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values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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cthickening_cthickening_subset (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (s : set α) :
cthickening ε (cthickening δ s) ⊆ cthickening (ε + δ) s | begin
intro x,
simp_rw [mem_cthickening_iff, ennreal.of_real_add hε hδ],
exact λ hx, inf_edist_le_inf_edist_cthickening_add.trans (add_le_add_right hx _),
end | lemma | metric.cthickening_cthickening_subset | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.of_real_add"
] | For the equality, see `cthickening_cthickening`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frontier_cthickening_disjoint (A : set α) :
pairwise (disjoint on (λ (r : ℝ≥0), frontier (cthickening r A))) | λ r₁ r₂ hr, ((disjoint_singleton.2 $ by simpa).preimage _).mono (frontier_cthickening_subset _)
(frontier_cthickening_subset _) | lemma | metric.frontier_cthickening_disjoint | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"disjoint",
"frontier",
"pairwise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
holder_with (C r : ℝ≥0) (f : X → Y) : Prop | ∀ x y, edist (f x) (f y) ≤ C * edist x y ^ (r : ℝ) | def | holder_with | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [] | A function `f : X → Y` between two `pseudo_emetric_space`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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
holder_on_with (C r : ℝ≥0) (f : X → Y) (s : set X) : Prop | ∀ (x ∈ s) (y ∈ s), edist (f x) (f y) ≤ C * edist x y ^ (r : ℝ) | def | holder_on_with | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [] | A function `f : X → Y` between two `pseudo_emeteric_space`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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
holder_on_with_empty (C r : ℝ≥0) (f : X → Y) : holder_on_with C r f ∅ | λ x hx, hx.elim | lemma | holder_on_with_empty | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
holder_on_with_singleton (C r : ℝ≥0) (f : X → Y) (x : X) : holder_on_with C r f {x} | by { rintro a (rfl : a = x) b (rfl : b = a), rw edist_self, exact zero_le _ } | lemma | holder_on_with_singleton | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.holder_on_with {s : set X} (hs : s.subsingleton) (C r : ℝ≥0) (f : X → Y) :
holder_on_with C r f s | hs.induction_on (holder_on_with_empty C r f) (holder_on_with_singleton C r f) | lemma | set.subsingleton.holder_on_with | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with",
"holder_on_with_empty",
"holder_on_with_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
holder_on_with_univ {C r : ℝ≥0} {f : X → Y} :
holder_on_with C r f univ ↔ holder_with C r f | by simp only [holder_on_with, holder_with, mem_univ, true_implies_iff] | lemma | holder_on_with_univ | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with",
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
holder_on_with_one {C : ℝ≥0} {f : X → Y} {s : set X} :
holder_on_with C 1 f s ↔ lipschitz_on_with C f s | by simp only [holder_on_with, lipschitz_on_with, nnreal.coe_one, ennreal.rpow_one] | lemma | holder_on_with_one | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"ennreal.rpow_one",
"holder_on_with",
"lipschitz_on_with",
"nnreal.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
holder_with_one {C : ℝ≥0} {f : X → Y} :
holder_with C 1 f ↔ lipschitz_with C f | holder_on_with_univ.symm.trans $ holder_on_with_one.trans lipschitz_on_univ | lemma | holder_with_one | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_with",
"lipschitz_on_univ",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
holder_with_id : holder_with 1 1 (id : X → X) | lipschitz_with.id.holder_with | lemma | holder_with_id | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
holder_with.holder_on_with {C r : ℝ≥0} {f : X → Y} (h : holder_with C r f)
(s : set X) :
holder_on_with C r f s | λ x _ y _, h x y | lemma | holder_with.holder_on_with | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with",
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le (h : holder_on_with C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) :
edist (f x) (f y) ≤ C * edist x y ^ (r : ℝ) | h x hx y hy | lemma | holder_on_with.edist_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_of_le (h : holder_on_with 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 * d ^ (r : ℝ) | (h.edist_le hx hy).trans (mul_le_mul_left' (ennreal.rpow_le_rpow hd r.coe_nonneg) _) | lemma | holder_on_with.edist_le_of_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"ennreal.rpow_le_rpow",
"holder_on_with",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {Cg rg : ℝ≥0} {g : Y → Z} {t : set Y} (hg : holder_on_with Cg rg g t)
{Cf rf : ℝ≥0} {f : X → Y} (hf : holder_on_with Cf rf f s) (hst : maps_to f s t) :
holder_on_with (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s | begin
intros 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)
end | lemma | holder_on_with.comp | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"ennreal.coe_mul",
"ennreal.coe_rpow_of_nonneg",
"ennreal.mul_rpow_of_nonneg",
"ennreal.rpow_mul",
"holder_on_with",
"mul_assoc",
"mul_comm",
"nnreal.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_holder_with {Cg rg : ℝ≥0} {g : Y → Z} {t : set Y} (hg : holder_on_with Cg rg g t)
{Cf rf : ℝ≥0} {f : X → Y} (hf : holder_with Cf rf f) (ht : ∀ x, f x ∈ t) :
holder_with (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) | holder_on_with_univ.mp $ hg.comp (hf.holder_on_with univ) (λ x _, ht x) | lemma | holder_on_with.comp_holder_with | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with",
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_on (hf : holder_on_with C r f s) (h0 : 0 < r) :
uniform_continuous_on f s | begin
refine emetric.uniform_continuous_on_iff.2 (λε εpos, _),
have : tendsto (λ d : ℝ≥0∞, (C : ℝ≥0∞) * d ^ (r : ℝ)) (𝓝 0) (𝓝 0),
from 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, λ ... | lemma | holder_on_with.uniform_continuous_on | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"ennreal.coe_ne_top",
"ennreal.tendsto_const_mul_rpow_nhds_zero_of_pos",
"gt_mem_nhds",
"holder_on_with",
"uniform_continuous_on"
] | A Hölder continuous function is uniformly continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on (hf : holder_on_with C r f s) (h0 : 0 < r) : continuous_on f s | (hf.uniform_continuous_on h0).continuous_on | lemma | holder_on_with.continuous_on | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"continuous_on",
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono (hf : holder_on_with C r f s) (ht : t ⊆ s) : holder_on_with C r f t | λ x hx y hy, hf.edist_le (ht hx) (ht hy) | lemma | holder_on_with.mono | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image_le_of_le (hf : holder_on_with C r f s) {d : ℝ≥0∞} (hd : emetric.diam s ≤ d) :
emetric.diam (f '' s) ≤ C * d ^ (r : ℝ) | emetric.diam_image_le_iff.2 $ λ x hx y hy, hf.edist_le_of_le hx hy $
(emetric.edist_le_diam_of_mem hx hy).trans hd | lemma | holder_on_with.ediam_image_le_of_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"emetric.diam",
"emetric.edist_le_diam_of_mem",
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image_le (hf : holder_on_with C r f s) :
emetric.diam (f '' s) ≤ C * emetric.diam s ^ (r : ℝ) | hf.ediam_image_le_of_le le_rfl | lemma | holder_on_with.ediam_image_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"emetric.diam",
"holder_on_with",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image_le_of_subset (hf : holder_on_with C r f s) (ht : t ⊆ s) :
emetric.diam (f '' t) ≤ C * emetric.diam t ^ (r : ℝ) | (hf.mono ht).ediam_image_le | lemma | holder_on_with.ediam_image_le_of_subset | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"emetric.diam",
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image_le_of_subset_of_le (hf : holder_on_with C r f s) (ht : t ⊆ s) {d : ℝ≥0∞}
(hd : emetric.diam t ≤ d) :
emetric.diam (f '' t) ≤ C * d ^ (r : ℝ) | (hf.mono ht).ediam_image_le_of_le hd | lemma | holder_on_with.ediam_image_le_of_subset_of_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"emetric.diam",
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image_inter_le_of_le (hf : holder_on_with C r f s) {d : ℝ≥0∞}
(hd : emetric.diam t ≤ d) :
emetric.diam (f '' (t ∩ s)) ≤ C * d ^ (r : ℝ) | hf.ediam_image_le_of_subset_of_le (inter_subset_right _ _) $
(emetric.diam_mono $ inter_subset_left _ _).trans hd | lemma | holder_on_with.ediam_image_inter_le_of_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"emetric.diam",
"emetric.diam_mono",
"holder_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image_inter_le (hf : holder_on_with C r f s) (t : set X) :
emetric.diam (f '' (t ∩ s)) ≤ C * emetric.diam t ^ (r : ℝ) | hf.ediam_image_inter_le_of_le le_rfl | lemma | holder_on_with.ediam_image_inter_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"emetric.diam",
"holder_on_with",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le (h : holder_with C r f) (x y : X) :
edist (f x) (f y) ≤ C * edist x y ^ (r : ℝ) | h x y | lemma | holder_with.edist_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_of_le (h : holder_with C r f) {x y : X} {d : ℝ≥0∞} (hd : edist x y ≤ d) :
edist (f x) (f y) ≤ C * d ^ (r : ℝ) | (h.holder_on_with univ).edist_le_of_le trivial trivial hd | lemma | holder_with.edist_le_of_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {Cg rg : ℝ≥0} {g : Y → Z} (hg : holder_with Cg rg g)
{Cf rf : ℝ≥0} {f : X → Y} (hf : holder_with Cf rf f) :
holder_with (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) | (hg.holder_on_with univ).comp_holder_with hf (λ _, trivial) | lemma | holder_with.comp | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_holder_on_with {Cg rg : ℝ≥0} {g : Y → Z} (hg : holder_with Cg rg g)
{Cf rf : ℝ≥0} {f : X → Y} {s : set X} (hf : holder_on_with Cf rf f s) :
holder_on_with (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s | (hg.holder_on_with univ).comp hf (λ _ _, trivial) | lemma | holder_with.comp_holder_on_with | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_on_with",
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous (hf : holder_with C r f) (h0 : 0 < r) : uniform_continuous f | uniform_continuous_on_univ.mp $ (hf.holder_on_with univ).uniform_continuous_on h0 | lemma | holder_with.uniform_continuous | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_with",
"uniform_continuous",
"uniform_continuous_on"
] | A Hölder continuous function is uniformly continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous (hf : holder_with C r f) (h0 : 0 < r) : continuous f | (hf.uniform_continuous h0).continuous | lemma | holder_with.continuous | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"continuous",
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image_le (hf : holder_with C r f) (s : set X) :
emetric.diam (f '' s) ≤ C * emetric.diam s ^ (r : ℝ) | emetric.diam_image_le_iff.2 $ λ x hx y hy, hf.edist_le_of_le $ emetric.edist_le_diam_of_mem hx hy | lemma | holder_with.ediam_image_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"emetric.diam",
"emetric.edist_le_diam_of_mem",
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_le_of_le (hf : holder_with C r f) {x y : X} {d : ℝ≥0} (hd : nndist x y ≤ d) :
nndist (f x) (f y) ≤ C * d ^ (r : ℝ) | begin
rw [← ennreal.coe_le_coe, ← edist_nndist, ennreal.coe_mul,
← ennreal.coe_rpow_of_nonneg _ r.coe_nonneg],
apply hf.edist_le_of_le,
rwa [edist_nndist, ennreal.coe_le_coe],
end | lemma | holder_with.nndist_le_of_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"edist_nndist",
"ennreal.coe_le_coe",
"ennreal.coe_mul",
"ennreal.coe_rpow_of_nonneg",
"holder_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_le (hf : holder_with C r f) (x y : X) :
nndist (f x) (f y) ≤ C * nndist x y ^ (r : ℝ) | hf.nndist_le_of_le le_rfl | lemma | holder_with.nndist_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_with",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_of_le (hf : holder_with C r f) {x y : X} {d : ℝ} (hd : dist x y ≤ d) :
dist (f x) (f y) ≤ C * d ^ (r : ℝ) | begin
lift d to ℝ≥0 using dist_nonneg.trans hd,
rw dist_nndist at hd ⊢,
norm_cast at hd ⊢,
exact hf.nndist_le_of_le hd
end | lemma | holder_with.dist_le_of_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"dist_nndist",
"holder_with",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le (hf : holder_with C r f) (x y : X) :
dist (f x) (f y) ≤ C * dist x y ^ (r : ℝ) | hf.dist_le_of_le le_rfl | lemma | holder_with.dist_le | topology.metric_space | src/topology/metric_space/holder.lean | [
"topology.metric_space.lipschitz",
"analysis.special_functions.pow.continuity"
] | [
"holder_with",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep [has_edist α] (s : set α) : ℝ≥0∞ | ⨅ (x ∈ s) (y ∈ s) (hxy : x ≠ y), edist x y | def | set.einfsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"has_edist"
] | The "extended infimum separation" of a set with an edist function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_einfsep_iff {d} : d ≤ s.einfsep ↔ ∀ (x y ∈ s) (hxy : x ≠ y), d ≤ edist x y | by simp_rw [einfsep, le_infi_iff] | lemma | set.le_einfsep_iff | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"le_infi_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_zero :
s.einfsep = 0 ↔ ∀ C (hC : 0 < C), ∃ (x y ∈ s) (hxy : x ≠ y), edist x y < C | by simp_rw [einfsep, ← bot_eq_zero, infi_eq_bot, infi_lt_iff] | theorem | set.einfsep_zero | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"infi_eq_bot",
"infi_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_pos :
0 < s.einfsep ↔ ∃ C (hC : 0 < C), ∀ (x y ∈ s) (hxy : x ≠ y), C ≤ edist x y | by { rw [pos_iff_ne_zero, ne.def, einfsep_zero], simp only [not_forall, not_exists, not_lt] } | theorem | set.einfsep_pos | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"not_exists",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_top : s.einfsep = ∞ ↔ ∀ (x y ∈ s) (hxy : x ≠ y), edist x y = ∞ | by simp_rw [einfsep, infi_eq_top] | lemma | set.einfsep_top | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"infi_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_lt_top : s.einfsep < ∞ ↔ ∃ (x y ∈ s) (hxy : x ≠ y), edist x y < ∞ | by simp_rw [einfsep, infi_lt_iff] | lemma | set.einfsep_lt_top | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"infi_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_ne_top : s.einfsep ≠ ∞ ↔ ∃ (x y ∈ s) (hxy : x ≠ y), edist x y ≠ ∞ | by simp_rw [←lt_top_iff_ne_top, einfsep_lt_top] | lemma | set.einfsep_ne_top | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_lt_iff {d} : s.einfsep < d ↔ ∃ (x y ∈ s) (h : x ≠ y), edist x y < d | by simp_rw [einfsep, infi_lt_iff] | lemma | set.einfsep_lt_iff | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"infi_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_of_einfsep_lt_top (hs : s.einfsep < ∞) : s.nontrivial | by { rcases einfsep_lt_top.1 hs with ⟨_, hx, _, hy, hxy, _⟩, exact ⟨_, hx, _, hy, hxy⟩ } | lemma | set.nontrivial_of_einfsep_lt_top | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_of_einfsep_ne_top (hs : s.einfsep ≠ ∞) : s.nontrivial | nontrivial_of_einfsep_lt_top (lt_top_iff_ne_top.mpr hs) | lemma | set.nontrivial_of_einfsep_ne_top | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.einfsep (hs : s.subsingleton) : s.einfsep = ∞ | by { rw einfsep_top, exact λ _ hx _ hy hxy, (hxy $ hs hx hy).elim } | lemma | set.subsingleton.einfsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_einfsep_image_iff {d} {f : β → α} {s : set β} :
d ≤ einfsep (f '' s) ↔ ∀ x y ∈ s, f x ≠ f y → d ≤ edist (f x) (f y) | by simp_rw [le_einfsep_iff, ball_image_iff] | lemma | set.le_einfsep_image_iff | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_edist_of_le_einfsep {d x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y)
(hd : d ≤ s.einfsep) : d ≤ edist x y | le_einfsep_iff.1 hd x hx y hy hxy | lemma | set.le_edist_of_le_einfsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_le_edist_of_mem {x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y) :
s.einfsep ≤ edist x y | le_edist_of_le_einfsep hx hy hxy le_rfl | lemma | set.einfsep_le_edist_of_mem | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_le_of_mem_of_edist_le {d x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y)
(hxy' : edist x y ≤ d) : s.einfsep ≤ d | le_trans (einfsep_le_edist_of_mem hx hy hxy) hxy' | lemma | set.einfsep_le_of_mem_of_edist_le | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_einfsep {d} (h : ∀ (x y ∈ s) (hxy : x ≠ y), d ≤ edist x y) :
d ≤ s.einfsep | le_einfsep_iff.2 h | lemma | set.le_einfsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_empty : (∅ : set α).einfsep = ∞ | subsingleton_empty.einfsep | lemma | set.einfsep_empty | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_singleton : ({x} : set α).einfsep = ∞ | subsingleton_singleton.einfsep | lemma | set.einfsep_singleton | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_Union_mem_option {ι : Type*} (o : option ι) (s : ι → set α) :
(⋃ i ∈ o, s i).einfsep = ⨅ i ∈ o, (s i).einfsep | by cases o; simp | lemma | set.einfsep_Union_mem_option | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_anti (hst : s ⊆ t) : t.einfsep ≤ s.einfsep | le_einfsep $ λ x hx y hy, einfsep_le_edist_of_mem (hst hx) (hst hy) | lemma | set.einfsep_anti | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_insert_le : (insert x s).einfsep ≤ ⨅ (y ∈ s) (hxy : x ≠ y), edist x y | begin
simp_rw le_infi_iff,
refine λ _ hy hxy, einfsep_le_edist_of_mem (mem_insert _ _) (mem_insert_of_mem _ hy) hxy
end | lemma | set.einfsep_insert_le | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"le_infi_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_einfsep_pair : edist x y ⊓ edist y x ≤ ({x, y} : set α).einfsep | begin
simp_rw [le_einfsep_iff, inf_le_iff, mem_insert_iff, mem_singleton_iff],
rintros a (rfl | rfl) b (rfl | rfl) hab; finish
end | lemma | set.le_einfsep_pair | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"inf_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_pair_le_left (hxy : x ≠ y) : ({x, y} : set α).einfsep ≤ edist x y | einfsep_le_edist_of_mem (mem_insert _ _) (mem_insert_of_mem _ (mem_singleton _)) hxy | lemma | set.einfsep_pair_le_left | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_pair_le_right (hxy : x ≠ y) : ({x, y} : set α).einfsep ≤ edist y x | by rw pair_comm; exact einfsep_pair_le_left hxy.symm | lemma | set.einfsep_pair_le_right | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_pair_eq_inf (hxy : x ≠ y) : ({x, y} : set α).einfsep = (edist x y) ⊓ (edist y x) | le_antisymm (le_inf (einfsep_pair_le_left hxy) (einfsep_pair_le_right hxy)) le_einfsep_pair | lemma | set.einfsep_pair_eq_inf | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_eq_infi : s.einfsep = ⨅ d : s.off_diag, (uncurry edist) (d : α × α) | begin
refine eq_of_forall_le_iff (λ _, _),
simp_rw [le_einfsep_iff, le_infi_iff, imp_forall_iff, set_coe.forall, subtype.coe_mk,
mem_off_diag, prod.forall, uncurry_apply_pair, and_imp]
end | lemma | set.einfsep_eq_infi | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"and_imp",
"eq_of_forall_le_iff",
"imp_forall_iff",
"le_infi_iff",
"set_coe.forall",
"subtype.coe_mk",
"uncurry_apply_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_of_fintype [decidable_eq α] [fintype s] :
s.einfsep = s.off_diag.to_finset.inf (uncurry edist) | begin
refine eq_of_forall_le_iff (λ _, _),
simp_rw [le_einfsep_iff, imp_forall_iff, finset.le_inf_iff, mem_to_finset, mem_off_diag,
prod.forall, uncurry_apply_pair, and_imp]
end | lemma | set.einfsep_of_fintype | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"and_imp",
"eq_of_forall_le_iff",
"finset.le_inf_iff",
"fintype",
"imp_forall_iff",
"uncurry_apply_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.einfsep (hs : s.finite) :
s.einfsep = hs.off_diag.to_finset.inf (uncurry edist) | begin
refine eq_of_forall_le_iff (λ _, _),
simp_rw [le_einfsep_iff, imp_forall_iff, finset.le_inf_iff, finite.mem_to_finset, mem_off_diag,
prod.forall, uncurry_apply_pair, and_imp]
end | lemma | set.finite.einfsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"and_imp",
"eq_of_forall_le_iff",
"finset.le_inf_iff",
"imp_forall_iff",
"uncurry_apply_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.coe_einfsep [decidable_eq α] {s : finset α} :
(s : set α).einfsep = s.off_diag.inf (uncurry edist) | by simp_rw [einfsep_of_fintype, ← finset.coe_off_diag, finset.to_finset_coe] | lemma | set.finset.coe_einfsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finset",
"finset.coe_off_diag",
"finset.to_finset_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.einfsep_exists_of_finite [finite s] (hs : s.nontrivial) :
∃ (x y ∈ s) (hxy : x ≠ y), s.einfsep = edist x y | begin
classical,
casesI nonempty_fintype s,
simp_rw einfsep_of_fintype,
rcases @finset.exists_mem_eq_inf _ _ _ _ (s.off_diag.to_finset) (by simpa) (uncurry edist)
with ⟨_, hxy, hed⟩,
simp_rw mem_to_finset at hxy,
refine ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩
end | lemma | set.nontrivial.einfsep_exists_of_finite | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finite",
"finset.exists_mem_eq_inf",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.einfsep_exists_of_nontrivial (hsf : s.finite) (hs : s.nontrivial) :
∃ (x y ∈ s) (hxy : x ≠ y), s.einfsep = edist x y | by { letI := hsf.fintype, exact hs.einfsep_exists_of_finite } | lemma | set.finite.einfsep_exists_of_nontrivial | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_pair (hxy : x ≠ y) : ({x, y} : set α).einfsep = edist x y | begin
nth_rewrite 0 [← min_self (edist x y)],
convert einfsep_pair_eq_inf hxy using 2,
rw edist_comm
end | lemma | set.einfsep_pair | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_insert :
einfsep (insert x s) = (⨅ (y ∈ s) (hxy : x ≠ y), edist x y) ⊓ (s.einfsep) | begin
refine le_antisymm (le_min einfsep_insert_le (einfsep_anti (subset_insert _ _))) _,
simp_rw [le_einfsep_iff, inf_le_iff, mem_insert_iff],
rintros y (rfl | hy) z (rfl | hz) hyz,
{ exact false.elim (hyz rfl) },
{ exact or.inl (infi_le_of_le _ (infi₂_le hz hyz)) },
{ rw edist_comm, exact or.inl (infi_le_... | lemma | set.einfsep_insert | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"inf_le_iff",
"infi_le_of_le",
"infi₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_triple (hxy : x ≠ y) (hyz : y ≠ z) (hxz : x ≠ z) :
einfsep ({x, y, z} : set α) = edist x y ⊓ edist x z ⊓ edist y z | by simp_rw [einfsep_insert, infi_insert, infi_singleton, einfsep_singleton,
inf_top_eq, cinfi_pos hxy, cinfi_pos hyz, cinfi_pos hxz] | lemma | set.einfsep_triple | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"cinfi_pos",
"inf_top_eq",
"infi_insert",
"infi_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_einfsep_pi_of_le {π : β → Type*} [fintype β] [∀ b, pseudo_emetric_space (π b)]
{s : Π (b : β), set (π b)} {c : ℝ≥0∞} (h : ∀ b, c ≤ einfsep (s b) ) :
c ≤ einfsep (set.pi univ s) | begin
refine le_einfsep (λ x hx y hy hxy, _),
rw mem_univ_pi at hx hy,
rcases function.ne_iff.mp hxy with ⟨i, hi⟩,
exact le_trans (le_einfsep_iff.1 (h i) _ (hx _) _ (hy _) hi) (edist_le_pi_edist _ _ i)
end | lemma | set.le_einfsep_pi_of_le | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"edist_le_pi_edist",
"fintype",
"pseudo_emetric_space",
"set.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_of_einfsep_eq_top (hs : s.einfsep = ∞) : s.subsingleton | begin
rw einfsep_top at hs,
exact λ _ hx _ hy, of_not_not (λ hxy, edist_ne_top _ _ (hs _ hx _ hy hxy))
end | theorem | set.subsingleton_of_einfsep_eq_top | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"edist_ne_top",
"of_not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_eq_top_iff : s.einfsep = ∞ ↔ s.subsingleton | ⟨subsingleton_of_einfsep_eq_top, subsingleton.einfsep⟩ | theorem | set.einfsep_eq_top_iff | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.einfsep_ne_top (hs : s.nontrivial) : s.einfsep ≠ ∞ | by { contrapose! hs, rw not_nontrivial_iff, exact subsingleton_of_einfsep_eq_top hs } | theorem | set.nontrivial.einfsep_ne_top | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.einfsep_lt_top (hs : s.nontrivial) : s.einfsep < ∞ | by { rw lt_top_iff_ne_top, exact hs.einfsep_ne_top } | theorem | set.nontrivial.einfsep_lt_top | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"lt_top_iff_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_lt_top_iff : s.einfsep < ∞ ↔ s.nontrivial | ⟨nontrivial_of_einfsep_lt_top, nontrivial.einfsep_lt_top⟩ | theorem | set.einfsep_lt_top_iff | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_ne_top_iff : s.einfsep ≠ ∞ ↔ s.nontrivial | ⟨nontrivial_of_einfsep_ne_top, nontrivial.einfsep_ne_top⟩ | theorem | set.einfsep_ne_top_iff | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_einfsep_of_forall_dist_le {d} (h : ∀ (x y ∈ s) (hxy : x ≠ y), d ≤ dist x y) :
ennreal.of_real d ≤ s.einfsep | le_einfsep $
λ x hx y hy hxy, (edist_dist x y).symm ▸ ennreal.of_real_le_of_real (h x hx y hy hxy) | lemma | set.le_einfsep_of_forall_dist_le | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"edist_dist",
"ennreal.of_real",
"ennreal.of_real_le_of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
einfsep_pos_of_finite [finite s] : 0 < s.einfsep | begin
casesI nonempty_fintype s,
by_cases hs : s.nontrivial,
{ rcases hs.einfsep_exists_of_finite with ⟨x, hx, y, hy, hxy, hxy'⟩,
exact hxy'.symm ▸ edist_pos.2 hxy },
{ rw not_nontrivial_iff at hs,
exact hs.einfsep.symm ▸ with_top.zero_lt_top }
end | lemma | set.einfsep_pos_of_finite | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finite",
"nonempty_fintype",
"with_top.zero_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relatively_discrete_of_finite [finite s] :
∃ C (hC : 0 < C), ∀ (x y ∈ s) (hxy : x ≠ y), C ≤ edist x y | by { rw ← einfsep_pos, exact einfsep_pos_of_finite } | lemma | set.relatively_discrete_of_finite | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.einfsep_pos (hs : s.finite) : 0 < s.einfsep | by { letI := hs.fintype, exact einfsep_pos_of_finite } | lemma | set.finite.einfsep_pos | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.relatively_discrete (hs : s.finite) :
∃ C (hC : 0 < C), ∀ (x y ∈ s) (hxy : x ≠ y), C ≤ edist x y | by { letI := hs.fintype, exact relatively_discrete_of_finite } | lemma | set.finite.relatively_discrete | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep [has_edist α] (s : set α) : ℝ | ennreal.to_real (s.einfsep) | def | set.infsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"ennreal.to_real",
"has_edist"
] | The "infimum separation" of a set with an edist function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infsep_zero : s.infsep = 0 ↔ s.einfsep = 0 ∨ s.einfsep = ∞ | by rw [infsep, ennreal.to_real_eq_zero_iff] | lemma | set.infsep_zero | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"ennreal.to_real_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_nonneg : 0 ≤ s.infsep | ennreal.to_real_nonneg | lemma | set.infsep_nonneg | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"ennreal.to_real_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_pos : 0 < s.infsep ↔ 0 < s.einfsep ∧ s.einfsep < ∞ | by simp_rw [infsep, ennreal.to_real_pos_iff] | lemma | set.infsep_pos | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"ennreal.to_real_pos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.infsep_zero (hs : s.subsingleton) : s.infsep = 0 | by { rw [infsep_zero, hs.einfsep], right, refl } | lemma | set.subsingleton.infsep_zero | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_of_infsep_pos (hs : 0 < s.infsep) : s.nontrivial | by { contrapose hs, rw not_nontrivial_iff at hs, exact hs.infsep_zero ▸ lt_irrefl _ } | lemma | set.nontrivial_of_infsep_pos | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_empty : (∅ : set α).infsep = 0 | subsingleton_empty.infsep_zero | lemma | set.infsep_empty | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_singleton : ({x} : set α).infsep = 0 | subsingleton_singleton.infsep_zero | lemma | set.infsep_singleton | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_pair_le_to_real_inf (hxy : x ≠ y) :
({x, y} : set α).infsep ≤ (edist x y ⊓ edist y x).to_real | by simp_rw [infsep, einfsep_pair_eq_inf hxy] | lemma | set.infsep_pair_le_to_real_inf | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_pair_eq_to_real : ({x, y} : set α).infsep = (edist x y).to_real | begin
by_cases hxy : x = y,
{ rw hxy, simp only [infsep_singleton, pair_eq_singleton, edist_self, ennreal.zero_to_real] },
{ rw [infsep, einfsep_pair hxy] }
end | lemma | set.infsep_pair_eq_to_real | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"ennreal.zero_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.le_infsep_iff {d} (hs : s.nontrivial) :
d ≤ s.infsep ↔ ∀ (x y ∈ s) (hxy : x ≠ y), d ≤ dist x y | by simp_rw [infsep, ← ennreal.of_real_le_iff_le_to_real (hs.einfsep_ne_top), le_einfsep_iff,
edist_dist, ennreal.of_real_le_of_real_iff (dist_nonneg)] | lemma | set.nontrivial.le_infsep_iff | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"dist_nonneg",
"edist_dist",
"ennreal.of_real_le_iff_le_to_real",
"ennreal.of_real_le_of_real_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.infsep_lt_iff {d} (hs : s.nontrivial) :
s.infsep < d ↔ ∃ (x y ∈ s) (hxy : x ≠ y), dist x y < d | by { rw ← not_iff_not, push_neg, exact hs.le_infsep_iff } | lemma | set.nontrivial.infsep_lt_iff | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.le_infsep {d} (hs : s.nontrivial) (h : ∀ (x y ∈ s) (hxy : x ≠ y), d ≤ dist x y) :
d ≤ s.infsep | hs.le_infsep_iff.2 h | lemma | set.nontrivial.le_infsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_edist_of_le_infsep {d x} (hx : x ∈ s) {y} (hy : y ∈ s)
(hxy : x ≠ y) (hd : d ≤ s.infsep) : d ≤ dist x y | begin
by_cases hs : s.nontrivial,
{ exact hs.le_infsep_iff.1 hd x hx y hy hxy },
{ rw not_nontrivial_iff at hs,
rw hs.infsep_zero at hd,
exact le_trans hd dist_nonneg }
end | lemma | set.le_edist_of_le_infsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"dist_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_le_dist_of_mem (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y) : s.infsep ≤ dist x y | le_edist_of_le_infsep hx hy hxy le_rfl | lemma | set.infsep_le_dist_of_mem | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_le_of_mem_of_edist_le {d x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y)
(hxy' : dist x y ≤ d) : s.infsep ≤ d | le_trans (infsep_le_dist_of_mem hx hy hxy) hxy' | lemma | set.infsep_le_of_mem_of_edist_le | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_pair : ({x, y} : set α).infsep = dist x y | by { rw [infsep_pair_eq_to_real, edist_dist], exact ennreal.to_real_of_real (dist_nonneg) } | lemma | set.infsep_pair | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"dist_nonneg",
"edist_dist",
"ennreal.to_real_of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_triple (hxy : x ≠ y) (hyz : y ≠ z) (hxz : x ≠ z) :
({x, y, z} : set α).infsep = dist x y ⊓ dist x z ⊓ dist y z | by simp only [infsep, einfsep_triple hxy hyz hxz, ennreal.to_real_inf, edist_ne_top x y,
edist_ne_top x z, edist_ne_top y z, dist_edist, ne.def, inf_eq_top_iff,
and_self, not_false_iff] | lemma | set.infsep_triple | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"dist_edist",
"edist_ne_top",
"ennreal.to_real_inf",
"inf_eq_top_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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