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

Goal state before:
pp: case const.refine_3.h0
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ ↑c ≠ 0 ∨ L ≠ 0

pp: case const.refine_3.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ ↑c ≠ ∞ ∨ L ≠ ∞

Tactic:
· simp only [c_ne_zero, Ne, ENNReal.coe_eq_zero, not_false_iff, true_or]

Goal state after:
case const.refine_3.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
s : Set α
hs : MeasurableSet s
L : ℝ≥0∞
hL : L < ↑c * μ s
c_ne_zero : c ≠ 0
this : L / ↑c < μ s
t : Set α
ht : MeasurableSet t
ts : t ⊆ s
mlt : L / ↑c < μ t
t_top : μ t < ∞
⊢ ↑c ≠ ∞ ∨ L ≠ ∞
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
ε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1
hC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty
this : Nonempty X
⊢ ε / 2 > 0

Tactic:
by simp_all [ε, ← IsClosed.notMem_iff_infDist_pos, mem_interior_iff_mem_nhds]

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

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type ?u.42} {β : Type ?u.41} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
⊢ ∀ ⦃f g : α × β → ℝ≥0∞⦄,
    Disjoint (support f) (support g) →
      Measurable f →
        Measurable g →
          (Measurable fun x ↦ ∫⁻ (y : β), f (x, y) ∂ν) →
            (Measurable fun x ↦ ∫⁻ (y : β), g (x, y) ∂ν) → Measurable fun x ↦ ∫⁻ (y : β), (f + g) (x, y) ∂ν

Tactic:
rintro f g - hf - h2f h2g
    simp only [Pi.add_apply]
    conv => enter [1, x]; erw [lintegral_add_left (hf.comp m)]
    exact h2f.add h2g

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
C : Set (Set α)
D : Set (Set β)
hC : generateFrom C = inst✝¹
hD : generateFrom D = inst✝
h2C : IsPiSystem C
h2D : IsPiSystem D
h3C : μ.FiniteSpanningSetsIn C
h3D : ν.FiniteSpanningSetsIn D
μν : Measure (α × β)
h₁ : ∀ s ∈ C, ∀ t ∈ D, μν (s ×ˢ t) = μ s * ν t
⊢ μ.prod ν = μν

Tactic:
refine
    (h3C.prod h3D).ext
      (generateFrom_eq_prod hC hD h3C.isCountablySpanning h3D.isCountablySpanning).symm
      (h2C.prod h2D) ?_
  rintro _ ⟨s, hs, t, ht, rfl⟩
  have := h3D.sigmaFinite
  rw [h₁ s hs t ht, prod_prod]

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S : Set G
g : G
h : IsComplement S {g}
x : G
x✝ : x ∈ ⊤
y : ↑S × ↑{g}
hy : (fun x ↦ ↑x.1 * ↑x.2) y = x * g
| x * g

Tactic:
← show y.2.1 = g from y.2.2

Goal state after:
G : Type u_1
inst✝ : Group G
S : Set G
g : G
h : IsComplement S {g}
x : G
x✝ : x ∈ ⊤
y : ↑S × ↑{g}
hy : (fun x ↦ ↑x.1 * ↑x.2) y = x * g
| x * ↑y.2
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
τ : Measure γ
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
this :
  (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod (sfiniteSeq τ p.2.2))) =
    sum fun p ↦ (sfiniteSeq μ p.1.1).prod ((sfiniteSeq ν p.1.2).prod (sfiniteSeq τ p.2))
i : (ℕ × ℕ) × ℕ
s : Set α
hs : MeasurableSet s
t : Set β
ht : MeasurableSet t
u : Set γ
hu : MeasurableSet u
⊢ (((sfiniteSeq μ i.1.1).prod (sfiniteSeq ν i.1.2)).prod (sfiniteSeq τ i.2))
      (⇑MeasurableEquiv.prodAssoc ⁻¹' s ×ˢ t ×ˢ u) =
    (sfiniteSeq μ i.1.1) s * ((sfiniteSeq ν i.1.2).prod (sfiniteSeq τ i.2)) (t ×ˢ u)

Tactic:
MeasurableEquiv.prodAssoc,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
τ : Measure γ
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
this :
  (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod (sfiniteSeq τ p.2.2))) =
    sum fun p ↦ (sfiniteSeq μ p.1.1).prod ((sfiniteSeq ν p.1.2).prod (sfiniteSeq τ p.2))
i : (ℕ × ℕ) × ℕ
s : Set α
hs : MeasurableSet s
t : Set β
ht : MeasurableSet t
u : Set γ
hu : MeasurableSet u
⊢ (((sfiniteSeq μ i.1.1).prod (sfiniteSeq ν i.1.2)).prod (sfiniteSeq τ i.2))
      (⇑{ toEquiv := Equiv.prodAssoc α β γ, measurable_toFun := ⋯, measurable_invFun := ⋯ } ⁻¹' s ×ˢ t ×ˢ u) =
    (sfiniteSeq μ i.1.1) s * ((sfiniteSeq ν i.1.2).prod (sfiniteSeq τ i.2)) (t ×ˢ u)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
a : α
ha : map (g a) μc = μd
⊢ SFinite μd

Tactic:
← ha

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
a : α
ha : map (g a) μc = μd
⊢ SFinite (map (g a) μc)
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝³ : MeasurableSpace G
inst✝² : Group G
inst✝¹ : MeasurableMul G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
g : G
⊢ MeasurePreserving (fun x ↦ x / g) μ μ

Tactic:
simp_rw [div_eq_mul_inv, measurePreserving_mul_right μ g⁻¹]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : IsZeroOrProbabilityMeasure ρ
⊢ IsZeroOrProbabilityMeasure ρ.fst

Tactic:
rcases eq_zero_or_isProbabilityMeasure ρ with h | h
  · simp only [h, fst_zero]
    infer_instance
  · infer_instance

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
X : α → β
Y : α → γ
μ : Measure α
hY : AEMeasurable Y μ
hX : AEMeasurable X μ
s : Set β
hs : MeasurableSet s
⊢ μ ((fun x ↦ (X x, Y x)) ⁻¹' Prod.fst ⁻¹' s) = (map X μ) s

Tactic:
Measure.map_apply_of_aemeasurable hX hs,

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
X : α → β
Y : α → γ
μ : Measure α
hY : AEMeasurable Y μ
hX : AEMeasurable X μ
s : Set β
hs : MeasurableSet s
⊢ μ ((fun x ↦ (X x, Y x)) ⁻¹' Prod.fst ⁻¹' s) = μ (X ⁻¹' s)
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : TopologicalSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : μ.IsMulLeftInvariant
K : Set G
hK : IsCompact K
U : Set G
hU : IsOpen[inst✝⁴] U
hne : U.Nonempty
h : μ U = 0
⊢ μ K ≤ 0

Tactic:
rw [← hU.interior_eq] at hne

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : TopologicalSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : μ.IsMulLeftInvariant
K : Set G
hK : IsCompact K
U : Set G
hU : IsOpen[inst✝⁴] U
hne : (interior U).Nonempty
h : μ U = 0
⊢ μ K ≤ 0
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
x✝ : G
⊢ (0 ⋆ₘₗ[μ] f) x✝ = 0 x✝

Tactic:
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
x✝ : G
⊢ (f ⋆ₘₗ[μ] 0) x✝ = 0 x✝

Tactic:
simp [mlconvolution]

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed[inst✝⁷] K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n ↦ (fun T ↦ T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
Lclosed : ∀ (n : ℕ), IsClosed[inst✝⁷] (L n)
⊢ μ univ = ∞

Tactic:
have M : ∀ n, μ (L n) = (n + 1 : ℕ) * μ K := fun n ↦ by
    induction n with
    | zero => simp only [L, one_mul, Nat.cast_one, iterate_zero, id, Nat.zero_add]
    | succ n IH =>
      calc
        μ (L (n + 1)) = μ (L n) + μ (g (L n) • K) := by
          simp_rw [L, iterate_succ']
          exact measure_union' (hg _ (Lcompact _)) (Lclosed _).measurableSet
        _ = (n + 1 + 1 : ℕ) * μ K := by
          simp only [IH, measure_smul, add_mul, Nat.cast_add, Nat.cast_one, one_mul]

Goal state after:
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed[inst✝⁷] K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n ↦ (fun T ↦ T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
Lclosed : ∀ (n : ℕ), IsClosed[inst✝⁷] (L n)
M : ∀ (n : ℕ), μ (L n) = ↑(n + 1) * μ K
⊢ μ univ = ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs₁ : MeasurableSet s
hs₂ : μ s = 0
⊢ (μ.withDensity f) s = 0

Tactic:
rw [withDensity_apply _ hs₁]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs₁ : MeasurableSet s
hs₂ : μ s = 0
⊢ ∫⁻ (a : α) in s, f a ∂μ = 0
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : T2Space α
inst✝ : μ.InnerRegularCompactLTTop
A : Set α
hA : MeasurableSet A
h'A : μ A ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ K ⊆ A, IsCompact K ∧ μ (A \ K) < ε

Tactic:
rcases hA.exists_isCompact_lt_add h'A hε with ⟨K, hKA, hKc, hK⟩
  exact ⟨K, hKA, hKc, measure_sdiff_lt_of_lt_add hKc.nullMeasurableSet hKA
    (ne_top_of_le_ne_top h'A <| measure_mono hKA) hK⟩

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

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

Tactic:
ext1 t ht
  rw [restrict_apply ht, withDensity_apply _ ht, withDensity_apply' _ (t ∩ s),
    restrict_restrict ht]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
p : α → Prop
f g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
a : α
ha : f a = g a
h : g a ≠ 0
⊢ {x | f x ≠ 0} a

Tactic:
← ha

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
p : α → Prop
f g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
a : α
ha : f a = g a
h : f a ≠ 0
⊢ {x | f 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∞
f_meas : Measurable f
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f ≤ ∫⁻ (a : α), (f * g) a ∂μ

Tactic:
← iSup_lintegral_measurable_le_eq_lintegral,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g : α → ℝ≥0∞
⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ g), ∫⁻ (a : α), g_1 a ∂μ.withDensity f ≤ ∫⁻ (a : α), (f * g) a ∂μ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ ∫⁻ (a : α), g a ∂μ.withDensity f

Tactic:
← iSup_lintegral_measurable_le_eq_lintegral,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ ≤ ∫⁻ (a : α), g a ∂μ.withDensity f
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
⊢ |x| ^ 2 < y ↔ |x| < √y

Tactic:
lt_sqrt (abs_nonneg _)

Goal state after:
x y : ℝ
⊢ |x| ^ 2 < y ↔ |x| ^ 2 < y
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z w : ℂ
⊢ √(normSq z / normSq w) = √(normSq z) / √(normSq w)

Tactic:
Real.sqrt_div (normSq_nonneg _)

Goal state after:
z w : ℂ
⊢ √(normSq z) / √(normSq w) = √(normSq z) / √(normSq w)
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z w : ℂ
h : z.re = w.re
⊢ |z.im - w.im| = dist z.im w.im

Tactic:
Real.dist_eq

Goal state after:
z w : ℂ
h : z.re = w.re
⊢ |z.im - w.im| = |z.im - w.im|
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z w : ℂ
h : z.im = w.im
⊢ |z.re - w.re| = dist z.re w.re

Tactic:
Real.dist_eq

Goal state after:
z w : ℂ
h : z.im = w.im
⊢ |z.re - w.re| = |z.re - w.re|
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: f : CauSeq ℂ fun x ↦ ‖x‖
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, (fun x ↦ ‖x‖) (↑f j - ↑f i) < ε
j : ℕ
hj : j ≥ i
⊢ ‖(starRingEnd ℂ) (↑f j) - (starRingEnd ℂ) (↑f i)‖ < ε

Tactic:
← map_sub,

Goal state after:
f : CauSeq ℂ fun x ↦ ‖x‖
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, (fun x ↦ ‖x‖) (↑f j - ↑f i) < ε
j : ℕ
hj : j ≥ i
⊢ ‖(starRingEnd ℂ) (↑f j - ↑f i)‖ < ε
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: z : ℂ
⊢ ↑z.im = (z - (starRingEnd ℂ) z) / (2 * I)

Tactic:
simp only [sub_conj, ofReal_mul, ofReal_ofNat, mul_right_comm,
    mul_div_cancel_left₀ _ (mul_ne_zero two_ne_zero I_ne_zero : 2 * I ≠ 0)]

Goal state after:
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: s t : Set ℝ
⊢ (s ×ℂ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty

Tactic:
by
  simp [Set.Nonempty, reProdIm, Complex.exists]

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

Goal state before:
pp: case neg.mp
ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R₂ M₂
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomInvPair σ' σ
L : SummationFilter ι
inst✝¹ : T2Space M
inst✝ : T2Space M₂
f : ι → M
e : M ≃SL[σ] M₂
hf : Summable f L
hL : ¬L.NeBot
⊢ e (∑ᶠ (z : ι), f z) = ∑ᶠ (z : ι), e (f z)

pp: case neg.mpr
ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R₂ M₂
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomInvPair σ' σ
L : SummationFilter ι
inst✝¹ : T2Space M
inst✝ : T2Space M₂
f : ι → M
e : M ≃SL[σ] M₂
hf : Summable f L
hL : ¬L.NeBot
⊢ ∑ᶠ (z : ι), e (f z) = e (∑ᶠ (z : ι), f z)

Tactic:
exacts [e.map_finsum f, (e.map_finsum f).symm]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
κ : Type u_6
R✝ : Type u_7
R₂ : Type u_8
M✝ : Type u_9
M₂ : Type u_10
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : Group α
inst✝ : MulAction α β
f : β → M
b₁ b₂ : β
a : α
ha : a • b₂ = b₁
⊢ ∑' (a : α), f (a • b₁) = ∑' (a : α), f (a • b₂)

Tactic:
← ha

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
κ : Type u_6
R✝ : Type u_7
R₂ : Type u_8
M✝ : Type u_9
M₂ : Type u_10
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : Group α
inst✝ : MulAction α β
f : β → M
b₁ b₂ : β
a : α
ha : a • b₂ = b₁
⊢ ∑' (a_1 : α), f (a_1 • a • b₂) = ∑' (a : α), f (a • b₂)
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℂ E
x : E
⊢ restrictScalars ℝ (toSpanSingleton ℂ x) = reCLM.smulRight x + I • imCLM.smulRight x

Tactic:
ext ⟨a, b⟩

Goal state after:
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℂ E
x : E
a b : ℝ
⊢ (restrictScalars ℝ (toSpanSingleton ℂ x)) { re := a, im := b } =
    (reCLM.smulRight x + I • imCLM.smulRight x) { re := a, im := b }
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: 𝕜 : Type u_2
𝕜' : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : RCLike 𝕜'
h : RCLike.im RCLike.I = 1
a : 𝕜
⊢ 0 ≤ (RCLike.map 𝕜 𝕜') a ↔ 0 ≤ a

Tactic:
rw [RCLike.nonneg_iff, RCLike.nonneg_iff (K := 𝕜)]

Goal state after:
𝕜 : Type u_2
𝕜' : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : RCLike 𝕜'
h : RCLike.im RCLike.I = 1
a : 𝕜
⊢ 0 ≤ RCLike.re ((RCLike.map 𝕜 𝕜') a) ∧ RCLike.im ((RCLike.map 𝕜 𝕜') a) = 0 ↔ 0 ≤ RCLike.re a ∧ RCLike.im a = 0
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: r : ℝ
s : Set ℂ
hs : s ⊆ sphere 0 r
⊢ (∃ x ∈ s, x ≤ 0) → -↑r ∈ s

Tactic:
rintro ⟨z, hzs, hz⟩

Goal state after:
r : ℝ
s : Set ℂ
hs : s ⊆ sphere 0 r
z : ℂ
hzs : z ∈ s
hz : z ≤ 0
⊢ -↑r ∈ s
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝ : RCLike K
z : K
h₀ : z ≠ 0
⊢ z⁻¹ = (starRingEnd K) z * ↑(‖z‖ ^ 2)⁻¹

Tactic:
· apply inv_eq_of_mul_eq_one_right
    rw [← mul_assoc, mul_conj, ofReal_inv, ofReal_pow, mul_inv_cancel₀]
    simpa

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

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
x : X
hx : x ∈ e.source
s : Set Y
⊢ map (↑e) (𝓝[↑e ⁻¹' s] x) = 𝓝[s] ↑e x

Tactic:
by
  rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.target_inter_inv_preimage_preimage,
    e.nhdsWithin_target_inter (e.map_source hx)]

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

Goal state before:
pp: X : Type u_1
Y : Type u_3
Z : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : OpenPartialHomeomorph X Y
s : Set X
t : Set Z
x : X
f : X → Z
hf : ContinuousWithinAt f s x
hxe : x ∈ e.source
ht : t ∈ 𝓝 (f x)
⊢ ∀ᶠ (x : Y) in 𝓝 (↑e x), x ∈ ↑e.symm ⁻¹' s ↔ x ∈ e.target ∩ ↑e.symm ⁻¹' (s ∩ f ⁻¹' t)

Tactic:
e.eventually_nhds _ hxe

Goal state after:
X : Type u_1
Y : Type u_3
Z : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : OpenPartialHomeomorph X Y
s : Set X
t : Set Z
x : X
f : X → Z
hf : ContinuousWithinAt f s x
hxe : x ∈ e.source
ht : t ∈ 𝓝 (f x)
⊢ ∀ᶠ (x : X) in 𝓝 x, ↑e x ∈ ↑e.symm ⁻¹' s ↔ ↑e x ∈ e.target ∩ ↑e.symm ⁻¹' (s ∩ f ⁻¹' t)
Module: Mathlib.Topology.OpenPartialHomeomorph.Continuity

Goal state before:
pp: X : Type u_1
Y : Type u_3
Z : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : OpenPartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
x : Y
hx : x ∈ s
⊢ ContinuousWithinAt (f ∘ ↑e) (↑e ⁻¹' s) (↑e.symm x) ↔ ContinuousWithinAt (f ∘ ↑e) (↑e.symm '' s) (↑e.symm x)

Tactic:
e.symm_image_eq_source_inter_preimage h,

Goal state after:
X : Type u_1
Y : Type u_3
Z : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : OpenPartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
x : Y
hx : x ∈ s
⊢ ContinuousWithinAt (f ∘ ↑e) (↑e ⁻¹' s) (↑e.symm x) ↔ ContinuousWithinAt (f ∘ ↑e) (e.source ∩ ↑e ⁻¹' s) (↑e.symm x)
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
c : ℝ
f : α → E
g : α → F
l : Filter α
hc : 0 < c
⊢ IsBigOWith c⁻¹ l f g ↔ ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖

Tactic:
simp only [IsBigOWith_def, ← div_eq_inv_mul, le_div_iff₀' hc]

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

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
R₃ : Type u_3
R₄ : Type u_4
E : Type u_5
E₂ : Type u_6
E₃ : Type u_7
E₄ : Type u_8
F : Type u_9
𝓕 : Type u_10
inst✝³⁶ : Semiring R
inst✝³⁵ : Semiring R₂
inst✝³⁴ : Semiring R₃
inst✝³³ : Semiring R₄
σ₁₂ : R →+* R₂
σ₂₁ : R₂ →+* R
σ₁₃ : R →+* R₃
σ₃₁ : R₃ →+* R
σ₁₄ : R →+* R₄
σ₄₁ : R₄ →+* R
σ₂₃ : R₂ →+* R₃
σ₃₂ : R₃ →+* R₂
σ₂₄ : R₂ →+* R₄
σ₄₂ : R₄ →+* R₂
σ₃₄ : R₃ →+* R₄
σ₄₃ : R₄ →+* R₃
inst✝³² : RingHomInvPair σ₁₂ σ₂₁
inst✝³¹ : RingHomInvPair σ₂₁ σ₁₂
inst✝³⁰ : RingHomInvPair σ₁₃ σ₃₁
inst✝²⁹ : RingHomInvPair σ₃₁ σ₁₃
inst✝²⁸ : RingHomInvPair σ₂₃ σ₃₂
inst✝²⁷ : RingHomInvPair σ₃₂ σ₂₃
inst✝²⁶ : RingHomInvPair σ₁₄ σ₄₁
inst✝²⁵ : RingHomInvPair σ₄₁ σ₁₄
inst✝²⁴ : RingHomInvPair σ₂₄ σ₄₂
inst✝²³ : RingHomInvPair σ₄₂ σ₂₄
inst✝²² : RingHomInvPair σ₃₄ σ₄₃
inst✝²¹ : RingHomInvPair σ₄₃ σ₃₄
inst✝²⁰ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁸ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁷ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁
inst✝¹⁵ : RingHomCompTriple σ₄₂ σ₂₁ σ₄₁
inst✝¹⁴ : RingHomCompTriple σ₄₃ σ₃₂ σ₄₂
inst✝¹³ : RingHomCompTriple σ₄₃ σ₃₁ σ₄₁
inst✝¹² : SeminormedAddCommGroup E
inst✝¹¹ : SeminormedAddCommGroup E₂
inst✝¹⁰ : SeminormedAddCommGroup E₃
inst✝⁹ : SeminormedAddCommGroup E₄
inst✝⁸ : Module R E
inst✝⁷ : Module R₂ E₂
inst✝⁶ : Module R₃ E₃
inst✝⁵ : Module R₄ E₄
inst✝⁴ : NormedAddCommGroup F
inst✝³ : Module R F
e : E ≃ₛₗᵢ[σ₁₂] E₂
α : Type u_11
inst✝² : TopologicalSpace α
inst✝¹ : Module R E₂
inst✝ : Module R E₃
⊢ ∀ (x : (E × E₂) × E₃), ‖(LinearEquiv.prodAssoc R E E₂ E₃) x‖ = ‖x‖

Tactic:
by
      rintro ⟨⟨e, f⟩, g⟩
      simp only [LinearEquiv.prodAssoc_apply, AddEquiv.toEquiv_eq_coe,
        Equiv.toFun_as_coe, EquivLike.coe_coe, AddEquiv.coe_prodAssoc,
        Equiv.prodAssoc_apply, Prod.norm_def, max_assoc]

Goal state after: