mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ μ K ≤ μ.outerMeasure ↑K

Tactic:
rw [Content.outerMeasure, inducedOuterMeasure_eq_iInf]
  · exact le_iInf fun U => le_iInf fun hU => le_iInf <| μ.le_innerContent K ⟨U, hU⟩
  · exact fun U hU => isOpen_iUnion hU
  · exact μ.innerContent_iUnion_nat
  · exact μ.innerContent_mono

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc : ∀ (x : Compacts G), K.carrier ⊆ interior x.carrier → μ K + ↑ε < μ x
lower_bound_iInf : μ K + ↑ε ≤ μ K
⊢ False

Tactic:
exact (lt_self_iff_false (μ K)).mp (lt_of_le_of_lt' lower_bound_iInf
    (ENNReal.lt_add_right (ne_top_of_lt (μ.lt_top K)) (ENNReal.coe_ne_zero.mpr hε)))

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : f.outer (Ioc a b) < f.length (t i) + ↑(ε' i)
⊢ f.outer (Ioc a b) < f.length (t i) + ofReal ↑(ε' i)

Tactic:
simpa using h₂

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

Goal state before:
pp: X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
⊢ s.indicator (const X c) * f = s.indicator (c • f)

Tactic:
ext1 x

Goal state after:
X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
x : X
⊢ (s.indicator (const X c) * f) x = s.indicator (c • f) x
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₄ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
have : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖) := by
          simp only [← inner_self_eq_norm_mul_norm, inner_add_add_self, mul_add, mul_comm, map_add]
          linarith

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₄ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
cd : Core 𝕜 F
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₄ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
have : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖) := by
          simp only [← inner_self_eq_norm_mul_norm, inner_add_add_self, mul_add, mul_comm, map_add]
          linarith

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
cd : Core 𝕜 F
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₄ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
cd : Core 𝕜 F
tF : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
h : ContinuousAt (fun v ↦ ⟪v, v⟫) 0
h' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫ < 1}
p : Seminorm 𝕜 F := normSeminorm 𝕜 F
this : p.ball 0 1 = {v | re ⟪v, v⟫ < 1}
⊢ WithSeminorms fun i ↦ p

Tactic:
withSeminorms_iff_mem_nhds_isVonNBounded,

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
cd : Core 𝕜 F
tF : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
h : ContinuousAt (fun v ↦ ⟪v, v⟫) 0
h' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫ < 1}
p : Seminorm 𝕜 F := normSeminorm 𝕜 F
this : p.ball 0 1 = {v | re ⟪v, v⟫ < 1}
⊢ p.ball 0 1 ∈ 𝓝 0 ∧ IsVonNBounded 𝕜 (p.ball 0 1)
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 y : E
h : ‖y‖ = 0
⊢ ⟪x, y⟫ = 0

Tactic:
by
  rw [inner_eq_zero_symm, inner_eq_zero_of_left _ h]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
⊢ ↑⟨chaar K₀ (K₁ ⊔ K₂), ⋯⟩ ≤ ↑⟨chaar K₀ K₁, ⋯⟩ + ↑⟨chaar K₀ K₂, ⋯⟩

Tactic:
simp only [NNReal.toReal, chaar_sup_le]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
⊢ 0 < (haarContent K₀).outerMeasure ↑K₀

Tactic:
by
  refine zero_lt_one.trans_le ?_
  rw [Content.outerMeasure_eq_iInf]
  refine le_iInf₂ fun U hU => le_iInf fun hK₀ => le_trans ?_ <| le_iSup₂ K₀.toCompacts hK₀
  exact haarContent_self.ge

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

Goal state before:
pp: case h.right.left
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this✝ : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n ↦ ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := fun n ↦ 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ↑(w p)
wq : F := ↑(w q)
a : F := u - wq
b : F := u - wp
half : ℝ := 1 / 2
div : ℝ := 1 / (↑N + 1)
this : 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖)
eq : δ ≤ ‖u - half • (wq + wp)‖
⊢ dist ↑(w p) ↑(w q) ≤ √(b✝ N)

Tactic:
have eq₂ : ‖a‖ ≤ δ + div := by grw [hw, Nat.one_div_le_one_div hq]

Goal state after:
case h.right.left
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this✝ : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n ↦ ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := fun n ↦ 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ↑(w p)
wq : F := ↑(w q)
a : F := u - wq
b : F := u - wp
half : ℝ := 1 / 2
div : ℝ := 1 / (↑N + 1)
this : 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖)
eq : δ ≤ ‖u - half • (wq + wp)‖
eq₂ : ‖a‖ ≤ δ + div
⊢ dist ↑(w p) ↑(w q) ≤ √(b✝ N)
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:
rwa [re_inner_self_nonpos, sub_eq_zero] at H

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

Goal state before:
pp: case a
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 := ⋯
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
⊢ ⨅ w, ‖u - ↑w‖ ≤ ‖u - v‖

Tactic:
change ⨅ w : K, ‖u - w‖ ≤ (fun w : K => ‖u - w‖) ⟨v, hv⟩

Goal state after:
case a
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⟩
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
⊢ ⨅ w, ‖u - ↑w‖ ≤ (fun w ↦ ‖u - ↑w‖) ⟨v, hv⟩
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
⊢ (∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0) → ‖u - v‖ = ⨅ w, ‖u - ↑w‖

Tactic:
by
      intro h
      have : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by
        intro w hw
        let w' := w - v
        have : w' ∈ K := Submodule.sub_mem _ hw hv
        have h₁ := h w' this
        exact le_of_eq h₁
      rwa [norm_eq_iInf_iff_real_inner_le_zero]
      exacts [Submodule.convex _, hv]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : RCLike 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
ι : Type u_4
ι' : Type u_5
ι'' : Type u_6
E' : Type u_7
inst✝³ : SeminormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
E'' : Type u_8
inst✝¹ : SeminormedAddCommGroup E''
inst✝ : InnerProductSpace 𝕜 E''
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
v' : Basis ι' 𝕜 E'
hv' : Orthonormal 𝕜 ⇑v'
e : ι ≃ ι'
h : ⇑(v.equiv v' e) ∘ ⇑v = ⇑v' ∘ ⇑e
⊢ Orthonormal 𝕜 (⇑v' ∘ ⇑e)

Tactic:
exact hv'.comp _ e.injective

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

Goal state before:
pp: case mp
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✝ : CompleteSpace E
f : (i : ι) → G i
⊢ (∀ ε > 0,
      ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε) →
    ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε

Tactic:
intro hf

Goal state after:
case mp
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✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε
⊢ ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: case mpr
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✝ : CompleteSpace E
f : (i : ι) → G i
⊢ (∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε) →
    ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε

Tactic:
intro hf

Goal state after:
case mpr
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✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
⊢ ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : Module R M
f : ι → ClosedSubmodule R M
⊢ ↑(sInf (Set.range fun i ↦ f i)) = ⨅ i, ↑(f i)

Tactic:
toSubmodule_sInf,

Goal state after:
ι : Sort u_1
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : Module R M
f : ι → ClosedSubmodule R M
⊢ ⨅ s ∈ Set.range fun i ↦ f i, ↑s = ⨅ i, ↑(f i)
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : ClosedSubmodule R M
⊢ (↑s).closure = s

Tactic:
by
  ext
  simp only [carrier_eq_coe, ClosedSubmodule.coe_toSubmodule, coe_closure, SetLike.mem_coe]
  rw [closure_eq_iff_isClosed.mpr]
  · rfl
  · exact s.isClosed'

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ[𝕜] E
f : E ≃ₗᵢ[𝕜] F
⊢ (↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv).IsSymmetric ↔ T.IsSymmetric

Tactic:
refine ⟨fun h x y => ?_, fun h x y => ?_⟩

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ[𝕜] E
f : E ≃ₗᵢ[𝕜] F
h : (↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv).IsSymmetric
x y : E
⊢ ⟪T x, y⟫ = ⟪x, T y⟫

case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ[𝕜] E
f : E ≃ₗᵢ[𝕜] F
h : T.IsSymmetric
x y : F
⊢ ⟪(↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv) x, y⟫ = ⟪x, (↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Projection.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
v w : E
⊢ (𝕜 ∙ v).starProjection w = (⟪v, w⟫ / ↑(‖v‖ ^ 2)) • v

Tactic:
by_cases hv : v = 0

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
hv : v = 0
⊢ (𝕜 ∙ v).starProjection w = (⟪v, w⟫ / ↑(‖v‖ ^ 2)) • v

case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
hv : ¬v = 0
⊢ (𝕜 ∙ v).starProjection w = (⟪v, w⟫ / ↑(‖v‖ ^ 2)) • v
Module: Mathlib.Analysis.InnerProductSpace.Projection.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
p : E →ₗ[𝕜] E
hp : p.IsSymmetricProjection
⊢ ∃ (x : p.range.HasOrthogonalProjection), p = ↑p.range.starProjection

Tactic:
have : (LinearMap.range p).HasOrthogonalProjection := hp.hasOrthogonalProjection_range

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
p : E →ₗ[𝕜] E
hp : p.IsSymmetricProjection
this : p.range.HasOrthogonalProjection
⊢ ∃ (x : p.range.HasOrthogonalProjection), p = ↑p.range.starProjection
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
ι : Type u_4
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
V : ι → Submodule 𝕜 E
inst✝ : ∀ (i : ι), CompleteSpace ↥(V i)
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
h : iSup V = ⊤
x : E
this : (i : ι) → DecidableEq ↥(V i) := fun i ↦ Classical.decEq ↥(V i)
⊢ ∑ i,
      (AddSubmonoidClass.subtype (V i))
        (DFinsupp.equivFunOnFintype (DFinsupp.equivFunOnFintype.symm fun i ↦ (V i).orthogonalProjectionOnto x) i) =
    x

Tactic:
Equiv.apply_symm_apply,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
ι : Type u_4
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
V : ι → Submodule 𝕜 E
inst✝ : ∀ (i : ι), CompleteSpace ↥(V i)
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
h : iSup V = ⊤
x : E
this : (i : ι) → DecidableEq ↥(V i) := fun i ↦ Classical.decEq ↥(V i)
⊢ ∑ x_1, (AddSubmonoidClass.subtype (V x_1)) ((V x_1).orthogonalProjectionOnto x) = x
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
C : ℝ
Cpos : C > 0
hC : ∀ (y : F), ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' s)

Tactic:
refine isOpen_iff.2 fun y yfs => ?_

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
C : ℝ
Cpos : C > 0
hC : ∀ (y : F), ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖
y : F
yfs : y ∈ ⇑f '' s
⊢ ∃ ε > 0, ball y ε ⊆ ⇑f '' s
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : IsBoundedSMul 𝕜 α
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊

Tactic:
rw [prod_nnnorm_eq_add hpt, prod_nnnorm_eq_add hpt, one_div, NNReal.rpow_inv_le_iff hp0,
        NNReal.mul_rpow, ← NNReal.rpow_mul, inv_mul_cancel₀ hp0.ne', NNReal.rpow_one, mul_add,
        ← NNReal.mul_rpow, ← NNReal.mul_rpow]

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : IsBoundedSMul 𝕜 α
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖(c • f).fst‖₊ ^ p.toReal + ‖(c • f).snd‖₊ ^ p.toReal ≤ (‖c‖₊ * ‖f.fst‖₊) ^ p.toReal + (‖c‖₊ * ‖f.snd‖₊) ^ p.toReal
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
A B : ↥(unitaryGroup n α)
⊢ ↑(toGL (A * B)) = ↑(toGL A * toGL B)

Tactic:
simp only [coe_toGL, toLin'_mul]

Goal state after:
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
A B : ↥(unitaryGroup n α)
⊢ toLin' A ∘ₗ toLin' B = ↑(toGL A * toGL B)
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝² : SeminormedAddCommGroup α
inst✝¹ : SeminormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : 0 < p.toReal
x : WithLp p (α × β)
⊢ ‖x‖ = (‖idemFst x‖ ^ p.toReal + ‖idemSnd x‖ ^ p.toReal) ^ (1 / p.toReal)

Tactic:
by
  rw [WithLp.prod_norm_eq_add hp, ← WithLp.norm_toLp_fst p α β x.fst,
    ← WithLp.norm_toLp_snd p α β x.snd]
  rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
⊢ spectrum 𝕜 f = spectrum 𝕜 ↑f

Tactic:
by
  ext μ
  rw [spectrum.mem_iff, spectrum.mem_iff, ContinuousLinearMap.isUnit_iff_isUnit_toLinearMap]
  rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: case inr.inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
α' : Type u_5
β' : Type u_6
hp✝ : Fact (1 ≤ p)
inst✝⁵ : PseudoEMetricSpace α
inst✝⁴ : PseudoEMetricSpace β
inst✝³ : PseudoEMetricSpace γ
inst✝² : PseudoEMetricSpace α'
inst✝¹ : PseudoEMetricSpace β'
inst✝ : Unique β
x y : WithLp p (α × β)
hp : 0 < p.toReal
⊢ edist x.fst y.fst = edist x y

Tactic:
WithLp.prod_edist_eq_add hp,

Goal state after:
case inr.inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
α' : Type u_5
β' : Type u_6
hp✝ : Fact (1 ≤ p)
inst✝⁵ : PseudoEMetricSpace α
inst✝⁴ : PseudoEMetricSpace β
inst✝³ : PseudoEMetricSpace γ
inst✝² : PseudoEMetricSpace α'
inst✝¹ : PseudoEMetricSpace β'
inst✝ : Unique β
x y : WithLp p (α × β)
hp : 0 < p.toReal
⊢ edist x.fst y.fst = (edist x.fst y.fst ^ p.toReal + edist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
a : ι → ℝ
r : ℝ
hr : 0 < r
⊢ volume (ball a r) = ofReal ((2 * r) ^ Fintype.card ι)

Tactic:
simp only [MeasureTheory.volume_pi_ball a hr, volume_ball, Finset.prod_const]

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
a : ι → ℝ
r : ℝ
hr : 0 < r
⊢ ofReal (2 * r) ^ Finset.univ.card = ofReal ((2 * r) ^ Fintype.card ι)