mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ 1 / √((|√(1 + ‖x‖ ^ 2)| ^ 2 - ‖x‖ ^ 2) / |√(1 + ‖x‖ ^ 2)| ^ 2) = √(1 + ‖x‖ ^ 2)

Tactic:
simp [sq_abs, Real.sq_sqrt (zero_lt_one_add_norm_sq x).le]

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✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : DecidableEq α
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
f : ↥(lp E p)
hp₀ : 0 < p
hp' : 0 < p.toReal
this :
  ∀ ε > 0,
    ∀ᶠ (x : Finset α) in (SummationFilter.unconditional α).filter,
      dist (∑ b ∈ x, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε
ε : ℝ
hε : ε > 0
s : Finset α
hs : dist (∑ b ∈ s, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε ^ p.toReal
⊢ dist (∑ b ∈ s, lp.single p b (↑f b)) f ^ p.toReal < ε ^ p.toReal

Tactic:
rw [dist_comm] at hs

Goal state after:
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : DecidableEq α
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
f : ↥(lp E p)
hp₀ : 0 < p
hp' : 0 < p.toReal
this :
  ∀ ε > 0,
    ∀ᶠ (x : Finset α) in (SummationFilter.unconditional α).filter,
      dist (∑ b ∈ x, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε
ε : ℝ
hε : ε > 0
s : Finset α
hs : dist (‖f‖ ^ p.toReal) (∑ b ∈ s, ‖↑f b‖ ^ p.toReal) < ε ^ p.toReal
⊢ dist (∑ b ∈ s, lp.single p b (↑f b)) f ^ p.toReal < ε ^ p.toReal
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)
p q : ℝ≥0∞
h : p ≤ q
f : ↥(lp E p)
⊢ ↑((linearMapOfLE 𝕜 E h) f) = ↑f

Tactic:
by
  ext; rfl

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)
p q : ℝ≥0∞
h : p ≤ q
⊢ (linearMapOfLE 𝕜 E h).toAddMonoidHom = AddSubgroup.inclusion ⋯

Tactic:
by
  ext; rfl

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)
p q r : ℝ≥0∞
hpq : p ≤ q
hqr : q ≤ r
⊢ linearMapOfLE 𝕜 E hqr ∘ₗ linearMapOfLE 𝕜 E hpq = linearMapOfLE 𝕜 E ⋯

Tactic:
by
  ext; rfl

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

Goal state before:
pp: case e'_10
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedSpace ℝ E
n : WithTop ℕ∞
x✝ : E
⊢ ‖x✝‖ ^ 2 = (⇑reCLM ∘ fun x ↦ ⟪id x, id x⟫) x✝

Tactic:
exact (inner_self_eq_norm_sq _).symm

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)
⊢ Integrable (fun p ↦ (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)

Tactic:
have h2_meas : AEStronglyMeasurable (fun y : G => ∫ x : G, ‖L (f y) (g (x - y))‖ ∂μ) ν :=
    h_meas.prod_swap.norm.integral_prod_right'

Goal state after:
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
⊢ Integrable (fun p ↦ (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: case pos.h_bound
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
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 : ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀))‖ ≤ b a + ‖F' a‖

Tactic:
replace ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ := ha_bound x hx

Goal state after:
case pos.h_bound
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
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‖
⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀))‖ ≤ b a + ‖F' a‖
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)
p : P
hp : p ∈ s
x : G
hx : g p x ≠ 0
A : support (g p) ⊆ k
⊢ LocallyCompactSpace G

Tactic:
have B : Continuous (g p) := by
      refine hg.comp_continuous (.prodMk_right _) fun x => ?_
      simpa only [prodMk_mem_set_prod_eq, mem_univ, and_true] using hp

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)
p : P
hp : p ∈ s
x : G
hx : g p x ≠ 0
A : support (g p) ⊆ k
B : Continuous[inst✝³, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (g p)
⊢ LocallyCompactSpace G
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
s : Set P
v : P → G
hv : ContinuousOn v s
g : P → G → E'
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)
⊢ ContinuousOn (fun x ↦ (f ⋆[L, μ] g x) (v x)) s

Tactic:
by
  apply
    (continuousOn_convolution_right_with_param L hk hgs hf hg).comp (continuousOn_id.prodMk hv)
  intro x hx
  simp only [hx, prodMk_mem_set_prod_eq, mem_univ, and_self_iff, _root_.id]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : OpensMeasurableSpace α
hs : MeasurableSet s
⊢ MeasurableSet (μ.everywherePosSubset s)

Tactic:
by
  rcases exists_isOpen_everywherePosSubset_eq_sdiff μ s with ⟨u, u_open, hu⟩
  rw [hu]
  exact hs.diff u_open.measurableSet

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
hφ : Tendsto (fun n ↦ support (φ n)) l (𝓝 0).smallSets
hmg : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (g i) μ
hcg : Tendsto (uncurry g) (l ×ˢ 𝓝 x₀) (𝓝 z₀)
hk : Tendsto k l (𝓝 x₀)
⊢ Tendsto (fun i ↦ (φ i ⋆[lsmul ℝ ℝ, μ] g i) (k i)) l (𝓝 z₀)

Tactic:
by
  simp_rw [tendsto_smallSets_iff] at hφ
  rw [Metric.tendsto_nhds] at hcg ⊢
  simp_rw [Metric.eventually_prod_nhds_iff] at hcg
  intro ε hε
  have h2ε : 0 < ε / 3 := div_pos hε (by simp)
  obtain ⟨p, hp, δ, hδ, hgδ⟩ := hcg _ h2ε
  dsimp only [uncurry] at hgδ
  have h2k := hk.eventually (ball_mem_nhds x₀ <| half_pos hδ)
  have h2φ := hφ (ball (0 : G) _) <| ball_mem_nhds _ (half_pos hδ)
  filter_upwards [hp, h2k, h2φ, hnφ, hiφ, hmg] with i hpi hki hφi hnφi hiφi hmgi
  have hgi : dist (g i (k i)) z₀ < ε / 3 := hgδ hpi (hki.trans <| half_lt_self hδ)
  have h1 : ∀ x' ∈ ball (k i) (δ / 2), dist (g i x') (g i (k i)) ≤ ε / 3 + ε / 3 := by
    intro x' hx'
    grw [dist_triangle_right, hgδ hpi ?_, hgi]
    grw [dist_triangle, hx'.out, hki, add_halves]
  have := dist_convolution_le (add_pos h2ε h2ε).le hφi hnφi hiφi hmgi h1
  refine ((dist_triangle _ _ _).trans_lt (add_lt_add_of_le_of_lt this hgi)).trans_eq ?_
  ring

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

Goal state before:
pp: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
D : G → ℝ := fun x ↦ ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous[inst✝⁹, _] D
x : G
C : Continuous[inst✝⁹, _] fun y ↦ g (y⁻¹ * x)
⊢ x * x₀⁻¹ ∈ support fun y ↦ g (y⁻¹ * x)

Tactic:
simpa using g_pos

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: E : Type uE
E' : Type uE'
F : Type uF
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : NormedSpace ℝ F
f : ℝ → E
g : ℝ → E'
L : E →L[ℝ] E' →L[ℝ] F
ν : Measure ℝ
inst✝ : NullSingletonClass ν
x : ℝ
⊢ (Ioi 0).indicator (fun x ↦ ∫ (t : ℝ) in 0..x, (L (f t)) (g (x - t)) ∂ν) x =
    ∫ (t : ℝ), (L ((Ioi 0).indicator f t)) ((Ioi 0).indicator g (x - t)) ∂ν

Tactic:
indicator

Goal state after:
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : NormedSpace ℝ F
f : ℝ → E
g : ℝ → E'
L : E →L[ℝ] E' →L[ℝ] F
ν : Measure ℝ
inst✝ : NullSingletonClass ν
x : ℝ
⊢ (if x ∈ Ioi 0 then ∫ (t : ℝ) in 0..x, (L (f t)) (g (x - t)) ∂ν else 0) =
    ∫ (t : ℝ), (L ((Ioi 0).indicator f t)) ((Ioi 0).indicator g (x - t)) ∂ν
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case pos
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : NormedSpace ℝ F
f : ℝ → E
g : ℝ → E'
L : E →L[ℝ] E' →L[ℝ] F
ν : Measure ℝ
inst✝ : NullSingletonClass ν
x : ℝ
h : x ∈ Ioi 0
⊢ ∫ (t : ℝ) in 0..x, (L (f t)) (g (x - t)) ∂ν = ∫ (t : ℝ), (L ((Ioi 0).indicator f t)) ((Ioi 0).indicator g (x - t)) ∂ν

Tactic:
rw [intervalIntegral.integral_of_le (le_of_lt h), integral_Ioc_eq_integral_Ioo, ←
      integral_indicator (measurableSet_Ioo : MeasurableSet (Ioo 0 x))]
    congr 1 with t : 1
    have : t ≤ 0 ∨ t ∈ Ioo 0 x ∨ x ≤ t := by
      rcases le_or_gt t 0 with (h | h)
      · exact Or.inl h
      · rcases lt_or_ge t x with (h' | h')
        exacts [Or.inr (Or.inl ⟨h, h'⟩), Or.inr (Or.inr h')]
    rcases this with (ht | ht | ht)
    · rw [indicator_of_notMem (notMem_Ioo_of_le ht), indicator_of_notMem (notMem_Ioi.mpr ht),
        map_zero, zero_apply]
    · rw [indicator_of_mem ht, indicator_of_mem (mem_Ioi.mpr ht.1),
          indicator_of_mem (mem_Ioi.mpr <| sub_pos.mpr ht.2)]
    · rw [indicator_of_notMem (notMem_Ioo_of_ge ht),
          indicator_of_notMem (notMem_Ioi.mpr (sub_nonpos_of_le ht)), map_zero]

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : HasContDiffBump E
inst✝ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
x : E
⊢ ↑f (c - x) / ∫ (x : E), ↑f x ∂μ = ↑f (c + x) / ∫ (x : E), ↑f x ∂μ

Tactic:
f.sub

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
⊢ 0 < ∫ (x : E), ↑f x ∂μ

Tactic:
refine (integral_pos_iff_support_of_nonneg f.nonneg' f.integrable).mpr ?_

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
⊢ 0 < μ (support ↑f)
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: case refine_3
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
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
eG : Type (max uG uE' uF uP) := ULift.{max uE' uF uP, uG} G
this✝¹ : MeasurableSpace eG := borel eG
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ULift.{max uG uF uP, uE'} E'
eF : Type (max uF uG uE' uP) := ULift.{max uG uE' uP, uF} F
eP : Type (max uP uG uE' uF) := ULift.{max uG uE' uF, uP} P
isoG : eG ≃L[𝕜] G := ContinuousLinearEquiv.ulift
isoE' : eE' ≃L[𝕜] E' := ContinuousLinearEquiv.ulift
isoF : eF ≃L[𝕜] F := ContinuousLinearEquiv.ulift
isoP : eP ≃L[𝕜] P := ContinuousLinearEquiv.ulift
ef : eG → E := f ∘ ⇑isoG
eμ : Measure eG := Measure.map (⇑isoG.symm) μ
eg : eP → eG → eE' := fun ep ex ↦ isoE'.symm (g (isoP ep) (isoG ex))
eL : E →L[𝕜] eE' →L[𝕜] eF := ↑(isoE'.arrowCongr isoF).symm ∘SL L
R : eP × eG → eF := fun q ↦ (ef ⋆[eL, eμ] eg q.1) q.2
hek : IsCompact (⇑isoG ⁻¹' k)
hes : IsOpen[ULift.topologicalSpace] (⇑isoP ⁻¹' s)
⊢ ContDiffOn 𝕜 (↑n) (↿eg) ((⇑isoP ⁻¹' s) ×ˢ univ)

Tactic:
apply isoE'.symm.contDiff.comp_contDiffOn

Goal state after:
case refine_3
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
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
eG : Type (max uG uE' uF uP) := ⋯
this✝¹ : MeasurableSpace eG := ⋯
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ⋯
eF : Type (max uF uG uE' uP) := ⋯
eP : Type (max uP uG uE' uF) := ⋯
isoG : eG ≃L[𝕜] G := ⋯
isoE' : eE' ≃L[𝕜] E' := ⋯
isoF : eF ≃L[𝕜] F := ⋯
isoP : eP ≃L[𝕜] P := ⋯
ef : eG → E := ⋯
eμ : Measure eG := ⋯
eg : eP → eG → eE' := ⋯
eL : E →L[𝕜] eE' →L[𝕜] eF := ⋯
R : eP × eG → eF := ⋯
hek : IsCompact (⇑isoG ⁻¹' k)
hes : IsOpen[ULift.topologicalSpace] (⇑isoP ⁻¹' s)
⊢ ContDiffOn 𝕜 (↑n) (fun x ↦ g (isoP x.1) (isoG x.2)) ((⇑isoP ⁻¹' s) ×ˢ univ)
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
i j : Fin N.succ
inej : i ≠ j
⊢ Pairwise fun i j ↦ a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨ a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j

Tactic:
simpa only [dist_eq_norm] using a.h

Goal state after:
Module: Mathlib.Analysis.Calculus.Conformal.InnerProduct

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : InnerProductSpace ℝ F
f : E → F
x : E
⊢ ConformalAt f x ↔ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪(fderiv ℝ f x) u, (fderiv ℝ f x) v⟫ = c * ⟪u, v⟫

Tactic:
conformalAt_iff_isConformalMap_fderiv,

Goal state after:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : InnerProductSpace ℝ F
f : E → F
x : E
⊢ IsConformalMap (fderiv ℝ f x) ↔ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪(fderiv ℝ f x) u, (fderiv ℝ f x) v⟫ = c * ⟪u, v⟫
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
i j : Fin N.succ
inej : i ≠ j
hi : ‖a.c i‖ ≤ 2
hj : 2 < ‖a.c j‖
⊢ Pairwise fun i j ↦ a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨ a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j

Tactic:
simpa only [dist_eq_norm] using a.h

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case inr.inr.refine_1
α : Type u_1
inst✝⁴ : MetricSpace α
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < μ s
h✝ : Nonempty α
Npos : N ≠ 0
o : Set α
so : s ⊆ o
omeas : MeasurableSet o
μo : μ o = μ s
a : BallPackage (↑s) α := { c := fun x ↦ ↑x, r := fun x ↦ r ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
u : Fin N → Set ↑s
hu : ∀ (i : Fin N), (u i).PairwiseDisjoint fun j ↦ closedBall (a.c j) (a.r j)
hu' : range a.c ⊆ ⋃ i, ⋃ j ∈ u i, ball (a.c j) (a.r j)
u_count : ∀ (i : Fin N), (u i).Countable
v : Fin N → Set α := fun i ↦ ⋃ x ∈ u i, closedBall (↑x) (r ↑x)
A : s = ⋃ i, s ∩ v i
S : ∑ _i, μ s / ↑N ≤ ∑ i, μ (s ∩ v i)
i : Fin N
hi : μ s / (↑N + 1) < μ (s ∩ v i)
B : μ (o ∩ v i) = ∑' (x : ↑(u i)), μ (o ∩ closedBall (↑↑x) (r ↑↑x))
w : Finset ↑(u i)
hw : μ s / (↑N + 1) < ∑ x ∈ w, μ (o ∩ closedBall (↑↑x) (r ↑↑x))
⊢ ↑(Finset.image (fun x ↦ ↑↑x) w) ⊆ s

Tactic:
simp only [image_subset_iff, Finset.coe_image]

Goal state after:
case inr.inr.refine_1
α : Type u_1
inst✝⁴ : MetricSpace α
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < μ s
h✝ : Nonempty α
Npos : N ≠ 0
o : Set α
so : s ⊆ o
omeas : MeasurableSet o
μo : μ o = μ s
a : BallPackage (↑s) α := { c := fun x ↦ ↑x, r := fun x ↦ r ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
u : Fin N → Set ↑s
hu : ∀ (i : Fin N), (u i).PairwiseDisjoint fun j ↦ closedBall (a.c j) (a.r j)
hu' : range a.c ⊆ ⋃ i, ⋃ j ∈ u i, ball (a.c j) (a.r j)
u_count : ∀ (i : Fin N), (u i).Countable
v : Fin N → Set α := fun i ↦ ⋃ x ∈ u i, closedBall (↑x) (r ↑x)
A : s = ⋃ i, s ∩ v i
S : ∑ _i, μ s / ↑N ≤ ∑ i, μ (s ∩ v i)
i : Fin N
hi : μ s / (↑N + 1) < μ (s ∩ v i)
B : μ (o ∩ v i) = ∑' (x : ↑(u i)), μ (o ∩ closedBall (↑↑x) (r ↑↑x))
w : Finset ↑(u i)
hw : μ s / (↑N + 1) < ∑ x ∈ w, μ (o ∩ closedBall (↑↑x) (r ↑↑x))
⊢ ↑w ⊆ (fun x ↦ ↑↑x) ⁻¹' s