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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Topology.IsInducing.comapPseudoMetricSpace {α β : Type*} [TopologicalSpace α]
[m : PseudoMetricSpace β] {f : α → β} (hf : IsInducing f) : PseudoMetricSpace α | .replaceTopology (.induced f m) hf.eq_induced | def | Topology.IsInducing.comapPseudoMetricSpace | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"PseudoMetricSpace",
"TopologicalSpace"
] | Pull back a pseudometric space structure by an inducing map. This is a version of
`PseudoMetricSpace.induced` useful in case if the domain already has a `TopologicalSpace`
structure. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.comapPseudoMetricSpace {α β} [UniformSpace α] [m : PseudoMetricSpace β]
(f : α → β) (h : IsUniformInducing f) : PseudoMetricSpace α | .replaceUniformity (.induced f m) h.comap_uniformity.symm | def | IsUniformInducing.comapPseudoMetricSpace | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"IsUniformInducing",
"PseudoMetricSpace",
"UniformSpace"
] | Pull back a pseudometric space structure by a uniform inducing map. This is a version of
`PseudoMetricSpace.induced` useful in case if the domain already has a `UniformSpace`
structure. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pseudoMetricSpace : PseudoMetricSpace (Subtype p) | PseudoMetricSpace.induced Subtype.val ‹_› | instance | Subtype.pseudoMetricSpace | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"PseudoMetricSpace",
"PseudoMetricSpace.induced"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_eq (x y : Subtype p) : dist x y = dist (x : α) y | rfl | lemma | Subtype.dist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_eq (x y : Subtype p) : nndist x y = nndist (x : α) y | rfl | lemma | Subtype.nndist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_ball (a : {a // p a}) (r : ℝ) : Subtype.val ⁻¹' (ball a.1 r) = ball a r | rfl | theorem | Subtype.preimage_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_closedBall {p : α → Prop} (a : {a // p a}) (r : ℝ) :
Subtype.val ⁻¹' (closedBall a.1 r) = closedBall a r | rfl | theorem | Subtype.preimage_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_ball {p : α → Prop} (a : {a // p a}) (r : ℝ) :
Subtype.val '' (ball a r) = ball a.1 r ∩ {a | p a} | by
rw [← preimage_ball, image_preimage_eq_inter_range, range_val_subtype] | theorem | Subtype.image_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_closedBall {p : α → Prop} (a : {a // p a}) (r : ℝ) :
Subtype.val '' (closedBall a r) = closedBall a.1 r ∩ {a | p a} | by
rw [← preimage_closedBall, image_preimage_eq_inter_range, range_val_subtype] | theorem | Subtype.image_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instPseudoMetricSpace : PseudoMetricSpace αᵐᵒᵖ | PseudoMetricSpace.induced MulOpposite.unop ‹_› | instance | MulOpposite.instPseudoMetricSpace | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"MulOpposite.unop",
"PseudoMetricSpace",
"PseudoMetricSpace.induced"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_unop (x y : αᵐᵒᵖ) : dist (unop x) (unop y) = dist x y | rfl | lemma | MulOpposite.dist_unop | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_op (x y : α) : dist (op x) (op y) = dist x y | rfl | lemma | MulOpposite.dist_op | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_unop (x y : αᵐᵒᵖ) : nndist (unop x) (unop y) = nndist x y | rfl | lemma | MulOpposite.nndist_unop | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_op (x y : α) : nndist (op x) (op y) = nndist x y | rfl | lemma | MulOpposite.nndist_op | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.dist_eq (a b : ℝ≥0) : dist a b = |(a : ℝ) - b| | rfl | lemma | NNReal.dist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) | eq_of_forall_ge_iff fun _ => by
simp only [max_le_iff, tsub_le_iff_right (α := ℝ≥0)]
simp only [← NNReal.coe_le_coe, coe_nndist, dist_eq, abs_sub_le_iff,
tsub_le_iff_right, NNReal.coe_add] | lemma | NNReal.nndist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"NNReal.coe_add",
"NNReal.coe_le_coe",
"coe_nndist",
"tsub_le_iff_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z | by
simp [NNReal.nndist_eq] | lemma | NNReal.nndist_zero_eq_val | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"NNReal.nndist_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z | by
rw [nndist_comm]
exact NNReal.nndist_zero_eq_val z | lemma | NNReal.nndist_zero_eq_val' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"NNReal.nndist_zero_eq_val",
"nndist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b | by
suffices (a : ℝ) ≤ (b : ℝ) + dist a b by
rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]
rw [← sub_le_iff_le_add']
exact le_of_abs_le (dist_eq a b).ge | lemma | NNReal.le_add_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"NNReal.coe_add",
"NNReal.coe_le_coe",
"coe_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.ball_zero_eq_Ico' (c : ℝ≥0) :
Metric.ball (0 : ℝ≥0) c.toReal = Set.Ico 0 c | by ext x; simp | lemma | NNReal.ball_zero_eq_Ico' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Metric.ball",
"Set.Ico"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.ball_zero_eq_Ico (c : ℝ) :
Metric.ball (0 : ℝ≥0) c = Set.Ico 0 c.toNNReal | by
by_cases! c_pos : 0 < c
· convert! NNReal.ball_zero_eq_Ico' (NNReal.mk c c_pos.le)
simp [Real.toNNReal, c_pos.le]
simp [c_pos] | lemma | NNReal.ball_zero_eq_Ico | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Metric.ball",
"NNReal.ball_zero_eq_Ico'",
"NNReal.mk",
"Real.toNNReal",
"Set.Ico"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.closedBall_zero_eq_Icc' (c : ℝ≥0) :
Metric.closedBall (0 : ℝ≥0) c.toReal = Set.Icc 0 c | by ext x; simp | lemma | NNReal.closedBall_zero_eq_Icc' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Metric.closedBall",
"Set.Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNReal.closedBall_zero_eq_Icc {c : ℝ} (c_nn : 0 ≤ c) :
Metric.closedBall (0 : ℝ≥0) c = Set.Icc 0 c.toNNReal | by
convert! NNReal.closedBall_zero_eq_Icc' (NNReal.mk c c_nn)
simp [Real.toNNReal, c_nn] | lemma | NNReal.closedBall_zero_eq_Icc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Metric.closedBall",
"NNReal.closedBall_zero_eq_Icc'",
"NNReal.mk",
"Real.toNNReal",
"Set.Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_eq (x y : ULift β) : dist x y = dist x.down y.down | rfl | lemma | ULift.dist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_eq (x y : ULift β) : nndist x y = nndist x.down y.down | rfl | lemma | ULift.nndist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_up_up (x y : β) : dist (ULift.up x) (ULift.up y) = dist x y | rfl | lemma | ULift.dist_up_up | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_up_up (x y : β) : nndist (ULift.up x) (ULift.up y) = nndist x y | rfl | lemma | ULift.nndist_up_up | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) | let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist
(fun x y : α × β => dist x.1 y.1 ⊔ dist x.2 y.2)
(fun x y ↦ by positivity) fun x y => by
simp only [ENNReal.ofReal_max, Prod.edist_eq, edist_dist]
i.replaceBornology fun s => by
simp only [← isBounded_image_fst_and_snd, isBounded_iff_eventually, ... | instance | Prod.pseudoMetricSpaceMax | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"ENNReal.ofReal_max",
"Prod.edist_eq",
"PseudoEMetricSpace.toPseudoMetricSpaceOfDist",
"PseudoMetricSpace",
"edist_dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Prod.dist_eq {x y : α × β} : dist x y = max (dist x.1 y.1) (dist x.2 y.2) | rfl | lemma | Prod.dist_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ | by
simp [Prod.dist_eq] | lemma | dist_prod_same_left | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Prod.dist_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_prod_same_right {x₁ x₂ : α} {y : β} : dist (x₁, y) (x₂, y) = dist x₁ x₂ | by
simp [Prod.dist_eq] | lemma | dist_prod_same_right | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Prod.dist_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_prod_same (x : α) (y : β) (r : ℝ) : ball x r ×ˢ ball y r = ball (x, y) r | ext fun z => by simp [Prod.dist_eq] | lemma | ball_prod_same | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Prod.dist_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_prod_same (x : α) (y : β) (r : ℝ) :
closedBall x r ×ˢ closedBall y r = closedBall (x, y) r | ext fun z => by simp [Prod.dist_eq] | lemma | closedBall_prod_same | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Prod.dist_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere_prod (x : α × β) (r : ℝ) :
sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r | by
obtain hr | rfl | hr := lt_trichotomy r 0
· simp [hr]
· cases x
simp_rw [← closedBall_eq_sphere_of_nonpos le_rfl, union_self, closedBall_prod_same]
· ext ⟨x', y'⟩
simp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq,
max_eq_iff]
refine or_congr (and_c... | lemma | sphere_prod | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Metric.mem_closedBall",
"Metric.mem_sphere",
"Prod.dist_eq",
"Set.mem_prod",
"Set.mem_union",
"closedBall_prod_same",
"le_rfl",
"lt_trichotomy"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1 p.2 | Metric.uniformContinuous_iff.2 fun ε ε0 =>
⟨ε / 2, half_pos ε0, fun {a b} h =>
calc dist (dist a.1 a.2) (dist b.1 b.2) ≤ dist a.1 b.1 + dist a.2 b.2 :=
dist_dist_dist_le _ _ _ _
_ ≤ dist a b + dist a b := add_le_add (le_max_left _ _) (le_max_right _ _)
_ < ε / 2 + ε / 2 := add_lt_add h h
... | lemma | uniformContinuous_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"UniformContinuous",
"add_halves",
"dist_dist_dist_le",
"half_pos"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
(hg : UniformContinuous g) : UniformContinuous fun b => dist (f b) (g b) | uniformContinuous_dist.comp (hf.prodMk hg) | lemma | UniformContinuous.dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_dist : Continuous fun p : α × α ↦ dist p.1 p.2 | uniformContinuous_dist.continuous | lemma | continuous_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.dist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
(hg : Continuous g) : Continuous fun b => dist (f b) (g b) | continuous_dist.comp₂ hf hg | lemma | Continuous.dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Continuous",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.dist {f g : β → α} {x : Filter β} {a b : α}
(hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) :
Tendsto (fun x => dist (f x) (g x)) x (𝓝 (dist a b)) | (continuous_dist.tendsto (a, b)).comp (hf.prodMk_nhds hg) | lemma | Filter.Tendsto.dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
Continuous f ↔ Continuous fun x : β × β => dist (f x.1) (f x.2) | ⟨fun h => h.fst'.dist h.snd', fun h =>
continuous_iff_continuousAt.2 fun _ => tendsto_iff_dist_tendsto_zero.2 <|
(h.comp (.prodMk_left _)).tendsto' _ _ <| dist_self _⟩ | lemma | continuous_iff_continuous_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Continuous",
"TopologicalSpace",
"dist_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_nndist : UniformContinuous fun p : α × α => nndist p.1 p.2 | uniformContinuous_dist.subtype_mk _ | lemma | uniformContinuous_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.nndist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
(hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b) | uniformContinuous_nndist.comp (hf.prodMk hg) | lemma | UniformContinuous.nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_nndist : Continuous fun p : α × α => nndist p.1 p.2 | uniformContinuous_nndist.continuous | lemma | continuous_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.nndist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
(hg : Continuous g) : Continuous fun b => nndist (f b) (g b) | continuous_nndist.comp₂ hf hg | lemma | Continuous.nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Continuous",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.nndist {f g : β → α} {x : Filter β} {a b : α}
(hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) :
Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b)) | (continuous_nndist.tendsto (a, b)).comp (hf.prodMk_nhds hg) | lemma | Filter.Tendsto.nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.ofDist_aux (ε : ℝ) (hε : 0 < ε) : ∃ δ > (0 : ℝ), ∀ x < δ, ∀ y < δ, x + y < ε | ⟨ε / 2, half_pos hε, fun _x hx _y hy => add_halves ε ▸ add_lt_add hx hy⟩ | theorem | UniformSpace.ofDist_aux | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"add_halves",
"half_pos"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : UniformSpace α | .ofFun dist dist_self dist_comm dist_triangle ofDist_aux | def | UniformSpace.ofDist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"UniformSpace",
"dist_comm",
"dist_self",
"dist_triangle"
] | Construct a uniform structure from a distance function and metric space axioms | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Bornology.ofDist {α : Type*} (dist : α → α → ℝ) (dist_comm : ∀ x y, dist x y = dist y x)
(dist_triangle : ∀ x y z, dist x z ≤ dist x y + dist y z) : Bornology α | Bornology.ofBounded { s : Set α | ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C }
⟨0, fun _ hx _ => hx.elim⟩ (fun _ ⟨c, hc⟩ _ h => ⟨c, fun _ hx _ hy => hc (h hx) (h hy)⟩)
(fun s hs t ht => by
rcases s.eq_empty_or_nonempty with rfl | ⟨x, hx⟩
· rwa [empty_union]
rcases t.eq_empty_or_nonempty w... | abbrev | Bornology.ofDist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Bornology",
"Bornology.ofBounded",
"Set",
"dist_comm",
"dist_triangle",
"le_rfl",
"trans"
] | Construct a bornology from a distance function and metric space axioms. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dist (α : Type*) where
/-- Distance between two points -/
dist : α → α → ℝ | class | Dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | The distance function (given an ambient metric space on `α`), which returns
a nonnegative real number `dist x y` given `x y : α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_nonneg' {α} {x y : α} (dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : 0 ≤ dist x y | have : 0 ≤ 2 * dist x y :=
calc 0 = dist x x := (dist_self _).symm
_ ≤ dist x y + dist y x := dist_triangle _ _ _
_ = 2 * dist x y := by rw [two_mul, dist_comm]
nonneg_of_mul_nonneg_right this two_pos | theorem | dist_nonneg' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm",
"dist_self",
"dist_triangle",
"nonneg_of_mul_nonneg_right",
"symm",
"two_mul"
] | This is an internal lemma used inside the default of `PseudoMetricSpace.edist`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetricSpace (α : Type u) : Type u extends Dist α where
dist_self : ∀ x : α, dist x x = 0
dist_comm : ∀ x y : α, dist x y = dist y x
dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z
/-- Extended distance between two points -/
edist : α → α → ℝ≥0∞ | fun x y => ENNReal.ofNNReal (.mk (dist x y) (dist_nonneg' _ ‹_› ‹_› ‹_›))
edist_dist : ∀ x y : α, edist x y = ENNReal.ofReal (dist x y) := by
intro x y; exact ENNReal.coe_nnreal_eq _
toUniformSpace : UniformSpace α := .ofDist dist dist_self dist_comm dist_triangle
uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α ... | class | PseudoMetricSpace | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Bornology",
"Bornology.ofDist",
"Dist",
"ENNReal.coe_nnreal_eq",
"ENNReal.ofNNReal",
"ENNReal.ofReal",
"UniformSpace",
"dist_comm",
"dist_nonneg'",
"dist_self",
"dist_triangle",
"edist_dist"
] | A pseudometric space is a type endowed with a `ℝ`-valued distance `dist` satisfying
reflexivity `dist x x = 0`, commutativity `dist x y = dist y x`, and the triangle inequality
`dist x z ≤ dist x y + dist y z`.
Note that we do not require `dist x y = 0 → x = y`. See metric spaces (`MetricSpace`) for the
similar class ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetricSpace.ext {α : Type*} {m m' : PseudoMetricSpace α}
(h : m.toDist = m'.toDist) : m = m' | by
let d := m.toDist
obtain ⟨_, _, _, _, hed, _, hU, _, hB⟩ := m
let d' := m'.toDist
obtain ⟨_, _, _, _, hed', _, hU', _, hB'⟩ := m'
obtain rfl : d = d' := h
congr
· ext x y : 2
rw [hed, hed']
· exact UniformSpace.ext (hU.trans hU'.symm)
· ext : 2
rw [← Filter.mem_sets, ← Filter.mem_sets, hB, ... | theorem | PseudoMetricSpace.ext | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter.mem_sets",
"PseudoMetricSpace",
"UniformSpace.ext"
] | Two pseudometric space structures with the same distance function coincide. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
(H : ∀ s : Set α, IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s) :
... | { dist := dist
dist_self := dist_self
dist_comm := dist_comm
dist_triangle := dist_triangle
toUniformSpace :=
(UniformSpace.ofDist dist dist_self dist_comm dist_triangle).replaceTopology <|
TopologicalSpace.ext_iff.2 fun s ↦ (H s).trans <| forall₂_congr fun x _ ↦
((UniformSpace.h... | def | PseudoMetricSpace.ofDistTopology | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Bornology.ofDist",
"IsOpen",
"PseudoMetricSpace",
"Set",
"TopologicalSpace",
"UniformSpace.hasBasis_ofFun",
"UniformSpace.ofDist",
"UniformSpace.ofDist_aux",
"comap",
"dist_comm",
"dist_self",
"dist_triangle",
"trans"
] | Construct a pseudo-metric space structure whose underlying topological space structure
(definitionally) agrees which a pre-existing topology which is compatible with a given distance
function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dist_self (x : α) : dist x x = 0 | PseudoMetricSpace.dist_self x | theorem | dist_self | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_comm (x y : α) : dist x y = dist y x | PseudoMetricSpace.dist_comm x y | theorem | dist_comm | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_dist (x y : α) : edist x y = ENNReal.ofReal (dist x y) | PseudoMetricSpace.edist_dist x y | theorem | edist_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.ofReal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z | PseudoMetricSpace.dist_triangle x y z | theorem | dist_triangle | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y | by
rw [dist_comm z]; apply dist_triangle | theorem | dist_triangle_left | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm",
"dist_triangle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z | by
rw [dist_comm y]; apply dist_triangle | theorem | dist_triangle_right | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm",
"dist_triangle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist z w | calc
dist x w ≤ dist x z + dist z w := dist_triangle x z w
_ ≤ dist x y + dist y z + dist z w := by gcongr; apply dist_triangle x y z | theorem | dist_triangle4 | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) | by
rw [add_left_comm, dist_comm x₁, ← add_assoc]
apply dist_triangle4 | theorem | dist_triangle4_left | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm",
"dist_triangle4"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ | by
rw [add_right_comm, dist_comm y₁]
apply dist_triangle4 | theorem | dist_triangle4_right | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm",
"dist_triangle4"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_triangle8 (a b c d e f g h : α) : dist a h ≤ dist a b + dist b c + dist c d
+ dist d e + dist e f + dist f g + dist g h | by
apply le_trans (dist_triangle4 a f g h)
gcongr
apply le_trans (dist_triangle4 a d e f)
gcongr
exact dist_triangle4 a b c d | theorem | dist_triangle8 | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle4",
"le_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
swap_dist : Function.swap (@dist α _) = dist | by funext x y; exact dist_comm _ _ | theorem | swap_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Function.swap",
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y | abs_sub_le_iff.2
⟨sub_le_iff_le_add.2 (dist_triangle _ _ _), sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩ | theorem | abs_dist_sub_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle",
"dist_triangle_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_nonneg {x y : α} : 0 ≤ dist x y | dist_nonneg' dist dist_self dist_comm dist_triangle | theorem | dist_nonneg | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm",
"dist_nonneg'",
"dist_self",
"dist_triangle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
evalDist : PositivityExt | where eval {u α} _zα _pα e := do
match u, α, e with
| 0, ~q(ℝ), ~q(@Dist.dist $β $inst $a $b) =>
let _inst ← synthInstanceQ q(PseudoMetricSpace $β)
assertInstancesCommute
pure (.nonnegative q(dist_nonneg))
| _, _, _ => throwError "not dist" | def | Mathlib.Meta.Positivity.evalDist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"dist_nonneg"
] | Extension for the `positivity` tactic: distances are nonnegative. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
abs_dist {a b : α} : |dist a b| = dist a b | abs_of_nonneg dist_nonneg | theorem | abs_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_nonneg"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NNDist (α : Type*) where
/-- Nonnegative distance between two points -/
nndist : α → α → ℝ≥0 | class | NNDist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | A version of `Dist` that takes value in `ℝ≥0`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_nndist (x y : α) : dist x y = nndist x y | rfl | theorem | dist_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | Express `dist` in terms of `nndist` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_nndist (x y : α) : ↑(nndist x y) = dist x y | rfl | theorem | coe_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_nndist (x y : α) : edist x y = nndist x y | by
rw [edist_dist, dist_nndist, ENNReal.ofReal_coe_nnreal] | theorem | edist_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.ofReal_coe_nnreal",
"dist_nndist",
"edist_dist"
] | Express `edist` in terms of `nndist` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nndist_edist (x y : α) : nndist x y = (edist x y).toNNReal | by
simp [edist_nndist] | theorem | nndist_edist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"edist_nndist"
] | Express `nndist` in terms of `edist` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y | (edist_nndist x y).symm | theorem | coe_nnreal_ennreal_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"edist_nndist",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c | by
rw [edist_nndist, ENNReal.coe_lt_coe] | theorem | edist_lt_coe | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.coe_lt_coe",
"edist_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c | by
rw [edist_nndist, ENNReal.coe_le_coe] | theorem | edist_le_coe | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.coe_le_coe",
"edist_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_lt_top {α : Type*} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ | (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top | theorem | edist_lt_top | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.ofReal_lt_top",
"PseudoMetricSpace",
"edist_dist",
"symm"
] | In a pseudometric space, the extended distance is always finite | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
edist_ne_top (x y : α) : edist x y ≠ ⊤ | (edist_lt_top x y).ne | theorem | edist_ne_top | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"edist_lt_top"
] | In a pseudometric space, the extended distance is always finite | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nndist_self (a : α) : nndist a a = 0 | NNReal.coe_eq_zero.1 (dist_self a) | theorem | nndist_self | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_self"
] | `nndist x x` vanishes | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dist_lt_coe {x y : α} {c : ℝ≥0} : dist x y < c ↔ nndist x y < c | Iff.rfl | theorem | dist_lt_coe | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤ c | Iff.rfl | theorem | dist_le_coe | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r | by
rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg] | theorem | edist_lt_ofReal | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.ofReal",
"ENNReal.ofReal_lt_ofReal_iff_of_nonneg",
"dist_nonneg",
"edist_dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) :
edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r | by
rw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr] | theorem | edist_le_ofReal | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.ofReal",
"ENNReal.ofReal_le_ofReal_iff",
"edist_dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_dist (x y : α) : nndist x y = Real.toNNReal (dist x y) | by
rw [dist_nndist, Real.toNNReal_coe] | theorem | nndist_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Real.toNNReal",
"Real.toNNReal_coe",
"dist_nndist"
] | Express `nndist` in terms of `dist` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nndist_comm (x y : α) : nndist x y = nndist y x | NNReal.eq <| dist_comm x y | theorem | nndist_comm | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"NNReal.eq",
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z | dist_triangle _ _ _ | theorem | nndist_triangle | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle"
] | Triangle inequality for the nonnegative distance | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y | dist_triangle_left _ _ _ | theorem | nndist_triangle_left | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z | dist_triangle_right _ _ _ | theorem | nndist_triangle_right | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_edist (x y : α) : dist x y = (edist x y).toReal | by
rw [edist_dist, ENNReal.toReal_ofReal dist_nonneg] | theorem | dist_edist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"ENNReal.toReal_ofReal",
"dist_nonneg",
"edist_dist"
] | Express `dist` in terms of `edist` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ball (x : α) (ε : ℝ) : Set α | { y | dist y x < ε } | def | Metric.ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set"
] | `ball x ε` is the set of all points `y` with `dist y x < ε` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_ball : y ∈ ball x ε ↔ dist y x < ε | Iff.rfl | theorem | Metric.mem_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_ball' : y ∈ ball x ε ↔ dist x y < ε | by rw [dist_comm, mem_ball] | theorem | Metric.mem_ball' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε | dist_nonneg.trans_lt hy | theorem | Metric.pos_of_mem_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_ball_self (h : 0 < ε) : x ∈ ball x ε | by
rwa [mem_ball, dist_self] | theorem | Metric.mem_ball_self | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty_ball : (ball x ε).Nonempty ↔ 0 < ε | ⟨fun ⟨_x, hx⟩ => pos_of_mem_ball hx, fun h => ⟨x, mem_ball_self h⟩⟩ | theorem | Metric.nonempty_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 | by
rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt] | theorem | Metric.ball_eq_empty | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"not_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_zero : ball x 0 = ∅ | by rw [ball_eq_empty] | theorem | Metric.ball_zero | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' | by
simpa [mem_ball] using exists_between' h | theorem | Metric.exists_lt_mem_ball_of_mem_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | If a point belongs to an open ball, then there is a strictly smaller radius whose ball also
contains it.
See also `exists_lt_subset_ball`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ball_eq_ball (ε : ℝ) (x : α) :
UniformSpace.ball x { p | dist p.2 p.1 < ε } = Metric.ball x ε | rfl | theorem | Metric.ball_eq_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.ball",
"UniformSpace.ball"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_eq_ball' (ε : ℝ) (x : α) :
UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε | by
ext
simp [dist_comm, UniformSpace.ball] | theorem | Metric.ball_eq_ball' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.ball",
"UniformSpace.ball",
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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