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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
hBU : ⋃ n, F ∩ s n = F
⊢ ∃ K ⊆ F, p K ∧ r < μ K

Tactic:
have : μ F = ⨆ n, μ (F ∩ s n) := by
    rw [← (monotone_const.inter hmono).measure_iUnion, hBU]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
⊢ ∃ K ⊆ F, p K ∧ r < μ K
Module: Mathlib.Topology.ContinuousMap.CocompactMap

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : CocompactMap α β
f' : α → β
h : f' = ⇑f
⊢ Continuous f'

Tactic:
h

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : CocompactMap α β
f' : α → β
h : f' = ⇑f
⊢ Continuous ⇑f
Module: Mathlib.Topology.ContinuousMap.CocompactMap

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : CocompactMap α β
f' : α → β
h : f' = ⇑f
⊢ Tendsto f' (cocompact α) (cocompact β)

Tactic:
h

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : CocompactMap α β
f' : α → β
h : f' = ⇑f
⊢ Tendsto (⇑f) (cocompact α) (cocompact β)
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
⊢ ∀ (s : Set α),
    MeasurableSet s →
      ∀ (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ s, ∃ U ⊇ s, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ s ≤ μ F + ε ∧ μ U ≤ μ s + ε

Tactic:
apply MeasurableSet.induction_on_open

Goal state after:
case isOpen
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
⊢ ∀ (U : Set α),
    IsOpen[inst✝²] U →
      ∀ (ε : ℝ≥0∞),
        ε ≠ 0 → ∃ F ⊆ U, ∃ U_1 ⊇ U, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U_1 ∧ μ U ≤ μ F + ε ∧ μ U_1 ≤ μ U + ε

case compl
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
⊢ ∀ (t : Set α),
    MeasurableSet t →
      (∀ (ε : ℝ≥0∞), ε ≠ 0 → ∃ F ⊆ t, ∃ U ⊇ t, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ t ≤ μ F + ε ∧ μ U ≤ μ t + ε) →
        ∀ (ε : ℝ≥0∞),
          ε ≠ 0 → ∃ F ⊆ tᶜ, ∃ U ⊇ tᶜ, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ tᶜ ≤ μ F + ε ∧ μ U ≤ μ tᶜ + ε

case iUnion
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
⊢ ∀ (f : ℕ → Set α),
    Pairwise (Function.onFun Disjoint f) →
      (∀ (i : ℕ), MeasurableSet (f i)) →
        (∀ (i : ℕ) (ε : ℝ≥0∞),
            ε ≠ 0 →
              ∃ F ⊆ f i, ∃ U ⊇ f i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (f i) ≤ μ F + ε ∧ μ U ≤ μ (f i) + ε) →
          ∀ (ε : ℝ≥0∞),
            ε ≠ 0 →
              ∃ F ⊆ ⋃ i, f i,
                ∃ U ⊇ ⋃ i, f i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (⋃ i, f i) ≤ μ F + ε ∧ μ U ≤ μ (⋃ i, f i) + ε
Module: Mathlib.MeasureTheory.Group.LIntegral

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → ℝ≥0∞
g : G
⊢ ∫⁻ (x : G), f (g * x⁻¹) ∂μ = ∫⁻ (x : G), f x ∂μ

Tactic:
lintegral_inv_eq_self (f <| g * ·),

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → ℝ≥0∞
g : G
⊢ ∫⁻ (x : G), f (g * x) ∂μ = ∫⁻ (x : G), f x ∂μ
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
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
⊢ EventuallyConst (Inv.inv ⁻¹' s) (ae μ) ↔ EventuallyConst s (ae μ)

Tactic:
eventuallyConst_preimage,

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
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
⊢ EventuallyConst s (Filter.map Inv.inv (ae μ)) ↔ EventuallyConst s (ae μ)
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
a : α
ha : f a = g a
h : {x | f x = 0} a
⊢ {x | g x = 0} a

Tactic:
h

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
a : α
ha : 0 = g a
h : {x | f x = 0} a
⊢ {x | g x = 0} a
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
a : α
ha : f a = g a
h : {x | g x = 0} a
⊢ {x | f x = 0} a

Tactic:
h

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
a : α
ha : f a = 0
h : {x | g x = 0} a
⊢ {x | f x = 0} a
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case add
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
g : α → ℝ≥0∞
⊢ ∀ ⦃g_1 : α → ℝ≥0∞⦄,
    Disjoint (Function.support g) (Function.support g_1) →
      Measurable g →
        Measurable g_1 →
          ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ →
            ∫⁻ (a : α), g_1 a ∂μ.withDensity f = ∫⁻ (a : α), (f * g_1) a ∂μ →
              ∫⁻ (a : α), (g + g_1) a ∂μ.withDensity f = ∫⁻ (a : α), (f * (g + g_1)) a ∂μ

Tactic:
h

Goal state after:
case add
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
g h : α → ℝ≥0∞
⊢ Disjoint (Function.support g) (Function.support h) →
    Measurable g →
      Measurable h →
        ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ →
          ∫⁻ (a : α), h a ∂μ.withDensity f = ∫⁻ (a : α), (f * h) a ∂μ →
            ∫⁻ (a : α), (g + h) a ∂μ.withDensity f = ∫⁻ (a : α), (f * (g + h)) a ∂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
h : HasFiniteIntegral f μ
c : ℝ≥0∞
hc : c ≠ ∞
⊢ HasFiniteIntegral f (c • μ)

Tactic:
simp only [HasFiniteIntegral, lintegral_smul_measure] at *
  exact mul_lt_top hc.lt_top h

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (x : α), f x ∂μ ≠ ∞
h : ∀ (x : α), ‖(f x).toReal‖ₑ = ENNReal.ofReal (f x).toReal
⊢ ∫⁻ (a : α), ‖(f a).toReal‖ₑ ∂μ < ∞

Tactic:
h

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (x : α), f x ∂μ ≠ ∞
h : ∀ (x : α), ‖(f x).toReal‖ₑ = ENNReal.ofReal (f x).toReal
⊢ ∫⁻ (a : α), ENNReal.ofReal (f a).toReal ∂μ < ∞
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ≥0
⊢ x * y = (sqrt x * sqrt y) ^ 2

Tactic:
mul_pow,

Goal state after:
x y : ℝ≥0
⊢ x * y = sqrt x ^ 2 * sqrt y ^ 2
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ √x / x = (√x)⁻¹

Tactic:
sqrt_div_self',

Goal state after:
x : ℝ
⊢ 1 / √x = (√x)⁻¹
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
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
a✝ : α
⊢ ‖F 0 a✝‖ ≤ bound a✝ → 0 a✝ ≤ bound a✝

Tactic:
h

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
h✝ : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
a✝ : α
h : ‖F 0 a✝‖ ≤ bound a✝
⊢ 0 a✝ ≤ bound a✝
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f : α → ℝ≥0∞
hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
this : f =ᵐ[μ] fun x ↦ ↑(f x).toNNReal
⊢ SigmaFinite (μ.withDensity f)

Tactic:
rw [withDensity_congr_ae this]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f : α → ℝ≥0∞
hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
this : f =ᵐ[μ] fun x ↦ ↑(f x).toNNReal
⊢ SigmaFinite (μ.withDensity fun x ↦ ↑(f x).toNNReal)
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable α
s : Set α
h : μ = sum fun x ↦ μ (measurableAtom ↑x) • dirac ↑x
⊢ SFinite μ

Tactic:
h

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable α
s : Set α
h : μ = sum fun x ↦ μ (measurableAtom ↑x) • dirac ↑x
⊢ SFinite (sum fun x ↦ μ (measurableAtom ↑x) • dirac ↑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 g : NormedAddGroupHom V₁ V₂
h : f.toAddMonoidHom = g.toAddMonoidHom
⊢ ⇑f.toAddMonoidHom = ⇑g.toAddMonoidHom

Tactic:
h

Goal state after:
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f g : NormedAddGroupHom V₁ V₂
h : f.toAddMonoidHom = g.toAddMonoidHom
⊢ ⇑g.toAddMonoidHom = ⇑g.toAddMonoidHom
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✝ f g : NormedAddGroupHom V₁ V₂
⊢ ∃ C, ∀ (v : V₁), ‖(↑(f.toAddMonoidHom - g.toAddMonoidHom)).toFun v‖ ≤ C * ‖v‖

Tactic:
simp only [AddMonoidHom.toFun_eq_coe, sub_eq_add_neg]

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✝ f g : NormedAddGroupHom V₁ V₂
⊢ ∃ C, ∀ (v : V₁), ‖(f.toAddMonoidHom + -g.toAddMonoidHom) v‖ ≤ C * ‖v‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f : NormedAddGroupHom V₁ V₂
g : NormedAddGroupHom V₂ V₃
C₁ C₂ C₃ : ℝ
h : C₃ = C₂ * C₁
hg : ‖g‖ ≤ C₂
hf : ‖f‖ ≤ C₁
⊢ ‖g.comp f‖ ≤ C₃

Tactic:
h

Goal state after:
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f : NormedAddGroupHom V₁ V₂
g : NormedAddGroupHom V₂ V₃
C₁ C₂ C₃ : ℝ
h : C₃ = C₂ * C₁
hg : ‖g‖ ≤ C₂
hf : ‖f‖ ≤ C₁
⊢ ‖g.comp f‖ ≤ C₂ * C₁
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_3
V₂ : Type u_4
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
v : V₁
⊢ f.toAddMonoidHom v = 0 ↔ f v = 0

Tactic:
coe_toAddMonoidHom

Goal state after:
V₁ : Type u_3
V₂ : Type u_4
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
v : V₁
⊢ f v = 0 ↔ f v = 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
⊢ ‖f‖ ≤ 1

Tactic:
refine opNorm_le_bound _ zero_le_one fun v => ?_
    simpa [one_mul] using h v

Goal state after:
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : StarRing R
inst✝¹ : TrivialStar R
inst✝ : NoZeroDivisors R
a : R
⊢ a ∈ unitary R ↔ a = 1 ∨ a = -1

Tactic:
simp [mem_iff, mul_self_eq_one_iff]

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : AddCommMonoid α
f : β → α
L : SummationFilter β
γ : Type u_5
inst✝³ : DivisionSemiring γ
inst✝² : Module γ α
inst✝¹ : ContinuousConstSMul γ α
inst✝ : T2Space α
g : γ
hg : g ≠ 0
⊢ ∑'[L] (i : β), g • f i = g • ∑'[L] (i : β), f i

Tactic:
exact tsum_const_smul' (Units.mk0 g hg)

Goal state after:
Module: Mathlib.Topology.PartialHomeomorph.Basic

Goal state before:
pp: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
h : e.source = univ
h' : e.target = univ
x : X
⊢ x ∈ e.source

Tactic:
h

Goal state after:
X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
h : e.source = univ
h' : e.target = univ
x : X
⊢ x ∈ univ
Module: Mathlib.Topology.PartialHomeomorph.Basic

Goal state before:
pp: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
h : e.source = univ
h' : e.target = univ
x : Y
⊢ x ∈ e.target

Tactic:
rw [h']
      exact mem_univ _

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

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ z < 0 ↔ ∃ x < 0, ↑x = z

Tactic:
by
  simp_rw [neg_iff (K := K), ext_iff (K := K)]; aesop

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Opens X
hs : Nonempty ↥s
⊢ (s.openPartialHomeomorphSubtypeCoe hs).target = ↑s

Tactic:
by
  simp [openPartialHomeomorphSubtypeCoe]

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

Goal state before:
pp: case refine_2
K : Type u_1
inst✝ : RCLike K
z : K
h : z = ↑‖z‖
⊢ ‖z‖ ≤ re z

Tactic:
h

Goal state after:
case refine_2
K : Type u_1
inst✝ : RCLike K
z : K
h : z = ↑‖z‖
⊢ ‖↑‖z‖‖ ≤ re ↑‖z‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x : K
hx : IsSelfAdjoint x
⊢ 0 ≤ re x ↔ 0 ≤ x

Tactic:
by
  simp [nonneg_iff (K := K), conj_eq_iff_im.mp hx]

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

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z w : K
hz : 0 < z
hw : 0 < w
⊢ 0 < z * w

Tactic:
rw [lt_iff_re_im, map_zero] at hz hw ⊢
    simp [mul_re, mul_im, ← hz.2, ← hw.2, mul_pos hz.1 hw.1]

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

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
h : I = 0
z : K
⊢ im (↑(re z) + ↑(im z) * I) = 0

Tactic:
h

Goal state after:
K : Type u_1
inst✝ : RCLike K
h : I = 0
z : K
⊢ im (↑(re z) + ↑(im z) * 0) = 0
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ ‖z‖ ≤ im z ↔ z = I * ↑‖z‖

Tactic:
by
  obtain (h | h) := I_eq_zero_or_im_I_eq_one (K := K)
  · simp [h, im_eq_zero]
  · have : (I : K) ≠ 0 := fun _ ↦ by simp_all
    rw [← mul_right_inj' (neg_ne_zero.mpr this)]
    convert! norm_le_re_iff_eq_norm (z := -I * z) using 2
    all_goals simp [neg_mul, ← mul_assoc, I_mul_I_of_nonzero this, norm_I_of_ne_zero this]

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

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
hk : NormedField 𝕜
h : IsRCLikeNormedField 𝕜
p : RCLike 𝕜
hp : hk = p.toNormedField
⊢ RCLike 𝕜

Tactic:
exact p.copy_of_normedField hk hp

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
l : Filter α
h : f =o[l] g
hg : ∀ᶠ (x : α) in l, 0 < ‖g x‖
⊢ ∀ᶠ (x : α) in l, ‖f x‖ < ‖g x‖

Tactic:
refine ((h.def (show 0 < 2⁻¹ by simp)).and hg).mono fun x ⟨hx₁, hx₂⟩ ↦ hx₁.trans_lt ?_

Goal state after:
α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
l : Filter α
h : f =o[l] g
hg : ∀ᶠ (x : α) in l, 0 < ‖g x‖
x : α
x✝ : ‖f x‖ ≤ 2⁻¹ * ‖g x‖ ∧ 0 < ‖g x‖
hx₁ : ‖f x‖ ≤ 2⁻¹ * ‖g x‖
hx₂ : 0 < ‖g x‖
⊢ 2⁻¹ * ‖g x‖ < ‖g x‖
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: case succ
α : Type u_1
R : Type u_13
inst✝² : SeminormedRing R
S : Type u_17
inst✝¹ : NormedRing S
inst✝ : NormMulClass S
l : Filter α
f : α → R
g : α → S
h : f =o[l] g
n : ℕ
ihn : (fun x ↦ f x ^ n.succ) =o[l] fun x ↦ g x ^ n.succ
⊢ (fun x ↦ f x ^ (n + 1).succ) =o[l] fun x ↦ g x ^ (n + 1).succ

Tactic:
convert! ihn.mul h

Goal state after:
case e'_7
α : Type u_1
R : Type u_13
inst✝² : SeminormedRing R
S : Type u_17
inst✝¹ : NormedRing S
inst✝ : NormMulClass S
l : Filter α
f : α → R
g : α → S
h : f =o[l] g
n : ℕ
ihn : (fun x ↦ f x ^ n.succ) =o[l] fun x ↦ g x ^ n.succ
x✝ : α
⊢ f x✝ ^ (n + 1).succ = f x✝ ^ n.succ * f x✝

case e'_8
α : Type u_1
R : Type u_13
inst✝² : SeminormedRing R
S : Type u_17
inst✝¹ : NormedRing S
inst✝ : NormMulClass S
l : Filter α
f : α → R
g : α → S
h : f =o[l] g
n : ℕ
ihn : (fun x ↦ f x ^ n.succ) =o[l] fun x ↦ g x ^ n.succ
x✝ : α
⊢ g x✝ ^ (n + 1).succ = g x✝ ^ n.succ * g x✝
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: case mp
α : Type u_1
E' : Type u_6
E'' : Type u_9
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : NormedAddCommGroup E''
f' : α → E'
l : Filter α
c : E''
hc : c ≠ 0
h : (fun _x ↦ c) =O[l] f'
C : ℝ
hC₀ : C > 0
hC : IsBigOWith C l (fun _x ↦ c) f'
⊢ ∀ᶠ (x : α) in l, ‖c‖ / C ≤ ‖f' x‖

Tactic:
exact hC.bound.mono fun x => (div_le_iff₀' hC₀).2

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

Goal state before:
pp: x : ℂ
h : cexp x = 0
⊢ x + -x = cexp x * cexp (-x)

Tactic:
h

Goal state after:
x : ℂ
h : cexp x = 0
⊢ x + -x = 0 * cexp (-x)
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: case hgf
α : Type u_1
𝕜 : Type u_17
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
h : Tendsto (fun x ↦ g x / f x) l atTop
x : α
⊢ 1 ≤ g x / f x → g x = 0 → f x = 0

Tactic:
h

Goal state after:
case hgf
α : Type u_1
𝕜 : Type u_17
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
h✝ : Tendsto (fun x ↦ g x / f x) l atTop
x : α
h : 1 ≤ g x / f x
⊢ g x = 0 → f x = 0
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ sinh x - cosh x = -cexp (-x)

Tactic:
by rw [← neg_sub, cosh_sub_sinh]

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

Goal state before:
pp: x : ℂ
⊢ sinh (x * I) / cos x = tan x * I

Tactic:
sinh_mul_I,

Goal state after:
x : ℂ
⊢ sin x * I / cos x = tan x * I
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x y : ℂ
⊢ cos x * cos y + sinh (x * I) * sinh (y * I) = cos x * cos y - sin x * sin y

Tactic:
sinh_mul_I,

Goal state after:
x y : ℂ
⊢ cos x * cos y + sin x * I * sinh (y * I) = cos x * cos y - sin x * sin y
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x y : ℂ
⊢ cos x * cos y + sin x * I * sinh (y * I) = cos x * cos y - sin x * sin y

Tactic:
sinh_mul_I,

Goal state after:
x y : ℂ
⊢ cos x * cos y + sin x * I * (sin y * I) = cos x * cos y - sin x * sin y
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x : ℂ
n : ℕ
hx : ‖x‖ / ↑n.succ ≤ 1 / 2
j : ℕ
hj : j ≥ n
k : ℕ := j - n
⊢ ‖∑ m ∈ range j, x ^ m / ↑m.factorial - ∑ m ∈ range n, x ^ m / ↑m.factorial‖ ≤ ‖x‖ ^ n / ↑n.factorial * 2

Tactic:
have hj : j = n + k := (add_tsub_cancel_of_le hj).symm

Goal state after:
x : ℂ
n : ℕ
hx : ‖x‖ / ↑n.succ ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ ‖∑ m ∈ range j, x ^ m / ↑m.factorial - ∑ m ∈ range n, x ^ m / ↑m.factorial‖ ≤ ‖x‖ ^ n / ↑n.factorial * 2
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ (starRingEnd ℂ) (-sinh (x * I)) = (starRingEnd ℂ) (sin x * -I)

Tactic:
sinh_mul_I,

Goal state after:
x : ℂ
⊢ (starRingEnd ℂ) (-(sin x * I)) = (starRingEnd ℂ) (sin x * -I)
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ cos x + sin x * I = cosh (x * I) + sinh (x * I)

Tactic:
sinh_mul_I,

Goal state after:
x : ℂ
⊢ cos x + sin x * I = cosh (x * I) + sin x * I
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ cos x - sin x * I = cosh (x * I) - sinh (x * I)

Tactic:
sinh_mul_I,

Goal state after:
x : ℂ
⊢ cos x - sin x * I = cosh (x * I) - sin x * I
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ sin x ^ 2 + cos x ^ 2 = cos x ^ 2 - sinh (x * I) ^ 2

Tactic:
sinh_mul_I,

Goal state after:
x : ℂ
⊢ sin x ^ 2 + cos x ^ 2 = cos x ^ 2 - (sin x * I) ^ 2
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ sin x ^ 2 + cos x ^ 2 = cos x ^ 2 - (sin x * I) ^ 2

Tactic:
mul_pow,

Goal state after:
x : ℂ
⊢ sin x ^ 2 + cos x ^ 2 = cos x ^ 2 - sin x ^ 2 * I ^ 2
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: E : Type u_3
E'' : Type u_9
inst✝¹ : Norm E
inst✝ : NormedAddCommGroup E''
f : ℕ → E
g'' : ℕ → E''
h : f =O[atTop] g''
⊢ f =O[cofinite] g''

Tactic:
Nat.cofinite_eq_atTop

Goal state after:
E : Type u_3
E'' : Type u_9
inst✝¹ : Norm E
inst✝ : NormedAddCommGroup E''
f : ℕ → E
g'' : ℕ → E''
h : f =O[atTop] g''
⊢ f =O[atTop] g''