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

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
⊢ (iteratedFDeriv 𝕜 (i + 1) fun z ↦ f (a • z)) = fun x ↦ a ^ (i + 1) • iteratedFDeriv 𝕜 (i + 1) f (a • x)

Tactic:
| succ i hi =>

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
i : ℕ
a : 𝕜
hf : ContDiff 𝕜 (↑i) f
⊢ (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)

Tactic:
induction i with

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

Goal state before:
pp: case inl
R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
hne : (↑C).Nonempty
r : R
y : M
hy : y ∈ ↑C
z : M
hz : z ∈ ↑C
hr : r < 0
⊢ -(r • z) - -(r • y) = r • (y - z)

Tactic:
neg_sub_neg,

Goal state after:
case inl
R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
hne : (↑C).Nonempty
r : R
y : M
hy : y ∈ ↑C
z : M
hz : z ∈ ↑C
hr : r < 0
⊢ r • y - r • z = r • (y - z)
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case neg.refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
int_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
hc : ¬c = 0
μs_lt_top : μ s < ∞
this : 0 < ε / ↑c
F : Set α
Fs : F ⊆ s
F_closed : IsClosed F
μF : μ s < μ F + ε / ↑c
x : α
⊢ F.indicator (fun x ↦ c) x ≤ (piecewise s hs (const α c) (const α 0)) x

pp: case neg.refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
int_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
hc : ¬c = 0
μs_lt_top : μ s < ∞
this : 0 < ε / ↑c
F : Set α
Fs : F ⊆ s
F_closed : IsClosed F
μF : μ s < μ F + ε / ↑c
⊢ ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≤ ∫⁻ (x : α), ↑(F.indicator (fun x ↦ c) x) ∂μ + ε

Tactic:
· simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
        Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, ← Function.const_def]
      grw [Fs]

Goal state after:
case neg.refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
int_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
hc : ¬c = 0
μs_lt_top : μ s < ∞
this : 0 < ε / ↑c
F : Set α
Fs : F ⊆ s
F_closed : IsClosed F
μF : μ s < μ F + ε / ↑c
⊢ ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≤ ∫⁻ (x : α), ↑(F.indicator (fun x ↦ c) x) ∂μ + ε
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:

Tactic:
(c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := by grw [μF]
        _ = c * μ F + ε := by
          simp_rw [mul_add]
          rw [ENNReal.mul_div_cancel _ ENNReal.coe_ne_top]
          simpa using hc

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

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
a : ℝ
absorbs : Absorbent ℝ s
h : gauge s x < a
⊢ ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s

Tactic:
by
  obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h
  exact ⟨b, hb, hba, hx⟩

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.Extension

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E →ₗ.[ℝ] ℝ
N : E → ℝ
N_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x
N_add : ∀ (x y : E), N (x + y) ≤ N x + N y
hf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x
N_0 : N 0 = 0
s : PointedCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
f' : E × ℝ →ₗ.[ℝ] ℝ := (-f).coprod (LinearMap.id.toPMap ⊤)
hf'_nonneg : ∀ (x : ↥f'.domain), ↑x ∈ s → 0 ≤ ↑f' x
⊢ ∃ g, (∀ (x : ↥f.domain), g ↑x = ↑f x) ∧ ∀ (x : E), g x ≤ N x

Tactic:
have hf'_dense : ∀ y : E × ℝ, ∃ x : f'.domain, ↑x + y ∈ s := by
    rintro ⟨x, y⟩
    exact ⟨⟨(0, N x - y), ⟨f.domain.zero_mem, trivial⟩⟩, by simp [s]⟩

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E →ₗ.[ℝ] ℝ
N : E → ℝ
N_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x
N_add : ∀ (x y : E), N (x + y) ≤ N x + N y
hf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x
N_0 : N 0 = 0
s : PointedCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
f' : E × ℝ →ₗ.[ℝ] ℝ := (-f).coprod (LinearMap.id.toPMap ⊤)
hf'_nonneg : ∀ (x : ↥f'.domain), ↑x ∈ s → 0 ≤ ↑f' x
hf'_dense : ∀ (y : E × ℝ), ∃ x, ↑x + y ∈ s
⊢ ∃ g, (∀ (x : ↥f.domain), g ↑x = ↑f x) ∧ ∀ (x : E), g x ≤ N x
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] 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 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
k_gt : n e < k
⊢ ‖f (x + y) - f x - f' y‖ ≤ ε * ‖y‖

Tactic:
set m := k - 1

Goal state after:
case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] 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 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < ‖y‖
h'k : ‖y‖ ≤ (1 / 2) ^ k
k_gt : n e < k
m : ℕ := k - 1
⊢ ‖f (x + y) - f x - f' y‖ ≤ ε * ‖y‖
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁸ : Field R
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace V
inst✝⁴ : IsTopologicalRing R
inst✝³ : Module R V
inst✝² : SeparatingDual R V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul R V
x y : V
hx : x ≠ 0
hy : y ≠ 0
G : StrongDual R V
Gx : G x = 1
Gy : G y ≠ 0
z : V
⊢ (fun z ↦ z + ((G y)⁻¹ * G z) • (x - y)) ((fun z ↦ z + G z • (y - x)) z) = z

Tactic:
simp only [RingHom.id_apply, smul_eq_mul,
        -- Note: https://github.com/leanprover-community/mathlib4/pull/8386 had to change `map_smulₛₗ` into `map_smulₛₗ _`
        map_add, map_smulₛₗ _, map_sub, Gx, mul_sub, mul_one, add_sub_cancel]

Goal state after:
R : Type u_1
V : Type u_2
inst✝⁸ : Field R
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace V
inst✝⁴ : IsTopologicalRing R
inst✝³ : Module R V
inst✝² : SeparatingDual R V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul R V
x y : V
hx : x ≠ 0
hy : y ≠ 0
G : StrongDual R V
Gx : G x = 1
Gy : G y ≠ 0
z : V
⊢ z + G z • (y - x) + ((G y)⁻¹ * (G z * G y)) • (x - y) = z
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
s t : Set E
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
this : ContinuousSMul ℝ E
⊢ ∃ f u v, (∀ a ∈ s, re (f a) < u) ∧ u < v ∧ ∀ b ∈ t, v < re (f b)

Tactic:
obtain ⟨g, u, v, h1⟩ := _root_.geometric_hahn_banach_compact_closed hs₁ hs₂ ht₁ ht₂ disj

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
s t : Set E
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
this : ContinuousSMul ℝ E
g : StrongDual ℝ E
u v : ℝ
h1 : (∀ a ∈ s, g a < u) ∧ u < v ∧ ∀ b ∈ t, v < g b
⊢ ∃ f u v, (∀ a ∈ s, re (f a) < u) ∧ u < v ∧ ∀ b ∈ t, v < re (f b)
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
f_zero : f 0 = 0
a✝ : X
⊢ ↑(mkD f g) a✝ = (ContinuousMap.mkD f ↑g) a✝

Tactic:
by_cases f_cont : Continuous f

Goal state after:
case pos
X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
f_zero : f 0 = 0
a✝ : X
f_cont : Continuous[inst✝², inst✝¹] f
⊢ ↑(mkD f g) a✝ = (ContinuousMap.mkD f ↑g) a✝

case neg
X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
f_zero : f 0 = 0
a✝ : X
f_cont : ¬Continuous[inst✝², inst✝¹] f
⊢ ↑(mkD f g) a✝ = (ContinuousMap.mkD f ↑g) a✝
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos
X : Type u_1
E : Type u_3
Fₗ : Type u_5
inst✝⁹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup Fₗ
inst✝⁴ : NormedSpace 𝕜 Fₗ
inst✝³ : NormedSpace ℝ Fₗ
inst✝² : CompleteSpace Fₗ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
L : E →L[𝕜] Fₗ
K : ℝ≥0
hL : AntilipschitzWith K ⇑L
φ : X → E
h : Integrable φ μ
⊢ ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

pp: case neg
X : Type u_1
E : Type u_3
Fₗ : Type u_5
inst✝⁹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup Fₗ
inst✝⁴ : NormedSpace 𝕜 Fₗ
inst✝³ : NormedSpace ℝ Fₗ
inst✝² : CompleteSpace Fₗ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
L : E →L[𝕜] Fₗ
K : ℝ≥0
hL : AntilipschitzWith K ⇑L
φ : X → E
h : ¬Integrable φ μ
⊢ ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

Tactic:
· exact integral_comp_comm L h

Goal state after:
case neg
X : Type u_1
E : Type u_3
Fₗ : Type u_5
inst✝⁹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup Fₗ
inst✝⁴ : NormedSpace 𝕜 Fₗ
inst✝³ : NormedSpace ℝ Fₗ
inst✝² : CompleteSpace Fₗ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
L : E →L[𝕜] Fₗ
K : ℝ≥0
hL : AntilipschitzWith K ⇑L
φ : X → E
h : ¬Integrable φ μ
⊢ ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case inr
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)
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < y - x
h'k : y - x ≤ (1 / 2) ^ k
k_gt : n e < k
⊢ ‖f y - f x - (y - x) • f'‖ ≤ ε * ‖y - x‖

Tactic:
set m := k - 1

Goal state after:
case inr
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)
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
hk : (1 / 2) ^ (k + 1) < y - x
h'k : y - x ≤ (1 / 2) ^ k
k_gt : n e < k
m : ℕ := k - 1
⊢ ‖f y - f x - (y - x) • f'‖ ≤ ε * ‖y - x‖
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case pos
ε : Type u_3
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
h : IntervalIntegrable f μ a b
hab : a ≤ b
⊢ AEStronglyMeasurable f (μ.restrict (Ioc a b))

Tactic:
exact h.aestronglyMeasurable

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
⊢ ContinuousOn (fun x ↦ ∫ (t : ℝ) in Ioc a x, f t ∂μ) (Icc a b)

pp: case neg
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
⊢ ContinuousOn (fun x ↦ ∫ (t : ℝ) in Ioc a x, f t ∂μ) (Icc a b)

Tactic:
· have : ∀ x ∈ Icc a b, ∫ t in Ioc a x, f t ∂μ = ∫ t in a..x, f t ∂μ := by
      intro x x_in
      simp_rw [integral_of_le x_in.1]
    rw [continuousOn_congr this]
    intro x₀ _
    refine continuousWithinAt_primitive (measure_singleton x₀) ?_
    simp only [intervalIntegrable_iff_integrableOn_Ioc_of_le, max_eq_right, h, min_self]
    exact h_int.mono Ioc_subset_Icc_self le_rfl

Goal state after:
case neg
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
⊢ ContinuousOn (fun x ↦ ∫ (t : ℝ) in Ioc a x, f t ∂μ) (Icc a b)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
t : ℝ
f : ℝ → ℂ
p : ℝ≥0∞
hLp : MemLp f p (volume.restrict (Ioc t (t + T)))
⊢ MeasurePreserving (fun x ↦ ↑((AddCircle.equivIoc T t) x)) volume (volume.restrict (Ioc t (t + T)))

Tactic:
refine .comp (measurePreserving_subtype_coe measurableSet_Ioc) ?_

Goal state after:
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : ℝ → ℂ
p : ℝ≥0∞
hLp : MemLp f p (volume.restrict (Ioc t (t + T)))
⊢ MeasurePreserving (⇑(AddCircle.equivIoc T t)) volume (Measure.comap Subtype.val volume)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: E✝ : Type u_1
inst✝¹ : NormedAddCommGroup E✝
f✝ : ℝ → E✝
T✝ : ℝ
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
hT : 0 < T
n₁ : ℕ
hn₁ : (t - min a₁ a₂) / T ≤ ↑n₁
n₂ : ℕ
hn₂ : (max a₁ a₂ - t) / T ≤ ↑n₂
⊢ t + ↑n₂ * T ≤ max (t - ↑n₁ * T) (t + ↑n₂ * T)

Tactic:
by apply le_max_right

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

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
f₁ f₂ : ℂ → E
hf : f₁ =ᶠ[codiscreteWithin (sphere c |R|)] f₂
hf₁ : CircleIntegrable f₁ c R
hR : ¬R = 0
⊢ (fun θ ↦ f₁ (circleMap c R θ)) =ᶠ[codiscreteWithin (Ι 0 (2 * π))] fun θ ↦ f₂ (circleMap c R θ)

Tactic:
rw [eventuallyEq_iff_exists_mem]

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
f₁ f₂ : ℂ → E
hf : f₁ =ᶠ[codiscreteWithin (sphere c |R|)] f₂
hf₁ : CircleIntegrable f₁ c R
hR : ¬R = 0
⊢ ∃ s ∈ codiscreteWithin (Ι 0 (2 * π)), EqOn (fun θ ↦ f₁ (circleMap c R θ)) (fun θ ↦ f₂ (circleMap c R θ)) s
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: c w : ℂ
R : ℝ
n : ℤ
⊢ CircleIntegrable (fun z ↦ (z - w) ^ n) c R ↔ R = 0 ∨ 0 ≤ n ∨ w ∉ sphere c |R|

Tactic:
constructor
  · intro h; contrapose! h; rcases h with ⟨hR, hn, hw⟩
    simp only [circleIntegrable_iff R, deriv_circleMap]
    rw [← image_circleMap_Ioc] at hw; rcases hw with ⟨θ, hθ, rfl⟩
    replace hθ : θ ∈ [[0, 2 * π]] := Icc_subset_uIcc (Ioc_subset_Icc_self hθ)
    refine not_intervalIntegrable_of_sub_inv_isBigO_punctured ?_ Real.two_pi_pos.ne hθ
    set f : ℝ → ℂ := fun θ' => circleMap c R θ' - circleMap c R θ
    have : ∀ᶠ θ' in 𝓝[≠] θ, f θ' ∈ ball (0 : ℂ) 1 \ {0} := by
      suffices ∀ᶠ z in 𝓝[≠] circleMap c R θ, z - circleMap c R θ ∈ ball (0 : ℂ) 1 \ {0} from
        ((differentiable_circleMap c R θ).hasDerivAt.tendsto_nhdsNE
          (deriv_circleMap_ne_zero hR)).eventually this
      filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (ball_mem_nhds _ zero_lt_one)]
      simp_all [dist_eq, sub_eq_zero]
    refine (((hasDerivAt_circleMap c R θ).isBigO_sub.mono inf_le_left).inv_rev
      (this.mono fun θ' h₁ h₂ => absurd h₂ h₁.2)).trans ?_
    refine IsBigO.of_bound |R|⁻¹ (this.mono fun θ' hθ' => ?_)
    set x := ‖f θ'‖
    suffices x⁻¹ ≤ x ^ n by
      simp only [smul_eq_mul, norm_mul,
        norm_inv, norm_I, mul_one]
      simpa only [norm_circleMap_zero, norm_zpow, Ne, abs_eq_zero.not.2 hR, not_false_iff,
        inv_mul_cancel_left₀] using this
    have : x ∈ Ioo (0 : ℝ) 1 := by simpa [x, and_comm] using hθ'
    rw [← zpow_neg_one]
    refine (zpow_right_strictAnti₀ this.1 this.2).le_iff_ge.2 (Int.lt_add_one_iff.1 ?_); exact hn
  · rintro (rfl | H)
    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.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝² : NormedAddCommGroup E
inst✝¹ : SFinite ν
inst✝ : NormedSpace ℝ E
f : α × β → E
hf : AEStronglyMeasurable f (μ.prod ν)
⊢ (fun x ↦ ∫ (y : β), f (x, y) ∂ν) =ᵐ[μ] fun x ↦ ∫ (y : β), AEStronglyMeasurable.mk f hf (x, y) ∂ν

Tactic:
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ hx using integral_congr_ae hx

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

Goal state before:
pp: case refine_1
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 : ∀ (i : Fin (n + 1)), ContinuousOn (fun y ↦ f y i) (Box.Icc I)
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
hE : CompleteSpace E
A :
  HasIntegral I ⊥ (fun x ↦ ∑ i, (f' x) (e i) i) volume.toBoxAdditive.toSMul
    (∫ (x : Fin (n + 1) → ℝ) in ↑I, ∑ i, (f' x) (e i) i ∂volume)
B :
  HasIntegral I IntegrationParams.GP (fun x ↦ ∑ i, (f' x) (e i) i) BoxAdditiveMap.volume
    (∑ i,
      (BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.upper i) x) i)
          BoxAdditiveMap.volume -
        BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.lower i) x) i)
          BoxAdditiveMap.volume))
i : Fin (n + 1)
x✝ : i ∈ Finset.univ
⊢ BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.upper i) x) i) BoxAdditiveMap.volume =
    ∫ (x : Fin n → ℝ) in ↑(I.face i), f (i.insertNth (I.upper i) x) i

Tactic:
have := Box.continuousOn_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i))
    have := (this.integrableOn_compact (μ := volume) (Box.isCompact_Icc _)).mono_set
      Box.coe_subset_Icc
    exact (this.hasBoxIntegral ⊥ rfl).integral_eq

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

Goal state before:
pp: case h₂
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)
⊢ HasFiniteIntegral (fun x ↦ ∫ (y : β), ‖AEStronglyMeasurable.mk f h1f (x, y)‖ ∂ν) μ ↔
    HasFiniteIntegral (fun x ↦ ∫ (y : β), ‖f (x, y)‖ ∂ν) μ

Tactic:
· apply hasFiniteIntegral_congr
    filter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm] with _ hx using
      integral_congr_ae (EventuallyEq.fun_comp 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✝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))
this : ∀ (i : ↥(Lp E 1 (μ.prod ν))), Measurable fun z ↦ ‖↑↑i z - ↑↑g z‖ₑ
⊢ Tendsto (fun i ↦ ‖i - g‖) (𝓝 g) (𝓝 0)

Tactic:
rw [← tendsto_iff_norm_sub_tendsto_zero]

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))
this : ∀ (i : ↥(Lp E 1 (μ.prod ν))), Measurable fun z ↦ ‖↑↑i z - ↑↑g z‖ₑ
⊢ Tendsto (fun i ↦ i) (𝓝 g) (𝓝 g)
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
f : α × β → E
s : Set α
t : Set β
hf : IntegrableOn f (s ×ˢ t) (μ.prod ν)
⊢ ∫ (z : α × β) in s ×ˢ t, f z ∂μ.prod ν = ∫ (x : α) in s, ∫ (y : β) in t, f (x, y) ∂ν ∂μ

Tactic:
by
  simp only [← Measure.prod_restrict s t, IntegrableOn] at hf ⊢
  exact integral_prod f hf

Goal state after: