mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: case smul_mem'
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : M⦄, x ∈ ⋃ c, ⋃ (_ : 0 < c), c • s → c • x ∈ ⋃ c, ⋃ (_ : 0 < c), c • s

Tactic:
simp only [mem_iUnion, mem_smul_set]

Goal state after:
case smul_mem'
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : M⦄, (∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = x) → ∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = c • x
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: case add_mem'
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ ∀ ⦃x : M⦄,
    x ∈ ⋃ c, ⋃ (_ : 0 < c), c • s → ∀ ⦃y : M⦄, y ∈ ⋃ c, ⋃ (_ : 0 < c), c • s → x + y ∈ ⋃ c, ⋃ (_ : 0 < c), c • s

Tactic:
simp only [mem_iUnion, mem_smul_set]

Goal state after:
case add_mem'
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ ∀ ⦃x : M⦄,
    (∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = x) →
      ∀ ⦃y : M⦄, (∃ i, ∃ (_ : 0 < i), ∃ y_1 ∈ s, i • y_1 = y) → ∃ i, ∃ (_ : 0 < i), ∃ y_1 ∈ s, i • y_1 = x + y
Module: Mathlib.Geometry.Convex.Cone.Pointed

Goal state before:
pp: R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁴ : Ring R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
C : PointedCone R E
r : R
x✝ : E
hx : x✝ ∈ C.support.carrier
⊢ r • x✝ ∈ C.support.carrier

Tactic:
by
    by_cases hr : 0 ≤ r
    · simpa using And.intro (C.smul_mem hr hx.1) (C.smul_mem hr hx.2)
    · have hr := le_of_lt <| neg_pos_of_neg <| lt_of_not_ge hr
      simpa using And.intro (C.smul_mem hr hx.2) (C.smul_mem hr hx.1)

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
inst✝² : RCLike 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsScalarTower ℝ 𝕜 E
hs : Balanced 𝕜 s
r : 𝕜
x : E
⊢ sInf {r_1 | 0 < r_1 ∧ ‖r‖ • x ∈ r_1 • s} = sInf {r_1 | 0 < r_1 ∧ r • x ∈ r_1 • s}

Tactic:
congr with θ

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
inst✝² : RCLike 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsScalarTower ℝ 𝕜 E
hs : Balanced 𝕜 s
r : 𝕜
x : E
θ : ℝ
⊢ 0 < θ ∧ ‖r‖ • x ∈ θ • s ↔ 0 < θ ∧ r • x ∈ θ • s
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: E : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
x y : E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul ℝ E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : T1Space E
hxy : x ≠ y
f : StrongDual ℝ E
s t : ℝ
hs : ∀ a ∈ {x}, f a < s
st : s < t
ht : ∀ b ∈ {y}, t < f b
⊢ f x < f y

Tactic:
linarith [hs x rfl, ht y rfl]

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
r ε : ℝ
L : F
x y z : ℝ
hy : y ∈ Icc x (x + r / 2)
hz : z ∈ Icc x (x + r / 2)
r' : ℝ
r'mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r
A : x + r / 2 ≤ x + r'
⊢ ‖f z - f y - (z - y) • L‖ ≤ ε * r

Tactic:
exact hr' _ ((Icc_subset_Icc le_rfl A) hy) _ ((Icc_subset_Icc le_rfl A) hz)

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
x y : E
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : T1Space E
hxy : x ≠ y
f : StrongDual 𝕜 E
s t : ℝ
hs : ∀ a ∈ {x}, re (f a) < s
st : s < t
ht : ∀ b ∈ {y}, t < re (f b)
⊢ re (f x) < re (f y)

Tactic:
linarith [hs x rfl, ht y rfl]

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
a b : ℝ
μ : Measure ℝ
⊢ ‖SubtractionMonoid.toSubNegZeroMonoid.toNegZeroClass.toZero.1‖ₑ ≠ ∞

Tactic:
by finiteness

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)
M :
  ∀ (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
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := k - 1
h'k : y - x ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < y - x
m_ge : n e ≤ m
km : k = m + 1
J : ‖f y - f x - (y - x) • L e (n e) m‖ ≤ 4 * (1 / 2) ^ e * ‖y - x‖
⊢ ‖(y - x) • (L e (n e) m - f')‖ ≤ ‖y - x‖ * (12 * (1 / 2) ^ e)

Tactic:
by rw [norm_smul]; gcongr; exact Lf' _ _ m_ge

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : ℝ → F
inst✝ : CompleteSpace F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
⊢ IsSeparable (range fun x ↦ derivWithin f (Ici x) x)

Tactic:
obtain ⟨t, t_count, ht⟩ : ∃ t : Set ℝ, t.Countable ∧ Dense t := exists_countable_dense ℝ

Goal state after:
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : ℝ → F
inst✝ : CompleteSpace F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
t : Set ℝ
t_count : t.Countable
ht : Dense t
⊢ IsSeparable (range fun x ↦ derivWithin f (Ici x) x)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → E
c : ℝ
hc : c ≠ 0
h✝ : ‖f (min a b)‖ₑ ≠ ∞
h' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞
h : IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b / c)
⊢ IntervalIntegrable f volume a b

Tactic:
simpa [hc] using h.comp_mul_left (c := c⁻¹) h' (by simp)

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

Goal state before:
pp: case refine_2
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
⊢ ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ‖{x_1 | x_1 ≤ x}.indicator f t‖ ≤ ‖f t‖

Tactic:
filter_upwards with x; filter_upwards with t
      dsimp [indicator]
      split_ifs <;> simp

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

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ IntervalIntegrable (fun x ↦ f (c + x)) volume (a - c) (b - c) ↔ IntervalIntegrable f volume a b

Tactic:
simp_rw [add_comm c]
  rw [IntervalIntegrable.comp_add_right_iff (by grind)]
  simp

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✝¹ :
  (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
this : ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, {t | t ≤ x}.indicator f x₀ = f x₀
⊢ ContinuousWithinAt (fun x ↦ {x_1 | x_1 ≤ x}.indicator f x₀) (Icc b₁ b₂) b₀

Tactic:
apply continuousWithinAt_const.congr_of_eventuallyEq this

Goal state after:
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✝¹ :
  (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
this : ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, {t | t ≤ x}.indicator f x₀ = f x₀
⊢ {t | t ≤ b₀}.indicator f x₀ = f x₀
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_3.inr
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✝¹ :
  (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 : b₀ < x₀
this : ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, {t | t ≤ x}.indicator f x₀ = 0
⊢ ContinuousWithinAt (fun x ↦ {x_1 | x_1 ≤ x}.indicator f x₀) (Icc b₁ b₂) b₀

Tactic:
apply continuousWithinAt_const.congr_of_eventuallyEq this

Goal state after:
case refine_3.inr
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✝¹ :
  (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 : b₀ < x₀
this : ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, {t | t ≤ x}.indicator f x₀ = 0
⊢ {t | t ≤ b₀}.indicator f x₀ = 0
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
μ : Measure ℝ
f : ℝ → E
⊢ ‖∫ (x : ℝ) in min a b..max a b, f x ∂μ‖ = ‖∫ (x : ℝ) in a..b, f x ∂μ‖

Tactic:
by
  cases le_total a b <;> simp [*, integral_symm a b]

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

Goal state before:
pp: case refine_1
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
⊢ ContinuousAt (fun x ↦ ∫ (s : ℝ) in a₀..b₀, F x.1 s ∂μ) (x₀, b₀)

Tactic:
exact (intervalIntegral.continuousAt_of_dominated_interval
        (Eventually.of_forall fun x ↦ (hF_meas x).mono_set <| hsub ha₀ hb₀)
          (h_bound.mono fun x hx ↦
            ae_imp_of_ae_restrict <| ae_restrict_of_ae_restrict_of_subset (hsub ha₀ hb₀) hx)
          (bound_integrable.mono_set_ae <| Eventually.of_forall <| hsub ha₀ hb₀) <|
          ae_imp_of_ae_restrict <| ae_restrict_of_ae_restrict_of_subset (hsub ha₀ hb₀) h_cont).fst'

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
this✝ :
  ∀ᶠ (p : X × ℝ) in 𝓝 (x₀, b₀), ‖∫ (s : ℝ) in b₀..p.2, F p.1 s - F x₀ s ∂μ‖ ≤ |∫ (s : ℝ) in b₀..p.2, 2 * bound s ∂μ|
this : ContinuousAt (fun t ↦ ∫ (s : ℝ) in b₀..t, 2 * bound s ∂μ) b₀
⊢ Tendsto (fun t ↦ ∫ (s : ℝ) in b₀..t, 2 * bound s ∂μ) (𝓝 b₀) (𝓝 0)

Tactic:
simpa [ContinuousAt] using this

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f (Icc a b) μ
h : a ≤ b
x : ℝ
⊢ x ∈ Icc a b → ∫ (t : ℝ) in Ioc a x, f t ∂μ = ∫ (t : ℝ) in a..x, f t ∂μ

Tactic:
x_in

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f (Icc a b) μ
h : a ≤ b
x : ℝ
x_in : x ∈ Icc a b
⊢ ∫ (t : ℝ) in Ioc a x, f t ∂μ = ∫ (t : ℝ) in a..x, f t ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

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

Tactic:
integral_deriv_eq_sub hdiff

Goal state after:
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f' f : ℝ → E
hderiv : deriv f = f'
hdiff : ∀ x ∈ [[a, b]], DifferentiableAt ℝ f x
hcont : ContinuousOn f' [[a, b]]
⊢ f b - f a = f b - f a

E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f' f : ℝ → E
hderiv : deriv f = f'
hdiff : ∀ x ∈ [[a, b]], DifferentiableAt ℝ f x
hcont : ContinuousOn f' [[a, b]]
⊢ IntervalIntegrable (deriv f) volume a b
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
n : ℤ
⊢ Tendsto floor (𝓝[≥] ↑n) (pure n)

Tactic:
by
  simpa only [floor_intCast] using tendsto_floor_right_pure_floor (n : α)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.IsMulRightInvariant
f : G → E
hf₁ : Integrable f μ
hf₂ : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
⊢ ∫ (x : G), f x ∂μ = ∫ (x : G ⧸ Γ), automorphize f x ∂Measure.map mk (μ.restrict 𝓕)

Tactic:
calc ∫ x : G, f x ∂μ = ∑' γ : Γ.op, ∫ x in 𝓕, f (γ • x) ∂μ :=
    h𝓕.integral_eq_tsum'' f hf₁
    _ = ∫ x in 𝓕, ∑' γ : Γ.op, f (γ • x) ∂μ := ?_
    _ = ∫ x : G ⧸ Γ, automorphize f x ∂μ_𝓕 :=
      (integral_map continuous_quotient_mk'.aemeasurable hf₂).symm

Goal state after:
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.IsMulRightInvariant
f : G → E
hf₁ : Integrable f μ
hf₂ : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
⊢ ∑' (γ : ↥Γ.op), ∫ (x : G) in 𝓕, f (γ • x) ∂μ = ∫ (x : G) in 𝓕, ∑' (γ : ↥Γ.op), f (γ • x) ∂μ
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ.IsMulRightInvariant
f : G → K
f_ℒ_1 : Integrable f μ
g : G ⧸ Γ → K
hg : AEStronglyMeasurable g (Measure.map mk (μ.restrict 𝓕))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ.restrict 𝓕)) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
⊢ ∫ (x : G), g ↑x * f x ∂μ = ∫ (x : G ⧸ Γ), g x * automorphize f x ∂Measure.map mk (μ.restrict 𝓕)

Tactic:
let π : G → G ⧸ Γ := QuotientGroup.mk

Goal state after:
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ.IsMulRightInvariant
f : G → K
f_ℒ_1 : Integrable f μ
g : G ⧸ Γ → K
hg : AEStronglyMeasurable g (Measure.map mk (μ.restrict 𝓕))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ.restrict 𝓕)) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
π : G → G ⧸ Γ := mk
⊢ ∫ (x : G), g ↑x * f x ∂μ = ∫ (x : G ⧸ Γ), g x * automorphize f x ∂Measure.map mk (μ.restrict 𝓕)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℝ → E
T : ℝ
inst✝ : NormedSpace ℝ E
hf : Periodic f T
t s : ℝ
h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable f volume t₁ t₂
⊢ ∫ (x : ℝ) in t..s + T, f x = (∫ (x : ℝ) in t..s, f x) + ∫ (x : ℝ) in t..t + T, f x

Tactic:
rw [hf.intervalIntegral_add_eq t s, integral_add_adjacent_intervals (h_int t s) (h_int s _)]

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

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ ≤ ε

Tactic:
obtain ⟨δ, δ0, hδ⟩ : ∃ δ > (0 : ℝ), ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π) :=
    ((nhdsWithin_hasBasis nhds_basis_closedBall _).tendsto_iff nhds_basis_ball).1 hy _
      (div_pos ε0 Real.two_pi_pos)

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ ≤ ε
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
h_bdd : Bornology.IsBounded (range f)
c' : F
hc' : f = const E c'
⊢ c = c'

Tactic:
exact tendsto_nhds_unique hb (by simpa [hc'] using! tendsto_const_nhds)

Goal state after:
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:
by
  simp_rw [isProperMap_iff_isCompact_preimage,
    hasBasis_cocompact.tendsto_right_iff, ← mem_preimage, eventually_mem_set, preimage_compl]
  refine and_congr_right fun f_cont ↦
    ⟨fun H K hK ↦ (H hK).compl_mem_cocompact, fun H K hK ↦ ?_⟩
  rcases mem_cocompact.mp (H K hK) with ⟨K', hK', hK'y⟩
  exact hK'.of_isClosed_subset (hK.isClosed.preimage f_cont)
    (compl_le_compl_iff_le.mp hK'y)

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

Goal state before:
pp: case refine_1
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : 0 < R
⊢ ContinuousOn f (closedBall c R)

pp: case refine_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : 0 < R
⊢ ∀ x ∈ ball c R \ ∅, DifferentiableAt ℂ f x

Tactic:
· simpa only [closure_ball c hR.ne.symm] using hf.continuousOn

Goal state after:
case refine_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : 0 < R
⊢ ∀ x ∈ ball c R \ ∅, DifferentiableAt ℂ f x
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : (map f p).natDegree ≠ 0
⊢ 0 ≤ B

Tactic:
Splits.natDegree_eq_card_roots h2

Goal state after:
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : (map f p).roots.card ≠ 0
⊢ 0 ≤ B
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case inr.inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : (map f p).natDegree < i
⊢ ‖0‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)

Tactic:
norm_zero

Goal state after:
case inr.inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : (map f p).Splits
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : (map f p).natDegree < i
⊢ 0 ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)