statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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ProperSpace (α : Type u) [PseudoMetricSpace α] : Prop where
isCompact_closedBall : ∀ x : α, ∀ r, IsCompact (closedBall x r) | class | ProperSpace | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ProperSpace.lean | [] | [
"IsCompact",
"PseudoMetricSpace"
] | A pseudometric space is proper if all closed balls are compact. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_sphere {α : Type*} [PseudoMetricSpace α] [ProperSpace α] (x : α) (r : ℝ) :
IsCompact (sphere x r) | (isCompact_closedBall x r).of_isClosed_subset isClosed_sphere sphere_subset_closedBall | theorem | isCompact_sphere | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ProperSpace.lean | [] | [
"IsCompact",
"ProperSpace",
"PseudoMetricSpace"
] | In a proper pseudometric space, all spheres are compact. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.sphere.compactSpace {α : Type*} [PseudoMetricSpace α] [ProperSpace α]
(x : α) (r : ℝ) : CompactSpace (sphere x r) | isCompact_iff_compactSpace.mp (isCompact_sphere _ _) | instance | Metric.sphere.compactSpace | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ProperSpace.lean | [] | [
"CompactSpace",
"ProperSpace",
"PseudoMetricSpace",
"isCompact_sphere"
] | In a proper pseudometric space, any sphere is a `CompactSpace` when considered as a subtype. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ProperSpace.of_isCompact_closedBall_of_le (R : ℝ)
(h : ∀ x : α, ∀ r, R ≤ r → IsCompact (closedBall x r)) : ProperSpace α | ⟨fun x r => IsCompact.of_isClosed_subset (h x (max r R) (le_max_right _ _)) isClosed_closedBall
(closedBall_subset_closedBall <| le_max_left _ _)⟩ | theorem | ProperSpace.of_isCompact_closedBall_of_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ProperSpace.lean | [] | [
"IsCompact",
"IsCompact.of_isClosed_subset",
"ProperSpace"
] | If all closed balls of large enough radius are compact, then the space is proper. Especially
useful when the lower bound for the radius is 0. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ProperSpace.of_seq_closedBall {β : Type*} {l : Filter β} [NeBot l] {x : α} {r : β → ℝ}
(hr : Tendsto r l atTop) (hc : ∀ᶠ i in l, IsCompact (closedBall x (r i))) :
ProperSpace α | where
isCompact_closedBall a r :=
let ⟨_i, hci, hir⟩ := (hc.and <| hr.eventually_ge_atTop <| r + dist a x).exists
hci.of_isClosed_subset isClosed_closedBall <| closedBall_subset_closedBall' hir | theorem | ProperSpace.of_seq_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ProperSpace.lean | [] | [
"Filter",
"IsCompact",
"ProperSpace"
] | If there exists a sequence of compact closed balls with the same center
such that the radii tend to infinity, then the space is proper. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prod_properSpace {α : Type*} {β : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
[ProperSpace α] [ProperSpace β] : ProperSpace (α × β) | where
isCompact_closedBall := by
rintro ⟨x, y⟩ r
rw [← closedBall_prod_same x y]
exact (isCompact_closedBall x r).prod (isCompact_closedBall y r) | instance | prod_properSpace | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ProperSpace.lean | [] | [
"ProperSpace",
"PseudoMetricSpace",
"closedBall_prod_same"
] | A binary product of proper spaces is proper. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pi_properSpace {X : β → Type*} [Fintype β] [∀ b, PseudoMetricSpace (X b)]
[h : ∀ b, ProperSpace (X b)] : ProperSpace (∀ b, X b) | by
refine .of_isCompact_closedBall_of_le 0 fun x r hr => ?_
rw [closedBall_pi _ hr]
exact isCompact_univ_pi fun _ => isCompact_closedBall _ _ | instance | pi_properSpace | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ProperSpace.lean | [] | [
"Fintype",
"ProperSpace",
"PseudoMetricSpace",
"closedBall_pi",
"isCompact_univ_pi"
] | A finite product of proper spaces is proper. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ProperSpace.of_isClosed {X : Type*} [PseudoMetricSpace X] [ProperSpace X]
{s : Set X} (hs : IsClosed s) :
ProperSpace s | ⟨fun x r ↦ Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
((isCompact_closedBall x.1 r).of_isClosed_subset
(hs.isClosedMap_subtype_val _ isClosed_closedBall) (Set.image_subset_iff.mpr subset_rfl))⟩ | lemma | ProperSpace.of_isClosed | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ProperSpace.lean | [] | [
"IsClosed",
"ProperSpace",
"PseudoMetricSpace",
"Set",
"subset_rfl"
] | A closed subspace of a proper space is proper.
This is true for any proper lipschitz map. See `LipschitzWith.properSpace`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_subseq_of_frequently_bounded (hs : IsBounded s) {x : ℕ → X}
(hx : ∃ᶠ n in atTop, x n ∈ s) :
∃ a ∈ closure s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) | have hcs : IsSeqCompact (closure s) := hs.isCompact_closure.isSeqCompact
have hu' : ∃ᶠ n in atTop, x n ∈ closure s := hx.mono fun _n hn => subset_closure hn
hcs.subseq_of_frequently_in hu' | theorem | tendsto_subseq_of_frequently_bounded | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Sequences.lean | [] | [
"IsSeqCompact",
"StrictMono",
"closure",
"subset_closure"
] | A version of **Bolzano-Weierstrass**: in a proper metric space (e.g. $ℝ^n$),
every bounded sequence has a converging subsequence. This version assumes only
that the sequence is frequently in some bounded set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_subseq_of_bounded (hs : IsBounded s) {x : ℕ → X} (hx : ∀ n, x n ∈ s) :
∃ a ∈ closure s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) | tendsto_subseq_of_frequently_bounded hs <| Frequently.of_forall hx | theorem | tendsto_subseq_of_bounded | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Sequences.lean | [] | [
"StrictMono",
"closure",
"tendsto_subseq_of_frequently_bounded"
] | A version of **Bolzano-Weierstrass**: in a proper metric space (e.g. $ℝ^n$),
every bounded sequence has a converging subsequence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_subset_iUnion_ball_radius_lt {r : ι → ℝ} (hs : IsClosed s)
(uf : ∀ x ∈ s, { i | x ∈ ball (c i) (r i) }.Finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
∃ r' : ι → ℝ, (s ⊆ ⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i < r i | by
rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with
⟨v, hsv, hvc, hcv⟩
have := fun i => exists_lt_subset_ball (hvc i) (hcv i)
choose r' hlt hsub using this
exact ⟨r', hsv.trans <| iUnion_mono <| hsub, hlt⟩ | theorem | exists_subset_iUnion_ball_radius_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | [] | [
"Finite",
"IsClosed",
"exists_lt_subset_ball",
"exists_subset_iUnion_closed_subset"
] | **Shrinking lemma** for coverings by open balls in a proper metric space. A point-finite open
cover of a closed subset of a proper metric space by open balls can be shrunk to a new cover by
open balls so that each of the new balls has strictly smaller radius than the old one. This version
assumes that `fun x ↦ ball (c ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_iUnion_ball_eq_radius_lt {r : ι → ℝ} (uf : ∀ x, { i | x ∈ ball (c i) (r i) }.Finite)
(uU : ⋃ i, ball (c i) (r i) = univ) :
∃ r' : ι → ℝ, ⋃ i, ball (c i) (r' i) = univ ∧ ∀ i, r' i < r i | let ⟨r', hU, hv⟩ := exists_subset_iUnion_ball_radius_lt isClosed_univ (fun x _ => uf x) uU.ge
⟨r', univ_subset_iff.1 hU, hv⟩ | theorem | exists_iUnion_ball_eq_radius_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | [] | [
"Finite",
"exists_subset_iUnion_ball_radius_lt",
"isClosed_univ"
] | Shrinking lemma for coverings by open balls in a proper metric space. A point-finite open cover
of a proper metric space by open balls can be shrunk to a new cover by open balls so that each of
the new balls has strictly smaller radius than the old one. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_subset_iUnion_ball_radius_pos_lt {r : ι → ℝ} (hr : ∀ i, 0 < r i) (hs : IsClosed s)
(uf : ∀ x ∈ s, { i | x ∈ ball (c i) (r i) }.Finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
∃ r' : ι → ℝ, (s ⊆ ⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i ∈ Ioo 0 (r i) | by
rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with
⟨v, hsv, hvc, hcv⟩
have := fun i => exists_pos_lt_subset_ball (hr i) (hvc i) (hcv i)
choose r' hlt hsub using this
exact ⟨r', hsv.trans <| iUnion_mono hsub, hlt⟩ | theorem | exists_subset_iUnion_ball_radius_pos_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | [] | [
"Finite",
"IsClosed",
"exists_pos_lt_subset_ball",
"exists_subset_iUnion_closed_subset"
] | Shrinking lemma for coverings by open balls in a proper metric space. A point-finite open cover
of a closed subset of a proper metric space by nonempty open balls can be shrunk to a new cover by
nonempty open balls so that each of the new balls has strictly smaller radius than the old one. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_iUnion_ball_eq_radius_pos_lt {r : ι → ℝ} (hr : ∀ i, 0 < r i)
(uf : ∀ x, { i | x ∈ ball (c i) (r i) }.Finite) (uU : ⋃ i, ball (c i) (r i) = univ) :
∃ r' : ι → ℝ, ⋃ i, ball (c i) (r' i) = univ ∧ ∀ i, r' i ∈ Ioo 0 (r i) | let ⟨r', hU, hv⟩ :=
exists_subset_iUnion_ball_radius_pos_lt hr isClosed_univ (fun x _ => uf x) uU.ge
⟨r', univ_subset_iff.1 hU, hv⟩ | theorem | exists_iUnion_ball_eq_radius_pos_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | [] | [
"Finite",
"exists_subset_iUnion_ball_radius_pos_lt",
"isClosed_univ"
] | Shrinking lemma for coverings by open balls in a proper metric space. A point-finite open cover
of a proper metric space by nonempty open balls can be shrunk to a new cover by nonempty open balls
so that each of the new balls has strictly smaller radius than the old one. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_locallyFinite_subset_iUnion_ball_radius_lt (hs : IsClosed s) {R : α → ℝ}
(hR : ∀ x ∈ s, 0 < R x) :
∃ (ι : Type u) (c : ι → α) (r r' : ι → ℝ),
(∀ i, c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
(LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) | by
have : ∀ x ∈ s, (𝓝 x).HasBasis (fun r : ℝ => 0 < r ∧ r < R x) fun r => ball x r := fun x hx =>
nhds_basis_uniformity (uniformity_basis_dist_lt (hR x hx))
rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs this with
⟨ι, c, r', hr', hsub', hfin⟩
rcases exists_subset_iUnion_ball_radius... | theorem | exists_locallyFinite_subset_iUnion_ball_radius_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | [] | [
"IsClosed",
"LocallyFinite",
"exists_subset_iUnion_ball_radius_pos_lt",
"nhds_basis_uniformity",
"refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set"
] | Let `R : α → ℝ` be a (possibly discontinuous) function on a proper metric space.
Let `s` be a closed set in `α` such that `R` is positive on `s`. Then there exists a collection of
pairs of balls `Metric.ball (c i) (r i)`, `Metric.ball (c i) (r' i)` such that
* all centers belong to `s`;
* for all `i` we have `0 < r i ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_locallyFinite_iUnion_eq_ball_radius_lt {R : α → ℝ} (hR : ∀ x, 0 < R x) :
∃ (ι : Type u) (c : ι → α) (r r' : ι → ℝ),
(∀ i, 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
(LocallyFinite fun i => ball (c i) (r' i)) ∧ ⋃ i, ball (c i) (r i) = univ | let ⟨ι, c, r, r', hlt, hfin, hsub⟩ :=
exists_locallyFinite_subset_iUnion_ball_radius_lt isClosed_univ fun x _ => hR x
⟨ι, c, r, r', fun i => (hlt i).2, hfin, univ_subset_iff.1 hsub⟩ | theorem | exists_locallyFinite_iUnion_eq_ball_radius_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | [] | [
"LocallyFinite",
"exists_locallyFinite_subset_iUnion_ball_radius_lt",
"isClosed_univ"
] | Let `R : α → ℝ` be a (possibly discontinuous) positive function on a proper metric space. Then
there exists a collection of pairs of balls `Metric.ball (c i) (r i)`, `Metric.ball (c i) (r' i)`
such that
* for all `i` we have `0 < r i < r' i < R (c i)`;
* the family of balls `Metric.ball (c i) (r' i)` is locally finite... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Similar (v₁ : ι → P₁) (v₂ : ι → P₂) : Prop | ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), (edist (v₁ i₁) (v₁ i₂) = r * edist (v₂ i₁) (v₂ i₂))
@[inherit_doc]
scoped[Similar] infixl:25 " ∼ " => Similar | def | Similar | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [] | Similarity between indexed sets of vertices v₁ and v₂.
Use `open scoped Similar` to access the `v₁ ∼ v₂` notation. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_iff_exists_edist_eq :
Similar v₁ v₂ ↔ (∃ r : ℝ≥0, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), (edist (v₁ i₁) (v₁ i₂) =
r * edist (v₂ i₁) (v₂ i₂))) | Iff.rfl | lemma | similar_iff_exists_edist_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Similar"
] | Similarity holds if and only if all extended distances are proportional. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_iff_exists_pairwise_edist_eq :
Similar v₁ v₂ ↔ (∃ r : ℝ≥0, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ (edist (v₁ i₁) (v₁ i₂) =
r * edist (v₂ i₁) (v₂ i₂))) | by
rw [similar_iff_exists_edist_eq]
refine ⟨?_, ?_⟩ <;> rintro ⟨r, hr, h⟩ <;> refine ⟨r, hr, fun i₁ i₂ ↦ ?_⟩
· exact fun _ ↦ h i₁ i₂
· by_cases hi : i₁ = i₂
· simp [hi]
· exact h hi | lemma | similar_iff_exists_pairwise_edist_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Pairwise",
"Similar",
"by_cases",
"similar_iff_exists_edist_eq"
] | Similarity holds if and only if all extended distances between points with different
indices are proportional. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Congruent.similar {v₁ : ι → P₁} {v₂ : ι → P₂} (h : Congruent v₁ v₂) : Similar v₁ v₂ | ⟨1, one_ne_zero, fun i₁ i₂ ↦ by simpa using h i₁ i₂⟩ | lemma | Congruent.similar | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Congruent",
"Similar",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
refl (v₁ : ι → P₁) : v₁ ∼ v₁ | ⟨1, one_ne_zero, fun _ _ => by {norm_cast; rw [one_mul]}⟩ | lemma | Similar.refl | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"one_mul",
"one_ne_zero",
"refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm (h : v₁ ∼ v₂) : v₂ ∼ v₁ | by
rcases h with ⟨r, hr, h⟩
refine ⟨r⁻¹, inv_ne_zero hr, fun _ _ => ?_⟩
rw [ENNReal.coe_inv hr, ← ENNReal.div_eq_inv_mul, ENNReal.eq_div_iff _ ENNReal.coe_ne_top, h]
norm_cast | lemma | Similar.symm | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"ENNReal.coe_inv",
"ENNReal.coe_ne_top",
"ENNReal.div_eq_inv_mul",
"ENNReal.eq_div_iff",
"inv_ne_zero",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.similar_comm : v₁ ∼ v₂ ↔ v₂ ∼ v₁ | ⟨Similar.symm, Similar.symm⟩ | lemma | similar_comm | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
trans (h₁ : v₁ ∼ v₂) (h₂ : v₂ ∼ v₃) : v₁ ∼ v₃ | by
rcases h₁ with ⟨r₁, hr₁, h₁⟩; rcases h₂ with ⟨r₂, hr₂, h₂⟩
refine ⟨r₁ * r₂, mul_ne_zero hr₁ hr₂, fun _ _ => ?_⟩
rw [ENNReal.coe_mul, mul_assoc, h₁, h₂] | lemma | Similar.trans | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"ENNReal.coe_mul",
"mul_assoc",
"mul_ne_zero",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
index_map (h : v₁ ∼ v₂) (f : ι' → ι) : (v₁ ∘ f) ∼ (v₂ ∘ f) | by
rcases h with ⟨r, hr, h⟩
refine ⟨r, hr, fun _ _ => ?_⟩
apply h | lemma | Similar.index_map | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [] | Change the index set ι to an index ι' that maps to ι. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
index_equiv (f : ι' ≃ ι) (v₁ : ι → P₁) (v₂ : ι → P₂) :
v₁ ∘ f ∼ v₂ ∘ f ↔ v₁ ∼ v₂ | by
refine ⟨fun h => ?_, fun h => Similar.index_map h f⟩
rcases h with ⟨r, hr, h⟩
refine ⟨r, hr, fun i₁ i₂ => ?_⟩
simpa [f.right_inv i₁, f.right_inv i₂] using h (f.symm i₁) (f.symm i₂) | lemma | Similar.index_equiv | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Similar.index_map"
] | Change between equivalent index sets ι and ι'. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
of_subsingleton_index [Subsingleton ι] : v₁ ∼ v₂ | Congruent.of_subsingleton_index.similar | lemma | Similar.of_subsingleton_index | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [] | Families with at most a single point are always similar. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
comp_left [FunLike F P₁ P₃] [DilationClass F P₁ P₃] (f : F) (h : v₁ ∼ v₂) :
f ∘ v₁ ∼ v₂ | .trans ⟨Dilation.ratio f, Dilation.ratio_ne_zero f, fun _ _ => Dilation.edist_eq f _ _⟩ h | lemma | Similar.comp_left | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Dilation.edist_eq",
"Dilation.ratio_ne_zero",
"DilationClass",
"FunLike",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_right [FunLike F P₂ P₃] [DilationClass F P₂ P₃] (f : F) (h : v₁ ∼ v₂) : v₁ ∼ f ∘ v₂ | .symm (h.symm.comp_left f) | lemma | Similar.comp_right | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"DilationClass",
"FunLike",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_left_iff [FunLike F P₁ P₃] [DilationClass F P₁ P₃] (f : F) : f ∘ v₁ ∼ v₂ ↔ v₁ ∼ v₂ | ⟨.trans <| .comp_right f (.refl _), .comp_left f⟩ | lemma | Similar.comp_left_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"DilationClass",
"FunLike",
"refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_right_iff [FunLike F P₂ P₃] [DilationClass F P₂ P₃] (f : F) : v₁ ∼ f ∘ v₂ ↔ v₁ ∼ v₂ | by
rw [similar_comm, comp_left_iff, similar_comm] | lemma | Similar.comp_right_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"DilationClass",
"FunLike",
"similar_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_isometry_left {f : P₁ → P₃} (hf : Isometry f) (h : v₁ ∼ v₂) : f ∘ v₁ ∼ v₂ | comp_left hf.toDilation h | lemma | Similar.comp_isometry_left | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Isometry"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_isometry_right {f : P₂ → P₃} (hf : Isometry f) (h : v₁ ∼ v₂) : v₁ ∼ f ∘ v₂ | comp_right hf.toDilation h | lemma | Similar.comp_isometry_right | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Isometry"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_isometry_left_iff {f : P₁ → P₃} (hf : Isometry f) : f ∘ v₁ ∼ v₂ ↔ v₁ ∼ v₂ | comp_left_iff hf.toDilation | lemma | Similar.comp_isometry_left_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Isometry"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_isometry_right_iff {f : P₂ → P₃} (hf : Isometry f) : v₁ ∼ f ∘ v₂ ↔ v₁ ∼ v₂ | comp_right_iff hf.toDilation | lemma | Similar.comp_isometry_right_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Isometry"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comm_left (h : ![a, b, c] ∼ ![a', b', c']) :
![b, a, c] ∼ ![b', a', c'] | by
have hl : ![b, a, c] = ![a, b, c] ∘ Equiv.swap 0 1 := by
ext i
fin_cases i <;> simp [Equiv.swap_apply_of_ne_of_ne]
have hr : ![b', a', c'] = ![a', b', c'] ∘ Equiv.swap 0 1 := by
ext i
fin_cases i <;> simp [Equiv.swap_apply_of_ne_of_ne]
grind [index_equiv] | theorem | Similar.comm_left | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Equiv.swap",
"Equiv.swap_apply_of_ne_of_ne"
] | Swapping the first two vertices preserves similarity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
comm_right (h : ![a, b, c] ∼ ![a', b', c']) :
![a, c, b] ∼ ![a', c', b'] | by
have hl : ![a, c, b] = ![a, b, c] ∘ Equiv.swap 1 2 := by
ext i
fin_cases i <;> simp [Equiv.swap_apply_of_ne_of_ne]
have hr : ![a', c', b'] = ![a', b', c'] ∘ Equiv.swap 1 2 := by
ext i
fin_cases i <;> simp [Equiv.swap_apply_of_ne_of_ne]
grind [index_equiv] | theorem | Similar.comm_right | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Equiv.swap",
"Equiv.swap_apply_of_ne_of_ne"
] | Swapping the last two vertices preserves similarity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
reverse_of_three (h : ![a, b, c] ∼ ![a', b', c']) :
![c, b, a] ∼ ![c', b', a'] | h.comm_left.comm_right.comm_left | theorem | Similar.reverse_of_three | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [] | Reversing the order of vertices preserves similarity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_iff_exists_nndist_eq :
Similar v₁ v₂ ↔ (∃ r : ℝ≥0, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), (nndist (v₁ i₁) (v₁ i₂) =
r * nndist (v₂ i₁) (v₂ i₂))) | exists_congr <| fun _ => and_congr Iff.rfl <| forall₂_congr <|
fun _ _ => by { rw [edist_nndist, edist_nndist]; norm_cast } | lemma | similar_iff_exists_nndist_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Similar",
"edist_nndist"
] | Similarity holds if and only if all non-negative distances are proportional. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_iff_exists_pairwise_nndist_eq :
Similar v₁ v₂ ↔ (∃ r : ℝ≥0, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ (nndist (v₁ i₁) (v₁ i₂) =
r * nndist (v₂ i₁) (v₂ i₂))) | by
simp_rw [similar_iff_exists_pairwise_edist_eq, edist_nndist]
exact_mod_cast Iff.rfl | lemma | similar_iff_exists_pairwise_nndist_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Pairwise",
"Similar",
"edist_nndist",
"similar_iff_exists_pairwise_edist_eq"
] | Similarity holds if and only if all non-negative distances between points with different
indices are proportional. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_iff_exists_dist_eq :
Similar v₁ v₂ ↔ (∃ r : ℝ≥0, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), (dist (v₁ i₁) (v₁ i₂) =
r * dist (v₂ i₁) (v₂ i₂))) | similar_iff_exists_nndist_eq.trans
(exists_congr <| fun _ => and_congr Iff.rfl <| forall₂_congr <|
fun _ _ => by { rw [dist_nndist, dist_nndist]; norm_cast }) | lemma | similar_iff_exists_dist_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Similar",
"dist_nndist"
] | Similarity holds if and only if all distances are proportional. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_iff_exists_pairwise_dist_eq :
Similar v₁ v₂ ↔ (∃ r : ℝ≥0, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ (dist (v₁ i₁) (v₁ i₂) =
r * dist (v₂ i₁) (v₂ i₂))) | by
simp_rw [similar_iff_exists_pairwise_nndist_eq, dist_nndist]
exact_mod_cast Iff.rfl | lemma | similar_iff_exists_pairwise_dist_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Pairwise",
"Similar",
"dist_nndist",
"similar_iff_exists_pairwise_nndist_eq"
] | Similarity holds if and only if all distances between points with different indices are
proportional. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_iff_exists_pos_dist_eq : Similar v₁ v₂ ↔
(∃ r : ℝ, 0 < r ∧ ∀ (i₁ i₂ : ι), (dist (v₁ i₁) (v₁ i₂) = r * dist (v₂ i₁) (v₂ i₂))) | by
rw [similar_iff_exists_dist_eq]
simp_rw [← pos_iff_ne_zero, NNReal.exists, ← NNReal.coe_pos, NNReal.coe_mk]
grind | lemma | similar_iff_exists_pos_dist_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"NNReal.coe_mk",
"NNReal.coe_pos",
"Similar",
"similar_iff_exists_dist_eq"
] | Similarity holds if and only if all distances are proportional with a positive real ratio. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_iff_exists_pos_pairwise_dist_eq :
Similar v₁ v₂ ↔ (∃ r : ℝ, 0 < r ∧ Pairwise fun i₁ i₂ ↦ (dist (v₁ i₁) (v₁ i₂) =
r * dist (v₂ i₁) (v₂ i₂))) | by
simp_rw [similar_iff_exists_pairwise_dist_eq]
simp_rw [← pos_iff_ne_zero, NNReal.exists, ← NNReal.coe_pos, NNReal.coe_mk]
grind | lemma | similar_iff_exists_pos_pairwise_dist_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"NNReal.coe_mk",
"NNReal.coe_pos",
"Pairwise",
"Similar",
"similar_iff_exists_pairwise_dist_eq"
] | Similarity holds iff pairwise distances are proportional with a positive ratio. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
similar_of_dist_mul_eq_dist_mul_eq (h_ne : dist a b ≠ 0) (h_ne' : dist a' b' ≠ 0)
(heq1 : dist a b * dist b' c' = dist b c * dist a' b')
(heq2 : dist a b * dist c' a' = dist c a * dist a' b') :
Similar ![a, b, c] ![a', b', c'] | by
set r : ℝ := (dist a b / dist a' b') with hr
have hr_pos : 0 < r := by positivity
apply Similar.of_exists_pos_pairwise_dist_eq
use r
refine ⟨hr_pos, ?_⟩
intro i j hij
fin_cases i <;> fin_cases j <;> try {rw [dist_self, dist_self, mul_zero]}
all_goals simp; grind [dist_comm]
alias similar_of_side_sid... | theorem | similar_of_dist_mul_eq_dist_mul_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Similarity.lean | [] | [
"Similar",
"dist_comm",
"dist_self"
] | If two triangles have two pairs of proportional adjacent sides, then the triangles are similar. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Snowflaking (X : Type*) (α : ℝ) (hα₀ : 0 < α) (hα₁ : α ≤ 1) where
/-- The value wrapped in `x : Snowflaking X α hα₀ hα₁`. -/
val : X | structure | Metric.Snowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | A copy of a type with metric given by `dist x y = (dist x.val y.val) ^ α`.
This is defined as a one-field structure. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofSnowflaking : Snowflaking X α hα₀ hα₁ ≃ X | where
toFun := val
invFun := mk
left_inv _ := rfl
right_inv _ := rfl | def | Metric.Snowflaking.ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | The natural equivalence between `Snowflaking X α hr₀ hr₁` and `X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toSnowflaking : X ≃ Snowflaking X α hα₀ hα₁ | ofSnowflaking.symm | def | Metric.Snowflaking.toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | The natural equivalence between `X` and `Snowflaking X α hr₀ hr₁`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toSnowflaking.sizeOf_spec [SizeOf X] (x : X) :
sizeOf (toSnowflaking x : Snowflaking X α hα₀ hα₁) = 1 + sizeOf x | rfl | theorem | Metric.Snowflaking.toSnowflaking.sizeOf_spec | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
casesOn_toSnowflaking {motive : Snowflaking X α hα₀ hα₁ → Sort*}
(toSnowflaking : ∀ x, motive (Snowflaking.toSnowflaking x)) (x : Snowflaking X α hα₀ hα₁) :
motive x | toSnowflaking x.val | def | Metric.Snowflaking.casesOn_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | This definition makes `cases x` and `induction x` use `toSnowflaking` instead of `mk`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mk_eq_toSnowflaking : (mk : X → Snowflaking X α hα₀ hα₁) = toSnowflaking | rfl | theorem | Metric.Snowflaking.mk_eq_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
val_eq_ofSnowflaking : (val : Snowflaking X α hα₀ hα₁ → X) = ofSnowflaking | rfl | theorem | Metric.Snowflaking.val_eq_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_toSnowflaking :
(toSnowflaking : X ≃ Snowflaking X α hα₀ hα₁).symm = ofSnowflaking | rfl | theorem | Metric.Snowflaking.symm_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_ofSnowflaking :
(ofSnowflaking : Snowflaking X α hα₀ hα₁ ≃ X).symm = toSnowflaking | rfl | theorem | Metric.Snowflaking.symm_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toSnowflaking_ofSnowflaking (x : Snowflaking X α hα₀ hα₁) :
toSnowflaking x.ofSnowflaking = x | rfl | theorem | Metric.Snowflaking.toSnowflaking_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofSnowflaking_toSnowflaking (x : X) :
(toSnowflaking x : Snowflaking X α hα₀ hα₁).ofSnowflaking = x | rfl | theorem | Metric.Snowflaking.ofSnowflaking_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofSnowflaking_comp_toSnowflaking :
(ofSnowflaking : Snowflaking X α hα₀ hα₁ → X) ∘ toSnowflaking = id | rfl | theorem | Metric.Snowflaking.ofSnowflaking_comp_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toSnowflaking_comp_ofSnowflaking :
(toSnowflaking : X → Snowflaking X α hα₀ hα₁) ∘ ofSnowflaking = id | rfl | theorem | Metric.Snowflaking.toSnowflaking_comp_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_toSnowflaking_eq_preimage (s : Set X) :
(toSnowflaking '' s : Set (Snowflaking X α hα₀ hα₁)) = ofSnowflaking ⁻¹' s | toSnowflaking.image_eq_preimage_symm _ | theorem | Metric.Snowflaking.image_toSnowflaking_eq_preimage | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_ofSnowflaking_eq_preimage (s : Set (Snowflaking X α hα₀ hα₁)) :
ofSnowflaking '' s = toSnowflaking ⁻¹' s | ofSnowflaking.image_eq_preimage_symm _ | theorem | Metric.Snowflaking.image_ofSnowflaking_eq_preimage | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_toSnowflaking_image_ofSnowflaking (s : Set (Snowflaking X α hα₀ hα₁)) :
toSnowflaking '' ofSnowflaking '' s = s | ofSnowflaking.symm_image_image _ | theorem | Metric.Snowflaking.image_toSnowflaking_image_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_ofSnowflaking_image_toSnowflaking (s : Set X) :
ofSnowflaking '' (toSnowflaking '' s : Set (Snowflaking X α hα₀ hα₁)) = s | ofSnowflaking.image_symm_image _ | theorem | Metric.Snowflaking.image_ofSnowflaking_image_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_ofSnowflaking : Continuous (ofSnowflaking : Snowflaking X α hα₀ hα₁ → X) | continuous_induced_dom | theorem | Metric.Snowflaking.continuous_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Continuous",
"continuous_induced_dom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_toSnowflaking : Continuous (toSnowflaking : X → Snowflaking X α hα₀ hα₁) | continuous_induced_rng.2 continuous_id | theorem | Metric.Snowflaking.continuous_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
homeomorph : Snowflaking X α hα₀ hα₁ ≃ₜ X | where
toEquiv := ofSnowflaking
continuous_invFun := continuous_toSnowflaking | def | Metric.Snowflaking.homeomorph | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | The natural homeomorphism between `Snowflaking X α hα₀ hα₁` and `X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isBounded_image_ofSnowflaking_iff {s : Set (Snowflaking X α hα₀ hα₁)} :
IsBounded (ofSnowflaking '' s) ↔ IsBounded s | isBounded_induced.symm | theorem | Metric.Snowflaking.isBounded_image_ofSnowflaking_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_preimage_toSnowflaking_iff {s : Set (Snowflaking X α hα₀ hα₁)} :
IsBounded (toSnowflaking ⁻¹' s) ↔ IsBounded s | by
rw [← image_ofSnowflaking_eq_preimage, isBounded_image_ofSnowflaking_iff] | theorem | Metric.Snowflaking.isBounded_preimage_toSnowflaking_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_image_toSnowflaking_iff {s : Set X} :
IsBounded (toSnowflaking '' s : Set (Snowflaking X α hα₀ hα₁)) ↔ IsBounded s | by
rw [← isBounded_image_ofSnowflaking_iff, image_ofSnowflaking_image_toSnowflaking] | theorem | Metric.Snowflaking.isBounded_image_toSnowflaking_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_preimage_ofSnowflaking_iff {s : Set X} :
IsBounded (ofSnowflaking ⁻¹' s : Set (Snowflaking X α hα₀ hα₁)) ↔ IsBounded s | by
rw [← image_toSnowflaking_eq_preimage, isBounded_image_toSnowflaking_iff] | theorem | Metric.Snowflaking.isBounded_preimage_ofSnowflaking_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_ofSnowflaking :
UniformContinuous (ofSnowflaking : Snowflaking X α hα₀ hα₁ → X) | uniformContinuous_comap | theorem | Metric.Snowflaking.uniformContinuous_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"UniformContinuous",
"uniformContinuous_comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_toSnowflaking :
UniformContinuous (toSnowflaking : X → Snowflaking X α hα₀ hα₁) | uniformContinuous_comap' uniformContinuous_id | theorem | Metric.Snowflaking.uniformContinuous_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"UniformContinuous",
"uniformContinuous_comap'",
"uniformContinuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquiv : Snowflaking X α hα₀ hα₁ ≃ᵤ X | where
toEquiv := ofSnowflaking
uniformContinuous_toFun := uniformContinuous_ofSnowflaking
uniformContinuous_invFun := uniformContinuous_toSnowflaking | def | Metric.Snowflaking.uniformEquiv | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | The natural uniform space equivalence between `Snowflaking X α hα hα₁`
and the underlying space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
edist_def (x y : Snowflaking X α hα₀ hα₁) :
edist x y = edist x.ofSnowflaking y.ofSnowflaking ^ α | rfl | theorem | Metric.Snowflaking.edist_def | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_toSnowflaking_toSnowflaking (x y : X) :
edist (toSnowflaking x : Snowflaking X α hα₀ hα₁) (toSnowflaking y) = edist x y ^ α | rfl | theorem | Metric.Snowflaking.edist_toSnowflaking_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_ofSnowflaking_ofSnowflaking (x y : Snowflaking X α hα₀ hα₁) :
edist x.ofSnowflaking y.ofSnowflaking = edist x y ^ α⁻¹ | by
rw [edist_def, ENNReal.rpow_rpow_inv hα₀.ne'] | theorem | Metric.Snowflaking.edist_ofSnowflaking_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"ENNReal.rpow_rpow_inv"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_ofSnowflaking_eball (x : X) (r : ℝ≥0∞) :
ofSnowflaking ⁻¹' Metric.eball x r =
Metric.eball (toSnowflaking x : Snowflaking X α hα₀ hα₁) (r ^ α) | by
ext ⟨y⟩
simp (disch := positivity) [ENNReal.rpow_lt_rpow_iff]
@[deprecated (since := "2026-01-24")]
alias preimage_ofSnowflaking_emetricBall := preimage_ofSnowflaking_eball | theorem | Metric.Snowflaking.preimage_ofSnowflaking_eball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"ENNReal.rpow_lt_rpow_iff",
"Metric.eball"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_toSnowflaking_eball (x : X) (r : ℝ≥0∞) :
toSnowflaking '' Metric.eball x r =
Metric.eball (toSnowflaking x : Snowflaking X α hα₀ hα₁) (r ^ α) | by
rw [image_toSnowflaking_eq_preimage, preimage_ofSnowflaking_eball]
@[deprecated (since := "2026-01-24")]
alias image_toSnowflaking_emetricBall := image_toSnowflaking_eball | theorem | Metric.Snowflaking.image_toSnowflaking_eball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Metric.eball"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_toSnowflaking_eball (x : Snowflaking X α hα₀ hα₁) (d : ℝ≥0∞) :
toSnowflaking ⁻¹' Metric.eball x d = Metric.eball x.ofSnowflaking (d ^ α⁻¹) | by
rw [toSnowflaking.preimage_eq_iff_eq_image, image_toSnowflaking_eball,
toSnowflaking_ofSnowflaking, ENNReal.rpow_inv_rpow hα₀.ne']
@[deprecated (since := "2026-01-24")]
alias preimage_toSnowflaking_emetricBall := preimage_toSnowflaking_eball | theorem | Metric.Snowflaking.preimage_toSnowflaking_eball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"ENNReal.rpow_inv_rpow",
"Metric.eball"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_ofSnowflaking_eball (x : Snowflaking X α hα₀ hα₁) (d : ℝ≥0∞) :
ofSnowflaking '' Metric.eball x d = Metric.eball x.ofSnowflaking (d ^ α⁻¹) | by
rw [image_ofSnowflaking_eq_preimage, preimage_toSnowflaking_eball]
@[deprecated (since := "2026-01-24")]
alias image_ofSnowflaking_emetricBall := image_ofSnowflaking_eball | theorem | Metric.Snowflaking.image_ofSnowflaking_eball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Metric.eball"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_ofSnowflaking_closedEBall (x : X) (r : ℝ≥0∞) :
ofSnowflaking ⁻¹' Metric.closedEBall x r =
Metric.closedEBall (toSnowflaking x : Snowflaking X α hα₀ hα₁) (r ^ α) | by
ext ⟨y⟩
simp (disch := positivity) [ENNReal.rpow_le_rpow_iff]
@[deprecated (since := "2026-01-24")]
alias preimage_ofSnowflaking_emetricClosedBall := preimage_ofSnowflaking_closedEBall | theorem | Metric.Snowflaking.preimage_ofSnowflaking_closedEBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"ENNReal.rpow_le_rpow_iff",
"Metric.closedEBall"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_toSnowflaking_closedEBall (x : X) (r : ℝ≥0∞) :
toSnowflaking '' Metric.closedEBall x r =
Metric.closedEBall (toSnowflaking x : Snowflaking X α hα₀ hα₁) (r ^ α) | by
rw [image_toSnowflaking_eq_preimage, preimage_ofSnowflaking_closedEBall]
@[deprecated (since := "2026-01-24")]
alias image_toSnowflaking_emetricClosedBall := image_toSnowflaking_closedEBall | theorem | Metric.Snowflaking.image_toSnowflaking_closedEBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Metric.closedEBall"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_toSnowflaking_closedEBall (x : Snowflaking X α hα₀ hα₁) (d : ℝ≥0∞) :
toSnowflaking ⁻¹' Metric.closedEBall x d = Metric.closedEBall x.ofSnowflaking (d ^ α⁻¹) | by
rw [toSnowflaking.preimage_eq_iff_eq_image, image_toSnowflaking_closedEBall,
toSnowflaking_ofSnowflaking, ENNReal.rpow_inv_rpow hα₀.ne']
@[deprecated (since := "2026-01-24")]
alias preimage_toSnowflaking_emetricClosedBall := preimage_toSnowflaking_closedEBall | theorem | Metric.Snowflaking.preimage_toSnowflaking_closedEBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"ENNReal.rpow_inv_rpow",
"Metric.closedEBall"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_ofSnowflaking_closedEBall (x : Snowflaking X α hα₀ hα₁) (d : ℝ≥0∞) :
ofSnowflaking '' Metric.closedEBall x d =
Metric.closedEBall x.ofSnowflaking (d ^ α⁻¹) | by
rw [image_ofSnowflaking_eq_preimage, preimage_toSnowflaking_closedEBall]
@[deprecated (since := "2026-01-24")]
alias image_ofSnowflaking_emetricClosedBall := image_ofSnowflaking_closedEBall | theorem | Metric.Snowflaking.image_ofSnowflaking_closedEBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Metric.closedEBall"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ediam_image_ofSnowflaking (s : Set (Snowflaking X α hα₀ hα₁)) :
ediam (ofSnowflaking '' s) = ediam s ^ α⁻¹ | by
refine eq_of_forall_ge_iff fun c ↦ ?_
simp only [ENNReal.rpow_inv_le_iff hα₀, ediam_le_iff, Set.forall_mem_image,
edist_ofSnowflaking_ofSnowflaking] | theorem | Metric.Snowflaking.ediam_image_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"ENNReal.rpow_inv_le_iff",
"Set",
"Set.forall_mem_image"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ediam_preimage_toSnowflaking (s : Set (Snowflaking X α hα₀ hα₁)) :
ediam (toSnowflaking ⁻¹' s) = ediam s ^ α⁻¹ | by
rw [← image_ofSnowflaking_eq_preimage, ediam_image_ofSnowflaking] | theorem | Metric.Snowflaking.ediam_preimage_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ediam_preimage_ofSnowflaking (s : Set X) :
ediam (ofSnowflaking ⁻¹' s : Set (Snowflaking X α hα₀ hα₁)) = ediam s ^ α | by
rw [← ENNReal.rpow_inv_rpow hα₀.ne' (ediam _), ← ediam_preimage_toSnowflaking,
← Set.preimage_comp, ofSnowflaking_comp_toSnowflaking, Set.preimage_id] | theorem | Metric.Snowflaking.ediam_preimage_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"ENNReal.rpow_inv_rpow",
"Set",
"Set.preimage_comp",
"Set.preimage_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ediam_image_toSnowflaking (s : Set X) :
ediam (toSnowflaking '' s : Set (Snowflaking X α hα₀ hα₁)) = ediam s ^ α | by
simp [image_toSnowflaking_eq_preimage] | theorem | Metric.Snowflaking.ediam_image_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_toSnowflaking_toSnowflaking [Dist X] (x y : X) :
dist (toSnowflaking x : Snowflaking X α hα₀ hα₁) (toSnowflaking y) = dist x y ^ α | rfl | theorem | Metric.Snowflaking.dist_toSnowflaking_toSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_ofSnowflaking_ofSnowflaking (x y : Snowflaking X α hα₀ hα₁) :
dist x.ofSnowflaking y.ofSnowflaking = dist x y ^ α⁻¹ | by
cases x; cases y
simp [Real.rpow_rpow_inv dist_nonneg hα₀.ne'] | theorem | Metric.Snowflaking.dist_ofSnowflaking_ofSnowflaking | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Real.rpow_rpow_inv",
"dist_nonneg"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_ofSnowflaking_ball (x : X) {r : ℝ} (hr : 0 ≤ r) :
ofSnowflaking ⁻¹' ball x r = ball (toSnowflaking x : Snowflaking X α hα₀ hα₁) (r ^ α) | by
ext ⟨y⟩
simp (disch := positivity) [Real.rpow_lt_rpow_iff] | theorem | Metric.Snowflaking.preimage_ofSnowflaking_ball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Real.rpow_lt_rpow_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_toSnowflaking_ball (x : X) {r : ℝ} (hr : 0 ≤ r) :
toSnowflaking '' ball x r = ball (toSnowflaking x : Snowflaking X α hα₀ hα₁) (r ^ α) | by
rw [image_toSnowflaking_eq_preimage, preimage_ofSnowflaking_ball x hr] | theorem | Metric.Snowflaking.image_toSnowflaking_ball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_toSnowflaking_ball (x : Snowflaking X α hα₀ hα₁) {r : ℝ} (hr : 0 ≤ r) :
toSnowflaking ⁻¹' ball x r = ball x.ofSnowflaking (r ^ α⁻¹) | by
rw [toSnowflaking.preimage_eq_iff_eq_image, image_toSnowflaking_ball _ (by positivity),
toSnowflaking_ofSnowflaking, Real.rpow_inv_rpow hr hα₀.ne'] | theorem | Metric.Snowflaking.preimage_toSnowflaking_ball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Real.rpow_inv_rpow"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_ofSnowflaking_ball (x : Snowflaking X α hα₀ hα₁) {r : ℝ} (hr : 0 ≤ r) :
ofSnowflaking '' ball x r = ball x.ofSnowflaking (r ^ α⁻¹) | by
rw [image_ofSnowflaking_eq_preimage, preimage_toSnowflaking_ball _ hr] | theorem | Metric.Snowflaking.image_ofSnowflaking_ball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_ofSnowflaking_closedBall (x : X) {r : ℝ} (hr : 0 ≤ r) :
ofSnowflaking ⁻¹' closedBall x r =
closedBall (toSnowflaking x : Snowflaking X α hα₀ hα₁) (r ^ α) | by
ext ⟨y⟩
simp (disch := positivity) [Real.rpow_le_rpow_iff] | theorem | Metric.Snowflaking.preimage_ofSnowflaking_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Real.rpow_le_rpow_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_toSnowflaking_closedBall (x : X) {r : ℝ} (hr : 0 ≤ r) :
toSnowflaking '' closedBall x r =
closedBall (toSnowflaking x : Snowflaking X α hα₀ hα₁) (r ^ α) | by
rw [image_toSnowflaking_eq_preimage, preimage_ofSnowflaking_closedBall x hr] | theorem | Metric.Snowflaking.image_toSnowflaking_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_toSnowflaking_closedBall (x : Snowflaking X α hα₀ hα₁) {r : ℝ} (hr : 0 ≤ r) :
toSnowflaking ⁻¹' closedBall x r = closedBall x.ofSnowflaking (r ^ α⁻¹) | by
rw [toSnowflaking.preimage_eq_iff_eq_image, image_toSnowflaking_closedBall _ (by positivity),
toSnowflaking_ofSnowflaking, Real.rpow_inv_rpow hr hα₀.ne'] | theorem | Metric.Snowflaking.preimage_toSnowflaking_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [
"Real.rpow_inv_rpow"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_ofSnowflaking_closedBall (x : Snowflaking X α hα₀ hα₁) {r : ℝ} (hr : 0 ≤ r) :
ofSnowflaking '' closedBall x r = closedBall x.ofSnowflaking (r ^ α⁻¹) | by
rw [image_ofSnowflaking_eq_preimage, preimage_toSnowflaking_closedBall _ hr] | theorem | Metric.Snowflaking.image_ofSnowflaking_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Snowflaking.lean | [
"Mathlib.Analysis.MeanInequalitiesPow"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux (δ : ℝ) (E : Set α) : α → ℝ≥0∞ | fun x : α => (1 : ℝ≥0∞) - infEDist x E / ENNReal.ofReal δ | def | thickenedIndicatorAux | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ENNReal.ofReal",
"Set"
] | The `δ`-thickened indicator of a set `E` is the function that equals `1` on `E`
and `0` outside a `δ`-thickening of `E` and interpolates (continuously) between
these values using `infEDist _ E`.
`thickenedIndicatorAux` is the unbundled `ℝ≥0∞`-valued function. See `thickenedIndicator`
for the (bundled) bounded continuo... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuous_thickenedIndicatorAux {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
Continuous (thickenedIndicatorAux δ E) | by
unfold thickenedIndicatorAux
let f := fun x : α => (⟨1, infEDist x E / ENNReal.ofReal δ⟩ : ℝ≥0 × ℝ≥0∞)
let sub := fun p : ℝ≥0 × ℝ≥0∞ => (p.1 : ℝ≥0∞) - p.2
rw [show (fun x : α => (1 : ℝ≥0∞) - infEDist x E / ENNReal.ofReal δ) = sub ∘ f by rfl]
apply (@ENNReal.continuous_nnreal_sub 1).comp
apply (ENNReal.co... | theorem | continuous_thickenedIndicatorAux | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Continuous",
"ENNReal.continuous_div_const",
"ENNReal.continuous_nnreal_sub",
"ENNReal.ofReal",
"Set",
"thickenedIndicatorAux"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_le_one (δ : ℝ) (E : Set α) (x : α) :
thickenedIndicatorAux δ E x ≤ 1 | by
apply tsub_le_self (α := ℝ≥0∞) | theorem | thickenedIndicatorAux_le_one | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"thickenedIndicatorAux",
"tsub_le_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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