) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := | ||
hf.ediam_image_le_of_subset_of_le (inter_subset_right _ _) <| | ||
(EMetric.diam_mono <| inter_subset_left _ _).trans hd | ||
#align holder_on_with.ediam_image_inter_le_of_le HolderOnWith.ediam_image_inter_le_of_le | ||
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theorem ediam_image_inter_le (hf : HolderOnWith C r f s) (t : Set X) : | ||
EMetric.diam (f '' (t ∩ s)) ≤ (C : ℝ≥0∞) * EMetric.diam t ^ (r : ℝ) := | ||
hf.ediam_image_inter_le_of_le le_rfl | ||
#align holder_on_with.ediam_image_inter_le HolderOnWith.ediam_image_inter_le | ||
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end HolderOnWith | ||
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namespace HolderWith | ||
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variable {C r : ℝ≥0} {f : X → Y} | ||
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theorem edist_le (h : HolderWith C r f) (x y : X) : | ||
edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) := | ||
h x y | ||
#align holder_with.edist_le HolderWith.edist_le | ||
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theorem edist_le_of_le (h : HolderWith C r f) {x y : X} {d : ℝ≥0∞} (hd : edist x y ≤ d) : | ||
edist (f x) (f y) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := | ||
(h.holderOnWith univ).edist_le_of_le trivial trivial hd | ||
#align holder_with.edist_le_of_le HolderWith.edist_le_of_le | ||
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theorem comp {Cg rg : ℝ≥0} {g : Y → Z} (hg : HolderWith Cg rg g) {Cf rf : ℝ≥0} {f : X → Y} | ||
(hf : HolderWith Cf rf f) : HolderWith (Cg * NNReal.rpow Cf rg) (rg * rf) (g ∘ f) := | ||
(hg.holderOnWith univ).comp_holderWith hf fun _ => trivial | ||
#align holder_with.comp HolderWith.comp | ||
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theorem comp_holderOnWith {Cg rg : ℝ≥0} {g : Y → Z} (hg : HolderWith Cg rg g) {Cf rf : ℝ≥0} | ||
{f : X → Y} {s : Set X} (hf : HolderOnWith Cf rf f s) : | ||
HolderOnWith (Cg * NNReal.rpow Cf rg) (rg * rf) (g ∘ f) s := | ||
(hg.holderOnWith univ).comp hf fun _ _ => trivial | ||
#align holder_with.comp_holder_on_with HolderWith.comp_holderOnWith | ||
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/-- A Hölder continuous function is uniformly continuous -/ | ||
protected theorem uniformContinuous (hf : HolderWith C r f) (h0 : 0 < r) : UniformContinuous f := | ||
uniformContinuousOn_univ.mp <| (hf.holderOnWith univ).uniformContinuousOn h0 | ||
#align holder_with.uniform_continuous HolderWith.uniformContinuous | ||
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protected theorem continuous (hf : HolderWith C r f) (h0 : 0 < r) : Continuous f := | ||
(hf.uniformContinuous h0).continuous | ||
#align holder_with.continuous HolderWith.continuous | ||
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theorem ediam_image_le (hf : HolderWith C r f) (s : Set X) : | ||
EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * EMetric.diam s ^ (r : ℝ) := | ||
EMetric.diam_image_le_iff.2 fun _ hx _ hy => | ||
hf.edist_le_of_le <| EMetric.edist_le_diam_of_mem hx hy | ||
#align holder_with.ediam_image_le HolderWith.ediam_image_le | ||
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end HolderWith | ||
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end Emetric | ||
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section Metric | ||
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variable [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : ℝ≥0} {f : X → Y} | ||
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namespace HolderWith | ||
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theorem nndist_le_of_le (hf : HolderWith C r f) {x y : X} {d : ℝ≥0} (hd : nndist x y ≤ d) : | ||
nndist (f x) (f y) ≤ C * d ^ (r : ℝ) := by | ||
norm_cast | ||
rw [← ENNReal.coe_le_coe, ← edist_nndist, ENNReal.coe_mul, ← | ||
ENNReal.coe_rpow_of_nonneg _ r.coe_nonneg] | ||
apply hf.edist_le_of_le | ||
rwa [edist_nndist, ENNReal.coe_le_coe] | ||
#align holder_with.nndist_le_of_le HolderWith.nndist_le_of_le | ||
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theorem nndist_le (hf : HolderWith C r f) (x y : X) : | ||
nndist (f x) (f y) ≤ C * nndist x y ^ (r : ℝ) := | ||
hf.nndist_le_of_le le_rfl | ||
#align holder_with.nndist_le HolderWith.nndist_le | ||
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theorem dist_le_of_le (hf : HolderWith C r f) {x y : X} {d : ℝ} (hd : dist x y ≤ d) : | ||
dist (f x) (f y) ≤ C * d ^ (r : ℝ) := by | ||
lift d to ℝ≥0 using dist_nonneg.trans hd | ||
rw [dist_nndist] at hd⊢ | ||
norm_cast at hd⊢ | ||
exact hf.nndist_le_of_le hd | ||
#align holder_with.dist_le_of_le HolderWith.dist_le_of_le | ||
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theorem dist_le (hf : HolderWith C r f) (x y : X) : dist (f x) (f y) ≤ C * dist x y ^ (r : ℝ) := | ||
hf.dist_le_of_le le_rfl | ||
#align holder_with.dist_le HolderWith.dist_le | ||
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end HolderWith | ||
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end Metric |