mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.RCLike.Extend

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
F : Type u_2
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : Module ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
g : StrongDual ℝ F
x : F
⊢ im (g.extendRCLike x) = -g (I • x)

Tactic:
obtain (h | h) := RCLike.I_eq_zero_or_im_I_eq_one (K := 𝕜)
  all_goals simp [h, extendRCLike_apply]

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
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set E
x : E
hx : x ∈ interior s
H₁ : Tendsto (fun r ↦ r⁻¹ • x) (𝓝[<] 1) (𝓝 (1⁻¹ • x))
⊢ x ∈ {x | gauge s x < 1}

Tactic:
inv_one,

Goal state after:
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set E
x : E
hx : x ∈ interior s
H₁ : Tendsto (fun r ↦ r⁻¹ • x) (𝓝[<] 1) (𝓝 (1 • x))
⊢ x ∈ {x | gauge s x < 1}
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : s ∈ 𝓝 0
⊢ {x | gauge s x < 1} = interior s

Tactic:
refine Subset.antisymm (fun x hx ↦ ?_) (interior_subset_gauge_lt_one s)

Goal state after:
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : s ∈ 𝓝 0
x : E
hx : x ∈ {x | gauge s x < 1}
⊢ x ∈ interior s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
r : ℝ
x : E
hs : Absorbent ℝ s
hr : 0 ≤ r
hsr : s ⊆ closedBall 0 r
⊢ ‖x‖ / r ≤ gauge s x

Tactic:
rw [← gauge_closedBall hr]

Goal state after:
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
r : ℝ
x : E
hs : Absorbent ℝ s
hr : 0 ≤ r
hsr : s ⊆ closedBall 0 r
⊢ gauge (closedBall 0 r) x ≤ gauge s x
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case right
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
L : F
r ε x r' : ℝ
rr' : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : ℝ
hx' : x' ∈ Ioo x (x + r' - s)
A : Icc x' (x' + s) ⊆ Icc x (x + r')
y : ℝ
hy : y ∈ Icc x' (x' + s)
⊢ ∀ z ∈ Icc x' (x' + s), ‖f z - f y - (z - y) • L‖ ≤ ε * r

Tactic:
z

Goal state after:
case right
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
L : F
r ε x r' : ℝ
rr' : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : ℝ
hx' : x' ∈ Ioo x (x + r' - s)
A : Icc x' (x' + s) ⊆ Icc x (x + r')
y : ℝ
hy : y ∈ Icc x' (x' + s)
z : ℝ
⊢ z ∈ Icc x' (x' + s) → ‖f z - f y - (z - y) • L‖ ≤ ε * r
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ ‖L₁ - L₂‖ ≤ 4 * ε

Tactic:
by
  suffices H : ‖(r / 2) • (L₁ - L₂)‖ ≤ r / 2 * (4 * ε) by
    rwa [norm_smul, Real.norm_of_nonneg (half_pos hr).le, mul_le_mul_iff_right₀ (half_pos hr)] at H
  calc
    ‖(r / 2) • (L₁ - L₂)‖ =
        ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂ -
            (f (x + r / 2) - f x - (x + r / 2 - x) • L₁)‖ := by
      simp [smul_sub]
    _ ≤ ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂‖ +
          ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₁‖ :=
      norm_sub_le _ _
    _ ≤ ε * r + ε * r := by
      apply add_le_add
      · apply le_of_mem_A h₂ <;> simp [(half_pos hr).le]
      · apply le_of_mem_A h₁ <;> simp [(half_pos hr).le]
    _ = r / 2 * (4 * ε) := by ring

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

Goal state before:
pp: case pos.refine_2
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
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
u u' : X → E
a✝ : Disjoint (Function.support u) (Function.support u')
u_int : Integrable u (μ.withDensity fun x ↦ ↑(f x))
u'_int : Integrable u' (μ.withDensity fun x ↦ ↑(f x))
⊢ (∫ (x : X), u x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • u x ∂μ →
    (∫ (x : X), u' x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • u' x ∂μ →
      (∫ (x : X), (u + u') x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • (u + u') x ∂μ

Tactic:
h

Goal state after:
case pos.refine_2
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
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
u u' : X → E
a✝ : Disjoint (Function.support u) (Function.support u')
u_int : Integrable u (μ.withDensity fun x ↦ ↑(f x))
u'_int : Integrable u' (μ.withDensity fun x ↦ ↑(f x))
h : (∫ (x : X), u x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • u x ∂μ
⊢ (∫ (x : X), u' x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • u' x ∂μ →
    (∫ (x : X), (u + u') x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • (u + u') x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case refine_2
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
μ : Measure ℝ
inst✝ : PseudoMetrizableSpace ε
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
p : ℕ
hp : m ≤ p
IH : (∀ k ∈ Ico m p, IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a p)
⊢ (∀ k ∈ Ico m (p + 1), IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a (p + 1))

Tactic:
h

Goal state after:
case refine_2
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
μ : Measure ℝ
inst✝ : PseudoMetrizableSpace ε
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
p : ℕ
hp : m ≤ p
IH : (∀ k ∈ Ico m p, IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a p)
h : ∀ k ∈ Ico m (p + 1), IntervalIntegrable f μ (a k) (a (k + 1))
⊢ IntervalIntegrable f μ (a m) (a (p + 1))
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : ℝ → F
inst✝ : CompleteSpace F
⊢ MeasurableSet {x | DifferentiableWithinAt ℝ f (Ici x) x}

Tactic:
by
  have : IsComplete (univ : Set F) := isComplete_univ
  convert! measurableSet_of_differentiableWithinAt_Ici_of_isComplete f this
  simp

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

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

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

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this✝ :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
this : IntervalIntegrable (fun x ↦ ‖f x‖) μ b₁ b₂
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ) (Icc b₁ b₂) b₀

Tactic:
refine continuousWithinAt_of_dominated_interval ?_ ?_ this ?_ <;> clear this

Goal state after:
case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
⊢ ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, AEStronglyMeasurable ({x_1 | x_1 ≤ x}.indicator f) (μ.restrict (Ι b₁ b₂))

case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
⊢ ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ‖{x_1 | x_1 ≤ x}.indicator f t‖ ≤ ‖f t‖

case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
⊢ ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ContinuousWithinAt (fun x ↦ {x_1 | x_1 ≤ x}.indicator f t) (Icc b₁ b₂) b₀
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: 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

Tactic:
h

Goal state after:
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 0
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: 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

Tactic:
h

Goal state after:
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 0
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
⊢ ‖∫ (x : ℝ) in min a b..max a b, f x ∂μ‖ = ‖∫ (x : ℝ) in a..b, f x ∂μ‖

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

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

Goal state before:
pp: case pos
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f (Icc a b) μ
h : a ≤ b
this : ∀ x ∈ Icc a b, ∫ (t : ℝ) in Ioc a x, f t ∂μ = ∫ (t : ℝ) in a..x, f t ∂μ
x₀ : ℝ
a✝ : x₀ ∈ Icc a b
⊢ IntervalIntegrable f μ (min a a) (max a b)

Tactic:
simp only [intervalIntegrable_iff_integrableOn_Ioc_of_le, max_eq_right, h, min_self]

Goal state after:
case pos
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
h_int : IntegrableOn f (Icc a b) μ
h : a ≤ b
this : ∀ x ∈ Icc a b, ∫ (t : ℝ) in Ioc a x, f t ∂μ = ∫ (t : ℝ) in a..x, f t ∂μ
x₀ : ℝ
a✝ : x₀ ∈ Icc a b
⊢ IntegrableOn f (Ioc a b) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case refine_2
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
p : ℕ
hmp : m ≤ p
IH :
  (∀ k ∈ Ico m p, IntervalIntegrable f μ (a k) (a (k + 1))) →
    ∑ k ∈ Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a p, f x ∂μ
⊢ (∀ k ∈ Ico m (p + 1), IntervalIntegrable f μ (a k) (a (k + 1))) →
    ∑ k ∈ Finset.Ico m (p + 1), ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a (p + 1), f x ∂μ

Tactic:
h

Goal state after:
case refine_2
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
μ : Measure ℝ
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
p : ℕ
hmp : m ≤ p
IH :
  (∀ k ∈ Ico m p, IntervalIntegrable f μ (a k) (a (k + 1))) →
    ∑ k ∈ Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a p, f x ∂μ
h : ∀ k ∈ Ico m (p + 1), IntervalIntegrable f μ (a k) (a (k + 1))
⊢ ∑ k ∈ Finset.Ico m (p + 1), ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a (p + 1), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inl
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hfi : IntervalIntegrable f μ a b
hab : a ≤ b
hf : 0 ≤ᵐ[μ.restrict (Ioc a b)] f
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b)] 0

Tactic:
exact integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi

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

Goal state before:
pp: case mp
a b c : ℝ
⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b → a = b ∨ c ∉ [[a, b]]

Tactic:
refine fun h => or_iff_not_imp_left.2 fun hne hc => ?_
    exact not_intervalIntegrable_of_sub_inv_isBigO_punctured (isBigO_refl _ _) hne hc h

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Floor

Goal state before:
pp: K : Type u_1
inst✝³ : Ring K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a c : K
d n : ℕ
⊢ ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)! → a * c ^ n < ↑(n - d)!

Tactic:
h

Goal state after:
K : Type u_1
inst✝³ : Ring K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a c : K
d n : ℕ
h : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!
⊢ a * c ^ n < ↑(n - d)!
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε < 0
n : ℕ
⊢ a * ε⁻¹ * c ^ n < ↑(n - d)! → ε < a * c ^ n / ↑(n - d)!

Tactic:
h

Goal state after:
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε < 0
n : ℕ
h : a * ε⁻¹ * c ^ n < ↑(n - d)!
⊢ ε < a * c ^ n / ↑(n - d)!
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε > 0
n : ℕ
⊢ a * ε⁻¹ * c ^ n < ↑(n - d)! → a * c ^ n / ↑(n - d)! < ε

Tactic:
h

Goal state after:
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε > 0
n : ℕ
h : a * ε⁻¹ * c ^ n < ↑(n - d)!
⊢ a * c ^ n / ↑(n - d)! < ε
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
hfi : Integrable f μ
y : ℝ
⊢ HasSum (fun n ↦ ∫ (x : ℝ) in y + ↑n..y + ↑n + 1, f x ∂μ) (∫ (x : ℝ), f x ∂μ)

Tactic:
simp_rw [integral_of_le (le_add_of_nonneg_right zero_le_one)]
  rw [← setIntegral_univ, ← iUnion_Ioc_add_intCast y]
  exact
    hasSum_integral_iUnion (fun i => measurableSet_Ioc) (pairwise_disjoint_Ioc_add_intCast y)
      hfi.integrableOn

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

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

Tactic:
by
    rw [← @cast_one α, ← cast_add]; exact pure_le_nhdsWithin le_rfl

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

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
t : α
ht : t ≠ ↑⌊t⌋
⊢ ContinuousAt (uncurry f ∘ Prod.map id fract) (s, t)

Tactic:
refine (h.continuousAt ?_).comp (continuousAt_id.prodMap (continuousAt_fract ht))

Goal state after:
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
t : α
ht : t ≠ ↑⌊t⌋
⊢ univ ×ˢ Icc 0 1 ∈ 𝓝 (Prod.map id fract (s, t))
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : μ.IsMulRightInvariant
π : G → G ⧸ Γ := ⋯
meas_π : Measurable π
s : Set (G ⧸ Γ)
s_meas : MeasurableSet s
hs : μ (π ⁻¹' s ∩ 𝓕) = 0
⊢ ∀ (g : ↥Γ.op), g • π ⁻¹' s = π ⁻¹' s

Tactic:
intro γ

Goal state after:
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : μ.IsMulRightInvariant
π : G → G ⧸ Γ := QuotientGroup.mk
meas_π : Measurable π
s : Set (G ⧸ Γ)
s_meas : MeasurableSet s
hs : μ (π ⁻¹' s ∩ 𝓕) = 0
γ : ↥Γ.op
⊢ γ • π ⁻¹' s = π ⁻¹' s
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case mpr.inl
c w : ℂ
n : ℤ
⊢ CircleIntegrable (fun z ↦ (z - w) ^ n) c 0

pp: case mpr.inr
c w : ℂ
R : ℝ
n : ℤ
H : 0 ≤ n ∨ w ∉ sphere c |R|
⊢ CircleIntegrable (fun z ↦ (z - w) ^ n) c R

Tactic:
exacts [circleIntegrable_zero_radius,
      ((continuousOn_id.sub continuousOn_const).zpow₀ _ fun z hz =>
        H.symm.imp_left fun (hw : w ∉ sphere c |R|) =>
          sub_ne_zero.2 <| ne_of_mem_of_not_mem hz hw).circleIntegrable']

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

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
I : Box (Fin (n + 1))
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 (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : ∀ (x : Fin (n + 1)), Tendsto (fun i ↦ (Box.lower ∘ ⇑J) i x) atTop (𝓝 (I.lower x))
hJu : ∀ (x : Fin (n + 1)), Tendsto (fun i ↦ (Box.upper ∘ ⇑J) i x) atTop (𝓝 (I.upper x))
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
  ∀ (k : ℕ),
    ∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i, (f' x) (e i) i =
      ∑ i,
        ((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
          ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
i : Fin (n + 1)
c : ℕ → ℝ
d : ℝ
hc : ∀ (k : ℕ), c k ∈ Set.Icc (I.lower i) (I.upper i)
⊢ Tendsto c atTop (𝓝 d) →
    Tendsto (fun k ↦ ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth (c k) x) i) atTop
      (𝓝 (∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth d x) i))

Tactic:
hcd

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
I : Box (Fin (n + 1))
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 (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : ∀ (x : Fin (n + 1)), Tendsto (fun i ↦ (Box.lower ∘ ⇑J) i x) atTop (𝓝 (I.lower x))
hJu : ∀ (x : Fin (n + 1)), Tendsto (fun i ↦ (Box.upper ∘ ⇑J) i x) atTop (𝓝 (I.upper x))
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
  ∀ (k : ℕ),
    ∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i, (f' x) (e i) i =
      ∑ i,
        ((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
          ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
i : Fin (n + 1)
c : ℕ → ℝ
d : ℝ
hc : ∀ (k : ℕ), c k ∈ Set.Icc (I.lower i) (I.upper i)
hcd : Tendsto c atTop (𝓝 d)
⊢ Tendsto (fun k ↦ ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth (c k) x) i) atTop
    (𝓝 (∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth d x) i))
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 z.im w.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
⊢ (((∫ (x : ℝ) in z.re..w.re, f (e (x, z.im))) - ∫ (x : ℝ) in z.re..w.re, f (e (x, w.im))) +
        I • ∫ (y : ℝ) in z.im..w.im, f (e (w.re, y))) -
      I • ∫ (y : ℝ) in z.im..w.im, f (e (z.re, y)) =
    ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • (f' (e (x, y))) 1 - (f' (e (x, y))) I

Tactic:
max_comm z.im

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
⊢ (((∫ (x : ℝ) in z.re..w.re, f (e (x, z.im))) - ∫ (x : ℝ) in z.re..w.re, f (e (x, w.im))) +
        I • ∫ (y : ℝ) in z.im..w.im, f (e (w.re, y))) -
      I • ∫ (y : ℝ) in z.im..w.im, f (e (z.re, y)) =
    ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • (f' (e (x, y))) 1 - (f' (e (x, y))) I