mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: A : Type u_7
inst✝ : NormedRing A
a b : ℝ
μ : Measure ℝ
f g : ℝ → A
hf : IntervalIntegrable f μ a b
hg : ContinuousOn g [[a, b]]
⊢ IntervalIntegrable (fun x ↦ g x * f x) μ a b

Tactic:
intervalIntegrable_iff

Goal state after:
A : Type u_7
inst✝ : NormedRing A
a b : ℝ
μ : Measure ℝ
f g : ℝ → A
hf : IntegrableOn f (Ι a b) μ
hg : ContinuousOn g [[a, b]]
⊢ IntegrableOn (fun x ↦ g x * f x) (Ι a b) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
inst✝³ : NormedAddCommGroup E
a b : ℝ
μ : Measure ℝ
f : ℝ → 𝕜
g : ℝ → E
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
hf : IntervalIntegrable f μ a b
hg : ContinuousOn g [[a, b]]
⊢ IntervalIntegrable (fun x ↦ f x • g x) μ a b

Tactic:
intervalIntegrable_iff

Goal state after:
𝕜 : Type u_2
E : Type u_5
inst✝³ : NormedAddCommGroup E
a b : ℝ
μ : Measure ℝ
f : ℝ → 𝕜
g : ℝ → E
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
hf : IntegrableOn f (Ι a b) μ
hg : ContinuousOn g [[a, b]]
⊢ IntegrableOn (fun x ↦ f x • g x) (Ι a b) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
inst✝³ : NormedAddCommGroup E
a b : ℝ
μ : Measure ℝ
f : ℝ → 𝕜
g : ℝ → E
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
hg : IntervalIntegrable g μ a b
hf : ContinuousOn f [[a, b]]
⊢ IntervalIntegrable (fun x ↦ f x • g x) μ a b

Tactic:
intervalIntegrable_iff

Goal state after:
𝕜 : Type u_2
E : Type u_5
inst✝³ : NormedAddCommGroup E
a b : ℝ
μ : Measure ℝ
f : ℝ → 𝕜
g : ℝ → E
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
hg : IntegrableOn g (Ι a b) μ
hf : ContinuousOn f [[a, b]]
⊢ IntegrableOn (fun x ↦ f x • g x) (Ι a b) μ
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
⊢ Continuous[instTopologicalSpaceProd, _] fun p ↦ ‖f a p.2 - f a p.1 - (L p.2 - L p.1)‖

Tactic:
fun_prop

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
b : ℝ
b_lt : b < s * r
hb : ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t, ‖f a z - f a y - (L z - L y)‖ ≤ b
ε : ℝ
εpos : 0 < ε
hε : b + 2 * ε < s * r
⊢ Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun p ↦ f a p.2

Tactic:
fun_prop

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

Goal state before:
pp: E : Type u_5
inst✝⁴ : NormedAddCommGroup E
μ : Measure ℝ
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : ConditionallyCompleteLinearOrder E
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
u : ℝ → E
a b : ℝ
hu : MonotoneOn u [[a, b]]
⊢ IntervalIntegrable u μ a b

Tactic:
intervalIntegrable_iff

Goal state after:
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
μ : Measure ℝ
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : ConditionallyCompleteLinearOrder E
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
u : ℝ → E
a b : ℝ
hu : MonotoneOn u [[a, b]]
⊢ IntegrableOn u (Ι a b) μ
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
b : ℝ
b_lt : b < s * r
hb : ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t, ‖f a z - f a y - (L z - L y)‖ ≤ b
ε : ℝ
εpos : 0 < ε
hε : b + 2 * ε < s * r
C : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun p ↦ f a p.2
D : EqOn (Function.uncurry f) (fun p ↦ f a p.2) ({a} ×ˢ closedBall x t)
v : Set (α × E)
v_open : IsOpen[instTopologicalSpaceProd] v
sub_v : {a} ×ˢ closedBall x t ⊆ v
hv : ∀ p ∈ v, dist (Function.uncurry f p) (f a p.2) < ε
w : Set α
w' : Set E
w_open : IsOpen[inst✝] w
sub_w : {a} ⊆ w
sub_w' : closedBall x t ⊆ w'
hww' : w ×ˢ w' ⊆ v
⊢ ∀ (p : α × E), p.1 ∈ w → p.2 ∈ closedBall x t → dist (Function.uncurry f p) (Function.uncurry f (a, p.2)) < ε

Tactic:
rintro ⟨b, y⟩ h hby

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
b✝ : ℝ
b_lt : b✝ < s * r
hb : ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t, ‖f a z - f a y - (L z - L y)‖ ≤ b✝
ε : ℝ
εpos : 0 < ε
hε : b✝ + 2 * ε < s * r
C : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun p ↦ f a p.2
D : EqOn (Function.uncurry f) (fun p ↦ f a p.2) ({a} ×ˢ closedBall x t)
v : Set (α × E)
v_open : IsOpen[instTopologicalSpaceProd] v
sub_v : {a} ×ˢ closedBall x t ⊆ v
hv : ∀ p ∈ v, dist (Function.uncurry f p) (f a p.2) < ε
w : Set α
w' : Set E
w_open : IsOpen[inst✝] w
sub_w : {a} ⊆ w
sub_w' : closedBall x t ⊆ w'
hww' : w ×ˢ w' ⊆ v
b : α
y : E
h : (b, y).1 ∈ w
hby : (b, y).2 ∈ closedBall x t
⊢ dist (Function.uncurry f (b, y)) (Function.uncurry f (a, (b, y).2)) < ε
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
μ : Measure ℝ
h : ¬IntervalIntegrable f μ a b
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0

Tactic:
intervalIntegrable_iff

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
h : ¬IntegrableOn f (Ι a b) μ
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 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
f : ℝ → E
c d : ℝ
⊢ c • ∫ (x : ℝ) in a..b, f (d - c * x) = ∫ (x : ℝ) in d - c * b..d - c * a, f x

Tactic:
by_cases hc : c = 0 <;> simp [hc, integral_comp_sub_mul]

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
x : X
t : ℝ
hx : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
ht : t ∈ Ioo a b
⊢ ∀ᵐ (t : ℝ) ∂μ.restrict (Ι b₀ t), ‖F x t - F x₀ t‖ ≤ 2 * bound t

Tactic:
apply (ae_restrict_of_ae_restrict_of_subset (hsub hb₀ ht) (hx.and hx₀)).mono

Goal state after:
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
x : X
t : ℝ
hx : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
ht : t ∈ Ioo a b
⊢ ∀ (x_1 : ℝ), ‖F x x_1‖ ≤ bound x_1 ∧ ‖F x₀ x_1‖ ≤ bound x_1 → ‖F x x_1 - F x₀ x_1‖ ≤ 2 * bound x_1
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ 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
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k volume
c : ℝ
hc : c ∈ s
d : ℝ
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hsub : [[c, d]] ⊆ k
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
hg_ae : ∀ᵐ (x : ℝ) ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖
hsub' : Ι c d ⊆ k
⊢ IntervalIntegrable (deriv f) volume c d

Tactic:
intervalIntegrable_iff

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ 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
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k volume
c : ℝ
hc : c ∈ s
d : ℝ
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hsub : [[c, d]] ⊆ k
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
hg_ae : ∀ᵐ (x : ℝ) ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖
hsub' : Ι c d ⊆ k
⊢ IntegrableOn (deriv f) (Ι c d) volume
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
hT : 0 < T
t : ℝ
μ : Measure ℝ
⊢ IsAddFundamentalDomain (↥(zmultiples T).op) (Ioc t (t + T)) μ

Tactic:
by
  refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_
  have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
    (Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_left_strictMono hT).injective).bijective
  refine (AddSubgroup.equivOp _).bijective.comp this |>.existsUnique_iff.2 ?_
  simpa using! existsUnique_add_zsmul_mem_Ioc hT x t

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

Goal state before:
pp: c : ℂ
R z : ℝ
hz : z ∈ univ
⊢ AnalyticAt ℝ (fun θ ↦ I) z

Tactic:
fun_prop

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
a₀ b₀ : ℝ
⊢ Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ ∫ (t : ℝ) in a₀..b₀, f x t ∂μ

Tactic:
fun_prop

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

Goal state before:
pp: R : ℝ
z w : ℂ
hw : w ∈ ball z R
this : ∀ (θ : ℝ), circleMap z R θ - w ≠ 0
⊢ Continuous fun θ ↦ circleMap z R θ - w

Tactic:
fun_prop

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

Goal state before:
pp: c : ℂ
R : ℝ
𝕜 : Type u_3
F : Type u_4
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
f : ℂ → F
g : ℂ → 𝕜
hf : CircleIntegrable f c R
hg : ContinuousOn g (sphere c |R|)
⊢ ContinuousOn (circleMap c R) [[0, 2 * π]]

Tactic:
fun_prop

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

Goal state before:
pp: c : ℂ
R : ℝ
𝕜 : Type u_3
F : Type u_4
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
f : ℂ → 𝕜
g : ℂ → F
hf : CircleIntegrable f c R
hg : ContinuousOn g (sphere c |R|)
⊢ ContinuousOn (circleMap c R) [[0, 2 * π]]

Tactic:
fun_prop

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

Goal state before:
pp: c : ℂ
R : ℝ
𝕜 : Type u_3
inst✝ : NormedRing 𝕜
f g : ℂ → 𝕜
hf : CircleIntegrable f c R
hg : ContinuousOn g (sphere c |R|)
⊢ ContinuousOn (circleMap c R) [[0, 2 * π]]

Tactic:
fun_prop

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

Goal state before:
pp: case hF.refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
ι : Type u_2
f : ι → ℂ → E
g : ℂ → E
c : ℂ
R : ℝ
l : Filter ι
inst✝ : l.IsCountablyGenerated
hR : 0 ≤ R
hf : ∀ᶠ (i : ι) in l, ContinuousOn (f i) (sphere c R)
h : TendstoUniformlyOn f g l (sphere c R)
i : ι
hi : ContinuousOn (f i) (sphere c R)
⊢ ContinuousOn (fun x ↦ circleMap 0 R x * I) [[0, 2 * π]]

Tactic:
fun_prop

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

Goal state before:
pp: case hF.refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
ι : Type u_2
f : ι → ℂ → E
g : ℂ → E
c : ℂ
R : ℝ
l : Filter ι
inst✝ : l.IsCountablyGenerated
hR : 0 ≤ R
hf : ∀ᶠ (i : ι) in l, ContinuousOn (f i) (sphere c R)
h : TendstoUniformlyOn f g l (sphere c R)
i : ι
hi : ContinuousOn (f i) (sphere c R)
⊢ ContinuousOn (circleMap c R) [[0, 2 * π]]

Tactic:
fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
μ : Measure α
p : ℝ≥0∞
f : α → ε
hf : MemLp f p μ
ν : Measure β
inst✝ : IsFiniteMeasure ν
hf' : MemLp f p (ν Set.univ • μ)
⊢ AEMeasurable Prod.fst (μ.prod ν)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
ν : Measure β
p : ℝ≥0∞
f : β → ε
hf : MemLp f p ν
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : SFinite ν
hf' : MemLp f p (μ Set.univ • ν)
⊢ AEMeasurable Prod.snd (μ.prod ν)

Tactic:
fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
w : ℂ
⊢ w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z =
    (2 * ↑π * I)⁻¹ • w ^ n • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z

Tactic:
rw [← smul_comm (w ^ n)]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
g : ↥(Lp E 1 (μ.prod ν))
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ‖∫ (y : β), ↑↑i (x, y) ∂ν - ∫ (y : β), ↑↑g (x, y) ∂ν‖ₑ ∂μ) (𝓝 g) (𝓝 0)

Tactic:
← lintegral_fn_integral_sub _ (L1.integrable_coeFn _) (L1.integrable_coeFn g)

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
g : ↥(Lp E 1 (μ.prod ν))
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ‖∫ (y : β), ↑↑i (x, y) - ↑↑g (x, y) ∂ν‖ₑ ∂μ) (𝓝 g) (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case refine_1.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
w : ℂ
hf : CircleIntegrable f c R
hw : ‖w‖ < R
hR : 0 < R
hwR : ‖w‖ / R ∈ Ico 0 1
n : ℕ
⊢ Measurable fun θ ↦ circleMap 0 R θ * I

Tactic:
fun_prop

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

Goal state before:
pp: case refine_1.hg.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
w : ℂ
hf : CircleIntegrable f c R
hw : ‖w‖ < R
hR : 0 < R
hwR : ‖w‖ / R ∈ Ico 0 1
n : ℕ
⊢ Measurable fun θ ↦ (w / (circleMap c R θ - c)) ^ n

Tactic:
fun_prop

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

Goal state before:
pp: case refine_1.hg.hg.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
w : ℂ
hf : CircleIntegrable f c R
hw : ‖w‖ < R
hR : 0 < R
hwR : ‖w‖ / R ∈ Ico 0 1
n : ℕ
⊢ Measurable fun θ ↦ (circleMap c R θ - c)⁻¹

Tactic:
fun_prop

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹⁶ : MeasurableSpace α
inst✝¹⁵ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹⁴ : SFinite ν
inst✝¹³ : SFinite μ
E : Type u_5
F : Type u_6
G : Type u_7
𝕜 : Type u_8
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : CompleteSpace F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
B : E →L[𝕜] F →L[𝕜] G
f : α → E
g : β → F
hf : Integrable f μ
hg : Integrable g ν
⊢ Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry fun x ↦ ⇑(B x))

Tactic:
fun_prop

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

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
n : ℕ
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
hp : ∀ z ∈ sphere c R, ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)
hq : iteratedDeriv n f c = n.factorial • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z
⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ (n + 1) • f z‖ ≤ R * (C / R ^ (n + 1))

Tactic:
exact circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const hR.le hp

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

Goal state before:
pp: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
⊢ f z = f w

Tactic:
by
  set g : ℂ → F := f ∘ fun t : ℂ => t • (w - z) + z
  suffices g 0 = g 1 by simpa [g]
  apply liouville_theorem_aux
  exacts [hf.comp ((differentiable_id.smul_const (w - z)).add_const z),
    hb.subset (range_comp_subset_range _ _)]

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case neg.inl
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : ¬p * q = 0
f g : (p * q).Gal
hfg :
  (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧
    (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g
x : (p * q).SplittingField
hx : x ∈ ↑(p.aroots (p * q).SplittingField + q.aroots (p * q).SplittingField).toFinset
h : x ∈ p.aroots (p * q).SplittingField
this : Fact (map (algebraMap F (p * q).SplittingField) p).Splits
⊢ f x = g x

Tactic:
have key :
      x =
        algebraMap p.SplittingField (p * q).SplittingField
          ((rootsEquivRoots p _).invFun
            ⟨x, (@Multiset.mem_toFinset _ (Classical.decEq _) _ _).mpr h⟩) :=
      Subtype.ext_iff.mp (Equiv.apply_symm_apply (rootsEquivRoots p _) ⟨x, _⟩).symm

Goal state after:
case neg.inl
F : Type u_1
inst✝ : Field F
p q : F[X]
hpq : ¬p * q = 0
f g : (p * q).Gal
hfg :
  (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧
    (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g
x : (p * q).SplittingField
hx : x ∈ ↑(p.aroots (p * q).SplittingField + q.aroots (p * q).SplittingField).toFinset
h : x ∈ p.aroots (p * q).SplittingField
this : Fact (map (algebraMap F (p * q).SplittingField) p).Splits
key :
  x = (algebraMap p.SplittingField (p * q).SplittingField) ↑((rootsEquivRoots p (p * q).SplittingField).invFun ⟨x, ⋯⟩)
⊢ f x = g x
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_1
R : Type u_3
S : Type u_4
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
φ : MvPolynomial σ R
hφ : φ.IsSymmetric
f : R →+* S
e : Perm σ
⊢ (rename ⇑e) ((MvPolynomial.map f) φ) = (MvPolynomial.map f) φ

Tactic:
← map_rename,

Goal state after:
σ : Type u_1
R : Type u_3
S : Type u_4
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
φ : MvPolynomial σ R
hφ : φ.IsSymmetric
f : R →+* S
e : Perm σ
⊢ (MvPolynomial.map f) ((rename ⇑e) φ) = (MvPolynomial.map f) φ
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : Semiring R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalSemiring R
inst✝ : Semiring S
p : S[X]
f : S →+* R
⊢ Continuous fun x ↦ eval₂ f x p

Tactic:
by
  simp only [eval₂_eq_sum]
  exact continuous_finsetSum _ fun c _ => continuous_const.mul (continuous_pow _)

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
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
⊢ w ∈ closure (ball c R \ s)

Tactic:
refine mem_closure_iff_nhds.2 fun t ht => ?_

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
⊢ (t ∩ (ball c R \ s)).Nonempty
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x ↦ w + ↑x
⊢ Continuous g

Tactic:
fun_prop

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

Goal state before:
pp: R : Type u_2
inst✝² : NormedRing R
inst✝¹ : IsAbsoluteValue norm
inst✝ : ProperSpace R
p : R[X]
h : 0 < p.degree
⊢ Continuous fun x ↦ eval x p

Tactic:
fun_prop

Goal state after: