mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : 𝕜
⊢ (iteratedDeriv 0 fun z ↦ f (s + z)) = fun t ↦ iteratedDeriv 0 f (s + t)

Tactic:
simp only [iteratedDeriv_zero]

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : 𝕜
⊢ (iteratedDeriv 0 fun z ↦ f (z + s)) = fun t ↦ iteratedDeriv 0 f (t + s)

Tactic:
simp only [iteratedDeriv_zero]

Goal state after:
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a ≠ b
hfc : ContinuousOn f (uIcc a b)
hfI : f a = f b
⊢ f (min a b) = f (max a b)

Tactic:
grind

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

Goal state before:
pp: case h.refine_3
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n✝ : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
n : ℕ
IH : ContDiffAt 𝕜 (↑↑n) (↑f) (↑f.symm a) → ContDiffAt 𝕜 (↑↑n) (↑f.symm) a
hf : ContDiffAt 𝕜 (↑↑n.succ) (↑f) (↑f.symm a)
f' : E → E →L[𝕜] F
hf' : ContDiffAt 𝕜 (↑n) f' (↑f.symm a)
u : Set E
hu : u ∈ 𝓝 (↑f.symm a)
hff' : ∀ x ∈ u, HasFDerivAt (↑f) (f' x) x
eq_f₀' : f' (↑f.symm a) = ↑f₀'
h_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑f.symm a)
t : Set E
htu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
htf : ↑f.symm a ∈ t
x : F
hx : x ∈ f.target ∩ ↑f.symm ⁻¹' t
⊢ HasFDerivAt (↑f.symm) ((inverse ∘ f' ∘ ↑f.symm) x) x

Tactic:
obtain ⟨hxu, e, he⟩ := htu hx.2

Goal state after:
case h.refine_3
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n✝ : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
n : ℕ
IH : ContDiffAt 𝕜 (↑↑n) (↑f) (↑f.symm a) → ContDiffAt 𝕜 (↑↑n) (↑f.symm) a
hf : ContDiffAt 𝕜 (↑↑n.succ) (↑f) (↑f.symm a)
f' : E → E →L[𝕜] F
hf' : ContDiffAt 𝕜 (↑n) f' (↑f.symm a)
u : Set E
hu : u ∈ 𝓝 (↑f.symm a)
hff' : ∀ x ∈ u, HasFDerivAt (↑f) (f' x) x
eq_f₀' : f' (↑f.symm a) = ↑f₀'
h_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑f.symm a)
t : Set E
htu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
htf : ↑f.symm a ∈ t
x : F
hx : x ∈ f.target ∩ ↑f.symm ⁻¹' t
hxu : ↑f.symm x ∈ u
e : E ≃L[𝕜] F
he : ↑e = f' (↑f.symm x)
⊢ HasFDerivAt (↑f.symm) ((inverse ∘ f' ∘ ↑f.symm) x) x
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
z : ℝ
hz : z ∈ Ioc a b
⊢ 0 < derivWithin f (Ioi a) z

Tactic:
simp_all only [and_imp, mem_Ioc, max_lt_iff]

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

Goal state before:
pp: case add
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), f₁ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f₁ x) ∂μ + ε
h₂ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), f₂ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f₂ x) ∂μ + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g,
    (∀ (x : α), (f₁ + f₂) x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑((f₁ + f₂) x) ∂μ + ε

Tactic:
| @add f₁ f₂ _ h₁ h₂ =>

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

Goal state before:
pp: case inr
R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
C✝ C₁ C₂ : PointedCone R E
x✝¹ : E
r : R
C : ConvexCone R E
hC : C.Pointed
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : E
hx : x ∈ ↑C
c : R
hc : 0 ≤ c
hpos : 0 < c
⊢ ⟨c, hc⟩ • x ∈ C

Tactic:
apply ConvexCone.smul_mem

Goal state after:
case inr.hc
R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
C✝ C₁ C₂ : PointedCone R E
x✝¹ : E
r : R
C : ConvexCone R E
hC : C.Pointed
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : E
hx : x ∈ ↑C
c : R
hc : 0 ≤ c
hpos : 0 < c
⊢ 0 < ↑⟨c, hc⟩

case inr.hx
R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
C✝ C₁ C₂ : PointedCone R E
x✝¹ : E
r : R
C : ConvexCone R E
hC : C.Pointed
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : E
hx : x ∈ ↑C
c : R
hc : 0 ≤ c
hpos : 0 < c
⊢ x ∈ C
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: y : ℝ
n : ℕ
hy₁ : -1 < y
hy₂ : y < 1
⊢ 1 - y ≠ 0

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: case refine_1
y : ℝ
n : ℕ
hy₁ : -1 < y
hy₂ : y < 1
⊢ 1 + y ≠ 0 ∧ 1 - y ≠ 0

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
hI : Set.Icc (-|x|) |x| ⊆ Set.Ioo (-1) 1
y : ℝ
hy : y ∈ Set.Ioo (-1) 1
⊢ -1 < y

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
hI : Set.Icc (-|x|) |x| ⊆ Set.Ioo (-1) 1
y : ℝ
hy : y ∈ Set.Ioo (-1) 1
⊢ y < 1

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h₀ : 0 ≤ x
h : x < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
y : ℝ
hy : y ∈ Set.Icc 0 x
⊢ -1 < y

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h₀ : 0 ≤ x
h : x < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
y : ℝ
hy : y ∈ Set.Icc 0 x
⊢ y < 1

Tactic:
grind

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

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
h : ∅ ⊆ 0
⊢ gauge ∅ = 0

pp: case inr
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
h : {0} ⊆ 0
⊢ gauge {0} = 0

Tactic:
exacts [gauge_empty, gauge_zero']

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
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ r, ⋃ (_ : 0 < r), ⋃ (_ : r < a), r • s

Tactic:
mem_ofPred,

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ gauge s x✝ < a ↔ x✝ ∈ ⋃ r, ⋃ (_ : 0 < r), ⋃ (_ : r < a), r • s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ r ∈ Ioo 0 a, r • s

Tactic:
mem_ofPred,

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ gauge s x✝ < a ↔ x✝ ∈ ⋃ r ∈ Ioo 0 a, r • s
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case add
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∫⁻ (x : α), ↑(f₁ x) ∂μ ≠ ∞ →
    ∀ {ε : ℝ≥0∞},
      ε ≠ 0 → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
h₂ :
  ∫⁻ (x : α), ↑(f₂ x) ∂μ ≠ ∞ →
    ∀ {ε : ℝ≥0∞},
      ε ≠ 0 → ∃ g, (∀ (x : α), g x ≤ f₂ x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
int_f : ∫⁻ (x : α), ↑((f₁ + f₂) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g,
    (∀ (x : α), g x ≤ (f₁ + f₂) x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑((f₁ + f₂) x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε

Tactic:
| @add f₁ f₂ _ h₁ h₂ =>

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

Goal state before:
pp: case inr.mp
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hx : a • x ∈ r • s
hr : 0 < r
this : 0 < a⁻¹ • r
⊢ (a⁻¹ • r)⁻¹ • x ∈ s

Tactic:
← mem_smul_set_iff_inv_smul_mem₀ this.ne',

Goal state after:
case inr.mp
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hx : a • x ∈ r • s
hr : 0 < r
this : 0 < a⁻¹ • r
⊢ x ∈ (a⁻¹ • r) • s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr.mpr
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hr : 0 < r
hx : x ∈ r • s
this : 0 < a • r
⊢ (a • r)⁻¹ • a • x ∈ s

Tactic:
← mem_smul_set_iff_inv_smul_mem₀ this.ne',

Goal state after:
case inr.mpr
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hr : 0 < r
hx : x ∈ r • s
this : 0 < a • r
⊢ a • x ∈ (a • r) • s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inl
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
ha : 0 ≤ 0
⊢ gauge (0 • s) = 0⁻¹ • gauge s

Tactic:
inv_zero,

Goal state after:
case inl
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
ha : 0 ≤ 0
⊢ gauge (0 • s) = 0 • gauge s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case left
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
x : E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : s ∈ 𝓝 0
ha : Absorbent ℝ s
ε : ℝ
hε₀ : 0 < ε
this : ε • s ∩ -(ε • s) ∈ 𝓝 0
y : E
hy : y ∈ ε • s ∩ -(ε • s)
⊢ gauge s x - ε ≤ gauge s (x + y)

Tactic:
rw [sub_le_iff_le_add]
    calc
      gauge s x = gauge s (x + y + (-y)) := by simp
      _ ≤ gauge s (x + y) + gauge s (-y) := gauge_add_le hc ha _ _
      _ ≤ gauge s (x + y) + ε := by grw [gauge_le_of_mem hε₀.le (mem_neg.1 hy.2)]

Goal state after:
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
x : E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : s ∈ 𝓝 0
⊢ x ∈ closure[inst✝²] s ∧ x ∉ interior s ↔ x ∈ frontier s

Tactic:
rfl

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

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hr : 0 ≤ 0
⊢ 0 x = 0

Tactic:
rfl

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 : StrongDual R V), f v = 0) → v = 0 ↔
    (∀ (x : V →L[R] R), ((ContinuousLinearMap.coeLM R).flip v) x = 0 x) → v = 0

Tactic:
rfl

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := NNReal.mk (ε / 2) ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := fun x ↦ (f x).toNNReal
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := fun x ↦ (-f x).toNNReal
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x ↦ ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
⊢ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), f x ∂μ + ε

Tactic:
show (∫ x : α, (g x).toReal ∂μ) < (∫ x : α, f x ∂μ) + ε
    exact
      calc
        (∫ x : α, (g x).toReal ∂μ) = ∫ x : α, EReal.toReal (gp x) - EReal.toReal (gm x) ∂μ :=
          integral_congr_ae ae_g
        _ = (∫ x : α, EReal.toReal (gp x) ∂μ) - ∫ x : α, ↑(gm x) ∂μ := by
          simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]
          exact integral_sub gp_integrable gm_integrable
        _ < (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ∫ x : α, ↑(gm x) ∂μ := by
          apply sub_lt_sub_right
          convert! gpint
          simp only [EReal.toReal_coe_ennreal]
        _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, ↑(fm x) ∂μ) - δ) := sub_le_sub_left gmint _
        _ = (∫ x : α, f x ∂μ) + 2 * δ := by
          simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring
        _ = (∫ x : α, f x ∂μ) + ε := by congr 1; simp [field, δ]

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

Goal state before:
pp: case neg
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))
⊢ ¬Integrable (fun x ↦ f x • g x) μ

Tactic:
← integrable_withDensity_iff_integrable_smul f_meas

Goal state after:
case neg
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))
⊢ ¬Integrable g (μ.withDensity fun x ↦ ↑(f x))
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos.refine_1
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))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ ∫ (x : X) in s, ↑(f x) • c ∂μ = ∫ (a : X) in s, f a • c ∂μ

Tactic:
rfl

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

Goal state before:
pp: case refine_4
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
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ (s n).indicator f a) atTop (𝓝 ((⋂ b, s b).indicator f a))

Tactic:
filter_upwards [] with a using le_trans (h_anti.tendsto_indicator _ _ _) (pure_le_nhds _)

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

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞

Tactic:
grind

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

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
c : ℝ
h : ‖f (min a b)‖ₑ ≠ ∞
⊢ ‖f (min (a + c) (b + c) + -c)‖ₑ ≠ ∞

Tactic:
grind

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

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

Tactic:
have C : Continuous (fun (p : α × E) ↦ f a p.2) := by fun_prop
    have D : ({a} ×ˢ closedBall x t).EqOn f.uncurry (fun p ↦ f a p.2) := by
      rintro ⟨b, y⟩ ⟨hb, -⟩
      simp only [mem_singleton_iff] at hb
      simp [hb]
    obtain ⟨v, v_open, sub_v, hv⟩ : ∃ v, IsOpen v ∧ {a} ×ˢ closedBall x t ⊆ v ∧
        ∀ p ∈ v, dist (Function.uncurry f p) (f a p.2) < ε :=
      Uniform.exists_is_open_mem_uniformity_of_forall_mem_eq (s := {a} ×ˢ closedBall x t)
        (fun p _ ↦ hf.continuousAt) (fun p _ ↦ C.continuousAt) D (dist_mem_uniformity εpos)
    obtain ⟨w, w', w_open, -, sub_w, sub_w', hww'⟩ : ∃ (w : Set α) (w' : Set E),
        IsOpen w ∧ IsOpen w' ∧ {a} ⊆ w ∧ closedBall x t ⊆ w' ∧ w ×ˢ w' ⊆ v :=
      generalized_tube_lemma isCompact_singleton (isCompact_closedBall x t) v_open sub_v
    refine ⟨w, w_open, sub_w rfl, ?_⟩
    rintro ⟨b, y⟩ h hby
    exact hv _ (hww' ⟨h, sub_w' hby⟩)

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

Goal state before:
pp: E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
inst✝² : CompleteSpace E
𝕜 : Type u_8
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : ℝ → 𝕜
c : E
⊢ ∫ (x : ℝ) in a..b, f x • c ∂μ = (∫ (x : ℝ) in a..b, f x ∂μ) • c

Tactic:
by
  simp only [intervalIntegral_eq_integral_uIoc, integral_smul_const, smul_assoc]

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 : 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
I : Tendsto (fun δ ↦ b₀ - δ) (𝓝 0) (𝓝 (b₀ - 0))
J : Tendsto (fun δ ↦ b₀ + δ) (𝓝 0) (𝓝 (b₀ + 0))
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b

Tactic:
simp only [sub_zero, add_zero] at I J

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
I : Tendsto (fun δ ↦ b₀ - δ) (𝓝 0) (𝓝 b₀)
J : Tendsto (fun δ ↦ b₀ + δ) (𝓝 0) (𝓝 b₀)
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ha : IntegrableOn f (Ioi a) μ
hb : IntegrableOn f (Ioi b) μ
this :
  ∀ {a b : ℝ},
    IntegrableOn f (Ioi a) μ →
      IntegrableOn f (Ioi b) μ →
        a ≤ b → ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in Ioi b, f x ∂μ = ∫ (x : ℝ) in Ioi a, f x ∂μ
hab : ¬a ≤ b
⊢ -∫ (x : ℝ) in b..a, f x ∂μ + (∫ (x : ℝ) in b..a, f x ∂μ + ∫ (x : ℝ) in Ioi a, f x ∂μ) = ∫ (x : ℝ) in Ioi a, f x ∂μ

Tactic:
grind

Goal state after: