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 ⌀ |
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@[simp]
hausdorffDist_self_zero : hausdorffDist s s = 0 := by simp [hausdorffDist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_self_zero | The Hausdorff distance between a set and itself is zero. |
hausdorffDist_comm : hausdorffDist s t = hausdorffDist t s := by
simp [hausdorffDist, hausdorffEdist_comm] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_comm | The Hausdorff distances from `s` to `t` and from `t` to `s` coincide. |
@[simp]
hausdorffDist_empty : hausdorffDist s ∅ = 0 := by
rcases s.eq_empty_or_nonempty with h | h
· simp [h]
· simp [hausdorffDist, hausdorffEdist_empty h] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_empty | The Hausdorff distance to the empty set vanishes (if you want to have the more reasonable
value `∞` instead, use `EMetric.hausdorffEdist`, which takes values in `ℝ≥0∞`). |
@[simp]
hausdorffDist_empty' : hausdorffDist ∅ s = 0 := by simp [hausdorffDist_comm] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_empty' | The Hausdorff distance to the empty set vanishes (if you want to have the more reasonable
value `∞` instead, use `EMetric.hausdorffEdist`, which takes values in `ℝ≥0∞`). |
hausdorffDist_le_of_infDist {r : ℝ} (hr : 0 ≤ r) (H1 : ∀ x ∈ s, infDist x t ≤ r)
(H2 : ∀ x ∈ t, infDist x s ≤ r) : hausdorffDist s t ≤ r := by
rcases s.eq_empty_or_nonempty with hs | hs
· rwa [hs, hausdorffDist_empty']
rcases t.eq_empty_or_nonempty with ht | ht
· rwa [ht, hausdorffDist_empty]
have : hausdorffEdist s t ≤ ENNReal.ofReal r := by
apply hausdorffEdist_le_of_infEdist _ _
· simpa only [infDist, ← ENNReal.le_ofReal_iff_toReal_le (infEdist_ne_top ht) hr] using H1
· simpa only [infDist, ← ENNReal.le_ofReal_iff_toReal_le (infEdist_ne_top hs) hr] using H2
exact ENNReal.toReal_le_of_le_ofReal hr this | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_le_of_infDist | Bounding the Hausdorff distance by bounding the distance of any point
in each set to the other set |
hausdorffDist_le_of_mem_dist {r : ℝ} (hr : 0 ≤ r) (H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r)
(H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r) : hausdorffDist s t ≤ r := by
apply hausdorffDist_le_of_infDist hr
· intro x xs
rcases H1 x xs with ⟨y, yt, hy⟩
exact le_trans (infDist_le_dist_of_mem yt) hy
· intro x xt
rcases H2 x xt with ⟨y, ys, hy⟩
exact le_trans (infDist_le_dist_of_mem ys) hy | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_le_of_mem_dist | Bounding the Hausdorff distance by exhibiting, for any point in each set,
another point in the other set at controlled distance |
hausdorffDist_le_diam (hs : s.Nonempty) (bs : IsBounded s) (ht : t.Nonempty)
(bt : IsBounded t) : hausdorffDist s t ≤ diam (s ∪ t) := by
rcases hs with ⟨x, xs⟩
rcases ht with ⟨y, yt⟩
refine hausdorffDist_le_of_mem_dist diam_nonneg ?_ ?_
· exact fun z hz => ⟨y, yt, dist_le_diam_of_mem (bs.union bt) (subset_union_left hz)
(subset_union_right yt)⟩
· exact fun z hz => ⟨x, xs, dist_le_diam_of_mem (bs.union bt) (subset_union_right hz)
(subset_union_left xs)⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_le_diam | The Hausdorff distance is controlled by the diameter of the union. |
infDist_le_hausdorffDist_of_mem (hx : x ∈ s) (fin : hausdorffEdist s t ≠ ⊤) :
infDist x t ≤ hausdorffDist s t :=
toReal_mono fin (infEdist_le_hausdorffEdist_of_mem hx) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_le_hausdorffDist_of_mem | The distance to a set is controlled by the Hausdorff distance. |
exists_dist_lt_of_hausdorffDist_lt {r : ℝ} (h : x ∈ s) (H : hausdorffDist s t < r)
(fin : hausdorffEdist s t ≠ ⊤) : ∃ y ∈ t, dist x y < r := by
have r0 : 0 < r := lt_of_le_of_lt hausdorffDist_nonneg H
have : hausdorffEdist s t < ENNReal.ofReal r := by
rwa [hausdorffDist, ← ENNReal.toReal_ofReal (le_of_lt r0),
ENNReal.toReal_lt_toReal fin ENNReal.ofReal_ne_top] at H
rcases exists_edist_lt_of_hausdorffEdist_lt h this with ⟨y, hy, yr⟩
rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff r0] at yr
exact ⟨y, hy, yr⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | exists_dist_lt_of_hausdorffDist_lt | If the Hausdorff distance is `< r`, any point in one of the sets is at distance
`< r` of a point in the other set. |
exists_dist_lt_of_hausdorffDist_lt' {r : ℝ} (h : y ∈ t) (H : hausdorffDist s t < r)
(fin : hausdorffEdist s t ≠ ⊤) : ∃ x ∈ s, dist x y < r := by
rw [hausdorffDist_comm] at H
rw [hausdorffEdist_comm] at fin
simpa [dist_comm] using exists_dist_lt_of_hausdorffDist_lt h H fin | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | exists_dist_lt_of_hausdorffDist_lt' | If the Hausdorff distance is `< r`, any point in one of the sets is at distance
`< r` of a point in the other set. |
infDist_le_infDist_add_hausdorffDist (fin : hausdorffEdist s t ≠ ⊤) :
infDist x t ≤ infDist x s + hausdorffDist s t := by
refine toReal_le_add' infEdist_le_infEdist_add_hausdorffEdist (fun h ↦ ?_) (flip absurd fin)
rw [infEdist_eq_top_iff, ← not_nonempty_iff_eq_empty] at h ⊢
rw [hausdorffEdist_comm] at fin
exact mt (nonempty_of_hausdorffEdist_ne_top · fin) h | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_le_infDist_add_hausdorffDist | The infimum distance to `s` and `t` are the same, up to the Hausdorff distance
between `s` and `t` |
hausdorffDist_image (h : Isometry Φ) :
hausdorffDist (Φ '' s) (Φ '' t) = hausdorffDist s t := by
simp [hausdorffDist, hausdorffEdist_image h] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_image | The Hausdorff distance is invariant under isometries. |
hausdorffDist_triangle (fin : hausdorffEdist s t ≠ ⊤) :
hausdorffDist s u ≤ hausdorffDist s t + hausdorffDist t u := by
refine toReal_le_add' hausdorffEdist_triangle (flip absurd fin) (not_imp_not.1 fun h ↦ ?_)
rw [hausdorffEdist_comm] at fin
exact ne_top_of_le_ne_top (add_ne_top.2 ⟨fin, h⟩) hausdorffEdist_triangle | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_triangle | The Hausdorff distance satisfies the triangle inequality. |
hausdorffDist_triangle' (fin : hausdorffEdist t u ≠ ⊤) :
hausdorffDist s u ≤ hausdorffDist s t + hausdorffDist t u := by
rw [hausdorffEdist_comm] at fin
have I : hausdorffDist u s ≤ hausdorffDist u t + hausdorffDist t s :=
hausdorffDist_triangle fin
simpa [add_comm, hausdorffDist_comm] using I | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_triangle' | The Hausdorff distance satisfies the triangle inequality. |
@[simp]
hausdorffDist_self_closure : hausdorffDist s (closure s) = 0 := by simp [hausdorffDist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_self_closure | The Hausdorff distance between a set and its closure vanishes. |
@[simp]
hausdorffDist_closure₁ : hausdorffDist (closure s) t = hausdorffDist s t := by
simp [hausdorffDist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_closure₁ | Replacing a set by its closure does not change the Hausdorff distance. |
@[simp]
hausdorffDist_closure₂ : hausdorffDist s (closure t) = hausdorffDist s t := by
simp [hausdorffDist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_closure₂ | Replacing a set by its closure does not change the Hausdorff distance. |
hausdorffDist_closure : hausdorffDist (closure s) (closure t) = hausdorffDist s t := by
simp [hausdorffDist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_closure | The Hausdorff distances between two sets and their closures coincide. |
hausdorffDist_zero_iff_closure_eq_closure (fin : hausdorffEdist s t ≠ ⊤) :
hausdorffDist s t = 0 ↔ closure s = closure t := by
simp [← hausdorffEdist_zero_iff_closure_eq_closure, hausdorffDist,
ENNReal.toReal_eq_zero_iff, fin] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_zero_iff_closure_eq_closure | Two sets are at zero Hausdorff distance if and only if they have the same closures. |
_root_.IsClosed.hausdorffDist_zero_iff_eq (hs : IsClosed s) (ht : IsClosed t)
(fin : hausdorffEdist s t ≠ ⊤) : hausdorffDist s t = 0 ↔ s = t := by
simp [← hausdorffEdist_zero_iff_eq_of_closed hs ht, hausdorffDist, ENNReal.toReal_eq_zero_iff,
fin] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | _root_.IsClosed.hausdorffDist_zero_iff_eq | Two closed sets are at zero Hausdorff distance if and only if they coincide. |
HolderWith (C r : ℝ≥0) (f : X → Y) : Prop :=
∀ x y, edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) | def | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | HolderWith | 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`. |
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)
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | HolderOnWith | 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`. |
holderOnWith_empty (C r : ℝ≥0) (f : X → Y) : HolderOnWith C r f ∅ := fun _ hx => hx.elim
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | holderOnWith_empty | null |
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 _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | holderOnWith_singleton | null |
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) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | Set.Subsingleton.holderOnWith | null |
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]
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | holderOnWith_univ | null |
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]
alias ⟨_, LipschitzOnWith.holderOnWith⟩ := holderOnWith_one
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | holderOnWith_one | null |
holderWith_one {C : ℝ≥0} {f : X → Y} : HolderWith C 1 f ↔ LipschitzWith C f :=
holderOnWith_univ.symm.trans <| holderOnWith_one.trans lipschitzOnWith_univ
alias ⟨_, LipschitzWith.holderWith⟩ := holderWith_one | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | holderWith_one | null |
holderWith_id : HolderWith 1 1 (id : X → X) :=
LipschitzWith.id.holderWith | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | holderWith_id | null |
protected 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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | HolderWith.holderOnWith | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | edist_le | null |
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 <| by gcongr | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | edist_le_of_le | null |
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 := by
intro x hx y hy
rw [ENNReal.coe_mul, mul_comm rg, NNReal.coe_mul, ENNReal.rpow_mul, mul_assoc,
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) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | comp | null |
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) :=
holderOnWith_univ.mp <| hg.comp (hf.holderOnWith univ) fun x _ => ht x | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | comp_holderWith | null |
protected 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)⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | uniformContinuousOn | A Hölder continuous function is uniformly continuous |
protected continuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) : ContinuousOn f s :=
(hf.uniformContinuousOn h0).continuousOn | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | continuousOn | null |
protected 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) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | mono | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | ediam_image_le_of_le | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | ediam_image_le | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | ediam_image_le_of_subset | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | ediam_image_le_of_subset_of_le | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | ediam_image_inter_le_of_le | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | ediam_image_inter_le | null |
restrict_iff {s : Set X} : HolderWith C r (s.restrict f) ↔ HolderOnWith C r f s := by
simp [HolderWith, HolderOnWith]
protected alias ⟨_, _root_.HolderOnWith.holderWith⟩ := restrict_iff | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | restrict_iff | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | edist_le | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | edist_le_of_le | null |
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) :=
(hg.holderOnWith univ).comp_holderWith hf fun _ => trivial | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | comp | null |
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 :=
(hg.holderOnWith univ).comp hf fun _ _ => trivial | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | comp_holderOnWith | null |
protected uniformContinuous (hf : HolderWith C r f) (h0 : 0 < r) : UniformContinuous f :=
uniformContinuousOn_univ.mp <| (hf.holderOnWith univ).uniformContinuousOn h0 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | uniformContinuous | A Hölder continuous function is uniformly continuous |
protected continuous (hf : HolderWith C r f) (h0 : 0 < r) : Continuous f :=
(hf.uniformContinuous h0).continuous | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | continuous | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | ediam_image_le | null |
const {y : Y} :
HolderWith C r (Function.const X y) := fun x₁ x₂ => by
simp only [Function.const_apply, edist_self, zero_le] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | const | null |
zero [Zero Y] : HolderWith C r (0 : X → Y) := .const | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | zero | null |
of_isEmpty [IsEmpty X] : HolderWith C r f := isEmptyElim | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | of_isEmpty | null |
mono {C' : ℝ≥0} (hf : HolderWith C r f) (h : C ≤ C') :
HolderWith C' r f :=
fun x₁ x₂ ↦ (hf x₁ x₂).trans (mul_left_mono (coe_le_coe.2 h)) | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | mono | null |
nndist_le_of_le (hf : HolderOnWith C r f s) (hx : x ∈ s) (hy : y ∈ s)
{d : ℝ≥0} (hd : nndist x y ≤ d) : nndist (f x) (f y) ≤ C * d ^ (r : ℝ) := by
rw [← ENNReal.coe_le_coe, ← edist_nndist, ENNReal.coe_mul,
ENNReal.coe_rpow_of_nonneg _ r.coe_nonneg]
apply hf.edist_le_of_le hx hy
rwa [edist_nndist, ENNReal.coe_le_coe] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | nndist_le_of_le | null |
nndist_le (hf : HolderOnWith C r f s) (hx : x ∈ s) (hy : y ∈ s) :
nndist (f x) (f y) ≤ C * nndist x y ^ (r : ℝ) :=
hf.nndist_le_of_le hx hy le_rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | nndist_le | null |
dist_le_of_le (hf : HolderOnWith C r f s) (hx : x ∈ s) (hy : y ∈ s)
{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 hx hy hd | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | dist_le_of_le | null |
dist_le (hf : HolderOnWith C r f s) (hx : x ∈ s) (hy : y ∈ s) :
dist (f x) (f y) ≤ C * dist x y ^ (r : ℝ) :=
hf.dist_le_of_le hx hy le_rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | dist_le | null |
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) :=
(hf.holderOnWith univ).nndist_le_of_le (mem_univ x) (mem_univ y) hd | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | nndist_le_of_le | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | nndist_le | null |
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) :=
(hf.holderOnWith univ).dist_le_of_le (mem_univ x) (mem_univ y) hd | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | dist_le_of_le | null |
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 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | dist_le | null |
@[simp]
holderWith_zero_iff : HolderWith 0 r f ↔ ∀ x₁ x₂, f x₁ = f x₂ := by
refine ⟨fun h x₁ x₂ => ?_, fun h x₁ x₂ => h x₁ x₂ ▸ ?_⟩
· specialize h x₁ x₂
simp [ENNReal.coe_zero, zero_mul, nonpos_iff_eq_zero, edist_eq_zero] at h
assumption
· simp only [edist_self, ENNReal.coe_zero, zero_mul, le_refl] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | holderWith_zero_iff | null |
add (hf : HolderWith C r f) (hg : HolderWith C' r g) :
HolderWith (C + C') r (f + g) := by
intro x₁ x₂
simp only [Pi.add_apply, coe_add]
grw [edist_add_add_le, hf x₁ x₂, hg x₁ x₂]
rw [add_mul] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | add | null |
smul {α} [SeminormedAddCommGroup α] [SMulZeroClass α Y] [IsBoundedSMul α Y] (a : α)
(hf : HolderWith C r f) : HolderWith (C * ‖a‖₊) r (a • f) := fun x₁ x₂ => by
refine edist_smul_le _ _ _ |>.trans ?_
rw [coe_mul, ENNReal.smul_def, smul_eq_mul, mul_comm (C : ℝ≥0∞), mul_assoc]
gcongr
exact hf x₁ x₂ | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | smul | null |
smul_iff {α} [SeminormedRing α] [Module α Y] [NormSMulClass α Y] (a : α)
(ha : ‖a‖₊ ≠ 0) :
HolderWith (C * ‖a‖₊) r (a • f) ↔ HolderWith C r f := by
simp_rw [HolderWith, coe_mul, Pi.smul_apply, edist_smul₀, ENNReal.smul_def, smul_eq_mul,
mul_comm (C : ℝ≥0∞), mul_assoc,
ENNReal.mul_le_mul_left (ENNReal.coe_ne_zero.mpr ha) ENNReal.coe_ne_top, mul_comm] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Analysis.SpecialFunctions.Pow.Continuity"
] | Mathlib/Topology/MetricSpace/Holder.lean | smul_iff | null |
noncomputable
eHolderNorm (r : ℝ≥0) (f : X → Y) : ℝ≥0∞ := ⨅ (C) (_ : HolderWith C r f), C | def | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm | The `r`-Hölder (semi-)norm in `ℝ≥0∞` of a function `f` is the least non-negative real
number `C` for which `f` is `r`-Hölder continuous with constant `C`. This is `∞` if no such
non-negative real exists. |
noncomputable
nnHolderNorm (r : ℝ≥0) (f : X → Y) : ℝ≥0 := (eHolderNorm r f).toNNReal | def | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | nnHolderNorm | The `r`-Hölder (semi)norm in `ℝ≥0`. |
MemHolder (r : ℝ≥0) (f : X → Y) : Prop := ∃ C, HolderWith C r f | def | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder | A function `f` is `MemHolder r f` if it is Hölder continuous. Namely, `f` has a finite
`r`-Hölder constant. This is equivalent to `f` having finite Hölder norm.
c.f. `memHolder_iff`. |
HolderWith.memHolder {C : ℝ≥0} (hf : HolderWith C r f) : MemHolder r f := ⟨C, hf⟩
@[simp] lemma eHolderNorm_lt_top : eHolderNorm r f < ∞ ↔ MemHolder r f := by
refine ⟨fun h => ?_,
fun hf => let ⟨C, hC⟩ := hf; iInf_lt_top.2 ⟨C, iInf_lt_top.2 ⟨hC, coe_lt_top⟩⟩⟩
simp_rw [eHolderNorm, iInf_lt_top] at h
let ⟨C, hC, _⟩ := h
exact ⟨C, hC⟩ | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | HolderWith.memHolder | null |
eHolderNorm_ne_top : eHolderNorm r f ≠ ∞ ↔ MemHolder r f := by
rw [← eHolderNorm_lt_top, lt_top_iff_ne_top]
@[simp] lemma eHolderNorm_eq_top : eHolderNorm r f = ∞ ↔ ¬ MemHolder r f := by
rw [← eHolderNorm_ne_top, not_not]
protected alias ⟨_, MemHolder.eHolderNorm_lt_top⟩ := eHolderNorm_lt_top
protected alias ⟨_, MemHolder.eHolderNorm_ne_top⟩ := eHolderNorm_ne_top | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm_ne_top | null |
coe_nnHolderNorm_le_eHolderNorm {r : ℝ≥0} {f : X → Y} :
(nnHolderNorm r f : ℝ≥0∞) ≤ eHolderNorm r f :=
coe_toNNReal_le_self
variable (X) in
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | coe_nnHolderNorm_le_eHolderNorm | null |
eHolderNorm_const (r : ℝ≥0) (c : Y) : eHolderNorm r (Function.const X c) = 0 := by
rw [eHolderNorm, ← ENNReal.bot_eq_zero, iInf₂_eq_bot]
exact fun C' hC' => ⟨0, .const, hC'⟩
variable (X) in
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm_const | null |
eHolderNorm_zero [Zero Y] (r : ℝ≥0) : eHolderNorm r (0 : X → Y) = 0 :=
eHolderNorm_const X r 0
variable (X) in
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm_zero | null |
nnHolderNorm_const (r : ℝ≥0) (c : Y) : nnHolderNorm r (Function.const X c) = 0 := by
refine le_antisymm (ENNReal.coe_le_coe.1 <|
le_trans coe_nnHolderNorm_le_eHolderNorm ?_) (zero_le _)
rw [eHolderNorm_const, ENNReal.coe_zero]
variable (X) in
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | nnHolderNorm_const | null |
nnHolderNorm_zero [Zero Y] (r : ℝ≥0) : nnHolderNorm r (0 : X → Y) = 0 :=
nnHolderNorm_const X r 0
attribute [simp] eHolderNorm_const eHolderNorm_zero | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | nnHolderNorm_zero | null |
eHolderNorm_of_isEmpty [hX : IsEmpty X] :
eHolderNorm r f = 0 := by
rw [eHolderNorm, ← ENNReal.bot_eq_zero, iInf₂_eq_bot]
exact fun ε hε => ⟨0, .of_isEmpty, hε⟩ | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm_of_isEmpty | null |
HolderWith.eHolderNorm_le {C : ℝ≥0} (hf : HolderWith C r f) :
eHolderNorm r f ≤ C :=
iInf₂_le C hf | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | HolderWith.eHolderNorm_le | null |
@[simp]
memHolder_const {c : Y} : MemHolder r (Function.const X c) :=
(HolderWith.const (C := 0)).memHolder | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | memHolder_const | See also `memHolder_const` for the version with the spelling `fun _ ↦ c`. |
@[simp]
memHolder_const' {c : Y} : MemHolder r (fun _ ↦ c : X → Y) :=
memHolder_const
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | memHolder_const' | Version of `memHolder_const` with the spelling `fun _ ↦ c` for the constant function. |
memHolder_zero [Zero Y] : MemHolder r (0 : X → Y) :=
memHolder_const | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | memHolder_zero | null |
eHolderNorm_eq_zero {r : ℝ≥0} {f : X → Y} :
eHolderNorm r f = 0 ↔ ∀ x₁ x₂, f x₁ = f x₂ := by
constructor
· refine fun h x₁ x₂ => ?_
by_cases hx : x₁ = x₂
· rw [hx]
· rw [eHolderNorm, ← ENNReal.bot_eq_zero, iInf₂_eq_bot] at h
rw [← edist_eq_zero, ← le_zero_iff]
refine le_of_forall_gt fun b hb => ?_
obtain ⟨C, hC, hC'⟩ := h (b / edist x₁ x₂ ^ (r : ℝ))
(ENNReal.div_pos hb.ne.symm (ENNReal.rpow_lt_top_of_nonneg zero_le_coe
(edist_lt_top x₁ x₂).ne).ne)
exact lt_of_le_of_lt (hC x₁ x₂) <| ENNReal.mul_lt_of_lt_div hC'
· intro h
rcases isEmpty_or_nonempty X with hX | hX
· exact eHolderNorm_of_isEmpty
· rw [← eHolderNorm_const X r (f hX.some)]
congr
simp [funext_iff, h _ hX.some] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm_eq_zero | null |
MemHolder.holderWith {r : ℝ≥0} {f : X → Y} (hf : MemHolder r f) :
HolderWith (nnHolderNorm r f) r f := by
intro x₁ x₂
by_cases hx : x₁ = x₂
· simp only [hx, edist_self, zero_le]
rw [nnHolderNorm, eHolderNorm, coe_toNNReal]
on_goal 2 => exact hf.eHolderNorm_lt_top.ne
have h₁ : edist x₁ x₂ ^ (r : ℝ) ≠ 0 :=
(Ne.symm <| ne_of_lt <| ENNReal.rpow_pos (edist_pos.2 hx) (edist_lt_top x₁ x₂).ne)
have h₂ : edist x₁ x₂ ^ (r : ℝ) ≠ ∞ := by
simp [(edist_lt_top x₁ x₂).ne]
rw [← ENNReal.div_le_iff h₁ h₂]
refine le_iInf₂ fun C hC => ?_
rw [ENNReal.div_le_iff h₁ h₂]
exact hC x₁ x₂ | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.holderWith | null |
memHolder_iff_holderWith {r : ℝ≥0} {f : X → Y} :
MemHolder r f ↔ HolderWith (nnHolderNorm r f) r f :=
⟨MemHolder.holderWith, HolderWith.memHolder⟩ | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | memHolder_iff_holderWith | null |
MemHolder.coe_nnHolderNorm_eq_eHolderNorm
{r : ℝ≥0} {f : X → Y} (hf : MemHolder r f) :
(nnHolderNorm r f : ℝ≥0∞) = eHolderNorm r f := by
rw [nnHolderNorm, coe_toNNReal]
exact ne_of_lt <| lt_of_le_of_lt hf.holderWith.eHolderNorm_le <| coe_lt_top | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.coe_nnHolderNorm_eq_eHolderNorm | null |
HolderWith.nnholderNorm_le {C r : ℝ≥0} {f : X → Y} (hf : HolderWith C r f) :
nnHolderNorm r f ≤ C := by
rw [← ENNReal.coe_le_coe, hf.memHolder.coe_nnHolderNorm_eq_eHolderNorm]
exact hf.eHolderNorm_le | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | HolderWith.nnholderNorm_le | null |
MemHolder.comp {r s : ℝ≥0} {Z : Type*} [MetricSpace Z] {f : Z → X} {g : X → Y}
(hf : MemHolder r f) (hg : MemHolder s g) : MemHolder (s * r) (g ∘ f) :=
(hg.holderWith.comp hf.holderWith).memHolder | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.comp | null |
MemHolder.nnHolderNorm_eq_zero {r : ℝ≥0} {f : X → Y} (hf : MemHolder r f) :
nnHolderNorm r f = 0 ↔ ∀ x₁ x₂, f x₁ = f x₂ := by
rw [← ENNReal.coe_eq_zero, hf.coe_nnHolderNorm_eq_eHolderNorm, eHolderNorm_eq_zero] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.nnHolderNorm_eq_zero | null |
MemHolder.add (hf : MemHolder r f) (hg : MemHolder r g) : MemHolder r (f + g) :=
(hf.holderWith.add hg.holderWith).memHolder | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.add | null |
MemHolder.smul {𝕜} [SeminormedRing 𝕜] [Module 𝕜 Y] [IsBoundedSMul 𝕜 Y]
{c : 𝕜} (hf : MemHolder r f) : MemHolder r (c • f) :=
(hf.holderWith.smul c).memHolder | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.smul | null |
MemHolder.smul_iff {𝕜} [SeminormedRing 𝕜] [Module 𝕜 Y] [NormSMulClass 𝕜 Y]
{c : 𝕜} (hc : ‖c‖₊ ≠ 0) : MemHolder r (c • f) ↔ MemHolder r f := by
refine ⟨fun ⟨h, hh⟩ => ⟨h * ‖c‖₊⁻¹, ?_⟩, .smul⟩
rw [← HolderWith.smul_iff _ hc, inv_mul_cancel_right₀ hc]
exact hh | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.smul_iff | null |
MemHolder.nsmul [NormedSpace ℝ Y] (n : ℕ) (hf : MemHolder r f) :
MemHolder r (n • f) := by
simp [← Nat.cast_smul_eq_nsmul (R := ℝ), hf.smul] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.nsmul | null |
MemHolder.nnHolderNorm_add_le (hf : MemHolder r f) (hg : MemHolder r g) :
nnHolderNorm r (f + g) ≤ nnHolderNorm r f + nnHolderNorm r g :=
(hf.add hg).holderWith.nnholderNorm_le.trans <|
coe_le_coe.2 (hf.holderWith.add hg.holderWith).nnholderNorm_le | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.nnHolderNorm_add_le | null |
eHolderNorm_add_le :
eHolderNorm r (f + g) ≤ eHolderNorm r f + eHolderNorm r g := by
by_cases hfg : MemHolder r f ∧ MemHolder r g
· obtain ⟨hf, hg⟩ := hfg
rw [← hf.coe_nnHolderNorm_eq_eHolderNorm, ← hg.coe_nnHolderNorm_eq_eHolderNorm,
← (hf.add hg).coe_nnHolderNorm_eq_eHolderNorm, ← coe_add, ENNReal.coe_le_coe]
exact hf.nnHolderNorm_add_le hg
· rw [Classical.not_and_iff_not_or_not, ← eHolderNorm_eq_top, ← eHolderNorm_eq_top] at hfg
obtain (h | h) := hfg
all_goals simp [h] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm_add_le | null |
eHolderNorm_smul {α} [NormedRing α] [Module α Y] [NormSMulClass α Y] (c : α) :
eHolderNorm r (c • f) = ‖c‖₊ * eHolderNorm r f := by
by_cases hc : ‖c‖₊ = 0
· rw [nnnorm_eq_zero] at hc
simp [hc]
by_cases hf : MemHolder r f
· refine le_antisymm ((hf.holderWith.smul c).eHolderNorm_le.trans ?_) <| mul_le_of_le_div' ?_
· rw [coe_mul, hf.coe_nnHolderNorm_eq_eHolderNorm, mul_comm]
· rw [← (hf.holderWith.smul c).memHolder.coe_nnHolderNorm_eq_eHolderNorm, ← coe_div hc]
refine HolderWith.eHolderNorm_le fun x₁ x₂ => ?_
rw [coe_div hc, ← ENNReal.mul_div_right_comm,
ENNReal.le_div_iff_mul_le (Or.inl <| coe_ne_zero.2 hc) <| Or.inl coe_ne_top,
mul_comm, ← smul_eq_mul, ← ENNReal.smul_def, ← edist_smul₀, ← Pi.smul_apply,
← Pi.smul_apply]
exact hf.smul.holderWith x₁ x₂
· rw [← eHolderNorm_eq_top] at hf
rw [hf, mul_top <| coe_ne_zero.2 hc, eHolderNorm_eq_top, MemHolder.smul_iff hc]
rw [nnnorm_eq_zero] at hc
intro h
exact h.eHolderNorm_lt_top.ne hf | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm_smul | null |
MemHolder.nnHolderNorm_smul {α} [NormedRing α] [Module α Y] [NormSMulClass α Y]
(hf : MemHolder r f) (c : α) :
nnHolderNorm r (c • f) = ‖c‖₊ * nnHolderNorm r f := by
rw [← ENNReal.coe_inj, coe_mul, hf.coe_nnHolderNorm_eq_eHolderNorm,
hf.smul.coe_nnHolderNorm_eq_eHolderNorm, eHolderNorm_smul] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.nnHolderNorm_smul | null |
eHolderNorm_nsmul [NormedSpace ℝ Y] (n : ℕ) :
eHolderNorm r (n • f) = n • eHolderNorm r f := by
simp [← Nat.cast_smul_eq_nsmul (R := ℝ), eHolderNorm_smul] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | eHolderNorm_nsmul | null |
MemHolder.nnHolderNorm_nsmul [NormedSpace ℝ Y] (n : ℕ) (hf : MemHolder r f) :
nnHolderNorm r (n • f) = n • nnHolderNorm r f := by
simp [← Nat.cast_smul_eq_nsmul (R := ℝ), hf.nnHolderNorm_smul] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Holder"
] | Mathlib/Topology/MetricSpace/HolderNorm.lean | MemHolder.nnHolderNorm_nsmul | null |
noncomputable einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x ∈ s) (y ∈ s) (_ : x ≠ y), edist x y | def | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep | The "extended infimum separation" of a set with an edist function. |
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