mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.CocompactMap

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : CocompactMap α β
s : Set β
hs : IsCompact s
h's : IsClosed s
t : Set α
ht : IsCompact t
hts : (⇑f ⁻¹' s)ᶜᶜ ⊆ t
⊢ ⇑f ⁻¹' s ⊆ t

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : CocompactMap α β
s : Set β
hs : IsCompact s
h's : IsClosed s
t : Set α
ht : IsCompact t
hts : (⇑f ⁻¹' s)ᶜᶜ ⊆ t
⊢ ⇑f ⁻¹' s ⊆ t
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
f : α × β → ℝ≥0∞
hf : AEMeasurable f (μ.prod ν)
⊢ ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν

Tactic:
← lintegral_prod_swap f

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
f : α × β → ℝ≥0∞
hf : AEMeasurable f (μ.prod ν)
⊢ ∫⁻ (z : β × α), f z.swap ∂ν.prod μ = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : BorelSpace G
μ : Measure G
inst✝³ : Group G
inst✝² : IsTopologicalGroup G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : μ.Regular
hμ : μ ≠ 0
s : Set G
hs : IsOpen[inst✝⁵] s
⊢ μ s ≠ 0 ↔ s.Nonempty

Tactic:
simpa using

Goal state after:
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : BorelSpace G
μ : Measure G
inst✝³ : Group G
inst✝² : IsTopologicalGroup G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : μ.Regular
hμ : μ ≠ 0
s : Set G
hs : IsOpen[inst✝⁵] s
⊢ ¬s = ∅ ↔ s.Nonempty
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z ↦ (z.2 * z.1, z.1⁻¹)
this : (map (fun z ↦ (z.2 * z.1, z.1⁻¹)) (μ.prod μ)) (s⁻¹ ×ˢ s⁻¹) = 0
⊢ μ s⁻¹ = 0

Tactic:
simpa using

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z ↦ (z.2 * z.1, z.1⁻¹)
this : (map (fun z ↦ (z.2 * z.1, z.1⁻¹)) (μ.prod μ)) (s⁻¹ ×ˢ s⁻¹) = 0
⊢ μ s⁻¹ = 0
Module: Mathlib.MeasureTheory.Group.LIntegral

Goal state before:
pp: G : Type u_1
inst✝³ : MeasurableSpace G
μ : Measure G
inst✝² : InvolutiveInv G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → ℝ≥0∞
⊢ ∫⁻ (x : G), f x⁻¹ ∂μ = ∫⁻ (x : G), f x ∂μ

Tactic:
simpa using

Goal state after:
G : Type u_1
inst✝³ : MeasurableSpace G
μ : Measure G
inst✝² : InvolutiveInv G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → ℝ≥0∞
⊢ ∫⁻ (x : G), f x⁻¹ ∂μ = ∫⁻ (x : G), f x ∂μ
Module: Mathlib.Analysis.LConvolution

Goal state before:
pp: G : Type u_1
mG : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : Inv G
f : G → ℝ≥0∞
μ : Measure G
⊢ 0 ⋆ₘₗ[μ] f = 0

Tactic:
ext; simp [mlconvolution]

Goal state after:
Module: Mathlib.Analysis.LConvolution

Goal state before:
pp: G : Type u_1
mG : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : Inv G
f : G → ℝ≥0∞
μ : Measure G
⊢ f ⋆ₘₗ[μ] 0 = 0

Tactic:
ext; simp [mlconvolution]

Goal state after:
Module: Mathlib.Analysis.LConvolution

Goal state before:
pp: case hf
G : Type u_1
mG : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SFinite μ
f g k : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hk : AEMeasurable k μ
x : G
⊢ AEMeasurable (Function.uncurry fun y x_1 ↦ f y * (g (y⁻¹ * x_1) * k ((y⁻¹ * x_1)⁻¹ * (y⁻¹ * x)))) (μ.prod μ)

Tactic:
simpa using

Goal state after:
case hf
G : Type u_1
mG : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SFinite μ
f g k : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hk : AEMeasurable k μ
x : G
⊢ AEMeasurable (Function.uncurry fun y x_1 ↦ f y * (g (y⁻¹ * x_1) * k (x_1⁻¹ * x))) (μ.prod μ)
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ≥0∞
s : Set α
⊢ (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ

Tactic:
apply le_antisymm ?_ (withDensity_apply_le f s)
  let t := toMeasurable μ s
  calc
  μ.withDensity f s ≤ μ.withDensity f t := measure_mono (subset_toMeasurable μ s)
  _ = ∫⁻ a in t, f a ∂μ := withDensity_apply f (measurableSet_toMeasurable μ s)
  _ = ∫⁻ a in s, f a ∂μ := by congr 1; exact restrict_toMeasurable_of_sFinite s

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hg : Measurable g
⊢ μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g

Tactic:
simpa using

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hg : Measurable g
⊢ μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
⊢ (μ.withDensity (r • f)) s = (r • μ.withDensity f) s

Tactic:
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
    smul_eq_mul, ← lintegral_const_mul r hf]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case a
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
⊢ ∀ (c : ℝ≥0∞), μ K < c → ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen[inst✝⁴] U), μ U < c

Tactic:
simpa using

Goal state after:
case a
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
⊢ ∀ (c : ℝ≥0∞), μ K < c → ∃ i, K ⊆ i ∧ IsOpen[inst✝⁴] i ∧ μ i < c
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t✝ ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
t : {x | f x = 0} =ᵐ[μ.withDensity f] {x | g x = 0}
⊢ (μ.withDensity f) {x | f x = 0} = 0

Tactic:
rw [measure_congr t, withDensity_congr_ae hfg]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t✝ ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
t : {x | f x = 0} =ᵐ[μ.withDensity f] {x | g x = 0}
⊢ (μ.withDensity g) {x | g x = 0} = 0
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f' x ≠ 0}
a : α
ha : ↑(f a) ≠ 0 → g a = g' a
h'a : f a = f' a
h_a_nonneg : f' a ≠ 0
⊢ ↑(f' a) ≠ 0

Tactic:
simpa using

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f' x ≠ 0}
a : α
ha : ↑(f a) ≠ 0 → g a = g' a
h'a : f a = f' a
h_a_nonneg : f' a ≠ 0
⊢ ¬f' a = 0
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
a : α
⊢ (dirac a).withDensity f = f a • dirac a

Tactic:
ext s hs
  classical
  simp [withDensity_apply f hs, setLIntegral_dirac, Set.indicator]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
ε : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
c : ε
hc : ‖c‖ₑ ≠ ∞
⊢ HasFiniteIntegral (fun x ↦ c) μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ

Tactic:
simpa using

Goal state after:
α : Type u_1
ε : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
c : ε
hc : ‖c‖ₑ ≠ ∞
⊢ (¬‖c‖ₑ = 0 → ¬μ univ = ∞) → ‖c‖ₑ = ∞ → μ = 0
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hfi : HasFiniteIntegral f μ
⊢ HasFiniteIntegral (-f) μ

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hfi : HasFiniteIntegral f μ
⊢ ∫⁻ (a : α), ‖f a‖ₑ ∂μ < ∞
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
ε : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hfi : HasFiniteIntegral f μ
⊢ HasFiniteIntegral (fun x ↦ ‖f x‖ₑ) μ

Tactic:
simpa using

Goal state after:
α : Type u_1
ε : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hfi : HasFiniteIntegral f μ
⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂μ < ∞
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hfi : HasFiniteIntegral f μ
⊢ HasFiniteIntegral (fun a ↦ ‖f a‖) μ

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hfi : HasFiniteIntegral f μ
⊢ ∫⁻ (a : α), ‖f a‖ₑ ∂μ < ∞
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: case mp
x y : ℝ
h : 0 ≤ y
⊢ x ^ 2 ≤ y → -√y ≤ x ∧ x ≤ √y

Tactic:
simpa using

Goal state after:
case mp
x y : ℝ
h : 0 ≤ y
⊢ x ^ 2 ≤ y → -√y ≤ x ∧ x ≤ √y
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
h : 0 ≤ x
⊢ √x = 0 ↔ x = 0

Tactic:
simpa using

Goal state after:
x : ℝ
h : 0 ≤ x
⊢ √x = 0 ↔ x = 0
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ

Tactic:
by
  rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul₀' hf.restrict]
  rw [← restrict_withDensity hs]
  exact hg.restrict

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 (ENNReal.ofReal 0))

Tactic:
refine h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp ?_

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
a : α
h : Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ Tendsto (fun n ↦ ‖F n a - f a‖) atTop (𝓝 0)
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
h : -1 ≤ x
⊢ √(1 + x) ≤ 1 + x / 2

Tactic:
by
  refine sqrt_le_iff.mpr ⟨by linarith, ?_⟩
  calc 1 + x
    _ ≤ 1 + x + (x / 2) ^ 2 := le_add_of_nonneg_right <| sq_nonneg _
    _ = _ := by ring

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: ι : Type u_2
s : Finset ι
f g : ι → ℝ≥0
⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)

Tactic:
simpa using

Goal state after:
ι : Type u_2
s : Finset ι
f g : ι → ℝ≥0
⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: ι : Type u_2
s : Finset ι
f g : ι → ℝ≥0
⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)

Tactic:
simpa [*] using sum_mul_le_sqrt_mul_sqrt _ (fun x ↦ sqrt (f x)) (fun x ↦ sqrt (g x))

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: ι : Type u_2
f g : ι → ℝ
s : Finset ι
hf : ∀ (i : ι), 0 ≤ f i
hg : ∀ (i : ι), 0 ≤ g i
⊢ ∑ i ∈ s, √(f i) * √(g i) ≤ √(∑ i ∈ s, f i) * √(∑ i ∈ s, g i)

Tactic:
simpa using

Goal state after:
ι : Type u_2
f g : ι → ℝ
s : Finset ι
hf : ∀ (i : ι), 0 ≤ f i
hg : ∀ (i : ι), 0 ≤ g i
⊢ ∑ i ∈ s, √(f i) * √(g i) ≤ √(∑ i ∈ s, f i) * √(∑ i ∈ s, g i)
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z : ℂ
⊢ ‖z‖ ≤ |z.re| + |z.im|

Tactic:
simpa using

Goal state after:
z : ℂ
⊢ ‖z‖ ≤ |z.re| + |z.im|
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z : ℂ
⊢ |z.im| < ‖z‖ ↔ z.re ≠ 0

Tactic:
simpa using

Goal state after:
z : ℂ
⊢ |z.im| < ‖z‖ ↔ ¬z.re = 0
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: f : CauSeq ℂ fun x ↦ ‖x‖
x✝ : ℝ
ε0 : x✝ > 0
i : ℕ
H : ∀ j ≥ i, ‖↑f j - ↑f i‖ < x✝
j : ℕ
ij : j ≥ i
⊢ |(fun n ↦ (↑f n).re) j - (fun n ↦ (↑f n).re) i| ≤ ‖↑f j - ↑f i‖

Tactic:
simpa using

Goal state after:
f : CauSeq ℂ fun x ↦ ‖x‖
x✝ : ℝ
ε0 : x✝ > 0
i : ℕ
H : ∀ j ≥ i, ‖↑f j - ↑f i‖ < x✝
j : ℕ
ij : j ≥ i
⊢ |(↑f j).re - (↑f i).re| ≤ ‖↑f j - ↑f i‖
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: f : CauSeq ℂ fun x ↦ ‖x‖
ε : ℝ
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, ‖↑f j - ↑f i‖ < ε
j : ℕ
ij : j ≥ i
⊢ |(fun n ↦ (↑f n).im) j - (fun n ↦ (↑f n).im) i| < ε

Tactic:
simpa using

Goal state after:
f : CauSeq ℂ fun x ↦ ‖x‖
ε : ℝ
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, ‖↑f j - ↑f i‖ < ε
j : ℕ
ij : j ≥ i
⊢ |(↑f j).im - (↑f i).im| < ε
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : HasFiniteIntegral f μ
x : α
⊢ ‖min (f x) 0‖ ≤ ‖f x‖

Tactic:
simpa using

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : HasFiniteIntegral f μ
x : α
⊢ -|f x| ≤ f x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ (μ.withDensity f).withDensity f⁻¹ ≤ μ

Tactic:
change (μ.withDensity f).withDensity (fun x ↦ (f x)⁻¹) ≤ μ
  rw [← withDensity_mul₀ hf hf.fun_inv]
  suffices (f * fun x ↦ (f x)⁻¹) ≤ᵐ[μ] 1 by
    refine (withDensity_mono this).trans ?_
    rw [withDensity_one]
  filter_upwards with x
  simp

Goal state after:
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z : ℂ
hz : z ∈ Metric.sphere 0 1
⊢ ‖z‖ = 1

Tactic:
simpa using

Goal state after:
z : ℂ
hz : z ∈ Metric.sphere 0 1
⊢ ‖z‖ = 1
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: x : ℝ
hx : ‖x‖ ≤ 1
⊢ normSq (-↑x + I * ↑√(1 - x ^ 2)) = 1

Tactic:
simpa using

Goal state after:
x : ℝ
hx : ‖x‖ ≤ 1
⊢ normSq (-↑x + I * ↑√(1 - x ^ 2)) = 1
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
𝕜 : Type u_7
inst✝² : NormedRing 𝕜
inst✝¹ : MulActionWithZero 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
f : α → β
c : 𝕜ˣ
h : HasFiniteIntegral (↑c • f) μ
⊢ HasFiniteIntegral f μ

Tactic:
simpa using

Goal state after:
case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
𝕜 : Type u_7
inst✝² : NormedRing 𝕜
inst✝¹ : MulActionWithZero 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
f : α → β
c : 𝕜ˣ
h : HasFiniteIntegral (↑c • f) μ
⊢ HasFiniteIntegral f μ
Module: Mathlib.Analysis.Complex.Order

Goal state before:
pp: z : ℂ
h : z.im = 0
⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0)

Tactic:
simpa using

Goal state after:
z : ℂ
h : z.im = 0
⊢ 0 ≤ z.re ^ 2
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
ε : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
f : α → ε
h : HasFiniteIntegral f μ
s : Set α
⊢ ∫⁻ (a : α) in s, ‖f a‖ₑ ∂μ ≤ ∫⁻ (a : α), ‖f a‖ₑ ∂μ

Tactic:
simpa using

Goal state after:
α : Type u_1
ε : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
f : α → ε
h : HasFiniteIntegral f μ
s : Set α
⊢ ∫⁻ (a : α) in s, ‖f a‖ₑ ∂μ ≤ ∫⁻ (a : α), ‖f a‖ₑ ∂μ
Module: Mathlib.Analysis.Complex.Order

Goal state before:
pp: z : ℂ
h : z.re = 0
⊢ z.re ^ 2 - z.im ^ 2 ≤ 0 ∧ (z.re = 0 ∨ z.im = 0)

Tactic:
simpa using

Goal state after:
z : ℂ
h : z.re = 0
⊢ 0 ≤ z.im ^ 2
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: m : ℤ
⊢ I ^ (4 * (m / 4) + m % 4) = I ^ (m % 4)

Tactic:
simp [zpow_add₀, zpow_mul, zpow_ofNat]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f✝ g : NormedAddGroupHom V₁ V₂
f : V₁ →+ V₂
K : ℝ≥0
h : LipschitzWith K ⇑f
x : V₁
⊢ ‖f x‖ ≤ ↑K * ‖x‖

Tactic:
simpa using

Goal state after:
V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f✝ g : NormedAddGroupHom V₁ V₂
f : V₁ →+ V₂
K : ℝ≥0
h : LipschitzWith K ⇑f
x : V₁
⊢ ‖f x‖ ≤ ↑K * ‖x‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : ℝ≥0
h : ∀ (x : V₁), ‖x‖ ≤ ↑K * ‖f x‖
x y : V₁
⊢ dist x y ≤ ↑K * dist (f x) (f y)

Tactic:
simpa using

Goal state after:
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : ℝ≥0
h : ∀ (x : V₁), ‖x‖ ≤ ↑K * ‖f x‖
x y : V₁
⊢ ‖x - y‖ ≤ ↑K * ‖f x - f y‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case pos
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ‖f g‖ = 0
⊢ ‖g‖ ≤ C' * ‖f g‖

Tactic:
simpa using

Goal state after:
case pos
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ‖f g‖ = 0
⊢ ‖g‖ ≤ 0
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : ℝ≥0
hf : LipschitzWith K ⇑f
x : V₁
⊢ ‖f x‖ ≤ ↑K * ‖x‖

Tactic:
simpa using

Goal state after:
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : ℝ≥0
hf : LipschitzWith K ⇑f
x : V₁
⊢ ‖f x‖ ≤ ↑K * ‖x‖
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : StarAddMonoid E
inst✝ : NormedStarGroup E
x : E
⊢ ‖x‖ ≤ ‖x⋆‖

Tactic:
simpa using

Goal state after:
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : StarAddMonoid E
inst✝ : NormedStarGroup E
x : E
⊢ ‖x‖ ≤ ‖x⋆‖
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : StarAddMonoid E
inst✝ : NormedStarGroup E
x : E
r : ℝ
⊢ (ball x r)⋆ = ball x⋆ r

Tactic:
simpa using

Goal state after:
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : StarAddMonoid E
inst✝ : NormedStarGroup E
x : E
r : ℝ
⊢ (ball x r)⋆ = ball x⋆ r
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : StarAddMonoid E
inst✝ : NormedStarGroup E
x : E
r : ℝ
⊢ (closedBall x r)⋆ = closedBall x⋆ r

Tactic:
simpa using

Goal state after:
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : StarAddMonoid E
inst✝ : NormedStarGroup E
x : E
r : ℝ
⊢ (closedBall x r)⋆ = closedBall x⋆ r
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : StarAddMonoid E
inst✝ : NormedStarGroup E
x : E
r : ℝ
⊢ (sphere x r)⋆ = sphere x⋆ r

Tactic:
simpa using

Goal state after:
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : StarAddMonoid E
inst✝ : NormedStarGroup E
x : E
r : ℝ
⊢ (sphere x r)⋆ = sphere x⋆ r
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: case inr
E : Type u_2
inst✝¹ : NonUnitalNormedRing E
inst✝ : StarRing E
h : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖

Tactic:
simpa using

Goal state after:
case inr
E : Type u_2
inst✝¹ : NonUnitalNormedRing E
inst✝ : StarRing E
h : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖

Tactic:
simpa using

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : IsSelfAdjoint x
⊢ ‖x * x‖ = ‖x‖ ^ 2

Tactic:
simpa using

Goal state after:
E : Type u_2
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : IsSelfAdjoint x
⊢ ‖x * x‖ = ‖x‖ * ‖x‖
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
⊢ x * x⋆ = 0 ↔ x = 0

Tactic:
simpa using

Goal state after:
E : Type u_2
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
⊢ x * x⋆ = 0 ↔ x = 0
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case refine_1
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : f.NormNoninc
v : V
⊢ ‖f v‖ ≤ 1 * ‖v‖

Tactic:
simpa using

Goal state after:
case refine_1
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : f.NormNoninc
v : V
⊢ ‖f v‖ ≤ ‖v‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case refine_2
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : ‖f‖ ≤ 1
v : V
⊢ ‖f v‖ ≤ ‖v‖

Tactic:
simpa using

Goal state after:
case refine_2
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : ‖f‖ ≤ 1
v : V
⊢ ‖f v‖ ≤ ‖v‖