mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hf : AEStronglyMeasurable f μ
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i) μ
hfs : Tendsto (fun i ↦ ∫⁻ (x : α), ‖fs i x - f x‖ₑ ∂μ) l (𝓝 0)
⊢ Tendsto (fun i ↦ setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))

Tactic:
rcases eq_or_neBot l with rfl | hl

Goal state after:
case inl
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hf : AEStronglyMeasurable f μ
fs : ι → α → E
hfsi : ∀ᶠ (i : ι) in ⊥, Integrable (fs i) μ
hfs : Tendsto (fun i ↦ ∫⁻ (x : α), ‖fs i x - f x‖ₑ ∂μ) ⊥ (𝓝 0)
⊢ Tendsto (fun i ↦ setToFun μ T hT (fs i)) ⊥ (𝓝 (setToFun μ T hT f))

case inr
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hf : AEStronglyMeasurable f μ
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i) μ
hfs : Tendsto (fun i ↦ ∫⁻ (x : α), ‖fs i x - f x‖ₑ ∂μ) l (𝓝 0)
hl : l.NeBot
⊢ Tendsto (fun i ↦ setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: case hbc
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : AddMonoid β
inst✝ : LipschitzAdd β
f g : α →ᵇ β
x✝ : α
C : ℝ
C_nonneg : 0 ≤ ↑(LipschitzAdd.C β)
f₁ g₁ f₂ g₂ : α →ᵇ β
x : α
⊢ dist (f₁ x, g₁ x) (f₂ x, g₂ x) ≤ dist (f₁, g₁) (f₂, g₂)

Tactic:
apply max_le_max <;> exact dist_coe_le_dist x

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
hF : ⊤.FG
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
h : ∀ (x : E), x ∈ F
this : ⊤ = F
⊢ FiniteDimensional 𝕜 E

Tactic:
exact .of_fg_top hF

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : α →ᵇ R
h : f * g = 0
hfg : ∀ (x : α), f x = 0 ∨ g x = 0
hfg' : ∀ (x : α), ‖(f + g) x‖ = max ‖f x‖ ‖g x‖
c : ℝ
hc : 0 ≤ c
⊢ ‖f + g‖ ≤ c ↔ max ‖f‖ ‖g‖ ≤ c

Tactic:
norm_le hc,

Goal state after:
α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : α →ᵇ R
h : f * g = 0
hfg : ∀ (x : α), f x = 0 ∨ g x = 0
hfg' : ∀ (x : α), ‖(f + g) x‖ = max ‖f x‖ ‖g x‖
c : ℝ
hc : 0 ≤ c
⊢ (∀ (x : α), ‖(f + g) x‖ ≤ c) ↔ max ‖f‖ ‖g‖ ≤ c
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : α →ᵇ R
h : f * g = 0
hfg : ∀ (x : α), f x = 0 ∨ g x = 0
hfg' : ∀ (x : α), ‖(f + g) x‖ = max ‖f x‖ ‖g x‖
c : ℝ
hc : 0 ≤ c
⊢ (∀ (x : α), ‖f x‖ ≤ c ∧ ‖g x‖ ≤ c) ↔ ‖f‖ ≤ c ∧ ‖g‖ ≤ c

Tactic:
norm_le hc,

Goal state after:
α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : α →ᵇ R
h : f * g = 0
hfg : ∀ (x : α), f x = 0 ∨ g x = 0
hfg' : ∀ (x : α), ‖(f + g) x‖ = max ‖f x‖ ‖g x‖
c : ℝ
hc : 0 ≤ c
⊢ (∀ (x : α), ‖f x‖ ≤ c ∧ ‖g x‖ ≤ c) ↔ (∀ (x : α), ‖f x‖ ≤ c) ∧ ∀ (x : α), ‖g x‖ ≤ c
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
R : Type u_1
inst✝ : SeminormedRing R
f : α →ᵇ R
n : ℕ
⊢ ⇑(npowRec (n + 1) f) = ⇑f ^ (n + 1)

Tactic:
rw [npowRec, pow_succ, coe_mul, coe_npowRec f n]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
m : MeasurableSpace α
hE : CompleteSpace E
μ : Measure α
f : α → E
hf : StronglyMeasurable f
a : α
⊢ ∫ (a : α) in {a}, f a ∂μ = μ.real {a} • f a

Tactic:
by
  simp only [Measure.restrict_singleton, integral_smul_measure, integral_dirac' f a hf,
    measureReal_def]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompactSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : C(α, β)
⊢ ‖f‖ * ‖f‖ ≤ ‖star f * f‖

Tactic:
by
    rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
      ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]
    intro x
    rw [Real.le_sqrt (norm_nonneg _) (norm_nonneg _), sq, ← CStarRing.norm_star_mul_self]
    exact ContinuousMap.norm_coe_le_norm (star f * f) x

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case h
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1 μ)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1 μ)
hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal
h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=
  fun f' ↦ Integrable.of_measure_le_smul hc' hμ'_le (L1.integrable_coeFn f')
⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε

Tactic:
have :
    eLpNorm (⇑(Integrable.toL1 g (h_int g)) - ⇑(Integrable.toL1 f (h_int f))) 1 μ' =
      eLpNorm (⇑g - ⇑f) 1 μ' :=
    eLpNorm_congr_ae ((Integrable.coeFn_toL1 _).sub (Integrable.coeFn_toL1 _))

Goal state after:
case h
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1 μ)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1 μ)
hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal
h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=
  fun f' ↦ Integrable.of_measure_le_smul hc' hμ'_le (L1.integrable_coeFn f')
this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'
⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace β
hα : TopologicalSpace α
h'α : OrderTopology α
inst✝ : T2Space β
f : α → β
a : α
y : β
h : (𝓝[<] a).NeBot
h' : Tendsto f (𝓝[<] a) (𝓝 y)
h'' : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y)
⊢ (𝓝[<] a).limUnder f = y

Tactic:
exact lim_eq h'

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
a : α
h_lin : Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
⊢ Tendsto (fun n ↦ x (n + k)) atTop l

Tactic:
tendsto_add_atTop_iff_nat

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
a : α
h_lin : Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
⊢ Tendsto x atTop l
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F✝
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
⊢ HasSum (fun n ↦ setToFun μ T hT (F n)) (setToFun μ T hT f)

Tactic:
have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] fun a => ∑' n, bound n a := by
    intro n
    filter_upwards [hb_nonneg, bound_summable]
      with _ ha0 ha_sum using ha_sum.le_tsum _ fun i _ => ha0 i

Goal state after:
α : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F✝
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
⊢ HasSum (fun n ↦ setToFun μ T hT (F n)) (setToFun μ T hT f)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
f✝ : StieltjesFunction R
inst✝ : OrderTopology R
f : R → ℝ
hf : Monotone f
x : R
s : Set ℝ
hs : s ∈ 𝓝 (rightLim f x)
l u : ℝ
hlu : rightLim f x ∈ Ioo l u
lus : Ioo l u ⊆ s
y₀ : R
hy₀ : x < y₀
y : R
xy : x < y
h'y : Ioo x y ⊆ f ⁻¹' Ioo l u
z : R
hz : z ∈ Ico x y
⊢ rightLim f z ∈ s

Tactic:
apply lus

Goal state after:
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
f✝ : StieltjesFunction R
inst✝ : OrderTopology R
f : R → ℝ
hf : Monotone f
x : R
s : Set ℝ
hs : s ∈ 𝓝 (rightLim f x)
l u : ℝ
hlu : rightLim f x ∈ Ioo l u
lus : Ioo l u ⊆ s
y₀ : R
hy₀ : x < y₀
y : R
xy : x < y
h'y : Ioo x y ⊆ f ⁻¹' Ioo l u
z : R
hz : z ∈ Ico x y
⊢ rightLim f z ∈ Ioo l u
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
f : StieltjesFunction R
inst✝ : IsEmpty R
s : Set R
⊢ f.length s = 0

Tactic:
simp only [length, if_pos]

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
A : ContinuousWithinAt (fun r ↦ ↑f r - ↑f a) (Ioi a) a
B : ↑f a - ↑f a < ↑δ
this : (𝓝[>] a).NeBot
⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'

Tactic:
exact (((tendsto_order.1 A).2 _ B).and self_mem_nhdsWithin).exists

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

Goal state before:
pp: case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
haux : StronglyMeasurable f
h'aux : IntegrableOn f t μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
x : X
hx : x ∈ t \ s → f x = 0
h'x : x ∉ k
xs : x ∉ s
⊢ x ∈ t ↔ x ∈ s

Tactic:
· simp only [xs, iff_false]
        intro xt
        exact h'x (hx ⟨xt, xs⟩)

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝⁴ : MeasurableConstSMul G α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : Countable G
β : Type u_6
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → β
hf : ∀ (g : G) (x : α), f (g • x) = f x
⊢ (∀ (g : G), AEStronglyMeasurable f (μ.restrict (g⁻¹ • (g⁻¹⁻¹ • s ∩ t)))) ↔
    ∀ (g : G), AEStronglyMeasurable f (μ.restrict (g⁻¹ • (g • s ∩ t)))

Tactic:
by simp only [inv_inv]

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
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ index (↑K) U ≤ index ↑K ↑K₀ * index (↑K₀) U

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ 0 < index (↑K₀) U

Tactic:
· apply le_index_mul K₀ K hU

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ 0 < index (↑K₀) U
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ENNReal.ofReal (∫ (x : X), g x ∂μ) ≤ μ s

Tactic:
rw [ofReal_integral_eq_lintegral_ofReal g_int (Eventually.of_forall (fun x ↦ le_max_right _ _))]

Goal state after:
case pos
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∫⁻ (x : X), ENNReal.ofReal (g x) ∂μ ≤ μ s
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ re ⟪x, x⟫ * re ⟪y, y⟫ = ‖x‖ * ‖y‖ * (‖x‖ * ‖y‖)

Tactic:
by simp only [inner_self_eq_norm_mul_norm]; ring

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

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
this : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ
⊢ ⟪x + y, x + y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ

Tactic:
simp only [inner_add_add_self, this, add_left_inj]

Goal state after:
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
this : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ
⊢ ⟪x, x⟫_ℝ + ⟪x, y⟫_ℝ + ⟪x, y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: case convert_3
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hz : ∀ (i : ι), v i ≠ 0
ho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
s : Finset ι
g : ι → 𝕜
hg : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
j : ι
_hj : j ∈ s
hji : j ≠ i
⊢ g j * ⟪v i, v j⟫ = 0

Tactic:
ho hji.symm,

Goal state after:
case convert_3
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hz : ∀ (i : ι), v i ≠ 0
ho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
s : Finset ι
g : ι → 𝕜
hg : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
j : ι
_hj : j ∈ s
hji : j ≠ i
⊢ g j * 0 = 0
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖

Tactic:
have h := @norm_add_mul_self ℝ _ _ _ _ x y
  simpa using h

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
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)
eq₁ : wp - wq = a - b
eq₂ : u + u - (wq + wp) = a + b
⊢ ‖u + u - (wq + wp)‖ * ‖u + u - (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = ‖a + b‖ * ‖a + b‖ + ‖a - b‖ * ‖a - b‖

Tactic:
rw [eq₁, eq₂]

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h₀ : x ≠ 0
h₀' : ¬↑‖x‖ = 0
h : (↑‖y‖ / ↑‖x‖) • x = y
⊢ ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖

Tactic:
conv_lhs => rw [← h, inner_smul_right, inner_self_eq_norm_sq_to_K]
    field

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 rw [map_smul, inner_smul_left]

Goal state after:
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
s t : Set E
⊢ (∀ i ∈ s, ∀ i_2 ∈ t, 𝕜 ∙ i ⟂ 𝕜 ∙ i_2) ↔ ∀ ⦃u : E⦄, u ∈ s → ∀ ⦃v : E⦄, v ∈ t → ⟪u, v⟫ = 0

Tactic:
isOrtho_iff_le,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s t : Set E
⊢ (∀ i ∈ s, ∀ i_2 ∈ t, 𝕜 ∙ i ≤ (𝕜 ∙ i_2)ᗮ) ↔ ∀ ⦃u : E⦄, u ∈ s → ∀ ⦃v : E⦄, v ∈ t → ⟪u, v⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
⊢ (∀ (x : E), ⟪T x, x⟫ = 0) ↔ T = 0

Tactic:
simp_rw [LinearMap.ext_iff, zero_apply]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
⊢ (∀ (x : E), ⟪T x, x⟫ = 0) ↔ ∀ (x : E), T x = 0
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x : WithLp 2 (α × β)
⊢ ‖x‖ = √(‖x.fst‖ ^ 2 + ‖x.snd‖ ^ 2)

Tactic:
rw [prod_norm_eq_of_nat 2 (by norm_cast) _, Real.sqrt_eq_rpow]
  norm_cast

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

Goal state before:
pp: α : Type u_1
m n : ℕ
P : Matrix (Fin (m + 1)) (Fin n) α → Prop
⊢ Forall P ↔ ∀ (x : Matrix (Fin (m + 1)) (Fin n) α), P x

Tactic:
by
    simp only [Forall, FinVec.forall_iff, forall_iff]
    exact Iff.symm Fin.forall_fin_succ_pi

Goal state after:
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).fst‖₊ ^ p.toReal + ‖(c • f).snd‖₊ ^ p.toReal ≤
    ‖c‖₊ ^ p.toReal * (‖f.fst‖₊ ^ p.toReal + ‖f.snd‖₊ ^ p.toReal) ^ (p.toReal⁻¹ * p.toReal)

Tactic:
inv_mul_cancel₀ hp0.ne',

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‖₊ ^ p.toReal * (‖f.fst‖₊ ^ p.toReal + ‖f.snd‖₊ ^ p.toReal) ^ 1
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 : NormSMulClass 𝕜 α
inst✝ : NormSMulClass 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‖₊ ^ p.toReal * (‖f.fst‖₊ ^ p.toReal + ‖f.snd‖₊ ^ p.toReal) ^ (p.toReal⁻¹ * p.toReal)

Tactic:
inv_mul_cancel₀ hp0.ne',

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 : NormSMulClass 𝕜 α
inst✝ : NormSMulClass 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‖₊ ^ p.toReal * (‖f.fst‖₊ ^ p.toReal + ‖f.snd‖₊ ^ p.toReal) ^ 1
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)
c : 𝕜
f : PiLp p β
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ∑ i, ‖(c • f).ofLp i‖₊ ^ p.toReal ≤ ‖c‖₊ ^ p.toReal * (∑ i, ‖f.ofLp i‖₊ ^ p.toReal) ^ (p.toReal⁻¹ * p.toReal)

Tactic:
inv_mul_cancel₀ hp0.ne',

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)
c : 𝕜
f : PiLp p β
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ∑ i, ‖(c • f).ofLp i‖₊ ^ p.toReal ≤ ‖c‖₊ ^ p.toReal * (∑ i, ‖f.ofLp i‖₊ ^ p.toReal) ^ 1