mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝¹² : Field R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSpace V
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : Module R V
inst✝⁶ : SeparatingDual R V
W : Type u_3
inst✝⁵ : AddCommGroup W
inst✝⁴ : TopologicalSpace W
inst✝³ : Module R W
inst✝² : Nontrivial W
inst✝¹ : ContinuousSMul R W
inst✝ : Nontrivial V
⊢ Nontrivial (V →L[R] W)

Tactic:
by
  obtain ⟨v, hv⟩ := exists_ne (0 : V)
  obtain ⟨w, hw⟩ := exists_ne (0 : W)
  obtain ⟨ψ, hψ⟩ := exists_ne_zero (R := R) hv
  exact ⟨ψ.smulRight w, 0, DFunLike.ne_iff.mpr ⟨v, by simp_all⟩⟩

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
x : α
⊢ -↑(f x) < g x

Tactic:
by simpa only [EReal.coe_neg] using g_lt_f x

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ n ≤ p

Tactic:
by assumption

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ n ≤ q

Tactic:
by assumption

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e

Tactic:
intro e p q e' p' q' hp hq hp' hq' he'

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
⊢ ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
J2 : ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e
J3 : ‖L e' p' r - L e' p' q'‖ ≤ 4 * (1 / 2) ^ e
⊢ ‖L e p q - L e' p' q'‖ = ‖L e p q - L e p r + (L e p r - L e' p' r) + (L e' p' r - L e' p' q')‖

Tactic:
by
        congr 1; abel

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

Goal state before:
pp: X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
hf : AEMeasurable f μ
g : X → E
f' : X → ℝ≥0 := AEMeasurable.mk f hf
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f' x)

Tactic:
by
      congr 1
      apply withDensity_congr_ae
      filter_upwards [hf.ae_eq_mk] with x hx
      rw [hx]

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

Goal state before:
pp: case refine_3.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this✝ :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
this : ∀ᵐ (t : ℝ) ∂μ, t < b₀ ∨ b₀ < t
x₀ : ℝ
hx₀ : x₀ < b₀
⊢ ContinuousWithinAt (fun x ↦ {x_1 | x_1 ≤ x}.indicator f x₀) (Icc b₁ b₂) b₀

Tactic:
have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = f x₀ := by
          apply mem_nhdsWithin_of_mem_nhds
          apply Eventually.mono (Ioi_mem_nhds hx₀)
          intro x hx
          simp [hx.le]
        apply continuousWithinAt_const.congr_of_eventuallyEq this
        simp [hx₀.le]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E
c : E
a b : ℝ
hf : IntervalIntegrable f volume a b
hmeas : StronglyMeasurableAtFilter f (𝓝 a) volume
ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)
⊢ HasStrictDerivAt (fun u ↦ ∫ (x : ℝ) in u..b, f x) (-c) a

Tactic:
by
  simpa only [← integral_symm] using
    (integral_hasStrictDerivAt_of_tendsto_ae_right hf.symm hmeas ha).fun_neg

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
⊢ ({t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ioc t v).Nonempty

Tactic:
obtain ⟨y, g'_lt_y', y_lt_G'⟩ : ∃ y : ℝ, (g' t : EReal) < y ∧ (y : EReal) < G' t :=
      EReal.lt_iff_exists_real_btwn.1 ((EReal.coe_le_coe_iff.2 (hφg t ht.2)).trans_lt (f_lt_G' t))

Goal state after:
g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
⊢ ({t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ioc t v).Nonempty
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
μ : Measure ℝ
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : ℝ
φ : ℝ → F →L[𝕜] E
hφ : IntervalIntegrable φ μ a b
v : F
⊢ ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, φ x ∂μ) v =
    (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, (φ x) v ∂μ

Tactic:
← integral_apply hφ.def' v,

Goal state after:
𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
μ : Measure ℝ
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : ℝ
φ : ℝ → F →L[𝕜] E
hφ : IntervalIntegrable φ μ a b
v : F
⊢ ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, φ x ∂μ) v =
    (if a ≤ b then 1 else -1) • (∫ (x : ℝ) in Ι a b, φ x ∂μ) v
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
d : ℝ
A : MeasurableEmbedding fun x ↦ x + d
⊢ ∫ (x : ℝ) in a + d..b + d, f x ∂Measure.map (fun x ↦ x + d) volume = ∫ (x : ℝ) in a + d..b + d, f x

Tactic:
rw [map_add_right_eq_self]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case inr
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ uIoo a b, DifferentiableAt ℝ f x
hint : IntervalIntegrable (deriv f) volume a b
hab : b ≤ a
⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a

Tactic:
simp only [uIcc_of_ge, hab, uIoo_of_ge] at hcont hderiv

Goal state after:
case inr
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f : ℝ → E
hint : IntervalIntegrable (deriv f) volume a b
hab : b ≤ a
hcont : ContinuousOn f (Icc b a)
hderiv : ∀ x ∈ Ioo b a, DifferentiableAt ℝ f x
⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: E✝ : Type u_1
inst✝¹ : NormedAddCommGroup E✝
f✝ : ℝ → E✝
T✝ : ℝ
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℝ → E
T t : ℝ
h₁f : Periodic f T
hT✝ : T ≠ 0
h₂f : IntervalIntegrable f volume t (t + T)
a₁ a₂ : ℝ
hT : 0 < T
n₁ : ℕ
hn₁ : (t - min a₁ a₂) / T ≤ ↑n₁
n₂ : ℕ
hn₂ : (max a₁ a₂ - t) / T ≤ ↑n₂
⊢ uIcc a₁ a₂ ⊆ uIcc (t - ↑n₁ * T) (t + ↑n₂ * T)

Tactic:
Set.uIcc_subset_uIcc_iff_le

Goal state after:
E✝ : Type u_1
inst✝¹ : NormedAddCommGroup E✝
f✝ : ℝ → E✝
T✝ : ℝ
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℝ → E
T t : ℝ
h₁f : Periodic f T
hT✝ : T ≠ 0
h₂f : IntervalIntegrable f volume t (t + T)
a₁ a₂ : ℝ
hT : 0 < T
n₁ : ℕ
hn₁ : (t - min a₁ a₂) / T ≤ ↑n₁
n₂ : ℕ
hn₂ : (max a₁ a₂ - t) / T ≤ ↑n₂
⊢ min (t - ↑n₁ * T) (t + ↑n₂ * T) ≤ min a₁ a₂ ∧ max a₁ a₂ ≤ max (t - ↑n₁ * T) (t + ↑n₂ * T)
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
hfc : ContinuousOn f (Box.Icc I)
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
Hi : ∀ x ∈ Set.Icc (I.lower i) (I.upper i), Integrable (I.face i) ⊥ (f ∘ e x) BoxAdditiveMap.volume
y : Fin n → ℝ
hy : y ∈ Box.Icc (I.face i)
g : (Fin (n + 1) → ℝ) → E := fun y ↦ f y - a - f' (y - x)
hg : g = fun y ↦ f y - a - f' (y - x)
hε : ∀ y ∈ Box.Icc I, ‖g y‖ ≤ ε * ‖y - x‖
⊢ ‖f' (Pi.single i (I.upper i - I.lower i)) - (f (e (I.upper i) y) - f (e (I.lower i) y))‖ ≤ 2 * ε * diam (Box.Icc I)

Tactic:
clear_value g

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
hfc : ContinuousOn f (Box.Icc I)
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
Hi : ∀ x ∈ Set.Icc (I.lower i) (I.upper i), Integrable (I.face i) ⊥ (f ∘ e x) BoxAdditiveMap.volume
y : Fin n → ℝ
hy : y ∈ Box.Icc (I.face i)
g : (Fin (n + 1) → ℝ) → E
hg : g = fun y ↦ f y - a - f' (y - x)
hε : ∀ y ∈ Box.Icc I, ‖g y‖ ≤ ε * ‖y - x‖
⊢ ‖f' (Pi.single i (I.upper i - I.lower i)) - (f (e (I.upper i) y) - f (e (I.lower i) y))‖ ≤ 2 * ε * diam (Box.Icc I)
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: c : ℂ
R : ℝ
⊢ circleMap c R '' Ioc 0 (2 * π) = sphere c |R|

Tactic:
by
  rw [← range_circleMap, ← (periodic_circleMap c R).image_Ioc Real.two_pi_pos 0, zero_add]

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

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
ι : Type u_3
s : Finset ι
f : ι → ℂ → E
h : ∀ i ∈ s, CircleIntegrable (f i) c R
⊢ CircleIntegrable (∑ i ∈ s, f i) c R

Tactic:
rw [CircleIntegrable, (by aesop : (fun θ ↦ (∑ i ∈ s, f i) (circleMap c R θ))
    = ∑ i ∈ s, fun θ ↦ f i (circleMap c R θ))] at *

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
ι : Type u_3
s : Finset ι
f : ι → ℂ → E
h : ∀ i ∈ s, CircleIntegrable (f i) c R
⊢ IntervalIntegrable (∑ i ∈ s, fun θ ↦ f i (circleMap c R θ)) volume 0 (2 * π)
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
y✝ : Fin n → ℝ
hy✝ : y✝ ∈ Box.Icc (I.face i)
g : (Fin (n + 1) → ℝ) → E
hε : ∀ y ∈ Box.Icc I, ‖g y‖ ≤ ε * ‖y - x‖
hfc : ContinuousOn (fun y ↦ a + f' (y - x) + g y) (Box.Icc I)
Hi :
  ∀ x_1 ∈ Set.Icc (I.lower i) (I.upper i),
    Integrable (I.face i) ⊥ ((fun y ↦ a + f' (y - x) + g y) ∘ e x_1) BoxAdditiveMap.volume
hg : g = fun y ↦ (fun y ↦ a + f' (y - x) + g y) y - a - f' (y - x)
this : ∀ z ∈ Set.Icc (I.lower i) (I.upper i), e z y✝ ∈ Box.Icc I
y : Fin (n + 1) → ℝ
hy : y ∈ Box.Icc I
⊢ dist y x ≤ diam (Box.Icc I)

Tactic:
exact dist_le_diam_of_mem I.isCompact_Icc.isBounded hy hxI

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
h'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int
ε : ℝ := Int.fract (t / T) * T
⊢ sInf ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x ≤
    (∫ (x : ℝ) in 0..ε, g x) + ⌊t / T⌋ • ∫ (x : ℝ) in ε..ε + T, g x

Tactic:
hg.intervalIntegral_add_eq ε 0,

Goal state after:
T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
h'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int
ε : ℝ := Int.fract (t / T) * T
⊢ sInf ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x ≤
    (∫ (x : ℝ) in 0..ε, g x) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..0 + T, g x
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case right
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Iio a₀) μ
a : ℝ
ha : a ≤ a₀
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a₀ ∩ Ici a) a

Tactic:
rw [Iic_inter_Ici]

Goal state after:
case right
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Iio a₀) μ
a : ℝ
ha : a ≤ a₀
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Icc a a₀) a
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
h'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int
ε : ℝ := Int.fract (t / T) * T
⊢ (∫ (x : ℝ) in 0..ε, g x) + ⌊t / T⌋ • ∫ (x : ℝ) in ε..ε + T, g x ≤
    sSup ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x

Tactic:
hg.intervalIntegral_add_eq ε 0,

Goal state after:
T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
h'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int
ε : ℝ := Int.fract (t / T) * T
⊢ (∫ (x : ℝ) in 0..ε, g x) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..0 + T, g x ≤
    sSup ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
δ₁ : ℝ
δ₁0 : 0 < δ₁
hδ₁ : ∀ x_1 ∈ closedBall x δ₁ ∩ Box.Icc I, f x_1 ∈ closedBall (f x) (ε / 2 / 2)
δ : ℝ
hδ : δ ∈ Set.Ioc 0 δ₁
y₁ : Fin (n + 1) → ℝ
hy₁ : y₁ ∈ closedBall x δ ∩ Box.Icc I
⊢ ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2

Tactic:
y₂

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
δ₁ : ℝ
δ₁0 : 0 < δ₁
hδ₁ : ∀ x_1 ∈ closedBall x δ₁ ∩ Box.Icc I, f x_1 ∈ closedBall (f x) (ε / 2 / 2)
δ : ℝ
hδ : δ ∈ Set.Ioc 0 δ₁
y₁ : Fin (n + 1) → ℝ
hy₁ : y₁ ∈ closedBall x δ ∩ Box.Icc I
y₂ : Fin (n + 1) → ℝ
⊢ y₂ ∈ closedBall x δ ∩ Box.Icc I → ‖f y₁ - f y₂‖ ≤ ε / 2
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
mF : MeasurableSpace F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
μ : Measure E
inst✝² : IsProbabilityMeasure μ
ν : Measure F
inst✝¹ : IsProbabilityMeasure ν
L : E × F →L[ℝ] G
inst✝ : CompleteSpace G
hLμ : Integrable (fun x ↦ (L ∘SL ContinuousLinearMap.inl ℝ E F) x.1) (μ.prod ν)
hLν : Integrable (fun x ↦ (L ∘SL ContinuousLinearMap.inr ℝ E F) x.2) (μ.prod ν)
⊢ ∫ (a : E × F), (L ∘SL ContinuousLinearMap.inl ℝ E F) a.1 ∂μ.prod ν +
      ∫ (a : E × F), (L ∘SL ContinuousLinearMap.inr ℝ E F) a.2 ∂μ.prod ν =
    ∫ (x : E), (L ∘SL ContinuousLinearMap.inl ℝ E F) x ∂μ + ∫ (y : F), (L ∘SL ContinuousLinearMap.inr ℝ E F) y ∂ν

Tactic:
integral_prod _ hLμ,

Goal state after:
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
mF : MeasurableSpace F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
μ : Measure E
inst✝² : IsProbabilityMeasure μ
ν : Measure F
inst✝¹ : IsProbabilityMeasure ν
L : E × F →L[ℝ] G
inst✝ : CompleteSpace G
hLμ : Integrable (fun x ↦ (L ∘SL ContinuousLinearMap.inl ℝ E F) x.1) (μ.prod ν)
hLν : Integrable (fun x ↦ (L ∘SL ContinuousLinearMap.inr ℝ E F) x.2) (μ.prod ν)
⊢ ∫ (x : E), ∫ (y : F), (L ∘SL ContinuousLinearMap.inl ℝ E F) (x, y).1 ∂ν ∂μ +
      ∫ (a : E × F), (L ∘SL ContinuousLinearMap.inr ℝ E F) a.2 ∂μ.prod ν =
    ∫ (x : E), (L ∘SL ContinuousLinearMap.inl ℝ E F) x ∂μ + ∫ (y : F), (L ∘SL ContinuousLinearMap.inr ℝ E F) y ∂ν
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
hp : p.degree ≠ 0
α : p.SplittingField := rootOfSplits ⋯ ⋯
hα : IsIntegral F α
⊢ p.natDegree ∣ finrank F p.SplittingField

Tactic:
use Module.finrank F⟮α⟯ p.SplittingField

Goal state after:
case h
F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
hp : p.degree ≠ 0
α : p.SplittingField := ⋯
hα : IsIntegral F α
⊢ finrank F p.SplittingField = p.natDegree * finrank (↥F⟮α⟯) p.SplittingField
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: τ : Type u_2
σ : Type u_5
R : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : Fintype σ
inst✝² : Fintype τ
inst✝¹ : DecidableEq σ
inst✝ : DecidableEq τ
n : ℕ
μ : n.Partition
e : σ ≃ τ
x✝ : { x // Nat.Partition.ofSym x = μ }
⊢ (Multiset.map (⇑(rename ⇑e) ∘ X) ↑↑x✝).prod = (Multiset.map X ↑↑((μ.ofSymShapeEquiv e) x✝)).prod

Tactic:
Nat.Partition.ofSymShapeEquiv

Goal state after:
τ : Type u_2
σ : Type u_5
R : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : Fintype σ
inst✝² : Fintype τ
inst✝¹ : DecidableEq σ
inst✝ : DecidableEq τ
n : ℕ
μ : n.Partition
e : σ ≃ τ
x✝ : { x // Nat.Partition.ofSym x = μ }
⊢ (Multiset.map (⇑(rename ⇑e) ∘ X) ↑↑x✝).prod =
    (Multiset.map X
        ↑↑({ toFun := fun x ↦ ⟨(Sym.equivCongr e) ↑x, ⋯⟩, invFun := fun x ↦ ⟨(Sym.equivCongr e.symm) ↑x, ⋯⟩,
                left_inv := ⋯, right_inv := ⋯ }
              x✝)).prod
Module: Mathlib.Topology.Maps.Proper.CompactlyGenerated

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
inst✝ : CompactlyCoherentSpace Y
f : X → Y
⊢ IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y)

Tactic:
isProperMap_iff_isCompact_preimage,

Goal state after:
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
inst✝ : CompactlyCoherentSpace Y
f : X → Y
⊢ (Continuous f ∧ ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)) ↔
    Continuous f ∧ Tendsto f (cocompact X) (cocompact Y)
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
⊢ k ≤ p.roots.card

Tactic:
rw [hroots]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
⊢ k ≤ p.natDegree
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_5
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
this : k ≤ p.roots.card
⊢ p.natDegree - k = p.roots.card - k

Tactic:
rw [hroots]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: case e'_6
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
k : ℕ
h : k ≤ p.natDegree
this : k ≤ p.roots.card
⊢ p.natDegree - k = p.roots.card - k

Tactic:
rw [hroots]

Goal state after: