mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}

Tactic:
let L0 : ℕ → F := fun e => L e (n e) (n e)

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
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)
⊢ x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
inst✝¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝ : RCLike 𝕜
f : X → 𝕜
hf : Integrable f μ
⊢ ∫ (x : X), ↑(re (f x)) ∂μ + I • ∫ (x : X), ↑(im (f x)) ∂μ = ∫ (x : X), f x ∂μ

Tactic:
← integral_smul,

Goal state after:
X : Type u_1
inst✝¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝ : RCLike 𝕜
f : X → 𝕜
hf : Integrable f μ
⊢ ∫ (x : X), ↑(re (f x)) ∂μ + ∫ (a : X), I • ↑(im (f a)) ∂μ = ∫ (x : X), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℝ → E
c : E
l l' : Filter ℝ
lt : Filter ι
μ : Measure ℝ
u v : ι → ℝ
inst✝² : CompleteSpace E
inst✝¹ : l'.IsMeasurablyGenerated
inst✝ : TendstoIxxClass Ioc l l'
hfm : StronglyMeasurableAtFilter f l' μ
hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)
hl : μ.FiniteAtFilter l'
hu : Tendsto u lt l
hv : Tendsto v lt l
huv : u ≤ᶠ[lt] v
x : ι
hx : u x ≤ v x
⊢ (fun t ↦ ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) x =
    (fun t ↦ ∫ (x : ℝ) in u t..v t, f x ∂μ - μ.real (Ioc (u t) (v t)) • c) x

Tactic:
by simp [integral_const', hx]

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

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℝ → E
c : E
l l' : Filter ℝ
lt : Filter ι
μ : Measure ℝ
u v : ι → ℝ
inst✝² : CompleteSpace E
inst✝¹ : l'.IsMeasurablyGenerated
inst✝ : TendstoIxxClass Ioc l l'
hfm : StronglyMeasurableAtFilter f l' μ
hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)
hl : μ.FiniteAtFilter l'
hu : Tendsto u lt l
hv : Tendsto v lt l
huv : u ≤ᶠ[lt] v
x : ι
hx : u x ≤ v x
⊢ (fun t ↦ ∫ (x : ℝ) in u t..v t, 1 ∂μ) x = (fun t ↦ μ.real (Ioc (u t) (v t))) x

Tactic:
by simp [integral_const', hx]

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
⊢ IntervalIntegrable f volume 0 t

Tactic:
rcases lt_trichotomy t 0 with h | h | h

Goal state after:
case inl
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : t < 0
⊢ IntervalIntegrable f volume 0 t

case inr.inl
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : t = 0
⊢ IntervalIntegrable f volume 0 t

case inr.inr
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : 0 < t
⊢ IntervalIntegrable f volume 0 t
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), -f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
⊢ IntervalIntegrable f volume 0 t

Tactic:
rcases lt_trichotomy t 0 with h | h | h

Goal state after:
case inl
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), -f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : t < 0
⊢ IntervalIntegrable f volume 0 t

case inr.inl
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), -f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : t = 0
⊢ IntervalIntegrable f volume 0 t

case inr.inr
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), -f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : 0 < t
⊢ IntervalIntegrable f volume 0 t
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

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

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

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
f : ℝ → E
a b : ℝ
μ : Measure ℝ
⊢ ‖∫ (x : ℝ) in a..b, f x ∂μ‖ = ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖

Tactic:
simp_rw [intervalIntegral_eq_integral_uIoc, norm_smul]

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
μ : Measure ℝ
⊢ ‖if a ≤ b then 1 else -1‖ * ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖ = ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖
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 + x / c) = c • ∫ (x : ℝ) in d + a / c..d + b / c, f x

Tactic:
simpa only [div_eq_inv_mul, inv_inv] using integral_comp_add_mul f (inv_ne_zero hc) d

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
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 : max a₀ b₀ < b
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist (∫ (t : ℝ) in a₀..x.2, f x.1 t ∂μ) (∫ (t : ℝ) in a₀..(q, b₀).2, f (q, b₀).1 t ∂μ) < ε

Tactic:
rw [max_lt_iff] at lt_b

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
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist (∫ (t : ℝ) in a₀..x.2, f x.1 t ∂μ) (∫ (t : ℝ) in a₀..(q, b₀).2, f (q, b₀).1 t ∂μ) < ε
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case e'_2
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : IsScalarTower ℝ 𝕜 E
f f' : 𝕜 → E
z₀ z₁ : 𝕜
hcont : ContinuousOn (fun t ↦ f' (z₀ + t • z₁)) (Icc 0 1)
hderiv : ∀ t ∈ Icc 0 1, HasDerivAt f (f' (z₀ + t • z₁)) (z₀ + t • z₁)
γ : ℝ → 𝕜 := fun t ↦ z₀ + t • z₁
hint : IntervalIntegrable (z₁ • f' ∘ γ) volume 0 1
hderiv' : ∀ t ∈ [[0, 1]], HasDerivAt (f ∘ γ) (z₁ • (f' ∘ γ) t) t
⊢ z₁ • ∫ (t : ℝ) in 0..1, f' (z₀ + t • z₁) = ∫ (y : ℝ) in 0..1, z₁ • (f' ∘ γ) y

Tactic:
← integral_smul,

Goal state after:
case e'_2
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : RCLike 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : IsScalarTower ℝ 𝕜 E
f f' : 𝕜 → E
z₀ z₁ : 𝕜
hcont : ContinuousOn (fun t ↦ f' (z₀ + t • z₁)) (Icc 0 1)
hderiv : ∀ t ∈ Icc 0 1, HasDerivAt f (f' (z₀ + t • z₁)) (z₀ + t • z₁)
γ : ℝ → 𝕜 := fun t ↦ z₀ + t • z₁
hint : IntervalIntegrable (z₁ • f' ∘ γ) volume 0 1
hderiv' : ∀ t ∈ [[0, 1]], HasDerivAt (f ∘ γ) (z₁ • (f' ∘ γ) t) t
⊢ ∫ (x : ℝ) in 0..1, z₁ • f' (z₀ + x • z₁) = ∫ (y : ℝ) in 0..1, z₁ • (f' ∘ γ) y
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hfi : IntervalIntegrable f volume a b
hpos : ∀ x ∈ Ioo a b, 0 < f x
hab : a < b
hsupp : Ioo a b ⊆ support f ∩ Ioc a b
h₀ : 0 ≤ᵐ[volume.restrict (Ι a b)] f
⊢ a < b ∧ 0 < volume (support f ∩ Ioc a b)

Tactic:
exact ⟨hab, ((Measure.measure_Ioo_pos _).mpr hab).trans_le (measure_mono hsupp)⟩

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

Goal state before:
pp: f g : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hgc : ContinuousOn g (Icc a b)
hle : ∀ x ∈ Ioc a b, f x ≤ g x
hlt : ∃ c ∈ Icc a b, f c < g c
⊢ ∫ (x : ℝ) in a..b, f x < ∫ (x : ℝ) in a..b, g x

Tactic:
apply integral_lt_integral_of_ae_le_of_measure_setOfPred_lt_ne_zero hab.le
    (hfc.intervalIntegrable_of_Icc hab.le) (hgc.intervalIntegrable_of_Icc hab.le)
  · simpa only [measurableSet_Ioc, ae_restrict_eq]
      using! (ae_restrict_mem measurableSet_Ioc).mono hle
  contrapose! hlt
  have h_eq : f =ᵐ[volume.restrict (Ioc a b)] g := by
    simp only [← not_le, ← ae_iff] at hlt
    exact EventuallyLE.antisymm ((ae_restrict_iff' measurableSet_Ioc).2 <|
      Eventually.of_forall hle) hlt
  rw [Measure.restrict_congr_set Ioc_ae_eq_Icc] at h_eq
  exact fun c hc ↦ (Measure.eqOn_Icc_of_ae_eq volume hab.ne h_eq hfc hgc hc).ge

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁴ : Ring α
inst✝³ : LinearOrder α
inst✝² : FloorRing α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
x : α
⊢ ↑⌊x⌋ ≤ x

Tactic:
exact floor_le _

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
⊢ ∫⁻ (a : ℝ) in Ioc t (t + T), f ↑a = ∫⁻ (b : AddCircle T), f b

Tactic:
rw [← (AddCircle.measurePreserving_mk T t).map_eq]

Goal state after:
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
⊢ ∫⁻ (a : ℝ) in Ioc t (t + T), f ↑a =
    ∫⁻ (b : AddCircle T), f b ∂Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T)))
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

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

Tactic:
intro y

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

Goal state before:
pp: case convert_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
y : Fin n → ℝ
hy : y ∈ Box.Icc (I.face i)
g : (Fin (n + 1) → ℝ) → E
hε : ∀ y ∈ Box.Icc I, ‖g y‖ ≤ ε * ‖y - x‖
hfc : ContinuousOn (fun y ↦ a + f' (y - x) + g y) (Box.Icc I)
Hi :
  ∀ x_1 ∈ Set.Icc (I.lower i) (I.upper i),
    Integrable (I.face i) ⊥ ((fun y ↦ a + f' (y - x) + g y) ∘ e x_1) BoxAdditiveMap.volume
hg : g = fun y ↦ (fun y ↦ a + f' (y - x) + g y) y - a - f' (y - x)
this : ∀ z ∈ Set.Icc (I.lower i) (I.upper i), e z y ∈ Box.Icc I
⊢ ‖g (e (I.lower i) y) - g (e (I.upper i) y)‖ ≤ 2 * ε * diam (Box.Icc I)

Tactic:
replace hε : ∀ y ∈ (Box.Icc I), ‖g y‖ ≤ ε * diam (Box.Icc I) := by
        intro y hy
        refine (hε y hy).trans (mul_le_mul_of_nonneg_left ?_ h0.le)
        rw [← dist_eq_norm]
        exact dist_le_diam_of_mem I.isCompact_Icc.isBounded hy hxI

Goal state after:
case convert_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
c : ℝ≥0
hc : I.distortion ≤ c
e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth
Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)
Hu : I.upper i ∈ Set.Icc (I.lower i) (I.upper i)
y : Fin n → ℝ
hy : y ∈ Box.Icc (I.face i)
g : (Fin (n + 1) → ℝ) → E
hfc : ContinuousOn (fun y ↦ a + f' (y - x) + g y) (Box.Icc I)
Hi :
  ∀ x_1 ∈ Set.Icc (I.lower i) (I.upper i),
    Integrable (I.face i) ⊥ ((fun y ↦ a + f' (y - x) + g y) ∘ e x_1) BoxAdditiveMap.volume
hg : g = fun y ↦ (fun y ↦ a + f' (y - x) + g y) y - a - f' (y - x)
this : ∀ z ∈ Set.Icc (I.lower i) (I.upper i), e z y ∈ Box.Icc I
hε : ∀ y ∈ Box.Icc I, ‖g y‖ ≤ ε * diam (Box.Icc I)
⊢ ‖g (e (I.lower i) y) - g (e (I.upper i) y)‖ ≤ 2 * ε * diam (Box.Icc I)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
⊢ sInf ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x ≤ ∫ (x : ℝ) in 0..t, g x

Tactic:
let h'_int := hg.intervalIntegrable₀ hT.ne' h_int

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

Goal state before:
pp: T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
⊢ ∫ (x : ℝ) in 0..t, g x ≤ sSup ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x

Tactic:
let h'_int := hg.intervalIntegrable₀ hT.ne' h_int

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : (μ.prod ν) s ≠ ∞
x : α
hx : ν (Prod.mk x ⁻¹' s) ≠ ∞
⊢ ENNReal.ofReal (ν (Prod.mk x ⁻¹' s)).toReal = ν (Prod.mk x ⁻¹' s)

Tactic:
simp [ofReal_toReal, hx]

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✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : AEStronglyMeasurable f (μ.prod ν)
x : α
hx : ∀ᵐ (y : β) ∂ν, AEStronglyMeasurable.mk f h1f (x, y) = f (x, y)
⊢ HasFiniteIntegral (fun y ↦ AEStronglyMeasurable.mk f h1f (x, y)) ν ↔ HasFiniteIntegral (fun y ↦ f (x, y)) ν

Tactic:
exact hasFiniteIntegral_congr hx

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

Goal state before:
pp: E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ∞)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ∞)
x : X
hx : x ∉ U
⊢ ∀ (y : Y), f x y = 0

Tactic:
intro y

Goal state after:
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ∞)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ∞)
x : X
hx : x ∉ U
y : Y
⊢ f x y = 0
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case inl
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.univ.pi fun i ↦ Set.Ioc (a i) (b i), ∑ i, (f' x) (e i) i ∂Measure.pi fun x ↦ volume) =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (b i) x) i) -
        ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (a i) x) i)

Tactic:
have : (pi Set.univ fun j => Ioc (a j) (b j)) = ∅ := univ_pi_eq_empty hi'

Goal state after:
case inl
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.univ.pi fun i ↦ Set.Ioc (a i) (b i), ∑ i, (f' x) (e i) i ∂Measure.pi fun x ↦ volume) =
    ∑ i,
      ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (b i) x) i) -
        ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (a i) x) i)
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
i : Fin n
hi : a (j.succAbove i) = b (j.succAbove i)
hi' : Set.Ioc (a (j.succAbove i)) (b (j.succAbove i)) = ∅
hne : j.succAbove i ≠ j
⊢ volume (Set.Icc (a ∘ j.succAbove) (b ∘ j.succAbove)) = 0

Tactic:
Real.volume_Icc_pi,

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
i : Fin n
hi : a (j.succAbove i) = b (j.succAbove i)
hi' : Set.Ioc (a (j.succAbove i)) (b (j.succAbove i)) = ∅
hne : j.succAbove i ≠ j
⊢ ∏ i, ENNReal.ofReal ((b ∘ j.succAbove) i - (a ∘ j.succAbove) i) = 0
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case right
F : Type u_1
inst✝² : Field F
p : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : Fact (map (algebraMap F E) p).Splits
⊢ Function.Surjective (mapRoots p E)

Tactic:
intro y

Goal state after:
case right
F : Type u_1
inst✝² : Field F
p : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : Fact (map (algebraMap F E) p).Splits
y : ↑(p.rootSet E)
⊢ ∃ a, mapRoots p E a = y
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
⊢ psum σ R 0 = ↑(Fintype.card σ)

Tactic:
by simp [psum]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
⊢ psum σ R 1 = ∑ i, X i

Tactic:
by simp [psum]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
σ : Type u_2
s : Finset σ
r : σ → R
k : ℕ
h : k ≤ #s
⊢ (∏ i ∈ s, (X + C (r i))).coeff k = ∑ t ∈ Finset.powersetCard (#s - k) s, ∏ i ∈ t, r i

Tactic:
Finset.prod,

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
σ : Type u_2
s : Finset σ
r : σ → R
k : ℕ
h : k ≤ #s
⊢ (map (fun i ↦ X + C (r i)) s.val).prod.coeff k = ∑ t ∈ Finset.powersetCard (#s - k) s, ∏ i ∈ t, r i