mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
t : Set α
⊢ ν t = ∑' (g : G), ν (t ∩ g • s)

Tactic:
have H : ν.restrict t ≪ μ := Measure.restrict_le_self.absolutelyContinuous.trans hν

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
t : Set α
H : ν.restrict t ≪ μ
⊢ ν t = ∑' (g : G), ν (t ∩ g • s)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a : R
⊢ f.measure {a} = ofReal (↑f a - leftLim (↑f) a)

Tactic:
by_cases ha : IsBot a
  · have : leftLim f a = f a := by
      apply leftLim_eq_of_eq_bot
      simp [nhdsLT_eq_bot_iff, ha]
    simp only [this, sub_self, ofReal_zero]
    apply eq_bot_iff.2
    rw [StieltjesFunction.measure]
    apply (outer_le_length _ _).trans
    rw [← length_sdiff_botSet]
    simp [subsingleton_botSet.eq_singleton_of_mem ha]
  obtain ⟨b, hb⟩ : ∃ b, b < a := by simpa only [IsBot, not_forall, not_le] using ha
  obtain ⟨u, u_mono, u_lt_a, u_lim⟩ :
    ∃ u : ℕ → R, StrictMono u ∧ (∀ n : ℕ, u n ∈ Ioo b a) ∧ Tendsto u atTop (𝓝 a) :=
    exists_seq_strictMono_tendsto' hb
  replace u_lt_a n : u n < a := (u_lt_a n).2
  have A : {a} = ⋂ n, Ioc (u n) a := by
    refine Subset.antisymm (fun x hx => by simp [mem_singleton_iff.1 hx, u_lt_a]) fun x hx => ?_
    replace hx : ∀ (i : ℕ), u i < x ∧ x ≤ a := by simpa using hx
    have : a ≤ x := le_of_tendsto' u_lim fun n => (hx n).1.le
    simp [le_antisymm this (hx 0).2]
  have L1 : Tendsto (fun n => f.measure (Ioc (u n) a)) atTop (𝓝 (f.measure {a})) := by
    rw [A]
    refine tendsto_measure_iInter_atTop (fun n => nullMeasurableSet_Ioc)
      (fun m n hmn => ?_) ?_
    · exact Ioc_subset_Ioc_left (u_mono.monotone hmn)
    · exact ⟨0, by simpa only [measure_Ioc] using ENNReal.ofReal_ne_top⟩
  have L2 :
      Tendsto (fun n => f.measure (Ioc (u n) a)) atTop (𝓝 (ofReal (f a - leftLim f a))) := by
    simp only [measure_Ioc]
    have : Tendsto (fun n => f (u n)) atTop (𝓝 (leftLim f a)) := by
      apply (f.mono.tendsto_leftLim a).comp
      exact
        tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ u_lim
          (Eventually.of_forall fun n => u_lt_a n)
    exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp (tendsto_const_nhds.sub this)
  exact tendsto_nhds_unique L1 L2

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 < μ ((support fun a ↦ f a - R) ∩ {x | R < f x})

Tactic:
rw [← pos_iff_ne_zero] at hμ
    rwa [Set.inter_eq_self_of_subset_right]
    exact fun x hx => Ne.symm (ne_of_lt <| sub_pos.2 hx)

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

Goal state before:
pp: case right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ ∃ x ∈ K₂.carrier, ∃ y, y⁻¹ ∈ V ∧ x * y = g₁⁻¹

Tactic:
refine ⟨_, h2g₃, (g₁ * g₃)⁻¹, ?_, ?_⟩
    · simp only [inv_inv, h1g₃]
    · simp only [mul_inv_rev, mul_inv_cancel_left]

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

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : MeasurableSet s
h : ∀ x ∈ s, f x ≤ g x
⊢ f ≤ᵐ[μ.restrict s] g

Tactic:
by simp [hs, EventuallyLE, eventually_inf_principal, ae_of_all _ h]

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

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
hf : Tendsto (↑f) atTop atTop
x : R
⊢ f.measure (Ioi x) = ∞

Tactic:
by
  have : Nonempty R := ⟨x⟩
  refine ENNReal.eq_top_of_forall_nnreal_le fun r ↦ ?_
  obtain ⟨N, hN⟩ := eventually_atTop.mp (tendsto_atTop.mp hf (r + f x))
  exact (f.measure_Ioc x (max x N) ▸ ENNReal.coe_nnreal_eq r ▸ (ENNReal.ofReal_le_ofReal <|
    le_tsub_of_add_le_right <| hN _ (le_max_right x N))).trans (measure_mono Ioc_subset_Ioi_self)

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

Goal state before:
pp: case hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ l ≤ ↑f x

Tactic:
exact Monotone.le_of_tendsto f.mono hfl x

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
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' {0}

Tactic:
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := ⋯
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' {0}
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
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' {1}

Tactic:
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := ⋯
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' {1}
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
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' Ici 0

Tactic:
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := ⋯
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' Ici 0
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
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' Ici 0

Tactic:
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := ⋯
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' Ici 0
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
this : leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a
⊢ ofReal ↑c * ofReal (↑f b - leftLim (↑f) a) = c • ofReal (↑f b - leftLim (↑f) a)

Tactic:
ofReal_coe_nnreal,

Goal state after:
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
this : leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a
⊢ ↑c * ofReal (↑f b - leftLim (↑f) a) = c • ofReal (↑f b - leftLim (↑f) a)
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
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 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h' : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : NormedAddCommGroup F := toNormedAddCommGroupOfTopology h h'
r : 𝕜
x : F
⊢ ‖r • x‖ ≤ ‖r‖ * ‖x‖

Tactic:
rw [norm_eq_sqrt_re_inner, inner_smul_left, inner_smul_right, ← mul_assoc]
      rw [RCLike.conj_mul, ← ofReal_pow, re_ofReal_mul, sqrt_mul, ← ofReal_normSq_eq_inner_self,
        ofReal_re]
      · simp [sqrt_normSq_eq_norm]
      · positivity

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
⊢ re ⟪x, x⟫ = ‖x‖ * ‖x‖

Tactic:
@norm_eq_sqrt_re_inner 𝕜,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ re ⟪x, x⟫ = √(re ⟪x, x⟫) * √(re ⟪x, x⟫)
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
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f (Compacts.map (fun x ↦ g * x) ⋯ K) - f K
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' {0}

Tactic:
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := ⋯
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' {0}
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
p : X
hp : p ∈ s
this : ContinuousOn (fun y ↦ (p, y)) k
⊢ ContinuousOn (f p) k

Tactic:
exact hf.comp this (by simp [MapsTo, hp])

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 * ((starRingEnd 𝕜) x + (starRingEnd 𝕜) y) = z * (starRingEnd 𝕜) x + z * (starRingEnd 𝕜) y

Tactic:
mul_add

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
x y z : 𝕜
⊢ z * (starRingEnd 𝕜) x + z * (starRingEnd 𝕜) y = z * (starRingEnd 𝕜) x + z * (starRingEnd 𝕜) y
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: case pos
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
⊢ ⟪u - v, w - v⟫_ℝ ≤ 0

Tactic:
have : p ≤ 0 := by
        have := this (1 : ℝ) (by simp) (by simp)
        linarith

Goal state after:
case pos
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✝¹ : 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
this : p ≤ 0
⊢ ⟪u - v, w - v⟫_ℝ ≤ 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: case neg
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)
⊢ False

Tactic:
have eq₁ : θ * q ≤ p :=
        calc
          θ * q ≤ p / q * q := mul_le_mul_of_nonneg_right (min_le_right _ _) (sq_nonneg _)
          _ = p := div_mul_cancel₀ _ hq

Goal state after:
case neg
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
⊢ False
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:
exact this θ (lt_min (by simp) (div_pos hp q_pos)) (by simp [θ])

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''
f : E →ₗ[𝕜] E'
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
hf : Orthonormal 𝕜 (⇑f ∘ ⇑v)
x y : E
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
hv.inner_finsupp_eq_sum_left,

Goal state after:
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''
f : E →ₗ[𝕜] E'
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
hf : Orthonormal 𝕜 (⇑f ∘ ⇑v)
x y : E
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
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''
f : E ≃ₗ[𝕜] E'
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
hf : Orthonormal 𝕜 (⇑↑f ∘ ⇑v)
x y : E
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
hv.inner_finsupp_eq_sum_left,

Goal state after:
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''
f : E ≃ₗ[𝕜] E'
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
hf : Orthonormal 𝕜 (⇑↑f ∘ ⇑v)
x y : E
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →L[𝕜] F
C : ℝ
hC : 0 ≤ C
h : ∀ (x : E) (y : F), ‖x‖ = 1 → ‖y‖ = 1 → re ⟪T x, y⟫_𝕜 ≤ C
x : E
hx : ‖x‖ = 1
hTx : ¬‖T x‖ = 0
⊢ ‖T x‖ ≤ C

Tactic:
specialize h x (((‖T x‖⁻¹ : ℝ) : 𝕜) • T x) hx (by simp [norm_smul, hTx])

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →L[𝕜] F
C : ℝ
hC : 0 ≤ C
x : E
h : re ⟪T x, ↑‖T x‖⁻¹ • T x⟫_𝕜 ≤ C
hx : ‖x‖ = 1
hTx : ¬‖T x‖ = 0
⊢ ‖T x‖ ≤ C
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
hT : (↑T).IsSymmetric
x : E
⊢ ∃ r, ⟪T x, x⟫ = ↑r

Tactic:
rw [← conj_eq_iff_real]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
hT : (↑T).IsSymmetric
x : E
⊢ (starRingEnd 𝕜) ⟪T x, x⟫ = ⟪T x, x⟫
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
h : Kᗮ = ⊤
this : ⊤ ⊓ K = ⊥
⊢ K = ⊥

Tactic:
top_inf_eq

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
h : Kᗮ = ⊤
this : K = ⊥
⊢ K = ⊥
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : ClosedSubmodule 𝕜 E
h : Kᗮ = ⊤
this : ⊤ ⊓ K = ⊥
⊢ K = ⊥

Tactic:
top_inf_eq

Goal state after:
𝕜 : Type u_4
E : Type u_5
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : ClosedSubmodule 𝕜 E
h : Kᗮ = ⊤
this : K = ⊥
⊢ K = ⊥
Module: Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : CompleteSpace E
⊢ Kᗮᗮ = K.topologicalClosure

Tactic:
refine le_antisymm ?_ ?_
  · convert Submodule.orthogonal_orthogonal_monotone K.le_topologicalClosure
    rw [K.topologicalClosure.orthogonal_orthogonal]
  · exact K.topologicalClosure_minimal K.le_orthogonal_orthogonal Kᗮ.isClosed_orthogonal

Goal state after:
Module: Mathlib.Data.Matrix.Reflection

Goal state before:
pp: l m : ℕ
α : Type u_1
inst✝ : NonUnitalNonAssocSemiring α
v : Fin l → α
A : Matrix (Fin l) (Fin m) α
⊢ vecMulᵣ v A = v ᵥ* A

Tactic:
simp [vecMulᵣ]
  rfl

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

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
f : PiLp p β
⊢ edist f f = 0

Tactic:
by
  rcases p.trichotomy with (rfl | rfl | h)
  · simp [edist_eq_card]
  · simp [edist_eq_iSup]
  · simp [edist_eq_sum h, ENNReal.zero_rpow_of_pos h, ENNReal.zero_rpow_of_pos (inv_pos.2 <| h)]

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

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x y : α × β
⊢ dist x y = ‖toLp p (-x + y)‖

Tactic:
dist_pseudoMetricSpaceToProd,

Goal state after:
p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x y : α × β
⊢ dist (toLp p x) (toLp p y) = ‖toLp p (-x + y)‖
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : NormedAddCommGroup α
inst✝ : NormedAddCommGroup β
x y : α × β
⊢ dist x y = ‖toLp p (-x + y)‖

Tactic:
dist_pseudoMetricSpaceToProd,

Goal state after:
p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : NormedAddCommGroup α
inst✝ : NormedAddCommGroup β
x y : α × β
⊢ dist (toLp p x) (toLp p y) = ‖toLp p (-x + y)‖
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : IsClosed[instTopologicalSpaceProd] ↑g.graph
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ⇑g

Tactic:
let : CompleteSpace g.graph := completeSpace_coe_iff_isComplete.mpr hg.isComplete
  let φ₀ : E →ₗ[𝕜] E × F := LinearMap.id.prod g
  have : Function.LeftInverse Prod.fst φ₀ := fun x => rfl
  let φ : E ≃ₗ[𝕜] g.graph :=
    (LinearEquiv.ofLeftInverse this).trans (LinearEquiv.ofEq _ _ g.graph_eq_range_prod.symm)
  let ψ : g.graph ≃L[𝕜] E :=
    φ.symm.toContinuousLinearEquivOfContinuous continuous_subtype_val.fst
  exact (continuous_subtype_val.comp ψ.symm.continuous).snd

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

Goal state before:
pp: a : ℝ
⊢ volume (Ici a) = ∞

Tactic:
by rw [← measure_congr Ioi_ae_eq_Ici]; simp

Goal state after: