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

Goal state before:
pp: R : Type u_1
E : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
s : Set (Submodule R E)
⊢ sSup (ofSubmodule '' s) = ⨆ S ∈ s, ↑S

Tactic:
sSup_eq_iSup,

Goal state after:
R : Type u_1
E : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
s : Set (Submodule R E)
⊢ ⨆ a ∈ ofSubmodule '' s, a = ⨆ S ∈ s, ↑S
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 : ¬Even m
⊢ term m = 0

Tactic:
← Nat.even_add_one

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 : Even (m + 1)
⊢ term m = 0
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
hs : Convex ℝ s
absorbs : Absorbent ℝ s
x y : E
⊢ gauge s (x + y) ≤ gauge s x + gauge s y

Tactic:
refine le_of_forall_pos_lt_add fun ε hε => ?_
  obtain ⟨a, ha, ha', x, hx, rfl⟩ :=
    exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s x) (half_pos hε))
  obtain ⟨b, hb, hb', y, hy, rfl⟩ :=
    exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s y) (half_pos hε))
  calc
    gauge s (a • x + b • y) ≤ a + b := gauge_le_of_mem (by positivity) <| by
      rw [hs.add_smul ha.le hb.le]
      exact add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy)
    _ < gauge s (a • x) + gauge s (b • y) + ε := by linarith

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

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
x : E
ε : ℝ
hε : 0 < ε
hs₂ : IsOpen[inst✝¹] s
hx : x ∈ ε • s
this : ε⁻¹ • x ∈ s
⊢ gauge s x < ε

Tactic:
have h_gauge_lt := gauge_lt_one_of_mem_of_isOpen hs₂ this

Goal state after:
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
x : E
ε : ℝ
hε : 0 < ε
hs₂ : IsOpen[inst✝¹] s
hx : x ∈ ε • s
this : ε⁻¹ • x ∈ s
h_gauge_lt : gauge s (ε⁻¹ • x) < 1
⊢ gauge s x < ε
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
s : Set E
inst✝⁴ : RCLike 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsScalarTower ℝ 𝕜 E
hs₀ : Balanced 𝕜 s
hs₁ : Convex ℝ s
hs₂ : Absorbent ℝ s
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : IsOpen[inst✝¹] s
⊢ (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s

Tactic:
rw [Seminorm.ball_zero_eq]
  exact setOfPred_gauge_lt_one_eq_self_of_isOpen hs₁ hs₂.zero_mem hs

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

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup V
inst✝³ : TopologicalSpace V
inst✝² : TopologicalSpace R
inst✝¹ : Module R V
inst✝ : SeparatingDual R V
x y : V
⊢ x = y ↔ ∀ (g : StrongDual R V), g x = g y

Tactic:
rw [← sub_eq_zero, eq_zero_iff_forall_dual_eq_zero (R := R) (x - y)]
  simp [sub_eq_zero]

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

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁵ : Field R
inst✝⁴ : AddCommGroup V
inst✝³ : TopologicalSpace R
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalRing R
inst✝ : Module R V
v : V
⊢ (¬∃ f, ¬f v = 0) → v = 0 ↔ (∀ (x : V →L[R] R), ((ContinuousLinearMap.coeLM R).flip v) x = 0 x) → v = 0

Tactic:
push Not

Goal state after:
R : Type u_1
V : Type u_2
inst✝⁵ : Field R
inst✝⁴ : AddCommGroup V
inst✝³ : TopologicalSpace R
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalRing R
inst✝ : Module R V
v : V
⊢ (∀ (f : StrongDual R V), f v = 0) → v = 0 ↔
    (∀ (x : V →L[R] R), ((ContinuousLinearMap.coeLM R).flip v) x = 0 x) → v = 0
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case neg
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs : Convex ℝ s
ht : Convex ℝ t
hst : Disjoint (interior s) t
hsint : (interior s).Nonempty
htne : ¬t.Nonempty
⊢ ∃ f u, (∀ a ∈ s, f a ≤ u) ∧ ∀ b ∈ t, u ≤ f b

Tactic:
· exact ⟨0, 0, by simp⟩

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

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
s t : Set E
inst✝⁴ : RCLike 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsScalarTower ℝ 𝕜 E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
hs : Convex ℝ s
ht : Convex ℝ t
hst : Disjoint (interior s) t
hsint : (interior s).Nonempty
htne : ¬t.Nonempty
⊢ ∃ f u, (∀ a ∈ s, re (f a) ≤ u) ∧ ∀ b ∈ t, u ≤ re (f b)

Tactic:
· exact ⟨0, 0, by simp⟩

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
R : Type u_3
inst✝⁵ : Zero X
inst✝⁴ : Zero Y
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace R
g : C(Y, R)₀
⊢ Continuous g.comp

Tactic:
rw [ContinuousMapZero.isEmbedding_toContinuousMap.continuous_iff]

Goal state after:
X : Type u_1
Y : Type u_2
R : Type u_3
inst✝⁵ : Zero X
inst✝⁴ : Zero Y
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace R
g : C(Y, R)₀
⊢ Continuous (_root_.toContinuousMap ∘ g.comp)
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
inst✝⁶ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
φ : X → H →L[𝕜] E
φ_int : Integrable φ μ
v : H
hE : ¬CompleteSpace E
hH : Nontrivial H
⊢ ¬CompleteSpace (H →L[𝕜] E)

Tactic:
SeparatingDual.completeSpace_continuousLinearMap_iff

Goal state after:
X : Type u_1
E : Type u_3
inst✝⁶ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
φ : X → H →L[𝕜] E
φ_int : Integrable φ μ
v : H
hE : ¬CompleteSpace E
hH : Nontrivial H
⊢ ¬CompleteSpace E
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_3
α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a ↦ ‖f a‖
h_int_eq : (fun i ↦ ∫ (a : α) in s i, f a ∂μ) = fun i ↦ ∫ (a : α), (s i).indicator f a ∂μ
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, (s n).indicator (fun a ↦ ‖f a‖) a ≤ bound a

Tactic:
refine fun n => Eventually.of_forall fun x => ?_

Goal state after:
case refine_3
α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a ↦ ‖f a‖
h_int_eq : (fun i ↦ ∫ (a : α) in s i, f a ∂μ) = fun i ↦ ∫ (a : α), (s i).indicator f a ∂μ
n : ℕ
x : α
⊢ (s n).indicator (fun a ↦ ‖f a‖) x ≤ bound x
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case e_μ
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
⊢ (μ.withDensity fun x ↦ ↑(f x)) = μ.withDensity fun x ↦ ↑(f' x)

Tactic:
apply withDensity_congr_ae

Goal state after:
case e_μ
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 := ⋯
⊢ (fun x ↦ ↑(f x)) =ᵐ[μ] fun x ↦ ↑(f' x)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case inr.h_bound
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ ∀ᶠ (n : ι) in l, ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → ‖F n x‖ ≤ C + 1

pp: case inr.bound_integrable
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ IntervalIntegrable (fun x ↦ C + 1) μ a b

pp: case inr.h_lim
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → Tendsto (fun n ↦ F n x) l (𝓝 (f x))

Tactic:
case h_bound =>
    have := uniformContinuous_norm.comp_tendstoUniformlyOn h_lim
      |>.eventually_forall_le (show C < C + 1 by simp) (by simpa [upperBounds] using hC)
    exact this.mono fun i hi ↦ .of_forall fun x hx ↦ hi x <| uIoc_subset_uIcc hx

Goal state after:
case inr.bound_integrable
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ IntervalIntegrable (fun x ↦ C + 1) μ a b

case inr.h_lim
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → Tendsto (fun n ↦ F n x) l (𝓝 (f x))
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
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ C * |b - a| = C * volume.real (Ι a b)

pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ volume (Ι a b) < ∞

Tactic:
· rw [uIoc, Real.volume_real_Ioc_of_le inf_le_sup, max_sub_min_eq_abs]

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ volume (Ι a b) < ∞
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
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, δ ∈ Ioo 0 1
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
⊢ Tendsto (fun δ ↦ (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b)) (𝓝 0)
    (𝓝 ((M + 1) * ENNReal.toReal 0 + 0 * μ.real (Icc a b)))

Tactic:
apply Tendsto.add (Tendsto.mul tendsto_const_nhds _)
        (Tendsto.mul tendsto_id tendsto_const_nhds)

Goal state after:
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
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, δ ∈ Ioo 0 1
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
⊢ Tendsto (fun x ↦ μ.real (Icc (b₀ - x) (b₀ + x))) (𝓝 0) (𝓝 (ENNReal.toReal 0))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

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

pp: case inr
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hint : IntervalIntegrable f' volume a b
hab : b ≤ a
⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a

Tactic:
· simp only [uIcc_of_le, min_eq_left, max_eq_right, hab] at hcont hderiv hint
    apply integral_eq_sub_of_hasDeriv_right_of_le hab hcont hderiv hint

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 f' : ℝ → E
hcont : ContinuousOn f [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hint : IntervalIntegrable f' volume a b
hab : b ≤ a
⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a
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
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
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
x : ℝ
hx : DifferentiableAt ℝ f x
⊢ deriv f' x = a (deriv f x)

Tactic:
fderiv_comp_deriv x _ hx

Goal state after:
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
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
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
x : ℝ
hx : DifferentiableAt ℝ f x
⊢ (fderiv ℝ (⇑a) (f x)) (deriv f x) = a (deriv f x)

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
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
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
x : ℝ
hx : DifferentiableAt ℝ f x
⊢ DifferentiableAt ℝ (⇑a) (f x)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a₁ a₂ a₃ : ℝ
h : a₂ ∈ Icc a₁ a₃
⊢ ∫ (x : ℝ) in a₁..a₃, {x | x ≤ a₂}.indicator f x ∂μ = ∫ (x : ℝ) in a₁..a₂, f x ∂μ

Tactic:
have : {x | x ≤ a₂} ∩ Ioc a₁ a₃ = Ioc a₁ a₂ := Iic_inter_Ioc_of_le h.2

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a₁ a₂ a₃ : ℝ
h : a₂ ∈ Icc a₁ a₃
this : {x | x ≤ a₂} ∩ Ioc a₁ a₃ = Ioc a₁ a₂
⊢ ∫ (x : ℝ) in a₁..a₃, {x | x ≤ a₂}.indicator f x ∂μ = ∫ (x : ℝ) in a₁..a₂, f x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: case inr.inr
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
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T t : ℝ},
    Periodic f T →
      T ≠ 0 → IntervalIntegrable f volume t (t + T) → ∀ (a₁ a₂ : ℝ), 0 < T → IntervalIntegrable f volume a₁ a₂
hT : ¬0 < T
h : T < 0
hnT : 0 < -T
⊢ IntervalIntegrable f volume a₁ a₂

Tactic:
(by ring : t = (t + T) + (-T))

Goal state after:
case inr.inr
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) (t + T)
a₁ a₂ : ℝ
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T t : ℝ},
    Periodic f T →
      T ≠ 0 → IntervalIntegrable f volume t (t + T) → ∀ (a₁ a₂ : ℝ), 0 < T → IntervalIntegrable f volume a₁ a₂
hT : ¬0 < T
h : T < 0
hnT : 0 < -T
⊢ IntervalIntegrable f volume a₁ a₂
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
⊢ MemLp (fun x ↦ f x.2) p (μ.prod ν)

Tactic:
have hf' : MemLp f p (μ .univ • ν) := hf.smul_measure (by simp)

Goal state after:
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)
⊢ MemLp (fun x ↦ f x.2) p (μ.prod ν)
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:

Tactic:
‖(∏ j, (I.upper j - I.lower j)) • f' (Pi.single i 1) -
            (integral (I.face i) ⊥ (f ∘ e (I.upper i)) BoxAdditiveMap.volume -
              integral (I.face i) ⊥ (f ∘ e (I.lower i)) BoxAdditiveMap.volume)‖ =
        ‖integral.{0, u, u} (I.face i) ⊥
            (fun x : Fin n → ℝ =>
              f' (Pi.single i (I.upper i - I.lower i)) -
                (f (e (I.upper i) x) - f (e (I.lower i) x)))
            BoxAdditiveMap.volume‖ := by
      rw [← integral_sub (Hi _ Hu) (Hi _ Hl), ← Box.volume_face_mul i, mul_smul, ← Box.volume_apply,
        ← BoxAdditiveMap.toSMul_apply, ← integral_const, ← BoxAdditiveMap.volume,
        ← integral_sub (integrable_const _) ((Hi _ Hu).sub (Hi _ Hl))]
      simp only [(· ∘ ·), Pi.sub_def, ← f'.map_smul, ← Pi.single_smul', smul_eq_mul, mul_one]
    _ ≤ (volume (I.face i : Set (Fin n → ℝ))).toReal * (2 * ε * c * (I.upper i - I.lower i)) := by
      -- The hard part of the estimate was done above, here we just replace `diam I.Icc`
      -- with `c * (I.upper i - I.lower i)`
      refine norm_integral_le_of_le_const (fun y hy => (this y hy).trans ?_) volume
      rw [mul_assoc (2 * ε)]
      gcongr
      exact I.diam_Icc_le_of_distortion_le i hc
    _ = 2 * ε * c * ∏ j, (I.upper j - I.lower j) := by
      rw [← measureReal_def, ← Measure.toBoxAdditive_apply, Box.volume_apply,
        ← I.volume_face_mul i]
      ac_rfl

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : s.Countable
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z ↦ I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) volume
e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdCLM.symm
he₁ : e (1, 0) = 1
he₂ : e (0, 1) = I
he : ∀ (x y : ℝ), True
F : ℝ × ℝ → E := f ∘ ⇑e
F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p ↦ f' (e p) ∘SL ↑e
hF' : ∀ (p : ℝ × ℝ), (-(I • F' p)) (1, 0) + (F' p) (0, 1) = -(I • (f' (e p)) 1 - (f' (e p)) I)
R : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]
t : Set (ℝ × ℝ) := ⇑e ⁻¹' s
hR : ⇑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R
htc : ContinuousOn F R
htd : ∀ p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \ t, HasFDerivAt F (F' p) p
⊢ (((∫ (x : ℝ) in z.re..w.re, f (e (x, z.im))) - ∫ (x : ℝ) in z.re..w.re, f (e (x, w.im))) +
        ∫ (x : ℝ) in w.im..z.im, -(I • f (e (w.re, x)))) -
      ∫ (x : ℝ) in w.im..z.im, -(I • f (e (z.re, x))) =
    ∫ (x : ℝ) in z.re..w.re, ∫ (x_1 : ℝ) in w.im..z.im, (-(I • F' (x, x_1))) (1, 0) + (F' (x, x_1)) (0, 1)

Tactic:
refine (integral2_divergence_prod_of_hasFDerivAt_off_countable (fun p => -(I • F p)) F
    (fun p => -(I • F' p)) F' z.re w.im w.re z.im t (hs.preimage e.injective)
    (htc.const_smul _).neg htc (fun p hp => ((htd p hp).const_smul I).neg) htd ?_).symm

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : s.Countable
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z ↦ I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) volume
e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdCLM.symm
he₁ : e (1, 0) = 1
he₂ : e (0, 1) = I
he : ∀ (x y : ℝ), True
F : ℝ × ℝ → E := f ∘ ⇑e
F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p ↦ f' (e p) ∘SL ↑e
hF' : ∀ (p : ℝ × ℝ), (-(I • F' p)) (1, 0) + (F' p) (0, 1) = -(I • (f' (e p)) 1 - (f' (e p)) I)
R : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]
t : Set (ℝ × ℝ) := ⇑e ⁻¹' s
hR : ⇑e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R
htc : ContinuousOn F R
htd : ∀ p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \ t, HasFDerivAt F (F' p) p
⊢ IntegrableOn (fun x ↦ (-(I • F' x)) (1, 0) + (F' x) (0, 1)) ([[z.re, w.re]] ×ˢ [[w.im, z.im]]) volume