Torsors of additive normed group actions.

This file defines torsors of additive normed group actions, with a metric space structure. The motivating case is Euclidean affine spaces.

class NormedAddTorsor (V : outParam (Type u_1)) (P : Type u_2) [SeminormedAddCommGroup V] [PseudoMetricSpace P] extends AddTorsor V P : Type (max u_1 u_2)

A NormedAddTorsor V P is a torsor of an additive seminormed group action by a SeminormedAddCommGroup V on points P. We bundle the pseudometric space structure and require the distance to be the same as results from the norm (which in fact implies the distance yields a pseudometric space, but bundling just the distance and using an instance for the pseudometric space results in type class problems).

• vadd : V → P → P
• zero_vadd (p : P) : 0 +ᵥ p = p
• add_vadd (g₁ g₂ : V) (p : P) : (g₁ + g₂) +ᵥ p = g₁ +ᵥ g₂ +ᵥ p
• vsub : P → P → V
• nonempty : Nonempty P
• vsub_vadd' (p₁ p₂ : P) : (p₁ -ᵥ p₂) +ᵥ p₂ = p₁
• vadd_vsub' (g : V) (p : P) : (g +ᵥ p) -ᵥ p = g
• dist_eq_norm' (x y : P) : dist x y = ‖x -ᵥ y‖

@[instance 100]

instance NormedAddTorsor.toAddTorsor' {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] : AddTorsor V P

Shortcut instance to help typeclass inference out.

Equations

• NormedAddTorsor.toAddTorsor' = NormedAddTorsor.toAddTorsor

@[instance 100]

instance NormedAddTorsor.to_isIsIsometricVAdd {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] : IsIsometricVAdd V P

@[instance 100]

instance SeminormedAddCommGroup.toNormedAddTorsor {V : Type u_2} [SeminormedAddCommGroup V] : NormedAddTorsor V V

A SeminormedAddCommGroup is a NormedAddTorsor over itself.

Equations

• SeminormedAddCommGroup.toNormedAddTorsor = { toAddTorsor := addGroupIsAddTorsor V, dist_eq_norm' := ⋯ }

instance AffineSubspace.toNormedAddTorsor {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [Module R V] (s : AffineSubspace R P) [Nonempty ↥s] : NormedAddTorsor ↥s.direction ↥s

A nonempty affine subspace of a NormedAddTorsor is itself a NormedAddTorsor.

Equations

• s.toNormedAddTorsor = { toAddTorsor := s.toAddTorsor, dist_eq_norm' := ⋯ }

theorem dist_eq_norm_vsub (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) : dist x y = ‖x -ᵥ y‖

The distance equals the norm of subtracting two points. In this lemma, it is necessary to have V as an explicit argument; otherwise rw dist_eq_norm_vsub sometimes doesn't work.

theorem nndist_eq_nnnorm_vsub (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) : nndist x y = ‖x -ᵥ y‖₊

theorem dist_eq_norm_vsub' (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) : dist x y = ‖y -ᵥ x‖

The distance equals the norm of subtracting two points. In this lemma, it is necessary to have V as an explicit argument; otherwise rw dist_eq_norm_vsub' sometimes doesn't work.

theorem nndist_eq_nnnorm_vsub' (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) : nndist x y = ‖y -ᵥ x‖₊

theorem dist_vadd_cancel_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x y : P) : dist (v +ᵥ x) (v +ᵥ y) = dist x y

theorem nndist_vadd_cancel_left {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

@[simp]

theorem dist_vadd_cancel_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v₁ v₂ : V) (x : P) : dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂

@[simp]

theorem nndist_vadd_cancel_right {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

@[simp]

theorem dist_vadd_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P) : dist (v +ᵥ x) x = ‖v‖

@[simp]

theorem nndist_vadd_left {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‖₊

@[simp]

theorem dist_vadd_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P) : dist x (v +ᵥ x) = ‖v‖

@[simp]

theorem nndist_vadd_right {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‖₊

def IsometryEquiv.vaddConst {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : V ≃ᵢ P

Isometry between the tangent space V of a (semi)normed add torsor P and P given by addition/subtraction of x : P.

Equations

• IsometryEquiv.vaddConst x = { toEquiv := Equiv.vaddConst x, isometry_toFun := ⋯ }

@[simp]

theorem IsometryEquiv.vaddConst_apply {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (v : V) : (vaddConst x) v = v +ᵥ x

@[simp]

theorem IsometryEquiv.vaddConst_symm_apply {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x p' : P) : (vaddConst x).symm p' = p' -ᵥ x

@[simp]

theorem dist_vsub_cancel_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P) : dist (x -ᵥ y) (x -ᵥ z) = dist y z

@[simp]

theorem nndist_vsub_cancel_left {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

def IsometryEquiv.constVSub {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : P ≃ᵢ V

Isometry between the tangent space V of a (semi)normed add torsor P and P given by subtraction from x : P.

Equations

• IsometryEquiv.constVSub x = { toEquiv := Equiv.constVSub x, isometry_toFun := ⋯ }

@[simp]

theorem IsometryEquiv.constVSub_symm_apply {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (x✝ : V) : (constVSub x).symm x✝ = -x✝ +ᵥ x

@[simp]

theorem IsometryEquiv.constVSub_apply {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x x✝ : P) : (constVSub x) x✝ = x -ᵥ x✝

@[simp]

theorem dist_vsub_cancel_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P) : dist (x -ᵥ z) (y -ᵥ z) = dist x y

@[simp]

theorem nndist_vsub_cancel_right {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

theorem dist_vadd_vadd_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P) : dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p'

theorem nndist_vadd_vadd_le {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'

theorem dist_vsub_vsub_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P) : dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄

theorem nndist_vsub_vsub_le {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

theorem edist_vadd_vadd_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P) : edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p'

theorem edist_vsub_vsub_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P) : edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄

def pseudoMetricSpaceOfNormedAddCommGroupOfAddTorsor (V : Type u_6) (P : Type u_7) [SeminormedAddCommGroup V] [AddTorsor V P] : PseudoMetricSpace P

The pseudodistance defines a pseudometric space structure on the torsor. This is not an instance because it depends on V to define a MetricSpace P.

Equations

• One or more equations did not get rendered due to their size.

def metricSpaceOfNormedAddCommGroupOfAddTorsor (V : Type u_6) (P : Type u_7) [NormedAddCommGroup V] [AddTorsor V P] : MetricSpace P

The distance defines a metric space structure on the torsor. This is not an instance because it depends on V to define a MetricSpace P.

Equations

• One or more equations did not get rendered due to their size.

theorem LipschitzWith.vadd {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [PseudoEMetricSpace α] {f : α → V} {g : α → P} {Kf Kg : NNReal} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) : LipschitzWith (Kf + Kg) (f +ᵥ g)

theorem LipschitzWith.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [PseudoEMetricSpace α] {f g : α → P} {Kf Kg : NNReal} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) : LipschitzWith (Kf + Kg) (f -ᵥ g)

theorem uniformContinuous_vadd {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] : UniformContinuous fun (x : V × P) => x.1 +ᵥ x.2

theorem uniformContinuous_vsub {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] : UniformContinuous fun (x : P × P) => x.1 -ᵥ x.2

@[instance 100]

instance NormedAddTorsor.to_continuousVAdd {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] : ContinuousVAdd V P

theorem continuous_vsub {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] : Continuous fun (x : P × P) => x.1 -ᵥ x.2

theorem Filter.Tendsto.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {l : Filter α} {f g : α → P} {x y : P} (hf : Tendsto f l (nhds x)) (hg : Tendsto g l (nhds y)) : Tendsto (f -ᵥ g) l (nhds (x -ᵥ y))

theorem Continuous.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} (hf : Continuous f) (hg : Continuous g) : Continuous fun (x : α) => f x -ᵥ g x

theorem ContinuousAt.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun (x : α) => f x -ᵥ g x) x

theorem ContinuousWithinAt.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (fun (x : α) => f x -ᵥ g x) s x

theorem ContinuousOn.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (fun (x : α) => f x -ᵥ g x) s

theorem Filter.Tendsto.lineMap {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [TopologicalSpace R] [Module R V] [ContinuousSMul R V] {l : Filter α} {f₁ f₂ : α → P} {g : α → R} {p₁ p₂ : P} {c : R} (h₁ : Tendsto f₁ l (nhds p₁)) (h₂ : Tendsto f₂ l (nhds p₂)) (hg : Tendsto g l (nhds c)) : Tendsto (fun (x : α) => (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)) l (nhds ((AffineMap.lineMap p₁ p₂) c))

theorem Filter.Tendsto.midpoint {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [TopologicalSpace R] [Module R V] [ContinuousSMul R V] [Invertible 2] {l : Filter α} {f₁ f₂ : α → P} {p₁ p₂ : P} (h₁ : Tendsto f₁ l (nhds p₁)) (h₂ : Tendsto f₂ l (nhds p₂)) : Tendsto (fun (x : α) => _root_.midpoint R (f₁ x) (f₂ x)) l (nhds (_root_.midpoint R p₁ p₂))

theorem IsClosed.vadd_right_of_isCompact {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : Set V} {t : Set P} (hs : IsClosed s) (ht : IsCompact t) : IsClosed (s +ᵥ t)