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

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
c : OrderedFinpartition n
x✝ : c ∈ Finset.univ
H₁ : (fun x ↦ p₁ x c.length) =O[l] 1
H₂ : (fun x m ↦ q₁ x (c.partSize m)) =O[l] 1
H₃ : (fun x m ↦ q₂ x (c.partSize m)) =O[l] 1
H₄ : (fun a m ↦ q₁ a (c.partSize m) - q₂ a (c.partSize m)) =O[l] f
⊢ (fun x ↦
      ‖p₁ x c.length‖ * ↑c.length * max ‖fun m ↦ q₁ x (c.partSize m)‖ ‖fun m ↦ q₂ x (c.partSize m)‖ ^ (c.length - 1) *
        ‖(fun m ↦ q₁ x (c.partSize m)) - fun m ↦ q₂ x (c.partSize m)‖) =O[l]
    f

Tactic:
have H₅ := ((H₂.prod_left H₃).norm_left.pow (c.length - 1)).mul H₄.norm_norm

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
c : OrderedFinpartition n
x✝ : c ∈ Finset.univ
H₁ : (fun x ↦ p₁ x c.length) =O[l] 1
H₂ : (fun x m ↦ q₁ x (c.partSize m)) =O[l] 1
H₃ : (fun x m ↦ q₂ x (c.partSize m)) =O[l] 1
H₄ : (fun a m ↦ q₁ a (c.partSize m) - q₂ a (c.partSize m)) =O[l] f
H₅ :
  (fun x ↦
      ‖(fun m ↦ q₁ x (c.partSize m), fun m ↦ q₂ x (c.partSize m))‖ ^ (c.length - 1) *
        ‖fun m ↦ q₁ x (c.partSize m) - q₂ x (c.partSize m)‖) =O[l]
    fun x ↦ 1 x ^ (c.length - 1) * ‖f x‖
⊢ (fun x ↦
      ‖p₁ x c.length‖ * ↑c.length * max ‖fun m ↦ q₁ x (c.partSize m)‖ ‖fun m ↦ q₂ x (c.partSize m)‖ ^ (c.length - 1) *
        ‖(fun m ↦ q₁ x (c.partSize m)) - fun m ↦ q₂ x (c.partSize m)‖) =O[l]
    f
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

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
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
i : Fin c.length
e : E
v : Fin m → E
j : Fin c.length
hij : j ≠ i
⊢ update c.partSize i (c.partSize i + 1) j = c.partSize j

Tactic:
simp [hij]

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

Goal state before:
pp: case neg.hf
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hab : a < b
⊢ ∀ x ∈ Ico a b, ∃ y, Tendsto f (𝓝[Ioo a b] x) (𝓝 y)

Tactic:
· intro x x_in
    rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl | h)
    · use la
      simpa [hab]
    · exact ⟨f x, hf x h⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
⊢ ∃ c ∈ Ioo a b, deriv f c = slope f a b

Tactic:
rw [slope_def_field]
  exact exists_deriv_eq_slope f hab hfc hfd

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
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
ne_top : μ s ≠ ∞
this : μ s < μ s + ε / ↑c
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u < μ s + ε / ↑c
⊢ ↑c * μ s + ↑c * (ε / ↑c) = ↑c * μ s + ε

Tactic:
rw [ENNReal.mul_div_cancel _ ENNReal.coe_ne_top]

Goal state after:
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
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : ¬c = 0
ne_top : μ s ≠ ∞
this : μ s < μ s + ε / ↑c
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u < μ s + ε / ↑c
⊢ ↑c ≠ 0
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : ∃ i < a, ∀ ⦃x : ℝ⦄, x ∈ Ioo i a → deriv f x ∈ Iic (-1)
f' : ℝ → ℝ := f ∘ Neg.neg
⊢ ∀ᶠ (x : ℝ) in 𝓝[>] (-a), deriv f' x = (-deriv f ∘ Neg.neg) x

Tactic:
(nhdsGT_basis (-a)).eventually_iff

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : ∃ i < a, ∀ ⦃x : ℝ⦄, x ∈ Ioo i a → deriv f x ∈ Iic (-1)
f' : ℝ → ℝ := f ∘ Neg.neg
⊢ ∃ i, -a < i ∧ ∀ ⦃x : ℝ⦄, x ∈ Ioo (-a) i → deriv f' x = (-deriv f ∘ Neg.neg) x
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C✝ C₁ C₂ C : ConvexCone R G
h₁ : C.Pointed
x : G
⊢ 0 ∈ C

Tactic:
exact h₁

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

Goal state before:
pp: case hf₀'
n : ℕ∞ω
x : ℝ
hx : x ≠ 0
y : ℝ
hy : 0 < y
⊢ HasDerivAt (↑expPartialHomeomorph) y (↑expPartialHomeomorph.symm y)

Tactic:
convert! hasDerivAt_exp (log y)
      rw [exp_log hy]

Goal state after:
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
C : ConvexCone 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ (∀ ⦃a : M⦄, a ∈ s → a ∉ C) → ∀ ⦃a : M⦄, a ∈ hull 𝕜 s → a ∉ C

Tactic:
mem_hull_of_convex hs

Goal state after:
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
C : ConvexCone 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ (∀ ⦃a : M⦄, a ∈ s → a ∉ C) → ∀ ⦃a : M⦄, (∃ r, 0 < r ∧ a ∈ r • s) → a ∉ C
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h₀ : 0 ≤ x
h : x < 1
n : ℕ
h₁ :
  1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1) ≤
    |x| ^ (2 * n + 1) / (1 - x ^ 2)
⊢ 1 / 2 * log ((1 + x) / (1 - x)) ≤ ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1) + x ^ (2 * n + 1) / (1 - x ^ 2)

Tactic:
abs_of_nonneg h₀,

Goal state after:
x : ℝ
h₀ : 0 ≤ x
h : x < 1
n : ℕ
h₁ :
  1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1) ≤ x ^ (2 * n + 1) / (1 - x ^ 2)
⊢ 1 / 2 * log ((1 + x) / (1 - x)) ≤ ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1) + x ^ (2 * n + 1) / (1 - x ^ 2)
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
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
⊢ ↑c * μ F + ↑c * (ε / ↑c) = ↑c * μ F + ε

Tactic:
rw [ENNReal.mul_div_cancel _ ENNReal.coe_ne_top]

Goal state after:
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
⊢ ↑c ≠ 0
Module: Mathlib.Analysis.Convex.Cone.Extension

Goal state before:
pp: case refine_2.inr.inr
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x
x : E
hx : x ∈ f.domain
r : ℝ
hy' : r • y ∈ (LinearPMap.mkSpanSingleton y (-c) ⋯).domain
hz : x + r • y ∈ (f.supSpanSingleton y (-c) hy).domain
hzs : x + r • y ∈ s
hr : 0 < r
⊢ (RingHom.id ℝ) r • c ≤ ↑f ⟨x, hx⟩

Tactic:
have : r⁻¹ • x + y ∈ s := by
        rwa [← s.smul_mem_iff hr, smul_add, smul_smul, mul_inv_cancel₀ hr.ne', one_smul]
      replace := c_le (r⁻¹ • ⟨x, hx⟩) this
      rwa [← mul_le_mul_iff_right₀ hr, f.map_smul, smul_eq_mul, ← mul_assoc, mul_inv_cancel₀ hr.ne',
        one_mul] at this

Goal state after:
Module: Mathlib.Analysis.RCLike.Extend

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
F : Type u_2
inst✝³ : AddCommGroup F
inst✝² : Module ℝ F
inst✝¹ : Module 𝕜 F
inst✝ : IsScalarTower ℝ 𝕜 F
fr : Dual ℝ F
fc : F → 𝕜 := fun x ↦ ↑(fr x) - I * ↑(fr (I • x))
add : ∀ (x y : F), fc (x + y) = fc x + fc y
A : ∀ (c : ℝ) (x : F), ↑(fr (↑c • x)) = ↑c * ↑(fr x)
smul_ℝ : ∀ (c : ℝ) (x : F), fc (↑c • x) = ↑c * fc x
x : F
⊢ fc (I • x) = I * fc x

Tactic:
obtain (h | h) := @I_mul_I_ax 𝕜 _
    · simp [fc, h]
    · simp [fc, mul_sub, ← mul_assoc, smul_smul, h, add_comm]

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
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs₂ : IsOpen[inst✝¹] s
x : E
hx : x ∈ s
⊢ x ∈ interior s

Tactic:
hs₂.interior_eq

Goal state after:
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs₂ : IsOpen[inst✝¹] s
x : E
hx : x ∈ s
⊢ x ∈ s
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case refine_1
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
⊢ (RingHom.id ℝ) y ≤ gauge s (y • x₀)

Tactic:
obtain h | h := le_or_gt y 0

Goal state after:
case refine_1.inl
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 : y ≤ 0
⊢ (RingHom.id ℝ) y ≤ gauge s (y • x₀)

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)
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
s : Set E
inst✝⁴ : RCLike 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsScalarTower ℝ 𝕜 E
hs₀ : Balanced 𝕜 s
hs₁ : Convex ℝ s
hs₂ : Absorbent ℝ s
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : IsOpen[inst✝¹] s
⊢ {y | (gaugeSeminorm hs₀ hs₁ hs₂) y < 1} = s

Tactic:
exact setOfPred_gauge_lt_one_eq_self_of_isOpen hs₁ hs₂.zero_mem hs

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

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁹ : Field R
inst✝⁸ : AddCommGroup V
inst✝⁷ : TopologicalSpace R
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalRing R
inst✝⁴ : Module R V
inst✝³ : SeparatingDual R V
W : Type u_3
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : FiniteDimensional R W
f : W →ₗ[R] V
⊢ Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap) ↔ Injective ⇑f

Tactic:
by
  constructor <;> intro hf
  · exact LinearMap.dualMap_surjective_iff.mp hf.of_comp
  have := (separatingDual_iff_injective.mp ‹_›).comp hf
  rw [← LinearMap.coe_comp] at this
  exact LinearMap.flip_surjective_iff₁.mpr this

Goal state after:
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✝ : ∃ α, ⇑f = ⇑(α • g)
y : Rˣ
h : ⇑f = ⇑(y • g)
⊢ ↑y • 1 = ↑g.symm ∘SL ↑f

Tactic:
by ext; simp [h]

Goal state after:
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 z • (y - x)) ((fun z ↦ z + ((G y)⁻¹ * G z) • (x - y)) z) = z

Tactic:
simp only [map_add, map_smulₛₗ _, map_mul, map_inv₀, RingHom.id_apply, map_sub, Gx,
        smul_eq_mul, mul_sub, mul_one]

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 y)⁻¹ * G z) • (x - y) + (G z + ((G y)⁻¹ * G z - (G y)⁻¹ * G z * G y)) • (y - x) = z
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case h_ind
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
e : E
s : Set X
s_meas : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : X), L (s.indicator (fun x ↦ e) x) ∂μ = L (μ.real s • e)

Tactic:
← @smul_one_smul E ℝ 𝕜 _ _ _ _ _ (μ.real s) e,

Goal state after:
case h_ind
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
e : E
s : Set X
s_meas : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : X), L (s.indicator (fun x ↦ e) x) ∂μ = L ((μ.real s • 1) • e)
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case neg.inl
X : Type u_1
E : Type u_3
inst✝⁶ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
φ : X → H →L[𝕜] E
φ_int : Integrable φ μ
v : H
hE : ¬CompleteSpace E
hH : Subsingleton H
⊢ (∫ (x : X), φ x ∂μ) v = ∫ (x : X), (φ x) v ∂μ

Tactic:
simp [Subsingleton.eq_zero v]

Goal state after:
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
⊢ StronglyMeasurable fun x ↦ derivWithin f (Ici x) x

Tactic:
by
  borelize F
  apply stronglyMeasurable_iff_measurable_separable.2 ⟨measurable_derivWithin_Ici f, ?_⟩
  obtain ⟨t, t_count, ht⟩ : ∃ t : Set ℝ, t.Countable ∧ Dense t := exists_countable_dense ℝ
  suffices H : range (fun x ↦ derivWithin f (Ici x) x) ⊆ closure (Submodule.span ℝ (f '' t)) from
    IsSeparable.mono (t_count.image f).isSeparable.span.closure H
  rintro - ⟨x, rfl⟩
  suffices H' : range (fun y ↦ derivWithin f (Ici x) y) ⊆ closure (Submodule.span ℝ (f '' t)) from
    H' (mem_range_self _)
  apply range_derivWithin_subset_closure_span_image
  calc Ici x
    = closure (Ioi x ∩ closure t) := by simp [dense_iff_closure_eq.1 ht]
  _ ⊆ closure (closure (Ioi x ∩ t)) := by
      apply closure_mono
      simpa [inter_comm] using (isOpen_Ioi (a := x)).closure_inter (s := t)
  _ ⊆ closure (Ici x ∩ t) := by
      rw [closure_closure]
      exact closure_mono (inter_subset_inter_left _ Ioi_subset_Ici_self)

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

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

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

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

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
g : ℝ → ℝ
h : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖f t‖ ≤ g t
hbound : IntervalIntegrable g μ a b
⊢ ‖∫ (t : ℝ) in a..b, f t ∂μ‖ ≤ |∫ (t : ℝ) in a..b, g t ∂μ|

Tactic:
rw [norm_intervalIntegral_eq, abs_intervalIntegral_eq]

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
g : ℝ → ℝ
h : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖f t‖ ≤ g t
hbound : IntervalIntegrable g μ a b
⊢ ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖ ≤ |∫ (x : ℝ) in Ι a b, g x ∂μ|
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x ↦ ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
h'f : Tendsto (fun x ↦ ‖f' x‖) l atTop
x : ℝ
hx : DifferentiableAt ℝ f x
⊢ (fderiv ℝ (⇑a) (f x)) (deriv f x) = a (deriv f x)

Tactic:
have : fderiv ℝ a (f x) = a.toContinuousLinearMap := a.toContinuousLinearMap.fderiv
        simp only [this]
        rfl

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
⊢ ν (QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A ∩ 𝓕) = ν (QuotientGroup.mk ⁻¹' A ∩ g • 𝓕)

Tactic:
set π_preA := π ⁻¹' A

Goal state after:
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
π_preA : Set G := QuotientGroup.mk ⁻¹' A
⊢ ν (QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A ∩ 𝓕) = ν (π_preA ∩ g • 𝓕)
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 := QuotientGroup.mk
meas_π : Measurable π
⊢ map π μ ≪ map π (μ.restrict 𝓕)

Tactic:
apply AbsolutelyContinuous.mk

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 := ⋯
meas_π : Measurable π
⊢ ∀ ⦃s : Set (G ⧸ Γ)⦄, MeasurableSet s → (map π (μ.restrict 𝓕)) s = 0 → (map π μ) s = 0
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case h₂
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)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
⊢ ‖f p x - f q x‖ ≤ δ

Tactic:
· exact le_of_lt (hv _ hp _ (h'δ hx))

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case hf'
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.IsMulRightInvariant
f : G → E
hf₁ : Integrable f μ
hf₂ : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
⊢ ∫⁻ (x : G), ‖f x‖ₑ ∂μ ≠ ∞

Tactic:
exact ne_of_lt hf₁.2

Goal state after: