Loogle!

Result

Found 203 declarations mentioning NNDist.nndist. Of these, only the first 200 are shown.

• NNDist.nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u_3} [self : NNDist α] : α → α → NNReal
• nndist_self 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (a : α) : nndist a a = 0
• nndist_comm 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y : α) : nndist x y = nndist y x
• coe_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y : α) : ↑(nndist x y) = dist x y
• coe_nnreal_ennreal_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y : α) : ↑(nndist x y) = edist x y
• dist_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y : α) : dist x y = ↑(nndist x y)
• edist_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y : α) : edist x y = ↑(nndist x y)
• nndist_dist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y : α) : nndist x y = (dist x y).toNNReal
• nndist_edist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y : α) : nndist x y = (edist x y).toNNReal
• Metric.boundedSpace_iff_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] : BoundedSpace α ↔ ∃ C, ∀ (a b : α), nndist a b ≤ C
• dist_le_coe 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] {x y : α} {c : NNReal} : dist x y ≤ ↑c ↔ nndist x y ≤ c
• dist_lt_coe 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] {x y : α} {c : NNReal} : dist x y < ↑c ↔ nndist x y < c
• edist_le_coe 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] {x y : α} {c : NNReal} : edist x y ≤ ↑c ↔ nndist x y ≤ c
• edist_lt_coe 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] {x y : α} {c : NNReal} : edist x y < ↑c ↔ nndist x y < c
• Real.nndist_eq 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  (x y : ℝ) : nndist x y = Real.nnabs (x - y)
• Real.nndist_eq' 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  (x y : ℝ) : nndist x y = Real.nnabs (y - x)
• nndist_triangle 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y z : α) : nndist x z ≤ nndist x y + nndist y z
• nndist_triangle_left 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y z : α) : nndist x y ≤ nndist z x + nndist z y
• nndist_triangle_right 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] (x y z : α) : nndist x y ≤ nndist x z + nndist y z
• Metric.isBounded_iff_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] {s : Set α} : Bornology.IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → nndist x y ≤ C
• nndist_ofAdd 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {X : Type u_1} [PseudoMetricSpace X] (a b : X) : nndist (Multiplicative.ofAdd a) (Multiplicative.ofAdd b) = nndist a b
• nndist_ofMul 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {X : Type u_1} [PseudoMetricSpace X] (a b : X) : nndist (Additive.ofMul a) (Additive.ofMul b) = nndist a b
• nndist_toDual 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {X : Type u_1} [PseudoMetricSpace X] (a b : X) : nndist (OrderDual.toDual a) (OrderDual.toDual b) = nndist a b
• nndist_ofDual 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {X : Type u_1} [PseudoMetricSpace X] (a b : Xᵒᵈ) : nndist (OrderDual.ofDual a) (OrderDual.ofDual b) = nndist a b
• nndist_toAdd 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {X : Type u_1} [PseudoMetricSpace X] (a b : Multiplicative X) : nndist (Multiplicative.toAdd a) (Multiplicative.toAdd b) = nndist a b
• nndist_toMul 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {X : Type u_1} [PseudoMetricSpace X] (a b : Additive X) : nndist (Additive.toMul a) (Additive.toMul b) = nndist a b
• eq_of_nndist_eq_zero 📋 Mathlib.Topology.MetricSpace.Defs
  {γ : Type w} [MetricSpace γ] {x y : γ} : nndist x y = 0 → x = y
• nndist_eq_zero 📋 Mathlib.Topology.MetricSpace.Defs
  {γ : Type w} [MetricSpace γ] {x y : γ} : nndist x y = 0 ↔ x = y
• zero_eq_nndist 📋 Mathlib.Topology.MetricSpace.Defs
  {γ : Type w} [MetricSpace γ] {x y : γ} : 0 = nndist x y ↔ x = y
• Inseparable.nndist_eq_zero 📋 Mathlib.Topology.MetricSpace.Pseudo.Basic
  {α : Type u} [PseudoMetricSpace α] {x y : α} : Inseparable x y → nndist x y = 0
• Metric.inseparable_iff_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Basic
  {α : Type u} [PseudoMetricSpace α] {x y : α} : Inseparable x y ↔ nndist x y = 0
• NNReal.nndist_zero_eq_val 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  (z : NNReal) : nndist 0 z = z
• NNReal.nndist_zero_eq_val' 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  (z : NNReal) : nndist z 0 = z
• AddOpposite.nndist_op 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} [PseudoMetricSpace α] (x y : α) : nndist (AddOpposite.op x) (AddOpposite.op y) = nndist x y
• MulOpposite.nndist_op 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} [PseudoMetricSpace α] (x y : α) : nndist (MulOpposite.op x) (MulOpposite.op y) = nndist x y
• NNReal.le_add_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  (a b : NNReal) : a ≤ b + nndist a b
• ULift.nndist_up_up 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {β : Type u_2} [PseudoMetricSpace β] (x y : β) : nndist { down := x } { down := y } = nndist x y
• AddOpposite.nndist_unop 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} [PseudoMetricSpace α] (x y : αᵃᵒᵖ) : nndist (AddOpposite.unop x) (AddOpposite.unop y) = nndist x y
• MulOpposite.nndist_unop 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} [PseudoMetricSpace α] (x y : αᵐᵒᵖ) : nndist (MulOpposite.unop x) (MulOpposite.unop y) = nndist x y
• ULift.nndist_eq 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {β : Type u_2} [PseudoMetricSpace β] (x y : ULift.{u_3, u_2} β) : nndist x y = nndist x.down y.down
• NNReal.nndist_eq 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  (a b : NNReal) : nndist a b = max (a - b) (b - a)
• uniformContinuous_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} [PseudoMetricSpace α] : UniformContinuous fun p => nndist p.1 p.2
• Subtype.nndist_eq 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} [PseudoMetricSpace α] {p : α → Prop} (x y : Subtype p) : nndist x y = nndist ↑x ↑y
• continuous_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} [PseudoMetricSpace α] : Continuous fun p => nndist p.1 p.2
• UniformContinuous.nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b)
• Continuous.nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) : Continuous fun b => nndist (f b) (g b)
• Filter.Tendsto.nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Constructions
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] {f g : β → α} {x : Filter β} {a b : α} (hf : Filter.Tendsto f x (nhds a)) (hg : Filter.Tendsto g x (nhds b)) : Filter.Tendsto (fun x => nndist (f x) (g x)) x (nhds (nndist a b))
• nndist_pi_const 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [Fintype β] [Nonempty β] (a b : α) : (nndist (fun x => a) fun x => b) = nndist a b
• nndist_pi_const_le 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [Fintype β] (a b : α) : (nndist (fun x => a) fun x => b) ≤ nndist a b
• nndist_le_pi_nndist 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {β : Type u_2} {X : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (X b)] (f g : (b : β) → X b) (b : β) : nndist (f b) (g b) ≤ nndist f g
• nndist_pi_def 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {β : Type u_2} {X : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (X b)] (f g : (b : β) → X b) : nndist f g = Finset.univ.sup fun b => nndist (f b) (g b)
• dist_pi_def 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {β : Type u_2} {X : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (X b)] (f g : (b : β) → X b) : dist f g = ↑(Finset.univ.sup fun b => nndist (f b) (g b))
• nndist_pi_le_iff 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {β : Type u_2} {X : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (X b)] {f g : (b : β) → X b} {r : NNReal} : nndist f g ≤ r ↔ ∀ (b : β), nndist (f b) (g b) ≤ r
• nndist_pi_lt_iff 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {β : Type u_2} {X : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (X b)] {f g : (b : β) → X b} {r : NNReal} (hr : 0 < r) : nndist f g < r ↔ ∀ (b : β), nndist (f b) (g b) < r
• nndist_single_single 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {β : Type u_2} [Fintype β] {Y : Type u_4} [PseudoMetricSpace Y] [Zero Y] [DecidableEq β] (i j : β) (a b : Y) (h : i ≠ j) : nndist (Pi.single i a) (Pi.single j b) = max (nndist a 0) (nndist b 0)
• nndist_pi_eq_iff 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {β : Type u_2} {X : β → Type u_3} [Fintype β] [(b : β) → PseudoMetricSpace (X b)] {f g : (b : β) → X b} {r : NNReal} (hr : 0 < r) : nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r
• Fin.nndist_insertNth_insertNth 📋 Mathlib.Topology.MetricSpace.Pseudo.Pi
  {n : ℕ} {α : Fin (n + 1) → Type u_4} [(i : Fin (n + 1)) → PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : (j : Fin n) → α (i.succAbove j)) : nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g)
• nndist_congr_left 📋 Mathlib.Topology.MetricSpace.Basic
  {α : Type u} [PseudoMetricSpace α] {x y z : α} (h : nndist x y = 0) : nndist z x = nndist z y
• nndist_congr_right 📋 Mathlib.Topology.MetricSpace.Basic
  {α : Type u} [PseudoMetricSpace α] {x y z : α} (h : nndist x y = 0) : nndist x z = nndist y z
• nndist_congr 📋 Mathlib.Topology.MetricSpace.Basic
  {α : Type u} [PseudoMetricSpace α] {w x y z : α} (hl : nndist w x = 0) (hr : nndist y z = 0) : nndist w y = nndist x z
• nndist_one_left 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedGroup E] (a : E) : nndist 1 a = ‖a‖₊
• nndist_one_right 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedGroup E] (a : E) : nndist a 1 = ‖a‖₊
• nndist_zero_left 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedAddGroup E] (a : E) : nndist 0 a = ‖a‖₊
• nndist_zero_right 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedAddGroup E] (a : E) : nndist a 0 = ‖a‖₊
• nndist_eq_nnnorm 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedAddCommGroup E] (a b : E) : nndist a b = ‖a - b‖₊
• nndist_eq_nnnorm_div 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedCommGroup E] (a b : E) : nndist a b = ‖a / b‖₊
• nndist_eq_nnnorm_sub 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedAddCommGroup E] (a b : E) : nndist a b = ‖a - b‖₊
• nndist_eq_nnnorm_inv_mul 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedGroup E] (a b : E) : nndist a b = ‖a⁻¹ * b‖₊
• nndist_eq_nnnorm_neg_add 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedAddGroup E] (a b : E) : nndist a b = ‖-a + b‖₊
• nndist_nnnorm_nnnorm_le 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedAddCommGroup E] (a b : E) : nndist ‖a‖₊ ‖b‖₊ ≤ ‖a - b‖₊
• nndist_nnnorm_nnnorm_le' 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedCommGroup E] (a b : E) : nndist ‖a‖₊ ‖b‖₊ ≤ ‖a / b‖₊
• nndist_nnnorm_nnnorm_le_nnnorm_inv_mul 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedGroup E] (a b : E) : nndist ‖a‖₊ ‖b‖₊ ≤ ‖a⁻¹ * b‖₊
• nndist_nnnorm_nnnorm_le_nnnorm_neg_add 📋 Mathlib.Analysis.Normed.Group.Basic
  {E : Type u_4} [SeminormedAddGroup E] (a b : E) : nndist ‖a‖₊ ‖b‖₊ ≤ ‖-a + b‖₊
• nndist_indicator 📋 Mathlib.Analysis.Normed.Group.Basic
  {α : Type u_2} {E : Type u_4} [SeminormedAddGroup E] (s t : Set α) (f : α → E) (x : α) : nndist (s.indicator f x) (t.indicator f x) = ‖(symmDiff s t).indicator f x‖₊
• nndist_mulIndicator 📋 Mathlib.Analysis.Normed.Group.Basic
  {α : Type u_2} {E : Type u_4} [SeminormedGroup E] (s t : Set α) (f : α → E) (x : α) : nndist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(symmDiff s t).mulIndicator f x‖₊
• NNRat.nndist_eq 📋 Mathlib.Topology.Instances.Rat
  (p q : ℚ≥0) : nndist p q = nndist ↑p ↑q
• AntilipschitzWith.le_mul_nndist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : AntilipschitzWith K f → ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)
• AntilipschitzWith.of_le_mul_nndist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : (∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)) → AntilipschitzWith K f
• antilipschitzWith_iff_le_mul_nndist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} : AntilipschitzWith K f ↔ ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)
• AntilipschitzWith.mul_le_nndist 📋 Mathlib.Topology.MetricSpace.Antilipschitz
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} (hf : AntilipschitzWith K f) (x y : α) : K⁻¹ * nndist x y ≤ nndist (f x) (f y)
• Isometry.nndist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : Isometry f → ∀ (x y : α), nndist (f x) (f y) = nndist x y
• Isometry.of_nndist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : (∀ (x y : α), nndist (f x) (f y) = nndist x y) → Isometry f
• isometry_iff_nndist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : Isometry f ↔ ∀ (x y : α), nndist (f x) (f y) = nndist x y
• IsometryClass.nndist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {F : Type u_1} {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [IsometryClass F α β] (f : F) (x y : α) : nndist (f x) (f y) = nndist x y
• IsometryEquiv.nndist_eq 📋 Mathlib.Topology.MetricSpace.Isometry
  {α : Type u_3} {β : Type u_4} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x y : α) : nndist (h x) (h y) = nndist x y
• LipschitzWith.nndist_le 📋 Mathlib.Topology.MetricSpace.Lipschitz
  {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : NNReal} {f : α → β} (hf : LipschitzWith K f) (x y : α) : nndist (f x) (f y) ≤ K * nndist x y
• nndist_add_left 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [PseudoMetricSpace M] [Add M] [IsIsometricVAdd M M] (a b c : M) : nndist (a + b) (a + c) = nndist b c
• nndist_mul_left 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [PseudoMetricSpace M] [Mul M] [IsIsometricSMul M M] (a b c : M) : nndist (a * b) (a * c) = nndist b c
• nndist_add_right 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [PseudoMetricSpace M] [Add M] [IsIsometricVAdd Mᵃᵒᵖ M] (a b c : M) : nndist (a + c) (b + c) = nndist a b
• nndist_mul_right 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [PseudoMetricSpace M] [Mul M] [IsIsometricSMul Mᵐᵒᵖ M] (a b c : M) : nndist (a * c) (b * c) = nndist a b
• nndist_smul 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} {X : Type w} [PseudoMetricSpace X] [SMul M X] [IsIsometricSMul M X] (c : M) (x y : X) : nndist (c • x) (c • y) = nndist x y
• nndist_vadd 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} {X : Type w} [PseudoMetricSpace X] [VAdd M X] [IsIsometricVAdd M X] (c : M) (x y : X) : nndist (c +ᵥ x) (c +ᵥ y) = nndist x y
• nndist_div_right 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [DivInvMonoid M] [PseudoMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M] (a b c : M) : nndist (a / c) (b / c) = nndist a b
• nndist_sub_right 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {M : Type u} [SubNegMonoid M] [PseudoMetricSpace M] [IsIsometricVAdd Mᵃᵒᵖ M] (a b c : M) : nndist (a - c) (b - c) = nndist a b
• nndist_inv_inv 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {G : Type v} [Group G] [PseudoMetricSpace G] [IsIsometricSMul G G] [IsIsometricSMul Gᵐᵒᵖ G] (a b : G) : nndist a⁻¹ b⁻¹ = nndist a b
• nndist_neg_neg 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsIsometricVAdd G G] [IsIsometricVAdd Gᵃᵒᵖ G] (a b : G) : nndist (-a) (-b) = nndist a b
• nndist_div_left 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {G : Type v} [Group G] [PseudoMetricSpace G] [IsIsometricSMul G G] [IsIsometricSMul Gᵐᵒᵖ G] (a b c : G) : nndist (a / b) (a / c) = nndist b c
• nndist_sub_left 📋 Mathlib.Topology.MetricSpace.IsometricSMul
  {G : Type v} [AddGroup G] [PseudoMetricSpace G] [IsIsometricVAdd G G] [IsIsometricVAdd Gᵃᵒᵖ G] (a b c : G) : nndist (a - b) (a - c) = nndist b c
• nndist_add_add_le 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} [SeminormedAddCommGroup E] (a₁ a₂ b₁ b₂ : E) : nndist (a₁ + a₂) (b₁ + b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂
• nndist_mul_mul_le 📋 Mathlib.Analysis.Normed.Group.Uniform
  {E : Type u_2} [SeminormedCommGroup E] (a₁ a₂ b₁ b₂ : E) : nndist (a₁ * a₂) (b₁ * b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂
• nndist_inv_inv₀ 📋 Mathlib.Analysis.Normed.Field.Basic
  {α : Type u_1} [NormedDivisionRing α] {z w : α} (hz : z ≠ 0) (hw : w ≠ 0) : nndist z⁻¹ w⁻¹ = nndist z w / (‖z‖₊ * ‖w‖₊)
• Dilation.mkOfNNDistEq 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_5} {β : Type u_6} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), nndist (f x) (f y) = r * nndist x y) : α →ᵈ β
• Dilation.nndist_eq 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_5} {β : Type u_6} {F : Type u_7} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] (f : F) (x y : α) : nndist (f x) (f y) = Dilation.ratio f * nndist x y
• Dilation.coe_mkOfNNDistEq 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_5} {β : Type u_6} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), nndist (f x) (f y) = r * nndist x y) : ⇑(Dilation.mkOfNNDistEq f h) = f
• Dilation.ratio_unique_of_nndist_ne_zero 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_5} {β : Type u_6} {F : Type u_7} [PseudoMetricSpace α] [PseudoMetricSpace β] [FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : NNReal} (hxy : nndist x y ≠ 0) (hr : nndist (f x) (f y) = r * nndist x y) : r = Dilation.ratio f
• Dilation.mk_coe_of_nndist_eq 📋 Mathlib.Topology.MetricSpace.Dilation
  {α : Type u_5} {β : Type u_6} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α →ᵈ β) (h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), nndist (f x) (f y) = r * nndist x y) : Dilation.mkOfNNDistEq (⇑f) h = f
• nndist_smul_le 📋 Mathlib.Analysis.Normed.MulAction
  {α : Type u_1} {β : Type u_2} [SeminormedAddGroup α] [SeminormedAddGroup β] [SMulZeroClass α β] [IsBoundedSMul α β] (s : α) (x y : β) : nndist (s • x) (s • y) ≤ ‖s‖₊ * nndist x y
• nndist_smul₀ 📋 Mathlib.Analysis.Normed.MulAction
  {α : Type u_1} {β : Type u_2} [SeminormedRing α] [SeminormedAddCommGroup β] [Module α β] [NormSMulClass α β] (s : α) (x y : β) : nndist (s • x) (s • y) = ‖s‖₊ * nndist x y
• measurable_nndist 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
  {α : Type u_1} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [SecondCountableTopology α] : Measurable fun p => nndist p.1 p.2
• Measurable.nndist 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
  {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [SecondCountableTopology α] {f g : β → α} (hf : Measurable f) (hg : Measurable g) : Measurable fun b => nndist (f b) (g b)
• Complex.nndist_of_im_eq 📋 Mathlib.Analysis.Complex.Norm
  {z w : ℂ} (h : z.im = w.im) : nndist z w = nndist z.re w.re
• Complex.nndist_of_re_eq 📋 Mathlib.Analysis.Complex.Norm
  {z w : ℂ} (h : z.re = w.re) : nndist z w = nndist z.im w.im
• Complex.nndist_conj_self 📋 Mathlib.Analysis.Complex.Norm
  (z : ℂ) : nndist ((starRingEnd ℂ) z) z = 2 * Real.nnabs z.im
• Complex.nndist_self_conj 📋 Mathlib.Analysis.Complex.Norm
  (z : ℂ) : nndist z ((starRingEnd ℂ) z) = 2 * Real.nnabs z.im
• nndist_star_star 📋 Mathlib.Analysis.CStarAlgebra.Basic
  {E : Type u_2} [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] (x y : E) : nndist (star x) (star y) = nndist x y
• Complex.nndist_conj_comm 📋 Mathlib.Analysis.Complex.Basic
  (z w : ℂ) : nndist ((starRingEnd ℂ) z) w = nndist z ((starRingEnd ℂ) w)
• Complex.nndist_conj_conj 📋 Mathlib.Analysis.Complex.Basic
  (z w : ℂ) : nndist ((starRingEnd ℂ) z) ((starRingEnd ℂ) w) = nndist z w
• ContinuousLinearMap.nndist_le_opNNNorm 📋 Mathlib.Analysis.Normed.Operator.NNNorm
  {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) (x y : E) : nndist (f x) (f y) ≤ ‖f‖₊ * nndist x y
• MeasureTheory.nndist_integral_add_measure_le_lintegral 📋 Mathlib.MeasureTheory.Integral.Bochner.Basic
  {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → G} (h₁ : MeasureTheory.Integrable f μ) (h₂ : MeasureTheory.Integrable f ν) : ↑(nndist (∫ (x : α), f x ∂μ) (∫ (x : α), f x ∂(μ + ν))) ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν
• nndist_eq_nnnorm_vsub 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) : nndist x y = ‖x -ᵥ y‖₊
• nndist_eq_nnnorm_vsub' 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) : nndist x y = ‖y -ᵥ x‖₊
• nndist_vsub_cancel_left 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P) : nndist (x -ᵥ y) (x -ᵥ z) = nndist y z
• nndist_vsub_cancel_right 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P) : nndist (x -ᵥ z) (y -ᵥ z) = nndist x y
• nndist_vadd_left 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P) : nndist (v +ᵥ x) x = ‖v‖₊
• nndist_vadd_right 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P) : nndist x (v +ᵥ x) = ‖v‖₊
• nndist_vsub_vsub_le 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P) : nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄
• nndist_vadd_cancel_left 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x y : P) : nndist (v +ᵥ x) (v +ᵥ y) = nndist x y
• nndist_vadd_cancel_right 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v₁ v₂ : V) (x : P) : nndist (v₁ +ᵥ x) (v₂ +ᵥ x) = nndist v₁ v₂
• nndist_vadd_vadd_le 📋 Mathlib.Analysis.Normed.Group.AddTorsor
  {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P) : nndist (v +ᵥ p) (v' +ᵥ p') ≤ nndist v v' + nndist p p'
• AffineIsometry.nndist_map 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x y : P) : nndist (f x) (f y) = nndist x y
• BoundedContinuousFunction.nndist_set_exists 📋 Mathlib.Topology.ContinuousMap.Bounded.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} : ∃ C, ∀ (x : α), nndist (f x) (g x) ≤ C
• BoundedContinuousFunction.NNReal.upper_bound 📋 Mathlib.Topology.ContinuousMap.Bounded.Basic
  {α : Type u_2} [TopologicalSpace α] (f : BoundedContinuousFunction α NNReal) (x : α) : f x ≤ nndist f 0
• BoundedContinuousFunction.nndist_coe_le_nndist 📋 Mathlib.Topology.ContinuousMap.Bounded.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} (x : α) : nndist (f x) (g x) ≤ nndist f g
• BoundedContinuousFunction.nndist_eq_iSup 📋 Mathlib.Topology.ContinuousMap.Bounded.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} : nndist f g = ⨆ x, nndist (f x) (g x)
• BoundedContinuousFunction.nndist_eq 📋 Mathlib.Topology.ContinuousMap.Bounded.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} : nndist f g = sInf {C | ∀ (x : α), nndist (f x) (g x) ≤ C}
• BoundedContinuousFunction.nnnorm_def 📋 Mathlib.Topology.ContinuousMap.Bounded.Normed
  {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) : ‖f‖₊ = nndist f 0
• BoundedContinuousFunction.nndist_le_two_nnnorm 📋 Mathlib.Topology.ContinuousMap.Bounded.Normed
  {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (x y : α) : nndist (f x) (f y) ≤ 2 * ‖f‖₊
• ContinuousMap.nndist_eq_iSup 📋 Mathlib.Topology.ContinuousMap.Compact
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] {f g : C(α, β)} : nndist f g = ⨆ x, nndist (f x) (g x)
• WithLp.nndist_fst_le 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  {p : ENNReal} {α : Type u_2} {β : Type u_3} [hp : Fact (1 ≤ p)] [PseudoMetricSpace α] [PseudoMetricSpace β] (x y : WithLp p (α × β)) : nndist x.fst y.fst ≤ nndist x y
• WithLp.nndist_snd_le 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  {p : ENNReal} {α : Type u_2} {β : Type u_3} [hp : Fact (1 ≤ p)] [PseudoMetricSpace α] [PseudoMetricSpace β] (x y : WithLp p (α × β)) : nndist x.snd y.snd ≤ nndist x y
• WithLp.prod_nndist_eq_sup 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  {α : Type u_2} {β : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] (x y : WithLp ⊤ (α × β)) : nndist x y = max (nndist x.fst y.fst) (nndist x.snd y.snd)
• WithLp.nndist_toLp_fst 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  (p : ENNReal) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x₁ x₂ : α) : nndist (WithLp.toLp p (x₁, 0)) (WithLp.toLp p (x₂, 0)) = nndist x₁ x₂
• WithLp.nndist_toLp_snd 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  (p : ENNReal) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (y₁ y₂ : β) : nndist (WithLp.toLp p (0, y₁)) (WithLp.toLp p (0, y₂)) = nndist y₁ y₂
• WithLp.prod_nndist_eq_of_L1 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x y : WithLp 1 (α × β)) : nndist x y = nndist x.fst y.fst + nndist x.snd y.snd
• WithLp.prod_nndist_eq_add 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  {p : ENNReal} {α : Type u_2} {β : Type u_3} [hp : Fact (1 ≤ p)] [PseudoMetricSpace α] [PseudoMetricSpace β] : p ≠ ⊤ → ∀ (x y : WithLp p (α × β)), nndist x y = (nndist x.fst y.fst ^ p.toReal + nndist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal)
• WithLp.prod_nndist_eq_of_L2 📋 Mathlib.Analysis.Normed.Lp.ProdLp
  {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x y : WithLp 2 (α × β)) : nndist x y = NNReal.sqrt (nndist x.fst y.fst ^ 2 + nndist x.snd y.snd ^ 2)
• PiLp.nndist_apply_le 📋 Mathlib.Analysis.Normed.Lp.PiLp
  {p : ENNReal} {ι : Type u_2} {β : ι → Type u_4} [hp : Fact (1 ≤ p)] [Fintype ι] [(i : ι) → PseudoMetricSpace (β i)] (x y : PiLp p β) (i : ι) : nndist (x.ofLp i) (y.ofLp i) ≤ nndist x y
• PiLp.nndist_eq_iSup 📋 Mathlib.Analysis.Normed.Lp.PiLp
  {ι : Type u_2} [Fintype ι] {β : ι → Type u_5} [(i : ι) → PseudoMetricSpace (β i)] (x y : PiLp ⊤ β) : nndist x y = ⨆ i, nndist (x.ofLp i) (y.ofLp i)
• PiLp.nndist_eq_of_L1 📋 Mathlib.Analysis.Normed.Lp.PiLp
  {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] (x y : PiLp 1 β) : nndist x y = ∑ i, nndist (x.ofLp i) (y.ofLp i)
• PiLp.nndist_single_same 📋 Mathlib.Analysis.Normed.Lp.PiLp
  (p : ENNReal) {ι : Type u_2} (β : ι → Type u_4) [hp : Fact (1 ≤ p)] [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] [DecidableEq ι] (i : ι) (b₁ b₂ : β i) : nndist (PiLp.single p i b₁) (PiLp.single p i b₂) = nndist b₁ b₂
• PiLp.nndist_eq_sum 📋 Mathlib.Analysis.Normed.Lp.PiLp
  {ι : Type u_2} [Fintype ι] {p : ENNReal} [Fact (1 ≤ p)] {β : ι → Type u_5} [(i : ι) → PseudoMetricSpace (β i)] (hp : p ≠ ⊤) (x y : PiLp p β) : nndist x y = (∑ i, nndist (x.ofLp i) (y.ofLp i) ^ p.toReal) ^ (1 / p.toReal)
• PiLp.nndist_toLp_single_same 📋 Mathlib.Analysis.Normed.Lp.PiLp
  (p : ENNReal) {ι : Type u_2} (β : ι → Type u_4) [hp : Fact (1 ≤ p)] [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] [DecidableEq ι] (i : ι) (b₁ b₂ : β i) : nndist (WithLp.toLp p (Pi.single i b₁)) (WithLp.toLp p (Pi.single i b₂)) = nndist b₁ b₂
• PiLp.nndist_eq_of_L2 📋 Mathlib.Analysis.Normed.Lp.PiLp
  {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] (x y : PiLp 2 β) : nndist x y = NNReal.sqrt (∑ i, nndist (x.ofLp i) (y.ofLp i) ^ 2)
• EuclideanSpace.nndist_single_same 📋 Mathlib.Analysis.InnerProductSpace.PiL2
  {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] (i : ι) (a b : 𝕜) : nndist (EuclideanSpace.single i a) (EuclideanSpace.single i b) = nndist a b
• EuclideanSpace.nndist_eq 📋 Mathlib.Analysis.InnerProductSpace.PiL2
  {𝕜 : Type u_7} [RCLike 𝕜] {n : Type u_8} [Fintype n] (x y : EuclideanSpace 𝕜 n) : nndist x y = NNReal.sqrt (∑ i, nndist (x.ofLp i) (y.ofLp i) ^ 2)
• BoxIntegral.Box.nndist_le_distortion_mul 📋 Mathlib.Analysis.BoxIntegral.Box.Basic
  {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) (i : ι) : nndist I.lower I.upper ≤ I.distortion * nndist (I.lower i) (I.upper i)
• Unitization.nndist_inr 📋 Mathlib.Analysis.Normed.Algebra.Unitization
  {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (a b : A) : nndist ↑a ↑b = nndist a b
• nndist_midpoint_midpoint_le 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} [SeminormedAddCommGroup V] [NormedSpace ℝ V] (p₁ p₂ p₃ p₄ : V) : nndist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / 2
• nndist_center_homothety 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜) : nndist p₁ ((AffineMap.homothety p₁ c) p₂) = ‖c‖₊ * nndist p₁ p₂
• nndist_homothety_center 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜) : nndist ((AffineMap.homothety p₁ c) p₂) p₁ = ‖c‖₊ * nndist p₁ p₂
• nndist_left_midpoint 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : nndist p₁ (midpoint 𝕜 p₁ p₂) = ‖2‖₊⁻¹ * nndist p₁ p₂
• nndist_midpoint_left 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : nndist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖₊⁻¹ * nndist p₁ p₂
• nndist_midpoint_right 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : nndist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖₊⁻¹ * nndist p₁ p₂
• nndist_right_midpoint 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : nndist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖₊⁻¹ * nndist p₁ p₂
• lipschitzWith_lineMap 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) : LipschitzWith (nndist p₁ p₂) ⇑(AffineMap.lineMap p₁ p₂)
• nndist_left_lineMap 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜) : nndist p₁ ((AffineMap.lineMap p₁ p₂) c) = ‖c‖₊ * nndist p₁ p₂
• nndist_lineMap_left 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜) : nndist ((AffineMap.lineMap p₁ p₂) c) p₁ = ‖c‖₊ * nndist p₁ p₂
• nndist_homothety_self 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜) : nndist ((AffineMap.homothety p₁ c) p₂) p₂ = ‖1 - c‖₊ * nndist p₁ p₂
• nndist_self_homothety 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜) : nndist p₂ ((AffineMap.homothety p₁ c) p₂) = ‖1 - c‖₊ * nndist p₁ p₂
• antilipschitzWith_lineMap 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {W : Type u_3} {Q : Type u_4} [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 W] {p₁ p₂ : Q} (h : p₁ ≠ p₂) : AntilipschitzWith (nndist p₁ p₂)⁻¹ ⇑(AffineMap.lineMap p₁ p₂)
• nndist_midpoint_midpoint_le' 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ p₃ p₄ : P) : nndist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / ‖2‖₊
• nndist_lineMap_right 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜) : nndist ((AffineMap.lineMap p₁ p₂) c) p₂ = ‖1 - c‖₊ * nndist p₁ p₂
• nndist_right_lineMap 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c : 𝕜) : nndist p₂ ((AffineMap.lineMap p₁ p₂) c) = ‖1 - c‖₊ * nndist p₁ p₂
• nndist_homothety 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ p : P) (c : 𝕜) : nndist ((AffineMap.homothety p c) p₁) ((AffineMap.homothety p c) p₂) = ‖c‖₊ * nndist p₁ p₂
• nndist_lineMap_lineMap 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] (p₁ p₂ : P) (c₁ c₂ : 𝕜) : nndist ((AffineMap.lineMap p₁ p₂) c₁) ((AffineMap.lineMap p₁ p₂) c₂) = nndist c₁ c₂ * nndist p₁ p₂
• isClosedMap_nndist 📋 Mathlib.Topology.MetricSpace.ProperSpace.Lemmas
  {α : Type u_1} [PseudoMetricSpace α] [ProperSpace α] (x : α) : IsClosedMap (nndist x)
• isProperMap_nndist 📋 Mathlib.Topology.MetricSpace.ProperSpace.Lemmas
  {α : Type u_1} [PseudoMetricSpace α] [ProperSpace α] (x : α) : IsProperMap (nndist x)
• HolderWith.nndist_le 📋 Mathlib.Topology.MetricSpace.Holder
  {X : Type u_1} {Y : Type u_2} [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : NNReal} {f : X → Y} (hf : HolderWith C r f) (x y : X) : nndist (f x) (f y) ≤ C * nndist x y ^ ↑r
• HolderWith.nndist_le_of_le 📋 Mathlib.Topology.MetricSpace.Holder
  {X : Type u_1} {Y : Type u_2} [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : NNReal} {f : X → Y} (hf : HolderWith C r f) {x y : X} {d : NNReal} (hd : nndist x y ≤ d) : nndist (f x) (f y) ≤ C * d ^ ↑r
• HolderOnWith.nndist_le 📋 Mathlib.Topology.MetricSpace.Holder
  {X : Type u_1} {Y : Type u_2} [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : NNReal} {f : X → Y} {s : Set X} {x y : X} (hf : HolderOnWith C r f s) (hx : x ∈ s) (hy : y ∈ s) : nndist (f x) (f y) ≤ C * nndist x y ^ ↑r
• HolderOnWith.nndist_le_of_le 📋 Mathlib.Topology.MetricSpace.Holder
  {X : Type u_1} {Y : Type u_2} [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : NNReal} {f : X → Y} {s : Set X} {x y : X} (hf : HolderOnWith C r f s) (hx : x ∈ s) (hy : y ∈ s) {d : NNReal} (hd : nndist x y ≤ d) : nndist (f x) (f y) ≤ C * d ^ ↑r
• TensorProduct.nndist_tmul_le 📋 Mathlib.Analysis.InnerProductSpace.TensorProduct
  {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] (x x' : E) (y y' : F) : nndist (x ⊗ₜ[𝕜] y) (x' ⊗ₜ[𝕜] y') ≤ ‖x‖₊ * ‖y‖₊ + ‖x'‖₊ * ‖y'‖₊
• MeasureTheory.hausdorffMeasure_lineMap_image 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {E : Type u_5} {P : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace P] [MetricSpace P] [NormedAddTorsor E P] [BorelSpace P] (x y : P) (s : Set ℝ) : (MeasureTheory.Measure.hausdorffMeasure 1) (⇑(AffineMap.lineMap x y) '' s) = nndist x y • (MeasureTheory.Measure.hausdorffMeasure 1) s
• Finsupp.nndist_def 📋 Mathlib.Analysis.Normed.Lp.Finsupp
  {ι : Type u_1} {X : Type u_2} [Zero X] {p : NNReal} [Fact (1 ≤ p)] [PseudoMetricSpace X] (f g : WithLp (↑p) (ι →₀ X)) : nndist f g = ((Finsupp.zipWith nndist ⋯ f.ofLp g.ofLp).sum fun _i r => r ^ ↑p) ^ (↑p)⁻¹
• BoundedContinuousFunction.apply_le_nndist_zero 📋 Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
  {X : Type u_1} [TopologicalSpace X] (f : BoundedContinuousFunction X NNReal) (x : X) : f x ≤ nndist 0 f
• EuclideanGeometry.dist_orthogonalProjection_eq_infNndist 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] [s.direction.HasOrthogonalProjection] (p : P) : nndist p ↑((EuclideanGeometry.orthogonalProjection s) p) = Metric.infNndist p ↑s
• Congruent.nndist_eq 📋 Mathlib.Topology.MetricSpace.Congruence
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : Congruent v₁ v₂ → ∀ (i₁ i₂ : ι), nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)
• Congruent.of_nndist_eq 📋 Mathlib.Topology.MetricSpace.Congruence
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : (∀ (i₁ i₂ : ι), nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)) → Congruent v₁ v₂
• congruent_iff_nndist_eq 📋 Mathlib.Topology.MetricSpace.Congruence
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : Congruent v₁ v₂ ↔ ∀ (i₁ i₂ : ι), nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)
• Congruent.of_pairwise_nndist_eq 📋 Mathlib.Topology.MetricSpace.Congruence
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : (Pairwise fun i₁ i₂ => nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)) → Congruent v₁ v₂
• Congruent.pairwise_nndist_eq 📋 Mathlib.Topology.MetricSpace.Congruence
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : Congruent v₁ v₂ → Pairwise fun i₁ i₂ => nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)
• congruent_iff_pairwise_nndist_eq 📋 Mathlib.Topology.MetricSpace.Congruence
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : Congruent v₁ v₂ ↔ Pairwise fun i₁ i₂ => nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)
• Similar.exists_nndist_eq 📋 Mathlib.Topology.MetricSpace.Similarity
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : Similar v₁ v₂ → ∃ r, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)
• Similar.of_exists_nndist_eq 📋 Mathlib.Topology.MetricSpace.Similarity
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : (∃ r, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)) → Similar v₁ v₂
• similar_iff_exists_nndist_eq 📋 Mathlib.Topology.MetricSpace.Similarity
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : Similar v₁ v₂ ↔ ∃ r, r ≠ 0 ∧ ∀ (i₁ i₂ : ι), nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)
• Similar.exists_pairwise_nndist_eq 📋 Mathlib.Topology.MetricSpace.Similarity
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : Similar v₁ v₂ → ∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ => nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)
• Similar.of_exists_pairwise_nndist_eq 📋 Mathlib.Topology.MetricSpace.Similarity
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : (∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ => nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)) → Similar v₁ v₂
• similar_iff_exists_pairwise_nndist_eq 📋 Mathlib.Topology.MetricSpace.Similarity
  {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [PseudoMetricSpace P₁] [PseudoMetricSpace P₂] : Similar v₁ v₂ ↔ ∃ r, r ≠ 0 ∧ Pairwise fun i₁ i₂ => nndist (v₁ i₁) (v₁ i₂) = r * nndist (v₂ i₁) (v₂ i₂)
• Hamming.nndist_eq_hammingDist 📋 Mathlib.InformationTheory.Hamming
  {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] (x y : Hamming β) : nndist x y = ↑(hammingDist (Hamming.ofHamming x) (Hamming.ofHamming y))
• MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz_estimate 📋 Mathlib.MeasureTheory.Measure.FiniteMeasure
  {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (f g : BoundedContinuousFunction Ω NNReal) : μ.testAgainstNN f ≤ μ.testAgainstNN g + nndist f g * μ.mass

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c