mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: case hc₀
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : (𝓝[>] 0 ⊓ 𝓟 {x_1 | x + ↑x_1 • y ∈ s}).NeBot
⊢ Tendsto ?m.98 (𝓝[>] 0 ⊓ 𝓟 {t | x + ↑t • y ∈ s}) (𝓝[≠] 0)

pp: case hmem
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : (𝓝[>] 0 ⊓ 𝓟 {x_1 | x + ↑x_1 • y ∈ s}).NeBot
⊢ ∀ᶠ (n : ℝ≥0) in 𝓝[>] 0 ⊓ 𝓟 {t | x + ↑t • y ∈ s}, x + ?m.98 n • y ∈ s

pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : (𝓝[>] 0 ⊓ 𝓟 {x_1 | x + ↑x_1 • y ∈ s}).NeBot
⊢ ℝ≥0 → ℝ≥0

Tactic:
· exact tendsto_id'.mpr <| inf_le_left.trans <| nhdsGT_le_nhdsNE _

Goal state after:
case hmem
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : (𝓝[>] 0 ⊓ 𝓟 {x_1 | x + ↑x_1 • y ∈ s}).NeBot
⊢ ∀ᶠ (n : ℝ≥0) in 𝓝[>] 0 ⊓ 𝓟 {t | x + ↑t • y ∈ s}, x + id n • y ∈ s
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
s : Set E
f : E → F
hu : UniqueDiffOn 𝕜 s
i : ℕ
hi : ∀ {x : E}, x ∈ s → iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
x : E
hx : x ∈ s
h : Fin (i + 1) → E
⊢ ((-fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h) =
    -((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h)

Tactic:
neg_apply,

Goal state after:
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
s : Set E
f : E → F
hu : UniqueDiffOn 𝕜 s
i : ℕ
hi : ∀ {x : E}, x ∈ s → iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
x : E
hx : x ∈ s
h : Fin (i + 1) → E
⊢ (-(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h) =
    -((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s x) (h 0)) (tail h)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
⊢ iteratedDeriv n (fun a ↦ a) 0 = if n = 1 then 1 else 0

Tactic:
by
  simp +contextual [iteratedDeriv_fun_id]

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

Goal state before:
pp: case succ.h
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
v : E
x✝ : Fin (i + 1) → E
⊢ DifferentiableAt 𝕜 (iteratedFDeriv 𝕜 i f ∘ fun x ↦ a • x) v

Tactic:
apply DifferentiableAt.comp

Goal state after:
case succ.h.hg
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
v : E
x✝ : Fin (i + 1) → E
⊢ DifferentiableAt 𝕜 (iteratedFDeriv 𝕜 i f) (a • v)

case hf
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
v : E
x✝ : Fin (i + 1) → E
⊢ DifferentiableAt 𝕜 (fun x ↦ a • x) v
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: case neg
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 (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : f c = f a
hC : ¬f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

Tactic:
refine ⟨C, ⟨lt_of_le_of_ne Cmem.1 <| mt ?_ hC, lt_of_le_of_ne Cmem.2 <| mt ?_ hC⟩, Or.inr Cge⟩
      exacts [fun h => by rw [h], fun h => by rw [h, hfI]]

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≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Tactic:
rcases exists_lt_lowerSemicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
    ⟨g0, f_lt_g0, g0_cont, g0_int⟩

Goal state after:
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≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
M : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
c : 𝕜
hc : 0 < c
y : M
hy : y ∈ s
⊢ c⁻¹ • c • y ∈ s

Tactic:
by rwa [smul_smul, inv_mul_cancel₀ hc.ne', one_smul]

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)
A : ∀ y ∈ Set.Icc 0 x, HasDerivAt F (F' y) y
y : ℝ
hy : 0 < y ∧ y < x
⊢ 0 ≤ 1 - y ^ 2

Tactic:
calc
    0 ≤ 1 - x ^ 2 := by simp [abs_of_nonneg h₀, h.le]
    _ ≤ 1 - y ^ 2 := sub_le_sub_left (pow_le_pow_left₀ hy.1.le hy.2.le 2) 1

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

Goal state before:
pp: 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
⊢ ∫⁻ (x : α), ↑(f₁ x) ∂μ + ∫⁻ (x : α), ↑(f₂ x) ∂μ ≠ ∞

Tactic:
rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]

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

Goal state before:
pp: 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
⊢ x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}

Tactic:
obtain ⟨f', f'K, hf'⟩ : ∃ f' ∈ K, Tendsto L0 atTop (𝓝 f') :=
    cauchySeq_tendsto_of_isComplete hK (fun e => (hn e (n e) (n e) le_rfl le_rfl).1) this

Goal state after:
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')
⊢ x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
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 • x ∈ a • r • s

Tactic:
exact smul_mem_smul_set hx

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

Goal state before:
pp: case refine_1.inr
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
s : Set E
hs₀ : 0 ∈ s
hs₁ : Convex ℝ s
hs₂ : IsOpen s
x₀ : E
hx₀ : x₀ ∉ s
f : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯
y : ℝ
hx : y • x₀ ∈ f.domain
h : 0 < y
⊢ (RingHom.id ℝ) y ≤ gauge s (y • x₀)

Tactic:
rw [gauge_smul_of_nonneg h.le, smul_eq_mul, RingHom.id_apply, le_mul_iff_one_le_right h]

Goal state after:
case refine_1.inr
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
s : Set E
hs₀ : 0 ∈ s
hs₁ : Convex ℝ s
hs₂ : IsOpen s
x₀ : E
hx₀ : x₀ ∉ s
f : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯
y : ℝ
hx : y • x₀ ∈ f.domain
h : 0 < y
⊢ 1 ≤ gauge s x₀
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1} = ∅
h : ¬0 = p x
hpx : 0 < p x
hpx₂ : 0 < 2 * p x
⊢ False

Tactic:
refine hp.subset ⟨hpx₂, (2 * p x)⁻¹ • x, ?_, smul_inv_smul₀ hpx₂.ne' _⟩

Goal state after:
case inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1} = ∅
h : ¬0 = p x
hpx : 0 < p x
hpx₂ : 0 < 2 * p x
⊢ (2 * p x)⁻¹ • x ∈ p.ball 0 1
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
r : ℝ
x : E
hs : ball 0 r ⊆ s
hr : 0 < r
⊢ gauge s x ≤ ‖x‖ / r

Tactic:
← gauge_ball hr.le

Goal state after:
case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
r : ℝ
x : E
hs : ball 0 r ⊆ s
hr : 0 < r
⊢ gauge s x ≤ gauge (ball 0 r) x
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_4
V : Type u_5
W : Type u_6
inst✝¹³ : NormedField R
inst✝¹² : AddCommGroup V
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSpace V
inst✝⁸ : TopologicalSpace W
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : Module R V
inst✝⁵ : Module R W
inst✝⁴ : SeparatingDual R V
inst✝³ : IsTopologicalAddGroup V
inst✝² : IsTopologicalAddGroup W
inst✝¹ : ContinuousSMul R V
inst✝ : ContinuousSMul R W
f g : V ≃L[R] W
⊢ (∀ (x : V →L[R] V), g.conjContinuousAlgEquiv x = f.conjContinuousAlgEquiv x) ↔ ∃ α, f = α • g

Tactic:
conjContinuousAlgEquiv_apply,

Goal state after:
R : Type u_4
V : Type u_5
W : Type u_6
inst✝¹³ : NormedField R
inst✝¹² : AddCommGroup V
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSpace V
inst✝⁸ : TopologicalSpace W
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : Module R V
inst✝⁵ : Module R W
inst✝⁴ : SeparatingDual R V
inst✝³ : IsTopologicalAddGroup V
inst✝² : IsTopologicalAddGroup W
inst✝¹ : ContinuousSMul R V
inst✝ : ContinuousSMul R W
f g : V ≃L[R] W
⊢ (∀ (x : V →L[R] V), ↑g ∘SL x ∘SL ↑g.symm = ↑f ∘SL x ∘SL ↑f.symm) ↔ ∃ α, f = α • g
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
a b : V
⊢ a + b + G (a + b) • (y - x) = a + G a • (y - x) + (b + G b • (y - x))

Tactic:
simp [add_smul]

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
a b : V
⊢ a + b + (G a • (y - x) + G b • (y - x)) = a + G a • (y - x) + (b + G b • (y - x))
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 : 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 : IsClosed s
x : E
hx : x ∈ {x | ∀ (i : StrongDual 𝕜 E), ∃ y ∈ s, re (i x) ≤ re (i y)}
h : x ∉ s
⊢ False

Tactic:
obtain ⟨l, s, hlA, hl⟩ := geometric_hahn_banach_closed_point (𝕜 := 𝕜) hs₁ hs₂ h

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✝ : 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 : IsClosed s✝
x : E
hx : x ∈ {x | ∀ (i : StrongDual 𝕜 E), ∃ y ∈ s✝, re (i x) ≤ re (i y)}
h : x ∉ s✝
l : StrongDual 𝕜 E
s : ℝ
hlA : ∀ a ∈ s✝, re (l a) < s
hl : s < re (l x)
⊢ False
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
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
⊢ x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}

Tactic:
obtain ⟨f', f'K, hf'⟩ : ∃ f' ∈ K, Tendsto L0 atTop (𝓝 f') :=
    cauchySeq_tendsto_of_isComplete hK (fun e => (hn e (n e) (n e) le_rfl le_rfl).1) this

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)
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
⊢ x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
Module: Mathlib.Analysis.Calculus.TangentCone.Prod

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
x : E
s : Set E
y : F
t : Set F
ht : y ∈ closure[inst✝²] t
z : E
hz : z ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 z)
⊢ (LinearMap.inl 𝕜 E F) z ∈ tangentConeAt 𝕜 (closure[inst✝⁷] s ×ˢ closure[inst✝²] t) (x, y)

Tactic:
refine mem_tangentConeAt_of_seq l c (fun n ↦ (d n, 0)) (hd₀.prodMk_nhds tendsto_const_nhds)
    (hds.mono fun n hn ↦ by simp [ht, subset_closure hn]) ?_

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
x : E
s : Set E
y : F
t : Set F
ht : y ∈ closure[inst✝²] t
z : E
hz : z ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 z)
⊢ Tendsto (fun n ↦ c n • (d n, 0)) l (𝓝 ((LinearMap.inl 𝕜 E F) z))
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
x : ℝ
hx : x ∈ Icc b₁ b₂
⊢ AEStronglyMeasurable f (μ.restrict (Ioc b₁ x))

Tactic:
exact (h_int' hx).1.aestronglyMeasurable

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
⊢ ContinuousAt (fun p ↦ ∫ (t : ℝ) in a₀..p.2, F p.1 t ∂μ) (x₀, b₀)

Tactic:
have hx₀ : ∀ᵐ t : ℝ ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t := h_bound.self_of_nhds

Goal state after:
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
⊢ ContinuousAt (fun p ↦ ∫ (t : ℝ) in a₀..p.2, F p.1 t ∂μ) (x₀, b₀)
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
f : ℝ → E
μ : Measure ℝ
h : b ≤ a
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = -∫ (x : ℝ) in Ioc b a, f x ∂μ

Tactic:
simp only [integral_symm b, integral_of_le h]

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

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := ⋯
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
B : ∀ᶠ (u : ℝ) in 𝓝 t, ↑y < G' u
m M : ℝ
H : Ioo m M ⊆ {x | (fun u ↦ ↑y < G' u) x}
hm : m < t
hM : t < M
this : Ioo t (min M b) ∈ 𝓝[>] t
u : ℝ
hu : u ∈ Ioo t (min M b)
I : Icc t u ⊆ Icc a b
C1 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), G' x < ↑∞
C2 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), x ∈ Icc t u
x : ℝ
G'x : G' x < ↑∞
hx : x ∈ Icc t u
⊢ ↑y ≤ ↑(G' x).toReal

Tactic:
have : x ∈ Ioo m M := by
              simp only [hm.trans_le hx.left,
                (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self_iff]

Goal state after:
g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
B : ∀ᶠ (u : ℝ) in 𝓝 t, ↑y < G' u
m M : ℝ
H : Ioo m M ⊆ {x | (fun u ↦ ↑y < G' u) x}
hm : m < t
hM : t < M
this✝ : Ioo t (min M b) ∈ 𝓝[>] t
u : ℝ
hu : u ∈ Ioo t (min M b)
I : Icc t u ⊆ Icc a b
C1 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), G' x < ↑∞
C2 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), x ∈ Icc t u
x : ℝ
G'x : G' x < ↑∞
hx : x ∈ Icc t u
this : x ∈ Ioo m M
⊢ ↑y ≤ ↑(G' x).toReal