mathlib-initiative/mathlib-tactics huggingface dataset:
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
D_pos : ∀ (x : G), 0 < D x
⊢ ∫ (y : G), ∫ (x : G), f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ ∂ν = ∫ (y : G), ∫ (x : G), f (y * x) * (D (y * x))⁻¹ * g x ∂μ ∂ν

Tactic:
congr with y

Goal state after:
case e_f
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
D_pos : ∀ (x : G), 0 < D x
y : G
⊢ ∫ (x : G), f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ = ∫ (x : G), f (y * x) * (D (y * x))⁻¹ * g x ∂μ
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case inr.refine_1.h₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n✝ : ℕ∞
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : s.Nonempty
ι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 }
T : Set ι
T_count : T.Countable
hT : ⋃ f ∈ T, support ↑f = s
g0 : ℕ → ι
hg : T = range g0
g : ℕ → E → ℝ := fun n ↦ ↑(g0 n)
g_s : ∀ (n : ℕ), support (g n) ⊆ s
s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n)
g_smooth : ∀ (n : ℕ), ContDiff ℝ ∞ (g n)
g_comp_supp : ∀ (n : ℕ), HasCompactSupport (g n)
g_nonneg : ∀ (n : ℕ) (x : E), 0 ≤ g n x
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
δc : HasSum δ c
c_lt : c < 1
r : ℕ → ℝ
rpos : ∀ (n : ℕ), 0 < r n
hr : ∀ (n i : ℕ), i ≤ n → ∀ (x : E), ‖iteratedFDeriv ℝ i (r n • g n) x‖ ≤ ↑(δ n)
S : ∀ (x : E), Summable fun n ↦ (r n • g n) x
x : E
hx : x ∈ s
n : ℕ
hn : x ∈ support (g n)
⊢ x ∈ support fun x ↦ ∑' (n : ℕ), (r n • g n) x

Tactic:
have I : 0 < r n * g n x := mul_pos (rpos n) (lt_of_le_of_ne (g_nonneg n x) (Ne.symm hn))

Goal state after:
case inr.refine_1.h₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n✝ : ℕ∞
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : s.Nonempty
ι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 }
T : Set ι
T_count : T.Countable
hT : ⋃ f ∈ T, support ↑f = s
g0 : ℕ → ι
hg : T = range g0
g : ℕ → E → ℝ := fun n ↦ ↑(g0 n)
g_s : ∀ (n : ℕ), support (g n) ⊆ s
s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n)
g_smooth : ∀ (n : ℕ), ContDiff ℝ ∞ (g n)
g_comp_supp : ∀ (n : ℕ), HasCompactSupport (g n)
g_nonneg : ∀ (n : ℕ) (x : E), 0 ≤ g n x
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
δc : HasSum δ c
c_lt : c < 1
r : ℕ → ℝ
rpos : ∀ (n : ℕ), 0 < r n
hr : ∀ (n i : ℕ), i ≤ n → ∀ (x : E), ‖iteratedFDeriv ℝ i (r n • g n) x‖ ≤ ↑(δ n)
S : ∀ (x : E), Summable fun n ↦ (r n • g n) x
x : E
hx : x ∈ s
n : ℕ
hn : x ∈ support (g n)
I : 0 < r n * g n x
⊢ x ∈ support fun x ↦ ∑' (n : ℕ), (r n • g n) x
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: case top
𝕜 : Type u𝕜
E : Type uE
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
G P : Type uP
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : NormedSpace 𝕜 G
inst✝⁶ : NormedAddCommGroup P
inst✝⁵ : NormedSpace 𝕜 P
f : G → E
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hf : LocallyIntegrable f μ
ih :
  ∀ (n : ℕ) {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 E']
    [inst_3 : NormedSpace ℝ F] [inst_4 : NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'},
    (∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) →
      ContDiffOn 𝕜 (↑↑n) (↿g) (s ×ˢ univ) → ContDiffOn 𝕜 (↑↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)
E' F : Type uP
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hg : ContDiffOn 𝕜 (↑⊤) (↿g) (s ×ˢ univ)
⊢ ContDiffOn 𝕜 (↑⊤) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)

Tactic:
rw [contDiffOn_infty] at hg ⊢

Goal state after:
case top
𝕜 : Type u𝕜
E : Type uE
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
G P : Type uP
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : NormedSpace 𝕜 G
inst✝⁶ : NormedAddCommGroup P
inst✝⁵ : NormedSpace 𝕜 P
f : G → E
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hf : LocallyIntegrable f μ
ih :
  ∀ (n : ℕ) {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 E']
    [inst_3 : NormedSpace ℝ F] [inst_4 : NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'},
    (∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) →
      ContDiffOn 𝕜 (↑↑n) (↿g) (s ×ˢ univ) → ContDiffOn 𝕜 (↑↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)
E' F : Type uP
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hg : ∀ (n : ℕ), ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
⊢ ∀ (n : ℕ), ContDiffOn 𝕜 (↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ HasContDiffBump E

Tactic:
refine ⟨⟨?_⟩⟩
  borelize E
  have IR : ∀ R : ℝ, 1 < R → 0 < (R - 1) / (R + 1) := by intro R hR; apply div_pos <;> linarith
  exact
    { toFun := fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)⁻¹ • x) else 0
      mem_Icc := fun R x => by
        push _ ∈ _
        split_ifs with h
        · refine ⟨y_nonneg _ _, y_le_one _ (IR R h)⟩
        · simp only [le_refl, zero_le_one, and_self]
      symmetric := fun R x => by
        split_ifs
        · simp only [y_neg, smul_neg]
        · rfl
      smooth := by
        suffices
          ContDiffOn ℝ ∞
            (uncurry y ∘ fun p : ℝ × E => ((p.1 - 1) / (p.1 + 1), ((p.1 + 1) / 2)⁻¹ • p.2))
            (Ioi 1 ×ˢ univ) by
          apply this.congr
          rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
          simp only [hR, uncurry_apply_pair, if_true, Function.comp_apply]
        apply (y_smooth E).comp
        · apply ContDiffOn.prodMk
          · refine
              (contDiffOn_fst.sub contDiffOn_const).div (contDiffOn_fst.add contDiffOn_const) ?_
            rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
            apply ne_of_gt
            dsimp only
            linarith
          · apply ContDiffOn.smul _ contDiffOn_snd
            refine ((contDiffOn_fst.add contDiffOn_const).div_const _).inv ?_
            rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
            apply ne_of_gt
            dsimp only
            linarith
        · rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
          have A : 0 < (R - 1) / (R + 1) := by apply div_pos <;> linarith
          have B : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith
          simp only [prodMk_mem_set_prod_eq, mem_Ioo, mem_univ, and_true, A, B]
      eq_one := fun R hR x hx => by
        have A : 0 < R + 1 := by linarith
        simp only [hR, if_true]
        apply y_eq_one_of_mem_closedBall (IR R hR)
        simp only [norm_smul, inv_div, mem_closedBall_zero_iff, Real.norm_eq_abs, abs_div, abs_two,
          abs_of_nonneg A.le]
        calc
          2 / (R + 1) * ‖x‖ ≤ 2 / (R + 1) := mul_le_of_le_one_right (by positivity) hx
          _ = 1 - (R - 1) / (R + 1) := by field
      support := fun R hR => by
        have A : 0 < (R + 1) / 2 := by linarith
        have C : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith
        simp only [hR, if_true, support_comp_inv_smul₀ A.ne', y_support _ (IR R hR) C,
          _root_.smul_ball A.ne', Real.norm_of_nonneg A.le, smul_zero]
        refine congr (congr_arg ball (Eq.refl 0)) ?_
        field }

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ s.card ≤ 5 ^ finrank ℝ E

Tactic:
borelize E
  let μ : Measure E := Measure.addHaar
  let δ : ℝ := (1 : ℝ) / 2
  let ρ : ℝ := (5 : ℝ) / 2
  have ρpos : 0 < ρ := by norm_num
  set A := ⋃ c ∈ s, ball (c : E) δ with hA
  have D : Set.Pairwise (s : Set E) (Disjoint on fun c => ball (c : E) δ) := by
    rintro c hc d hd hcd
    apply ball_disjoint_ball
    rw [dist_eq_norm]
    convert! h c hc d hd hcd
    norm_num
  have A_subset : A ⊆ ball (0 : E) ρ := by
    refine iUnion₂_subset fun x hx => ?_
    apply ball_subset_ball'
    calc
      δ + dist x 0 ≤ δ + 2 := by rw [dist_zero_right]; exact add_le_add le_rfl (hs x hx)
      _ = 5 / 2 := by norm_num
  have I :
    (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) ≤
      ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) :=
    calc
      (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) = μ A := by
        rw [hA, measure_biUnion_finset D fun c _ => measurableSet_ball]
        have I : 0 < δ := by norm_num
        simp only [μ.addHaar_ball_of_pos _ I]
        simp only [Finset.sum_const, nsmul_eq_mul, mul_assoc]
      _ ≤ μ (ball (0 : E) ρ) := measure_mono A_subset
      _ = ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1) := by
        simp only [μ.addHaar_ball_of_pos _ ρpos]
  have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) :=
    (ENNReal.mul_le_mul_iff_left (measure_ball_pos _ _ zero_lt_one).ne' measure_ball_lt_top.ne).1 I
  have K : (s.card : ℝ) ≤ (5 : ℝ) ^ finrank ℝ E := by
    have := ENNReal.toReal_le_of_le_ofReal (pow_nonneg ρpos.le _) J
    simpa [ρ, δ, div_eq_mul_inv, mul_pow] using this
  exact mod_cast K

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

Goal state before:
pp: G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
f : ℕ → G
m_max : Maximal (fun x ↦ x ∈ A) (range f)
hms : range f ⊆ s
hdj : (range f).PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ range f, x • (k * k⁻¹)
hm : (range f).Nonempty
h'm : (range f).Countable
i : ℕ
⊢ MeasurableSet (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)

Tactic:
apply MeasurableSet.disjointed (fun j ↦ hs.inter ?_)

Goal state after:
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
f : ℕ → G
m_max : Maximal (fun x ↦ x ∈ A) (range f)
hms : range f ⊆ s
hdj : (range f).PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ range f, x • (k * k⁻¹)
hm : (range f).Nonempty
h'm : (range f).Countable
i j : ℕ
⊢ MeasurableSet (f j • (k * k⁻¹))
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : IsTopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : μ'.InnerRegularCompactLTTop
s✝ : Set G
hs : μ s✝ ≠ ∞
h's : μ' s✝ ≠ ∞
c : ℝ≥0 := μ'.haarScalarFactor μ
s : Set G
⊢ MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s

Tactic:
s_meas

Goal state after:
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : IsTopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : μ'.InnerRegularCompactLTTop
s✝ : Set G
hs : μ s✝ ≠ ∞
h's : μ' s✝ ≠ ∞
c : ℝ≥0 := μ'.haarScalarFactor μ
s : Set G
s_meas : MeasurableSet s
⊢ μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.Regular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
⊢ map Inv.inv (map Inv.inv μ) = c ^ 2 • μ

Tactic:
← inv_def μ,

Goal state after:
G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.Regular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
⊢ map Inv.inv μ.inv = c ^ 2 • μ
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.Regular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
⊢ c • map Inv.inv μ = c ^ 2 • μ

Tactic:
← inv_def μ,

Goal state after:
G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.Regular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
⊢ c • μ.inv = c ^ 2 • μ
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.Regular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
this : map Inv.inv (map Inv.inv μ) = c ^ 2 • μ
⊢ μ = c ^ 2 • μ

Tactic:
map_map continuous_inv.measurable continuous_inv.measurable

Goal state after:
G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.Regular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
this : map ((fun a ↦ a⁻¹) ∘ fun a ↦ a⁻¹) μ = c ^ 2 • μ
⊢ μ = c ^ 2 • μ
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
⊢ map Inv.inv (map Inv.inv μ) = c ^ 2 • μ

Tactic:
← inv_def μ,

Goal state after:
G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
⊢ map Inv.inv μ.inv = c ^ 2 • μ
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
⊢ c • map Inv.inv μ = c ^ 2 • μ

Tactic:
← inv_def μ,

Goal state after:
G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
⊢ c • μ.inv = c ^ 2 • μ
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
this : map Inv.inv (map Inv.inv μ) = c ^ 2 • μ
⊢ μ = c ^ 2 • μ

Tactic:
map_map continuous_inv.measurable continuous_inv.measurable

Goal state after:
G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
this : map ((fun a ↦ a⁻¹) ∘ fun a ↦ a⁻¹) μ = c ^ 2 • μ
⊢ μ = c ^ 2 • μ
Module: Mathlib.Analysis.Calculus.ContDiff.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝¹ : CompleteSpace 𝕜
f : D → E →L[𝕜] F
s : Set D
inst✝ : FiniteDimensional 𝕜 E
⊢ ContDiffOn 𝕜 n f s ↔ ∀ (y : E), ContDiffOn 𝕜 n (fun x ↦ (f x) y) s

Tactic:
refine ⟨fun h y => h.clm_apply contDiffOn_const, fun h => ?_⟩
  let d := finrank 𝕜 E
  have hd : d = finrank 𝕜 (Fin d → 𝕜) := (finrank_fin_fun 𝕜).symm
  let e₁ := ContinuousLinearEquiv.ofFinrankEq hd
  let e₂ := (e₁.arrowCongr (1 : F ≃L[𝕜] F)).trans (ContinuousLinearEquiv.piRing (Fin d))
  rw [← id_comp f, ← e₂.symm_comp_self]
  exact e₂.symm.contDiff.comp_contDiffOn (contDiffOn_pi.mpr fun i => h _)

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

Goal state before:
pp: case inr.inl
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
c' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fin N.succ), ‖c' i‖ ≤ 2
i j : Fin N.succ
inej : i ≠ j
hij : ‖a.c i‖ ≤ ‖a.c j‖
Hj : 2 < ‖a.c j‖
H'j : ‖a.c j‖ ≤ 2 ↔ False
Hi : ‖a.c i‖ ≤ 2
⊢ 1 - δ ≤ ‖c' i - c' j‖

Tactic:
simp_rw [c', Hi, if_true, H'j, if_false]
      exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj

Goal state after:
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C₁ C₂ s t₁ t₂ : ℝ≥0
A : Set X
hf₁ : HolderOnWith C₁ r f A
hf₂ : HolderOnWith C₂ s f A
ht : t₁ + t₂ = 1
x : X
hx : x ∈ A
y : X
hy : y ∈ A
⊢ edist (f x) (f y) ^ ↑t₁ * edist (f x) (f y) ^ ↑t₂ ≤ (↑C₁ * edist x y ^ ↑r) ^ ↑t₁ * (↑C₂ * edist x y ^ ↑s) ^ ↑t₂

Tactic:
nth_grw 1 [hf₁ x hx y hy, hf₂ x hx y hy]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
x : E
hfx : ContDiffAt ℝ (↑k) f x
hgx : ContDiffAt ℝ (↑k) g (f x)
⊢ iteratedFDeriv ℝ k (g ∘ f) x - iteratedFDeriv ℝ k (g ∘ f) a =
    (ftaylorSeries ℝ g (f x)).taylorComp (ftaylorSeries ℝ f x) k -
      (ftaylorSeries ℝ g (f a)).taylorComp (ftaylorSeries ℝ f a) k

Tactic:
iteratedFDeriv_comp hgx hfx le_rfl,

Goal state after:
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
x : E
hfx : ContDiffAt ℝ (↑k) f x
hgx : ContDiffAt ℝ (↑k) g (f x)
⊢ (ftaylorSeries ℝ g (f x)).taylorComp (ftaylorSeries ℝ f x) k - iteratedFDeriv ℝ k (g ∘ f) a =
    (ftaylorSeries ℝ g (f x)).taylorComp (ftaylorSeries ℝ f x) k -
      (ftaylorSeries ℝ g (f a)).taylorComp (ftaylorSeries ℝ f a) k
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoMetricSpace X
inst✝ : MetricSpace Y
r : ℝ≥0
f : X → Y
h : ∀ (x₁ x₂ : X), f x₁ = f x₂
x₁ x₂ : X
⊢ edist (f x₁) (f x₁) ≤ ↑0 * edist x₁ x₂ ^ ↑r

Tactic:
simp only [edist_self, ENNReal.coe_zero, zero_mul, le_refl]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: case neg
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
α : ↑I
f : E → F
a : E
g : F → G
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
i : ℕ
hfd : ¬DifferentiableAt ℝ f a
hg : ContDiffPointwiseHolderAt 0 α g (f a)
hf : ContDiffPointwiseHolderAt 0 α f a
hi : i ≤ 0
⊢ (fun a_1 ↦ ftaylorSeries ℝ g (f a_1) i - ftaylorSeries ℝ g (f a) i) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α

Tactic:
obtain rfl : i = 0 := by rwa [nonpos_iff_eq_zero] at hi

Goal state after:
case neg
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
α : ↑I
f : E → F
a : E
g : F → G
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
hfd : ¬DifferentiableAt ℝ f a
hg : ContDiffPointwiseHolderAt 0 α g (f a)
hf : ContDiffPointwiseHolderAt 0 α f a
hi : 0 ≤ 0
⊢ (fun a_1 ↦ ftaylorSeries ℝ g (f a_1) 0 - ftaylorSeries ℝ g (f a) 0) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α