mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
θ : ℝ
hθ₁ : 0 < θ
hθ₂ : θ ≤ 1
⊢ ‖u - v - θ • (w - v)‖ ^ 2 = ‖u - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ + θ * θ * ‖w - v‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
θ : ℝ
hθ₁ : 0 < θ
hθ₂ : θ ≤ 1
this : ‖u - v‖ ^ 2 ≤ ‖u - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ + θ * θ * ‖w - v‖ ^ 2
⊢ ‖u - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ + θ * θ * ‖w - v‖ ^ 2 =
    ‖u - v‖ ^ 2 + (θ * θ * ‖w - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
I : ∀ p ∈ s, IntegrableOn (fun y ↦ (L (g y)) (f p y)) k μ
p : X
hp : p ∈ v
h'p : p ∈ s
⊢ ∫ (x : Y) in k, ‖(L (g x)) (f p x) - (L (g x)) (f q x)‖ ∂μ ≤ ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = -1 ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hx : ‖x‖ = 1
hy : ‖y‖ = 1
⊢ ⟪x, y⟫ = 1 ↔ x = y

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
⊢ ContinuousOn (fun x ↦ ∫ (y : Y), (L (g y)) (f x y) ∂μ) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hx : ‖x‖ = 1
hy : ‖y‖ = 1
⊢ ⟪x, y⟫ = -1 ↔ x = -y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
hx : ‖x‖ = 1
hy : ‖y‖ = 1
⊢ ⟪x, y⟫_ℝ ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
hx : ‖x‖ = 1
hy : ‖y‖ = 1
⊢ -1 ≤ ⟪x, y⟫_ℝ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
X : Type u_5
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace Y
μ : Measure Y
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → Y → E
s : Set X
k : Set Y
inst✝ : IsFiniteMeasureOnCompacts μ
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
⊢ ContinuousOn (fun x ↦ ∫ (y : Y), f x y ∂μ) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
hx : ‖x‖ = 1
hy : ‖y‖ = 1
⊢ ⟪x, y⟫_ℝ < 1 ↔ x ≠ y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
H : re ⟪x - y, x - y⟫ ≤ 0
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
⊢ ‖x‖ ^ 2 ≤ ‖y‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
θ : ℝ
hθ₁ : 0 < θ
hθ₂ : θ ≤ 1
this : 0 ≤ θ * θ * ‖w - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ
eq₁ :
  ‖u - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ + θ * θ * ‖w - v‖ ^ 2 =
    ‖u - v‖ ^ 2 + (θ * θ * ‖w - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ)
⊢ θ * θ * ‖w - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ = θ * (θ * ‖w - v‖ ^ 2 - 2 * ⟪u - v, w - v⟫_ℝ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
H₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2
⊢ re ⟪y, x⟫ = ‖y‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
n : ℕ
v : E
vs : Fin n → E
⊢ Orthonormal 𝕜 (Matrix.vecCons v vs) ↔ ‖v‖ = 1 ∧ (∀ (i : Fin n), ⟪v, vs i⟫ = 0) ∧ Orthonormal 𝕜 vs

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
h : Orthonormal 𝕜 v
i : ι
this : ↑‖v i‖₊ = 1
⊢ ‖v i‖₊ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
h : Orthonormal 𝕜 v
i : ι
⊢ ‖v i‖₊ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
h : Orthonormal 𝕜 v
i : ι
⊢ ‖v i‖ₑ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ‖x + y‖ = ‖x‖ + ‖y‖ ↔ ‖y‖ • x = ‖x‖ • y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
θ : ℝ
hθ₁ : 0 < θ
hθ₂ : θ ≤ 1
⊢ 2 * p ≤ θ * q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁶ : RCLike 𝕜
G : Type u_4
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : AddCommGroup G
inst✝² : Module 𝕜 G
F : Type u_5
inst✝¹ : FunLike F G E
inst✝ : LinearMapClass F 𝕜 G E
f : F
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G E f
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G E f
x y z : G
⊢ ⟪f (x + y), f z⟫_𝕜 = ⟪f x, f z⟫_𝕜 + ⟪f y, f z⟫_𝕜

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁶ : RCLike 𝕜
G : Type u_4
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : AddCommGroup G
inst✝² : Module 𝕜 G
F : Type u_5
inst✝¹ : FunLike F G E
inst✝ : LinearMapClass F 𝕜 G E
f : F
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G E f
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G E f
x y : G
r : 𝕜
⊢ ⟪f (r • x), f y⟫_𝕜 = (starRingEnd 𝕜) r * ⟪f x, f y⟫_𝕜

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : DecidableEq ι
v : ι → E
h : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
i : ι
⊢ ‖v i‖ ^ 2 = 1 ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
x : 𝕜
⊢ ‖x‖ ^ 2 = re (x * star x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
x y : 𝕜
⊢ (starRingEnd 𝕜) (x * star y) = y * star x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
x y z : 𝕜
⊢ z * star (x + y) = z * star x + z * star y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
x y z : 𝕜
⊢ y * star (z • x) = (starRingEnd 𝕜) z * (y * star x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * 0
hq : q = 0
⊢ 0 < 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
G : Type u_4
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y z : E
⊢ ⟪x + y, z⟫_ℝ = ⟪x, z⟫_ℝ + ⟪y, z⟫_ℝ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * 0
hq : q = 0
⊢ 1 ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
G : Type u_4
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
r : ℝ
⊢ ⟪r • x, y⟫_ℝ = (starRingEnd ℝ) r * ⟪x, y⟫_ℝ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ ⟪x, I • x⟫_ℝ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * 0
hq : q = 0
⊢ p ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : DecidableEq ι
v : ι → E
⊢ Orthonormal 𝕜 v ↔ ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : Subsingleton ι
v : ι → E
⊢ Orthonormal 𝕜 v ↔ ∀ (i : ι), ‖v i‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : DecidableEq E
s : Set E
⊢ Orthonormal 𝕜 Subtype.val ↔ ∀ v ∈ s, ∀ w ∈ s, ⟪v, w⟫ = if v = w then 1 else 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: x y : ℝ
⊢ inner ℝ x y = x * y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x y : E
⊢ (RingHom.id ℝ) (((innerₗ E) x) y) = ((innerₗ E) y) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
⊢ 0 ≤ ((innerₗ E) x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι →₀ 𝕜
i : ι
⊢ ⟪v i, (linearCombination 𝕜 v) l⟫ = l i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι → 𝕜
s : Finset ι
i : ι
hi : i ∈ s
⊢ ⟪v i, ∑ i ∈ s, l i • v i⟫ = l i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
⊢ instInnerProductSpaceRealComplex.toSMul = Complex.instRCLike.toSMul

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι →₀ 𝕜
i : ι
⊢ ⟪(linearCombination 𝕜 v) l, v i⟫ = (starRingEnd 𝕜) (l i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι → 𝕜
s : Finset ι
i : ι
hi : i ∈ s
⊢ ⟪∑ i ∈ s, l i • v i, v i⟫ = (starRingEnd 𝕜) (l i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l₁ l₂ : ι →₀ 𝕜
⊢ ⟪(linearCombination 𝕜 v) l₁, (linearCombination 𝕜 v) l₂⟫ = l₁.sum fun i y ↦ (starRingEnd 𝕜) y * l₂ i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l₁ l₂ : ι →₀ 𝕜
⊢ ⟪(linearCombination 𝕜 v) l₁, (linearCombination 𝕜 v) l₂⟫ = l₂.sum fun i y ↦ (starRingEnd 𝕜) (l₁ i) * y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
i j : ι
hij : i ≠ j
a : (fun _i ↦ 𝕜) i
b : (fun _i ↦ 𝕜) j
⊢ ⟪((fun i ↦ LinearIsometry.toSpanSingleton 𝕜 E ⋯) i) a, ((fun i ↦ LinearIsometry.toSpanSingleton 𝕜 E ⋯) j) b⟫ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : DecidableEq ι
i j : ι
v : G i
w : G j
⊢ ⟪(V i) v, (V j) w⟫ = if i = j then ⟪(V i) v, (V j) w⟫ else 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝¹ : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : DecidableEq ι
l : ⨁ (i : ι), G i
i : ι
v : G i
⊢ (DFinsupp.sum l fun j w ↦ if i = j then ⟪(V i) v, (V j) w⟫ else 0) = ⟪v, l i⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l₁ l₂ : ι → 𝕜
s : Finset ι
⊢ ⟪∑ i ∈ s, l₁ i • v i, ∑ i ∈ s, l₂ i • v i⟫ = ∑ i ∈ s, (starRingEnd 𝕜) (l₁ i) * l₂ i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
s : Finset ι
v : ι → E
hv : Orthonormal 𝕜 v
a : ι → ι → 𝕜
⊢ ∑ i ∈ s, ∑ j ∈ s, a i j • ⟪v j, v i⟫ = ∑ k ∈ s, a k k

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * q
hq : ¬q = 0
q_pos : 0 < q
hp : 0 < ⟪u - v, w - v⟫_ℝ
θ : ℝ := min 1 (p / q)
eq₁ : θ * q ≤ p
⊢ 0 < 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : Fintype ι
l : (i : ι) → G i
i : ι
v : G i
⊢ ⟪(V i) v, ∑ j, (V j) (l j)⟫ = ∑ j, ⟪(V i) v, (V j) (l j)⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: V : Type u_4
inst✝¹ : SeminormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
x y : V
⊢ ⟪T y, x⟫_ℂ =
    (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ + Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ -
        Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /
      4

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * q
hq : ¬q = 0
q_pos : 0 < q
hp : 0 < ⟪u - v, w - v⟫_ℝ
θ : ℝ := min 1 (p / q)
eq₁ : θ * q ≤ p
⊢ θ ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι →₀ 𝕜
hl : (linearCombination 𝕜 v) l = 0
i : ι
⊢ ⟪v i, (linearCombination 𝕜 v) l⟫ = ⟪v i, 0⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : Fintype ι
l : (i : ι) → G i
i : ι
v : G i
⊢ (∑ j, if i = j then ⟪(V i) v, (V j) (l j)⟫ else 0) = ⟪v, l i⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
⊢ LinearIndependent 𝕜 v

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * q
hq : ¬q = 0
q_pos : 0 < q
hp : 0 < ⟪u - v, w - v⟫_ℝ
θ : ℝ := min 1 (p / q)
eq₁ : θ * q ≤ p
⊢ 2 * p ≤ θ * q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: V : Type u_4
inst✝¹ : SeminormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
x y : V
⊢ ⟪T x, y⟫_ℂ =
    (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ - Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ +
        Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /
      4

Tactic:
by

Goal state after: