mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ
u : Set E
H : u ∈ 𝓝 x
f' : E → E →L[𝕜] F
h_fderiv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
h_cont_diff : ContDiffWithinAt 𝕜 (↑n) f' u x
t : Set E
htu : t ⊆ u
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
hxt : x ∈ t
y : E
⊢ y ∈ t → HasFDerivAt f (f' y) y

Tactic:
hyt

Goal state after:
case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ
u : Set E
H : u ∈ 𝓝 x
f' : E → E →L[𝕜] F
h_fderiv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
h_cont_diff : ContDiffWithinAt 𝕜 (↑n) f' u x
t : Set E
htu : t ⊆ u
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
hxt : x ∈ t
y : E
hyt : y ∈ t
⊢ HasFDerivAt f (f' y) y
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case inr
𝔸 : Type u_2
inst✝³ : Ring 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : T2Space 𝔸
x : 𝔸
val✝ : Algebra ℚ 𝔸
⊢ ∑' (n : ℕ), (↑n !)⁻¹ • MulOpposite.op x ^ n = MulOpposite.op (∑' (n : ℕ), (↑n !)⁻¹ • x ^ n)

Tactic:
← MulOpposite.op_pow,

Goal state after:
case inr
𝔸 : Type u_2
inst✝³ : Ring 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : T2Space 𝔸
x : 𝔸
val✝ : Algebra ℚ 𝔸
⊢ ∑' (n : ℕ), (↑n !)⁻¹ • MulOpposite.op (x ^ n) = MulOpposite.op (∑' (n : ℕ), (↑n !)⁻¹ • x ^ n)
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝔸 : Type u_1
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
⊢ ContinuousOn exp (Metric.eball 0 ∞)

Tactic:
←
    expSeries_radius_eq_top ℚ 𝔸

Goal state after:
𝔸 : Type u_1
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
⊢ ContinuousOn exp (Metric.eball 0 (expSeries ℚ 𝔸).radius)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
f g : 𝕜 → F
s t : Set 𝕜
h : f =ᶠ[𝓝[s] x] g
ht : t ⊆ s
n : ℕ
⊢ iteratedDerivWithin n f t =ᶠ[𝓝[s] x] iteratedDerivWithin n g t

Tactic:
by
  unfold iteratedDerivWithin
  exact h.iteratedFDerivWithin' ht n |>.fun_comp (fun a ↦ a fun _ ↦ 1)

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
s : Set 𝕜
f : 𝕜 → F
c : 𝕜
⊢ iteratedDerivWithin n (fun z ↦ f (z - c)) s = fun x ↦ iteratedDerivWithin n f (-c +ᵥ s) (x - c)

Tactic:
by
  simpa only [sub_eq_add_neg] using iteratedDerivWithin_comp_add_const (-c)

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
⊢ (fun a ↦ (p₁ a).taylorComp (q₁ a) n - (p₂ a).taylorComp (q₂ a) n) =O[l] f

Tactic:
by
  simp only [FormalMultilinearSeries.taylorComp, ← Finset.sum_sub_distrib]
  refine .fun_sum fun c _ ↦ ?_
  refine .trans (.of_norm_le fun _ ↦
    c.norm_compAlongOrderedFinpartition_sub_compAlongOrderedFinpartition_le ..) ?_
  refine .add ?_ ?_
  · have H₁ : (p₁ · c.length) =O[l] (1 : α → ℝ) := (hp_bdd _ c.length_le).isBigO_one ℝ
    have H₂ : ∀ m, (q₁ · (c.partSize m)) =O[l] (1 : α → ℝ) := fun m ↦
      (hq₁_bdd _ <| c.partSize_le _).isBigO_one ℝ
    have H₃ : ∀ m, (q₂ · (c.partSize m)) =O[l] (1 : α → ℝ) := fun m ↦
      (hq₂_bdd _ <| c.partSize_le _).isBigO_one ℝ
    have H₄ : ∀ m, (fun a ↦ q₁ a (c.partSize m) - q₂ a (c.partSize m)) =O[l] f := fun m ↦
      hqf _ <| c.partSize_le _
    rw [← Asymptotics.isBigO_pi] at H₂ H₃ H₄
    have H₅ := ((H₂.prod_left H₃).norm_left.pow (c.length - 1)).mul H₄.norm_norm
    simpa [mul_assoc] using! H₁.norm_left.mul <| H₅.const_mul_left c.length
  · have H₁ : (fun a ↦ p₁ a c.length - p₂ a c.length) =O[l] f := hpf _ c.length_le
    have H₂ : ∀ i, (q₂ · (c.partSize i)) =O[l] (1 : α → ℝ) := fun i ↦
      (hq₂_bdd _ <| c.partSize_le i).isBigO_one ℝ
    simpa using H₁.norm_norm.mul <| .finsetProd fun i _ ↦ (H₂ i).norm_left

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
x : E
i : ℕ
f g : E → F
hf : ContDiffWithinAt 𝕜 (↑i) f s x
hg : ContDiffWithinAt 𝕜 (↑i) g s x
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ iteratedFDerivWithin 𝕜 i (f - g) s x = iteratedFDerivWithin 𝕜 i f s x - iteratedFDerivWithin 𝕜 i g s x

Tactic:
rw [sub_eq_add_neg, iteratedFDerivWithin_add_apply hf _ hu hx,
    iteratedFDerivWithin_neg_apply hu hx, sub_eq_add_neg]
  exact hg.neg

Goal state after:
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
l : Y
hab : a < b
hfc : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 l)
hb : Tendsto f (𝓝[<] b) (𝓝 l)
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Ioo a b) c

Tactic:
have h : EqOn (extendFrom (Ioo a b) f) f (Ioo a b) := extendFrom_extends hfc

Goal state after:
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
l : Y
hab : a < b
hfc : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 l)
hb : Tendsto f (𝓝[<] b) (𝓝 l)
h : EqOn (extendFrom (Ioo a b) f) f (Ioo a b)
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Ioo a b) c
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
C : ℝ
hf'_gt : ∀ x ∈ interior D, C < deriv f x
⊢ ∀ x ∈ D, ∀ y ∈ D, x < y → C * (y - x) < f y - f x

Tactic:
intro x hx y hy hxy
  have hxyD : Icc x y ⊆ D := hD.ordConnected.out hx hy
  have hxyD' : Ioo x y ⊆ interior D :=
    subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩
  obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
    exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')
  have : C < (f y - f x) / (y - x) := ha ▸ hf'_gt _ (hxyD' a_mem)
  exact (lt_div_iff₀ (sub_pos.2 hxy)).1 this

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
M : Type u_4
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 ∈ toCone s hs ↔ ∃ c, 0 < c ∧ ∃ y ∈ s, c • y = x

Tactic:
by
  simp only [toCone, ConvexCone.mem_mk, mem_iUnion, mem_smul_set, eq_comm, exists_prop]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
this : Tendsto (fun t ↦ |x| ^ (t + 1) / (1 - |x|)) atTop (𝓝 (|x| * 0 / (1 - |x|)))
⊢ Tendsto (fun t ↦ |x| ^ (t + 1) / (1 - |x|)) atTop (𝓝 0)

Tactic:
by
      simpa

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
h_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)
m : ℕ
hm : m ∉ Set.range fun x ↦ 2 * x
⊢ term m = 0

Tactic:
range_two_mul,

Goal state after:
x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
h_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)
m : ℕ
hm : m ∉ {a | Even a}
⊢ term m = 0
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: a : ℝ
h : 0 < a
h₁ : |1 / (2 * a + 1)| < 1
h₂ : 2 * a + 1 ≠ 0
h₃ : a ≠ 0
⊢ log (1 + a⁻¹) = log (1 + 1 / (2 * a + 1)) - log (1 - 1 / (2 * a + 1))

Tactic:
← log_div

Goal state after:
a : ℝ
h : 0 < a
h₁ : |1 / (2 * a + 1)| < 1
h₂ : 2 * a + 1 ≠ 0
h₃ : a ≠ 0
⊢ log (1 + a⁻¹) = log ((1 + 1 / (2 * a + 1)) / (1 - 1 / (2 * a + 1)))

case hx
a : ℝ
h : 0 < a
h₁ : |1 / (2 * a + 1)| < 1
h₂ : 2 * a + 1 ≠ 0
h₃ : a ≠ 0
⊢ 1 + 1 / (2 * a + 1) ≠ 0

case hy
a : ℝ
h : 0 < a
h₁ : |1 / (2 * a + 1)| < 1
h₂ : 2 * a + 1 ≠ 0
h₃ : a ≠ 0
⊢ 1 - 1 / (2 * a + 1) ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: case inr
a : ℝ
h : 0 ≤ a
ha0 : a ≠ 0
⊢ HasSum (fun k ↦ 2 * (1 / (2 * ↑k + 1)) * (a / (a + 2)) ^ (2 * k + 1)) (log (1 + a))

Tactic:
convert! hasSum_log_one_add_inv (inv_pos.mpr (lt_of_le_of_ne h ha0.symm)) using 4

Goal state after:
case e'_5
a : ℝ
h : 0 ≤ a
ha0 : a ≠ 0
x✝ : ℕ
⊢ a / (a + 2) = 1 / (2 * a⁻¹ + 1)

case e'_6
a : ℝ
h : 0 ≤ a
ha0 : a ≠ 0
⊢ a = a⁻¹⁻¹
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
⊢ gauge s 0 = 0

Tactic:
gauge_def'

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ s} = 0
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
x : E
⊢ gauge 0 x = 0 x

Tactic:
gauge_def'

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
x : E
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = 0 x
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
⊢ gauge (-s) (-x) = gauge s x

Tactic:
simp_rw [gauge_def', smul_neg, neg_mem_neg]

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
fint : Integrable (fun x ↦ ↑(f x)) μ
ε : ℝ≥0
εpos : 0 < ↑ε
If : ∫⁻ (x : α), ↑(f x) ∂μ < ∞
g : α → ℝ≥0
gf : ∀ (x : α), g x ≤ f x
gcont : UpperSemicontinuous g
gint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε
⊢ ∫⁻ (x : α), ↑(g x) ∂μ < ∞

Tactic:
refine lt_of_le_of_lt (lintegral_mono fun x => ?_) If

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
fint : Integrable (fun x ↦ ↑(f x)) μ
ε : ℝ≥0
εpos : 0 < ↑ε
If : ∫⁻ (x : α), ↑(f x) ∂μ < ∞
g : α → ℝ≥0
gf : ∀ (x : α), g x ≤ f x
gcont : UpperSemicontinuous g
gint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε
x : α
⊢ ↑(g x) ≤ ↑(f x)
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr.mp
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
r : ℝ
y : E
hy : y ∈ s
h : a • y = r⁻¹ • x
hr : 0 < r
⊢ a⁻¹ • r⁻¹ • x ∈ s

Tactic:
← h,

Goal state after:
case inr.mp
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
r : ℝ
y : E
hy : y ∈ s
h : a • y = r⁻¹ • x
hr : 0 < r
⊢ a⁻¹ • a • y ∈ s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : Module α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
symmetric : ∀ x ∈ s, -x ∈ s
a : α
⊢ gauge (a • s) = |a|⁻¹ • gauge s

Tactic:
rw [← gauge_smul_left_of_nonneg (abs_nonneg a)]
  obtain h | h := abs_choice a
  · rw [h]
  · rw [h, Set.neg_smul_set, ← Set.smul_set_neg]
    congr
    ext y
    refine ⟨symmetric _, fun hy => ?_⟩
    rw [← neg_neg y]
    exact symmetric _ hy

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

Goal state before:
pp: case inr.refine_1
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1}.Nonempty
r : ℝ
⊢ r ∈ {r | 0 < r ∧ x ∈ r • p.ball 0 1} → p x ≤ r

pp: case inr.refine_2
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1}.Nonempty
r : ℝ
hr : r ∈ lowerBounds {r | 0 < r ∧ x ∈ r • p.ball 0 1}
ε : ℝ
hε : 0 < ε
⊢ r ≤ p x + ε

Tactic:
· rintro ⟨hr, y, hy, rfl⟩
    rw [p.mem_ball_zero] at hy
    rw [map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos hr]
    exact mul_le_of_le_one_right hr.le hy.le

Goal state after:
case inr.refine_2
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1}.Nonempty
r : ℝ
hr : r ∈ lowerBounds {r | 0 < r ∧ x ∈ r • p.ball 0 1}
ε : ℝ
hε : 0 < ε
⊢ r ≤ p x + ε
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
δ : ℝ≥0 := NNReal.mk (ε / 2) ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := fun x ↦ (f x).toNNReal
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := fun x ↦ (-f x).toNNReal
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x ↦ ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
⊢ ∫ (x : α), (↑(gp x)).toReal - (↑↑(gm x)).toReal ∂μ = ∫ (x : α), (↑(gp x)).toReal ∂μ - ∫ (x : α), ↑(gm x) ∂μ

Tactic:
simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]
          exact integral_sub gp_integrable gm_integrable

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

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁸ : Field R
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace V
inst✝⁴ : IsTopologicalRing R
inst✝³ : Module R V
inst✝² : SeparatingDual R V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul R V
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ ∃ A, A x = y

Tactic:
by
  obtain ⟨G, Gx, Gy⟩ : ∃ G : StrongDual R V, G x = 1 ∧ G y ≠ 0 :=
    exists_eq_one_ne_zero_of_ne_zero_pair hx hy
  let A : V ≃L[R] V :=
  { toFun := fun z ↦ z + G z • (y - x)
    invFun := fun z ↦ z + ((G y)⁻¹ * G z) • (x - y)
    map_add' := fun a b ↦ by simp [add_smul]; abel
    map_smul' := by simp [smul_smul]
    left_inv := fun z ↦ by
      simp only [RingHom.id_apply, smul_eq_mul,
        -- Note: https://github.com/leanprover-community/mathlib4/pull/8386 had to change `map_smulₛₗ` into `map_smulₛₗ _`
        map_add, map_smulₛₗ _, map_sub, Gx, mul_sub, mul_one, add_sub_cancel]
      rw [mul_comm (G z), ← mul_assoc, inv_mul_cancel₀ Gy]
      simp only [smul_sub, one_mul]
      abel
    right_inv := fun z ↦ by
        -- Note: https://github.com/leanprover-community/mathlib4/pull/8386 had to change `map_smulₛₗ` into `map_smulₛₗ _`
      simp only [map_add, map_smulₛₗ _, map_mul, map_inv₀, RingHom.id_apply, map_sub, Gx,
        smul_eq_mul, mul_sub, mul_one]
      rw [mul_comm _ (G y), ← mul_assoc, mul_inv_cancel₀ Gy]
      simp only [smul_sub, one_mul, add_sub_cancel]
      abel }
  exact ⟨A, show x + G x • (y - x) = y by simp [Gx]⟩

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

Goal state before:
pp: case neg
X : Type u_1
E : Type u_3
inst✝⁷ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace ℝ E
ι : Type u_8
inst✝² : Fintype ι
M : ι → Type u_9
inst✝¹ : (i : ι) → NormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace 𝕜 (M i)
φ : X → ContinuousMultilinearMap 𝕜 M E
φ_int : Integrable φ μ
m : (i : ι) → M i
hE : ¬CompleteSpace E
i : ι
hi : m i = 0
⊢ (∫ (x : X), φ x ∂μ) m = ∫ (x : X), (φ x) m ∂μ

Tactic:
simp [ContinuousMultilinearMap.map_coord_zero _ i hi]

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

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
hc : c ≠ 0
d : ℝ
⊢ ∫ (x : ℝ) in a..b, f (d + c * x) = c⁻¹ • ∫ (x : ℝ) in d + c * a..d + c * b, f x

Tactic:
← integral_comp_add_left,

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
hc : c ≠ 0
d : ℝ
⊢ ∫ (x : ℝ) in a..b, f (d + c * x) = c⁻¹ • ∫ (x : ℝ) in c * a..c * b, f (d + x)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
this :
  ∀ {a b : ℝ},
    IntegrableOn f (Iic a) μ →
      IntegrableOn f (Iic b) μ →
        a ≤ b → ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ
hab : ¬a ≤ b
⊢ ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ

Tactic:
rw [integral_symm, ← this hb ha (le_of_not_ge hab), neg_sub]

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 ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ioo a b, φ x ≤ g' x
⊢ ∫ (y : ℝ) in a..b, φ y ≤ g b - g a

Tactic:
by
  rw [← neg_le_neg_iff]
  convert!
    sub_le_integral_of_hasDeriv_right_of_le hab hcont.fun_neg (fun x hx => (hderiv x hx).neg)
      φint.neg fun x hx => neg_le_neg (hφg x hx) using 1
  · abel
  · simp only [← integral_neg]; rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b : ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : [[a, b]] ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
⊢ ¬IntervalIntegrable g volume a b

Tactic:
rw [intervalIntegrable_iff']
  exact not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter _ hl hd hf hfg

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: case inr
E : Type u_1
F : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x ↦ ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
hlt : c < max a b
⊢ ∃ l, TendstoIxxClass Icc l l ∧ l.NeBot ∧ l ≤ 𝓝 c ∧ [[a, b]] \ {c} ∈ l

Tactic:
refine ⟨𝓝[>] c, inferInstance, inferInstance, inf_le_left, ?_⟩

Goal state after:
case inr
E : Type u_1
F : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x ↦ ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
hlt : c < max a b
⊢ [[a, b]] \ {c} ∈ 𝓝[>] c
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ

Tactic:
by
  simp only [integral_of_le hab, integral_Ioc_eq_integral_Ioo]
  apply setIntegral_mono_on
  · apply hf.1.mono Ioo_subset_Ioc_self le_rfl
  · apply hg.1.mono Ioo_subset_Ioc_self le_rfl
  · exact measurableSet_Ioo
  · exact h

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
inst✝⁵ : PolishSpace G
Γ : Subgroup G
inst✝⁴ : Γ.Normal
inst✝³ : T2Space (G ⧸ Γ)
inst✝² : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝¹ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ (map (fun x_1 ↦ x * x_1) μ) A = μ ((fun x ↦ x₁ • x) ⁻¹' A)

Tactic:
← h,

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
inst✝⁵ : PolishSpace G
Γ : Subgroup G
inst✝⁴ : Γ.Normal
inst✝³ : T2Space (G ⧸ Γ)
inst✝² : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝¹ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ (map (fun x ↦ ⟦x₁⟧ * x) μ) A = μ ((fun x ↦ x₁ • x) ⁻¹' A)
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ν s = 0
⊢ μ univ = 0

Tactic:
← h,

Goal state after:
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ν s = 0
⊢ ν s = 0