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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x ↦ ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
ψ : SimpleFunc α ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
n : ℕ
hn : ∀ (x : α), ψ x ≤ ↑n
this : ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ ≠ ∞
hsm : MeasurableSet (toMeasurable μ s)
⊢ ∫⁻ (a : α) in toMeasurable μ s, min (f a) ↑n ∂μ + ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ ≤
    ∫⁻ (a : α) in toMeasurable μ s, g n a ∂μ + ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ

Tactic:
lintegral_add_compl _ hsm,

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x ↦ ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
ψ : SimpleFunc α ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
n : ℕ
hn : ∀ (x : α), ψ x ≤ ↑n
this : ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ ≠ ∞
hsm : MeasurableSet (toMeasurable μ s)
⊢ ∫⁻ (x : α), min (f x) ↑n ∂μ ≤
    ∫⁻ (a : α) in toMeasurable μ s, g n a ∂μ + ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε

Tactic:
by
  /- Let `s` be a covering of `α` by pairwise disjoint measurable sets of finite measure. Let
    `δ : ℕ → ℝ≥0` be a positive function such that `∑' i, μ (s i) * δ i < ε`. Then the function that
     is equal to `δ n` on `s n` is a positive function with integral less than `ε`. -/
  set s : ℕ → Set α := disjointed (spanningSets μ)
  have : ∀ n, μ (s n) < ∞ := fun n =>
    (measure_mono <| disjointed_subset _ _).trans_lt (measure_spanningSets_lt_top μ n)
  obtain ⟨δ, δpos, δsum⟩ : ∃ δ : ℕ → ℝ≥0, (∀ i, 0 < δ i) ∧ (∑' i, μ (s i) * δ i) < ε :=
    ENNReal.exists_pos_tsum_mul_lt_of_countable ε0 _ fun n => (this n).ne
  set N : α → ℕ := spanningSetsIndex μ
  have hN_meas : Measurable N := measurableSet_spanningSetsIndex μ
  have hNs : ∀ n, N ⁻¹' {n} = s n := preimage_spanningSetsIndex_singleton μ
  refine ⟨δ ∘ N, fun x => δpos _, measurable_from_nat.comp hN_meas, ?_⟩
  simp_rw [Function.comp_apply, ← Function.comp_apply (f := (fun n ↦ (↑(δ n) : ℝ≥0∞))),
    lintegral_comp measurable_from_nat.coe_nnreal_ennreal hN_meas]
  simpa [N, hNs, lintegral_countable', measurableSet_spanningSetsIndex, mul_comm] using δsum

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : Measurable f
μ : Measure (Measure α)
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (a : Measure α), (map f a) s ∂μ = ∫⁻ (μ : Measure α), μ (f ⁻¹' s) ∂μ

Tactic:
map_apply hf hs

Goal state after:
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:
simp_all [ε, ← IsClosed.notMem_iff_infDist_pos, mem_interior_iff_mem_nhds]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
M : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)
u v : Set α
u_open : IsOpen[inst✝³] u
v_open : IsOpen[inst✝³] v
xu : f x ∈ u
yv : g y ∈ v
huv : Disjoint u v
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
εx : ℝ
εxpos : εx > 0
hεx : ball x εx ⊆ f ⁻¹' u
εy : ℝ
εypos : εy > 0
hεy : ball y εy ⊆ g ⁻¹' v
n : ℕ
hn : (1 / 2) ^ n < min εx εy
⊢ f '' cylinder x n ⊆ u

Tactic:
rw [image_subset_iff]
      apply Subset.trans _ hεx
      intro z hz
      rw [mem_cylinder_iff_dist_le] at hz
      exact hz.trans_lt (hn.trans_le (min_le_left _ _))

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

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝³ : Group G
inst✝² : MulAction G A
inst✝¹ : MeasurableSpace A
inst✝ : MeasurableConstSMul G A
μ✝ ν : Measure A
g✝ : G
g g' : Gᵈᵐᵃ
μ : Measure A
⊢ map (fun x ↦ DomMulAct.mk.symm (g * g')⁻¹ • x) μ =
    map (fun x ↦ DomMulAct.mk.symm g⁻¹ • x) (map (fun x ↦ DomMulAct.mk.symm g'⁻¹ • x) μ)

Tactic:
rw [map_map]
    · simp [Function.comp_def, mul_smul]
    · exact measurable_const_smul ..
    · exact measurable_const_smul ..

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
this : range f = ⋂ n, F n
⊢ ∀ (s : ↑b), MeasurableSet (E s)

Tactic:
intro b

Goal state after:
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b✝ : Set (Set γ)
b_count : b✝.Countable
b_nonempty : ∅ ∉ b✝
hb : IsTopologicalBasis b✝
this✝ : Encodable ↑b✝
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b✝ → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
this : range f = ⋂ n, F n
b : ↑b✝
⊢ MeasurableSet (E b)
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:
rw [← 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.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
hst : IsComplement S T
a : G
ha : a ∈ S
b : G
hb : b ∈ S
hab : a ≠ b
⊢ ∀ ⦃a_1 : G⦄, a_1 ∈ (fun x ↦ x • T) a → ∀ ⦃b_1 : G⦄, b_1 ∈ (fun x ↦ x • T) b → a_1 ≠ b_1

Tactic:
rintro _ ⟨c, hc, rfl⟩ _ ⟨d, hd, rfl⟩

Goal state after:
G : Type u_1
inst✝ : Group G
S T : Set G
hst : IsComplement S T
a : G
ha : a ∈ S
b : G
hb : b ∈ S
hab : a ≠ b
c : G
hc : c ∈ T
d : G
hd : d ∈ T
⊢ (fun x ↦ a • x) c ≠ (fun x ↦ b • x) d
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case inr
ι : Type u_2
γ : Type u_3
β : Type u_5
inst✝⁶ : MeasurableSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : IsCompletelyPseudoMetrizableSpace γ
inst✝³ : SecondCountableTopology γ
inst✝² : MeasurableSpace γ
hγ : OpensMeasurableSpace γ
inst✝¹ : Countable ι
l : Filter ι
inst✝ : l.IsCountablyGenerated
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
hl : l.NeBot
this : UpgradedIsCompletelyPseudoMetrizableSpace γ := upgradeIsCompletelyPseudoMetrizable γ
u : ℕ → Set ι
hu : l.HasAntitoneBasis u
⊢ MeasurableSet {x | ∃ c, Tendsto (fun n ↦ f n x) l (𝓝 c)}

Tactic:
simp_rw [← cauchy_map_iff_exists_tendsto]

Goal state after:
case inr
ι : Type u_2
γ : Type u_3
β : Type u_5
inst✝⁶ : MeasurableSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : IsCompletelyPseudoMetrizableSpace γ
inst✝³ : SecondCountableTopology γ
inst✝² : MeasurableSpace γ
hγ : OpensMeasurableSpace γ
inst✝¹ : Countable ι
l : Filter ι
inst✝ : l.IsCountablyGenerated
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
hl : l.NeBot
this : UpgradedIsCompletelyPseudoMetrizableSpace γ := upgradeIsCompletelyPseudoMetrizable γ
u : ℕ → Set ι
hu : l.HasAntitoneBasis u
⊢ MeasurableSet {x | Cauchy (map (fun n ↦ f n x) l)}
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✝ : SFinite ρ
⊢ SFinite ρ.fst

Tactic:
fst

Goal state after:
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✝ : SFinite ρ
⊢ SFinite (map Prod.fst ρ)
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✝ : IsFiniteMeasure ρ
⊢ IsFiniteMeasure ρ.fst

Tactic:
fst

Goal state after:
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✝ : IsFiniteMeasure ρ
⊢ IsFiniteMeasure (map Prod.fst ρ)
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✝ : IsProbabilityMeasure ν
⊢ (μ.prod ν).fst = μ

Tactic:
ext1 s hs
  rw [fst_apply hs, ← prod_univ, prod_prod, measure_univ, mul_one]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : StandardBorelSpace α
s : Set α
hs : MeasurableSet s
⊢ ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s

Tactic:
by
  let := upgradeStandardBorel α
  obtain ⟨t, hle, ht, s_clopen⟩ := hs.isClopenable
  refine ⟨t, ?_, ht, s_clopen⟩
  constructor
  rw [eq_borel_upgradeStandardBorel α, borel_eq_borel_of_le ht _ hle]
  infer_instance

Goal state after:
Module: Mathlib.Algebra.QuadraticDiscriminant

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NeZero 2
a b c : K
ha : a ≠ 0
h : ∃ s, discrim a b c = s * s
⊢ ∃ x, a * (x * x) + b * x + c = 0

Tactic:
by
  rcases h with ⟨s, hs⟩
  use (-b + s) / (2 * a)
  rw [quadratic_eq_zero_iff ha hs]
  simp

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✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : AddLeftMono β
f g : α → β
hg : HasFiniteIntegral g μ
hnonneg : ∀ᵐ (a : α) ∂μ, 0 ≤ f a
h : ∀ᵐ (a : α) ∂μ, f a ≤ g a
a : α
hn : 0 ≤ f a
ha : f a ≤ g a
⊢ |f a| ≤ |g a|

Tactic:
abs_of_nonneg hn,

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : AddLeftMono β
f g : α → β
hg : HasFiniteIntegral g μ
hnonneg : ∀ᵐ (a : α) ∂μ, 0 ≤ f a
h : ∀ᵐ (a : α) ∂μ, f a ≤ g a
a : α
hn : 0 ≤ f a
ha : f a ≤ g a
⊢ f a ≤ |g a|
Module: Mathlib.MeasureTheory.Function.Egorov

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : PseudoEMetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
δ : ℝ≥0∞
hδ : δ > 0
N : ℕ
hN : (↑N)⁻¹ < δ
n : ι
hn : notConvergentSeqLTIndex ⋯ hf hsm hs hfg N ≤ n
x : α
hx : x ∈ s \ iUnionNotConvergentSeq hε hf hsm hs hfg
⊢ edist (g x) (f n x) ≤ (↑N)⁻¹

Tactic:
have : edist (f n x) (g x) ≤ (N : ℝ≥0∞)⁻¹ :=
    not_lt.mp fun h ↦ hx.2 <| Set.mem_iUnion.2 ⟨N, hx.1, mem_notConvergentSeq_iff.2 ⟨n, hn, h⟩⟩

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : PseudoEMetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
δ : ℝ≥0∞
hδ : δ > 0
N : ℕ
hN : (↑N)⁻¹ < δ
n : ι
hn : notConvergentSeqLTIndex ⋯ hf hsm hs hfg N ≤ n
x : α
hx : x ∈ s \ iUnionNotConvergentSeq hε hf hsm hs hfg
this : edist (f n x) (g x) ≤ (↑N)⁻¹
⊢ edist (g x) (f n x) ≤ (↑N)⁻¹
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: a : ℕ
⊢ ⌊√↑a⌋ = ↑a.sqrt

Tactic:
by
  rw [Int.floor_eq_iff]
  exact ⟨nat_sqrt_le_real_sqrt, real_sqrt_lt_nat_sqrt_succ⟩

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

Goal state before:
pp: s t : Set ℝ
⊢ s ×ℂ t = ∅ ↔ s = ∅ ∨ t = ∅

Tactic:
by
  simp [← not_nonempty_iff_eq_empty, reProdIm_nonempty, -not_and, not_and_or]

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

Goal state before:
pp: case hf
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (Function.uncurry fun a x ↦ ((fun z ↦ f z.1 * g z.2) * fun z ↦ φ (z.1 * z.2)) (a, a⁻¹ * x)) (μ.prod μ)

Tactic:
first | fun_prop | dsimp; fun_prop

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

Goal state before:
pp: case hf
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable ((fun z ↦ f z.1 * g z.2) * fun z ↦ φ (z.1 * z.2)) (μ.prod μ)

Tactic:
first | fun_prop | dsimp; fun_prop

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

Goal state before:
pp: case hf
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (f ⋆ₘₗ[μ] g) μ

Tactic:
first | fun_prop | dsimp; fun_prop

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

Goal state before:
pp: case hg
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable φ μ

Tactic:
first | fun_prop | dsimp; fun_prop

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

Goal state before:
pp: case hf
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (fun z ↦ f z.1 * g z.2) (μ.prod μ)

Tactic:
first | fun_prop | dsimp; fun_prop

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

Goal state before:
pp: case hg
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (fun z ↦ φ (z.1 * z.2)) (μ.prod μ)

Tactic:
first | fun_prop | dsimp; fun_prop

Goal state after:
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 : 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‖
⊢ ∃ g_1, f g_1 = f g ∧ ‖g_1‖ ≤ C' * ‖f g‖

Tactic:
use g, rfl

Goal state after:
case right
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‖
⊢ ‖g‖ ≤ C' * ‖f g‖
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: case neg
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
y : M₂
hf : ¬Summable f L
hf' : ¬Summable (fun z ↦ e (f z)) L
⊢ 0 = y ↔ 0 = e.symm y

Tactic:
refine ⟨?_, fun H ↦ ?_⟩

Goal state after:
case neg.refine_1
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
y : M₂
hf : ¬Summable f L
hf' : ¬Summable (fun z ↦ e (f z)) L
⊢ 0 = y → 0 = e.symm y

case neg.refine_2
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
y : M₂
hf : ¬Summable f L
hf' : ¬Summable (fun z ↦ e (f z)) L
H : 0 = e.symm y
⊢ 0 = y
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : RCLike K
n : ℕ
inst✝ : n.AtLeastTwo
z : K
⊢ re (OfNat.ofNat n * z) = OfNat.ofNat n * re z

Tactic:
← ofReal_ofNat,

Goal state after:
K : Type u_1
inst✝¹ : RCLike K
n : ℕ
inst✝ : n.AtLeastTwo
z : K
⊢ re (↑(OfNat.ofNat n) * z) = OfNat.ofNat n * re z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : RCLike K
n : ℕ
inst✝ : n.AtLeastTwo
z : K
⊢ im (OfNat.ofNat n * z) = OfNat.ofNat n * im z

Tactic:
← ofReal_ofNat,

Goal state after:
K : Type u_1
inst✝¹ : RCLike K
n : ℕ
inst✝ : n.AtLeastTwo
z : K
⊢ im (↑(OfNat.ofNat n) * z) = OfNat.ofNat n * im z
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: case refine_2
A : Type u_1
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Module ℂ A
inst✝ : StarModule ℂ A
a : A
h : 0 ≤ ℜ a ∧ ℑ a = 0
⊢ 0 ≤ a

Tactic:
rw [← realPart_add_I_smul_imaginaryPart a, h.2]

Goal state after:
case refine_2
A : Type u_1
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Module ℂ A
inst✝ : StarModule ℂ A
a : A
h : 0 ≤ ℜ a ∧ ℑ a = 0
⊢ 0 ≤ ↑(ℜ a) + I • ↑0
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
hI : I ≠ 0
⊢ ‖1‖ = 1

Tactic:
norm_one

Goal state after:
K : Type u_1
inst✝ : RCLike K
hI : I ≠ 0
⊢ 1 = 1
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x : K
⊢ ‖2 * ↑(re x)‖ ^ 2 = re (2 * ↑(re x)) ^ 2

Tactic:
← ofReal_ofNat,

Goal state after:
K : Type u_1
inst✝ : RCLike K
x : K
⊢ ‖↑2 * ↑(re x)‖ ^ 2 = re (↑2 * ↑(re x)) ^ 2
Module: Mathlib.Topology.OpenPartialHomeomorph.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
h : IsOpenEmbedding f
inst✝ : Nonempty X
x : Y
hx : x ∈ range f
⊢ ↑(toOpenPartialHomeomorph f h) (↑(toOpenPartialHomeomorph f h).symm x) = x

Tactic:
OpenPartialHomeomorph.right_inv

Goal state after:
X : Type u_1
Y : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
h : IsOpenEmbedding f
inst✝ : Nonempty X
x : Y
hx : x ∈ range f
⊢ x = x

case h
X : Type u_1
Y : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
h : IsOpenEmbedding f
inst✝ : Nonempty X
x : Y
hx : x ∈ range f
⊢ x ∈ (toOpenPartialHomeomorph f h).target
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
⊢ 𝓝 (↑e.symm (↑e x)) = 𝓝 x

Tactic:
rw [e.left_inv 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
f : Z → X
s : Set Z
x : Z
hx : f x ∈ e.source
h : f ⁻¹' e.source ∈ 𝓝[s] x
fe_cont : ContinuousWithinAt (↑e ∘ f) (s ∩ f ⁻¹' e.source) x
this : ContinuousWithinAt (↑e.symm ∘ ↑e ∘ f) (s ∩ f ⁻¹' e.source) x
⊢ f x = (↑e.symm ∘ ↑e ∘ f) x

Tactic:
simp [e.left_inv hx]

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

Goal state before:
pp: α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
h : f =O[l] g
⊢ ∃ c > 0, ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖

Tactic:
rw [isBigO_iff'] at h
    obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
    refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩
    filter_upwards [hc] with x hx
    rwa [inv_mul_le_iff₀ (by positivity)]

Goal state after: