mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case hx
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p.toReal
f : ↥(lp E p)
⊢ 0 ≤ ∑' (i : α), ‖↑f i‖ ^ p.toReal

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E p)
hp : 0 < p.toReal
i : α
⊢ 0 ≤ ‖↑f i‖ ^ p.toReal

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E p)
h : ‖f‖ = 0
hp : 0 < p.toReal
hf : HasSum (fun i ↦ ‖↑f i‖ ^ p.toReal) 0
i : α
⊢ 0 ≤ ‖↑f i‖ ^ p.toReal

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E p)
⊢ f = 0 ↔ ↑f = 0

Tactic:
rw [lp.ext_iff, coeFn_zero]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : p ≠ 0
f : ↥(lp E p)
i✝ : α
hp' : p ≠ ∞
hp'' : 0 < p.toReal
i : α
⊢ 0 ≤ ‖↑f i‖ ^ p.toReal

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p.toReal
f : ↥(lp E p)
s : Finset α
i : α
⊢ 0 ≤ ‖↑f i‖ ^ p.toReal

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
F : α → Type u_5
inst✝ : (i : α) → NormedAddCommGroup (F i)
hp : ∞ ≠ 0
x : ↥(lp E ∞)
y : ↥(lp F ∞)
h : ∀ (i : α), ‖↑x i‖ ≤ ‖↑y i‖
⊢ 0 ≤ ‖y‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
c : 𝕜
hp : ∞ ≠ 0
f : ↥(lp E ∞)
h✝ : Nonempty α
hfc : IsLUB (Set.range fun x ↦ ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)
⊢ 0 ≤ ‖c‖ * ‖f‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E✝ : α → Type u_4
p q : ℝ≥0∞
inst✝⁵ : (i : α) → NormedAddCommGroup (E✝ i)
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedRing 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 E
inst✝ : CompleteSpace E
f : ↥(lp (fun x ↦ E) 1)
⊢ ‖{ toFun := fun f ↦ ∑' (i : α), ↑f i, map_add' := ⋯, map_smul' := ⋯ } f‖ ≤ 1 * ‖f‖

Tactic:
simpa using norm_tsum_le f

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p q : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
I : Type u_5
B : I → Type u_6
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : ↥(lp B ∞)
⊢ 0 ≤ ‖f‖ * ‖g‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p✝ q : ℝ≥0∞
inst✝⁴ : (i : α) → NormedAddCommGroup (E i)
inst✝³ : NormedRing 𝕜
inst✝² : (i : α) → Module 𝕜 (E i)
inst✝¹ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
inst✝ : DecidableEq α
p : ℝ≥0∞
i : α
a : E i
j : α
⊢ j ∈ {i_1 | Pi.single i a i_1 ≠ 0} → j ∈ {i}

Tactic:
simp only [Set.mem_singleton_iff, Ne,
      Set.mem_ofPred_eq]

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p✝ q : ℝ≥0∞
inst✝⁴ : (i : α) → NormedAddCommGroup (E i)
inst✝³ : NormedRing 𝕜
inst✝² : (i : α) → Module 𝕜 (E i)
inst✝¹ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
inst✝ : DecidableEq α
p : ℝ≥0∞
i : α
a : E i
j : α
⊢ ¬Pi.single i a j = 0 → j = i
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
x✝ : x ∈ univ
⊢ 0 < 1 + ‖x‖ ^ 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
⊢ 0 < 1 + ‖x‖ ^ 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : ↥(lp E p)
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 ↑f)
⊢ ‖f‖ ≤ C

Tactic:
by
  obtain ⟨i, hi⟩ := hCF.exists
  have hC : 0 ≤ C := (norm_nonneg _).trans hi
  rcases eq_top_or_lt_top p with (rfl | hp)
  · apply norm_le_of_forall_le hC
    exact norm_apply_le_of_tendsto hCF hf
  · have : 0 < p := zero_lt_one.trans_le _i.elim
    have hp' : 0 < p.toReal := ENNReal.toReal_pos this.ne' hp.ne
    apply norm_le_of_forall_sum_le hp' hC
    exact sum_rpow_le_of_tendsto hp.ne hCF hf

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
F : ι → ↥(lp E p)
hF : Bornology.IsBounded (Set.range F)
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
⊢ Memℓp f p

Tactic:
obtain ⟨C, hCF⟩ : ∃ C, ∀ k, ‖F k‖ ≤ C := hF.exists_norm_le.imp fun _ ↦ Set.forall_mem_range.1

Goal state after:
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
F : ι → ↥(lp E p)
hF : Bornology.IsBounded (Set.range F)
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
C : ℝ
hCF : ∀ (k : ι), ‖F k‖ ≤ C
⊢ Memℓp f p
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case mpr.hC
α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
f : α → ↥(lp (fun x ↦ ℝ) ∞)
s : Set α
K : ℝ≥0
hgl : ∀ (i : ι), ∀ x ∈ s, ∀ y ∈ s, dist (↑(f x) i) (↑(f y) i) ≤ ↑K * dist x y
x : α
hx : x ∈ s
y : α
hy : y ∈ s
⊢ 0 ≤ ↑K * dist x y

Tactic:
positivity

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

Goal state before:
pp: p : ℝ[X]
⊢ Tendsto (fun x ↦ if x ≤ 0 then 0 else Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0) (𝓝 0)

Tactic:
refine tendsto_const_nhds.if ?_

Goal state after:
p : ℝ[X]
⊢ Tendsto (fun x ↦ Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | ¬x ≤ 0}) (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: case inr.inr
p : ℝ[X]
x : ℝ
hx : 0 < x
⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x

Tactic:
have := ((p.hasDerivAt x⁻¹).mul (hasDerivAt_neg _).exp).comp x (hasDerivAt_inv hx.ne')

Goal state after:
case inr.inr
p : ℝ[X]
x : ℝ
hx : 0 < x
this :
  HasDerivAt (((fun x ↦ Polynomial.eval x p) * fun x ↦ rexp (-x)) ∘ Inv.inv)
    ((Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + Polynomial.eval x⁻¹ p * (rexp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x
⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x
Module: Mathlib.Analysis.Calculus.BumpFunction.InnerProduct

Goal state before:
pp: case funProp.discharger
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
R : ℝ
hR : 0 < R - 1
⊢ (R, 0).1 - 1 ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := fun a ↦ |bound a|
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
⊢ ∀ᵐ (a : α) ∂μ, ‖F' a‖ ≤ b a

Tactic:
by
      apply (h_diff.and h_lipsch).mono
      rintro a ⟨ha_diff, ha_lip⟩
      exact ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Calculus

Goal state before:
pp: n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
⊢ 0 < 1 + ‖x‖ ^ 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ ν : Measure G
inst✝⁵ : AddGroup G
inst✝⁴ : MeasurableAdd₂ G
inst✝³ : MeasurableNeg G
inst✝² : SFinite μ
inst✝¹ : μ.IsAddRightInvariant
inst✝ : SFinite ν
hf : Integrable f ν
hg : Integrable g μ
h_meas : AEStronglyMeasurable (fun p ↦ (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
h2_meas : AEStronglyMeasurable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν
t : G
⊢ 0 ≤ ∫ (x : G), ‖(L (f t)) (g (x - t))‖ ∂μ

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case refine_2
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this : 𝓝 0 = 𝓝 (0 + 0 + 0)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x_1 : ι × E) in l.curry (𝓝 x),
    dist ((↑‖x_1.2 - x‖)⁻¹ • (f x_1.1 x_1.2 - f x_1.1 x - ((f' x_1.1 x) x_1.2 - (f' x_1.1 x) x))) 0 < ε

Tactic:
rw [eventually_curry_iff]

Goal state after:
case refine_2
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this : 𝓝 0 = 𝓝 (0 + 0 + 0)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x_1 : ι) in l,
    ∀ᶠ (y : E) in 𝓝 x,
      dist
          ((↑‖(x_1, y).2 - x‖)⁻¹ •
            (f (x_1, y).1 (x_1, y).2 - f (x_1, y).1 x - ((f' (x_1, y).1 x) (x_1, y).2 - (f' (x_1, y).1 x) x)))
          0 <
        ε
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: case pos.h_bound.h₁.hb
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ 0 ≤ ‖x - x₀‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: case pos.h_bound.h₂.hb
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ 0 ≤ ‖x - x₀‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: case pos.h_bound.h₂.hc
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ 0 ≤ ‖F' a‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
g : P → G → E'
s : Set P
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContinuousOn (↿g) (s ×ˢ univ)
H : ∃ p ∈ s, ∃ x, g p x ≠ 0
this : LocallyCompactSpace G
q₀ : P
x₀ : G
hq₀ : (q₀, x₀).1 ∈ s
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := ⋯
k'_comp : IsCompact k'
g' : P × G → G → E' := ⋯
s' : Set (P × G) := ⋯
A : ContinuousOn (uncurry g') (s' ×ˢ univ)
⊢ ∀ (p : P × G) (x : G), p ∈ s' → x ∉ k' → g' p x = 0

Tactic:
rintro ⟨p, x⟩ y ⟨hp, hx⟩ hy

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
g : P → G → E'
s : Set P
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContinuousOn (↿g) (s ×ˢ univ)
H : ∃ p ∈ s, ∃ x, g p x ≠ 0
this : LocallyCompactSpace G
q₀ : P
x₀ : G
hq₀ : (q₀, x₀).1 ∈ s
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := -k +ᵥ t
k'_comp : IsCompact k'
g' : P × G → G → E' := fun p x ↦ g p.1 (p.2 - x)
s' : Set (P × G) := s ×ˢ t
A : ContinuousOn (uncurry g') (s' ×ˢ univ)
p : P
x y : G
hp : (p, x).1 ∈ s
hx : (p, x).2 ∈ t
hy : y ∉ k'
⊢ g' (p, x) y = 0
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
x₀ : H
s : Set H
inst✝ : NormedSpace ℝ H
μ : Measure ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[ℝ] E
a b : ℝ
bound : ℝ → ℝ
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ∀ x ∈ s, ‖F' x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ∀ x ∈ s, HasFDerivAt (fun x ↦ F x t) (F' x t) x
⊢ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Tactic:
rw [ae_restrict_uIoc_iff] at h_diff h_bound

Goal state after:
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
x₀ : H
s : Set H
inst✝ : NormedSpace ℝ H
μ : Measure ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[ℝ] E
a b : ℝ
bound : ℝ → ℝ
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound :
  (∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc a b), ∀ x_1 ∈ s, ‖F' x_1 x‖ ≤ bound x) ∧
    ∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc b a), ∀ x_1 ∈ s, ‖F' x_1 x‖ ≤ bound x
bound_integrable : IntervalIntegrable bound μ a b
h_diff :
  (∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc a b), ∀ x_1 ∈ s, HasFDerivAt (fun x_2 ↦ F x_2 x) (F' x_1 x) x_1) ∧
    ∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc b a), ∀ x_1 ∈ s, HasFDerivAt (fun x_2 ↦ F x_2 x) (F' x_1 x) x_1
⊢ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: case neg
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
F : 𝕜 → α → E
x₀ : 𝕜
s : Set 𝕜
F' : α → E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x ↦ F x a) s
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (L (F' a)) x₀
hm : AEStronglyMeasurable (⇑L ∘ F') μ
key : HasFDerivAt (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
hF'_int : Integrable F' μ
hE : ¬CompleteSpace E
⊢ HasDerivAt (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Tactic:
simpa [integral, hE] using! hasDerivAt_const x₀ 0

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

Goal state before:
pp: case neg
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
hF : ¬CompleteSpace F
⊢ ∫ (x : E), f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E), f x ∂μ

Tactic:
simp [integral, hF]

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : BorelSpace G
inst✝⁴ : SecondCountableTopology G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : SFinite μ
inst✝¹ : NormedSpace ℝ E'
inst✝ : CompleteSpace E'
ι : Type u_1
g : ι → G → E'
l : Filter ι
x₀ : G
z₀ : E'
φ : ι → G → ℝ
k : ι → G
hnφ : ∀ᶠ (i : ι) in l, ∀ (x : G), 0 ≤ φ i x
hiφ : ∀ᶠ (i : ι) in l, ∫ (x : G), φ i x ∂μ = 1
hmg : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (g i) μ
hk : Tendsto k l (𝓝 x₀)
hφ : ∀ t ∈ 𝓝 0, ∀ᶠ (x : ι) in l, support (φ x) ⊆ t
hcg :
  ∀ ε > 0,
    ∃ pa,
      (∀ᶠ (i : ι) in l, pa i) ∧ ∃ ε_1 > 0, ∀ ⦃i : ι⦄, pa i → ∀ ⦃x : G⦄, dist x x₀ < ε_1 → dist (uncurry g (i, x)) z₀ < ε
ε : ℝ
hε : ε > 0
h2ε : 0 < ε / 3
p : ι → Prop
hp : ∀ᶠ (i : ι) in l, p i
δ : ℝ
hδ : δ > 0
hgδ : ∀ ⦃i : ι⦄, p i → ∀ ⦃x : G⦄, dist x x₀ < δ → dist (g i x) z₀ < ε / 3
h2k : ∀ᶠ (x : ι) in l, dist (k x) x₀ < δ / 2
h2φ : ∀ᶠ (x : ι) in l, support (φ x) ⊆ ball 0 (δ / 2)
i : ι
hpi : p i
hki : dist (k i) x₀ < δ / 2
hφi : support (φ i) ⊆ ball 0 (δ / 2)
hnφi : ∀ (x : G), 0 ≤ φ i x
hiφi : ∫ (x : G), φ i x ∂μ = 1
hmgi : AEStronglyMeasurable (g i) μ
hgi : dist (g i (k i)) z₀ < ε / 3
⊢ ∀ x' ∈ ball (k i) (δ / 2), dist (g i x') (g i (k i)) ≤ ε / 3 + ε / 3

Tactic:
intro x' hx'

Goal state after:
G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : BorelSpace G
inst✝⁴ : SecondCountableTopology G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : SFinite μ
inst✝¹ : NormedSpace ℝ E'
inst✝ : CompleteSpace E'
ι : Type u_1
g : ι → G → E'
l : Filter ι
x₀ : G
z₀ : E'
φ : ι → G → ℝ
k : ι → G
hnφ : ∀ᶠ (i : ι) in l, ∀ (x : G), 0 ≤ φ i x
hiφ : ∀ᶠ (i : ι) in l, ∫ (x : G), φ i x ∂μ = 1
hmg : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (g i) μ
hk : Tendsto k l (𝓝 x₀)
hφ : ∀ t ∈ 𝓝 0, ∀ᶠ (x : ι) in l, support (φ x) ⊆ t
hcg :
  ∀ ε > 0,
    ∃ pa,
      (∀ᶠ (i : ι) in l, pa i) ∧ ∃ ε_1 > 0, ∀ ⦃i : ι⦄, pa i → ∀ ⦃x : G⦄, dist x x₀ < ε_1 → dist (uncurry g (i, x)) z₀ < ε
ε : ℝ
hε : ε > 0
h2ε : 0 < ε / 3
p : ι → Prop
hp : ∀ᶠ (i : ι) in l, p i
δ : ℝ
hδ : δ > 0
hgδ : ∀ ⦃i : ι⦄, p i → ∀ ⦃x : G⦄, dist x x₀ < δ → dist (g i x) z₀ < ε / 3
h2k : ∀ᶠ (x : ι) in l, dist (k x) x₀ < δ / 2
h2φ : ∀ᶠ (x : ι) in l, support (φ x) ⊆ ball 0 (δ / 2)
i : ι
hpi : p i
hki : dist (k i) x₀ < δ / 2
hφi : support (φ i) ⊆ ball 0 (δ / 2)
hnφi : ∀ (x : G), 0 ≤ φ i x
hiφi : ∫ (x : G), φ i x ∂μ = 1
hmgi : AEStronglyMeasurable (g i) μ
hgi : dist (g i (k i)) z₀ < ε / 3
x' : G
hx' : x' ∈ ball (k i) (δ / 2)
⊢ dist (g i x') (g i (k i)) ≤ ε / 3 + ε / 3