File: Mathlib/Topology/MetricSpace/Holder.lean

full_name: HolderWith
code: /-- 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 
goal: Prop

full_name: HolderOnWith
code: /-- 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 
goal: Prop

full_name: holderOnWith_empty
code: @[simp]
theorem holderOnWith_empty (C r : ℝ≥0) (f : X → Y) : HolderOnWith C r f ∅ 
goal: HolderOnWith C r f ∅

full_name: holderOnWith_singleton
code: @[simp]
theorem holderOnWith_singleton (C r : ℝ≥0) (f : X → Y) (x : X) : HolderOnWith C r f {x} 
goal: HolderOnWith C r f {x}

full_name: Set.Subsingleton.holderOnWith
code: theorem Set.Subsingleton.holderOnWith {s : Set X} (hs : s.Subsingleton) (C r : ℝ≥0) (f : X → Y) :
    HolderOnWith C r f s 
goal: HolderOnWith C r f s

full_name: holderOnWith_univ
code: theorem holderOnWith_univ {C r : ℝ≥0} {f : X → Y} : HolderOnWith C r f univ ↔ HolderWith C r f 
goal: HolderOnWith C r f Set.univ ↔ HolderWith C r f

full_name: holderOnWith_one
code: @[simp]
theorem holderOnWith_one {C : ℝ≥0} {f : X → Y} {s : Set X} :
    HolderOnWith C 1 f s ↔ LipschitzOnWith C f s 
goal: HolderOnWith C 1 f s ↔ LipschitzOnWith C f s

full_name: holderWith_one
code: @[simp]
theorem holderWith_one {C : ℝ≥0} {f : X → Y} : HolderWith C 1 f ↔ LipschitzWith C f 
goal: HolderWith C 1 f ↔ LipschitzWith C f

full_name: holderWith_id
code: theorem holderWith_id : HolderWith 1 1 (id : X → X) 
goal: HolderWith 1 1 id

full_name: HolderWith.holderOnWith
code: protected theorem HolderWith.holderOnWith {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f)
    (s : Set X) : HolderOnWith C r f s 
goal: HolderOnWith C r f s

full_name: HolderOnWith.edist_le
code: 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) 
goal: edist (f x) (f y) ≤ ↑C * edist x y ^ ↑r

full_name: HolderOnWith.edist_le_of_le
code: 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) 
goal: edist (f x) (f y) ≤ ↑C * d ^ ↑r

full_name: HolderOnWith.comp
code: 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 * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s 
goal: HolderOnWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f) s

full_name: HolderOnWith.comp_holderWith
code: 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 * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) 
goal: HolderWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f)

full_name: HolderOnWith.uniformContinuousOn
code: /-- A Hölder continuous function is uniformly continuous -/
protected theorem uniformContinuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) :
    UniformContinuousOn f s 
goal: UniformContinuousOn f s

full_name: HolderOnWith.continuousOn
code: protected theorem continuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) : ContinuousOn f s 
goal: ContinuousOn f s

full_name: HolderOnWith.mono
code: protected theorem mono (hf : HolderOnWith C r f s) (ht : t ⊆ s) : HolderOnWith C r f t 
goal: HolderOnWith C r f t

full_name: HolderOnWith.ediam_image_le_of_le
code: 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) 
goal: EMetric.diam (f '' s) ≤ ↑C * d ^ ↑r

full_name: HolderOnWith.ediam_image_le
code: theorem ediam_image_le (hf : HolderOnWith C r f s) :
    EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * EMetric.diam s ^ (r : ℝ) 
goal: EMetric.diam (f '' s) ≤ ↑C * EMetric.diam s ^ ↑r

full_name: HolderOnWith.ediam_image_le_of_subset
code: 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) 
goal: EMetric.diam (f '' t) ≤ ↑C * EMetric.diam t ^ ↑r

full_name: HolderOnWith.ediam_image_le_of_subset_of_le
code: 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) 
goal: EMetric.diam (f '' t) ≤ ↑C * d ^ ↑r

full_name: HolderOnWith.ediam_image_inter_le_of_le
code: 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) 
goal: EMetric.diam (f '' (t ∩ s)) ≤ ↑C * d ^ ↑r

full_name: HolderOnWith.ediam_image_inter_le
code: 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) 
goal: EMetric.diam (f '' (t ∩ s)) ≤ ↑C * EMetric.diam t ^ ↑r

full_name: HolderWith.edist_le
code: theorem edist_le (h : HolderWith C r f) (x y : X) :
    edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) 
goal: edist (f x) (f y) ≤ ↑C * edist x y ^ ↑r

full_name: HolderWith.edist_le_of_le
code: 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) 
goal: edist (f x) (f y) ≤ ↑C * d ^ ↑r

full_name: HolderWith.comp
code: 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 * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) 
goal: HolderWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f)

full_name: HolderWith.comp_holderOnWith
code: 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 * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s 
goal: HolderOnWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f) s

full_name: HolderWith.uniformContinuous
code: /-- A Hölder continuous function is uniformly continuous -/
protected theorem uniformContinuous (hf : HolderWith C r f) (h0 : 0 < r) : UniformContinuous f 
goal: UniformContinuous f

full_name: HolderWith.continuous
code: protected theorem continuous (hf : HolderWith C r f) (h0 : 0 < r) : Continuous f 
goal: Continuous f

full_name: HolderWith.ediam_image_le
code: theorem ediam_image_le (hf : HolderWith C r f) (s : Set X) :
    EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * EMetric.diam s ^ (r : ℝ) 
goal: EMetric.diam (f '' s) ≤ ↑C * EMetric.diam s ^ ↑r

full_name: HolderWith.nndist_le_of_le
code: 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) 
goal: nndist (f x) (f y) ≤ C * d ^ ↑r

full_name: HolderWith.nndist_le
code: theorem nndist_le (hf : HolderWith C r f) (x y : X) :
    nndist (f x) (f y) ≤ C * nndist x y ^ (r : ℝ) 
goal: nndist (f x) (f y) ≤ C * nndist x y ^ ↑r

full_name: HolderWith.dist_le_of_le
code: 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) 
goal: dist (f x) (f y) ≤ ↑C * d ^ ↑r

full_name: HolderWith.dist_le
code: theorem dist_le (hf : HolderWith C r f) (x y : X) : dist (f x) (f y) ≤ C * dist x y ^ (r : ℝ) 
goal: dist (f x) (f y) ≤ ↑C * dist x y ^ ↑r