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

Goal state before:
pp: case inr.right
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOn (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
ε : ℝ
hε : ε > 0
t : Set G
ht : t ∈ map (fun n ↦ f n x) l
ht' : ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
n : ι
hn : f n x ∈ t
n' : ι
hn' : f n' x ∈ t
z : E
a✝ : z ∈ Metric.ball x r
⊢ dist (f (n, n').1 x) (f (n, n').2 x) < ε

Tactic:
exact ht' _ hn _ hn'

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

Goal state before:
pp: 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 : ∀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
hf' : ∀ ε > 0, ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
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 : ι × E) in l ×ˢ 𝓝 x, dist (g' x.2 - f' x.1 x.2) 0 < ε

Tactic:
by simpa [dist_eq_norm] using hf' ε hε

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.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 : 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
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 1 (↿g) (s ×ˢ univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : P × G → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
⊢ HasFDerivAt (fun q ↦ (f ⋆[L, μ] g q.1) q.2) ((f ⋆[precompR (P × G) L, μ] fun x ↦ fderiv 𝕜 ↿g (q₀.1, x)) q₀.2) q₀

Tactic:
have A : ∀ p ∈ s, Continuous (g p) := fun p hp ↦ by
    refine hg.continuousOn.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 : 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
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 1 (↿g) (s ×ˢ univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : P × G → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
A :
  ∀ p ∈ s,
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
      (g p)
⊢ HasFDerivAt (fun q ↦ (f ⋆[L, μ] g q.1) q.2) ((f ⋆[precompR (P × G) L, μ] fun x ↦ fderiv 𝕜 ↿g (q₀.1, x)) q₀.2) q₀
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : AddGroup G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E'
hbg : BddAbove (range fun x ↦ ‖g x‖)
hf : Integrable f μ
hg : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
⊢ Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f ⋆[L, μ] g)

Tactic:
refine continuous_iff_continuousAt.mpr fun x₀ => ?_
  have : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t : G ∂μ, ‖L (f t) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖ := by
    filter_upwards with x; filter_upwards with t
    apply_rules [L.le_of_opNorm₂_le_of_le, le_rfl, le_ciSup hbg (x - t)]
  refine continuousAt_of_dominated ?_ this (by fun_prop) ?_
  · exact Eventually.of_forall fun x =>
      hf.aestronglyMeasurable.convolution_integrand_snd' L hg.aestronglyMeasurable
  · filter_upwards with t; fun_prop

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✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedSpace 𝕜 E
inst✝¹¹ : NormedSpace 𝕜 E'
inst✝¹⁰ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁹ : MeasurableSpace G
μ : Measure G
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : AddCommGroup G
inst✝⁶ : μ.IsAddLeftInvariant
inst✝⁵ : μ.IsNegInvariant
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E
hbf : BddAbove (range fun x ↦ ‖f x‖)
hf : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hg : Integrable g μ
⊢ Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f ⋆[L, μ] g)

Tactic:
rw [← convolution_flip]
  exact hbf.continuous_convolution_right_of_integrable L.flip hg hf

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
⊢ μ.IsEverywherePos (μ.everywherePosSubset s)

Tactic:
intro x hx n hn

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
⊢ 0 < μ n
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
⊢ ∀ᶠ (x : ι) in l, dist ((φ x ⋆[lsmul ℝ ℝ, μ] g x) (k x)) z₀ < ε

Tactic:
have h2ε : 0 < ε / 3 := div_pos hε (by simp)

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
⊢ ∀ᶠ (x : ι) in l, dist ((φ x ⋆[lsmul ℝ ℝ, μ] g x) (k x)) z₀ < ε
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegularCompactLTTop
hs : MeasurableSet s
h's : μ s ≠ ∞
⊢ μ.IsEverywherePos (μ.everywherePosSubset s)

Tactic:
intro x hx n hn

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegularCompactLTTop
hs : MeasurableSet s
h's : μ s ≠ ∞
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
⊢ 0 < μ n
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : μ.IsEverywherePos s
h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u
⊢ ν.IsEverywherePos s

Tactic:
intro x hx n hn

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : μ.IsEverywherePos s
h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u
x : α
hx : x ∈ s
n : Set α
hn : n ∈ 𝓝[s] x
⊢ 0 < ν n
Module: Mathlib.Analysis.Calculus.ContDiff.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 : 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
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 1 (↿g) (s ×ˢ univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : P × G → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
A :
  ∀ p ∈ s,
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
      (g p)
A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q
g'_zero : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g' (p, x) = 0
ε C : ℝ
εpos : 0 < ε
h₀ε : ball q₀.1 ε ⊆ s
hε : ∀ (p : P) (x : G), ‖p - q₀.1‖ < ε → ‖g' (p, x)‖ ≤ C
I1 : ∀ᶠ (x : P × G) in 𝓝 q₀, AEStronglyMeasurable (fun a ↦ (L (f a)) (g x.1 (x.2 - a))) μ
I2 : Integrable (fun a ↦ (L (f a)) (g q₀.1 (q₀.2 - a))) μ
I3 : AEStronglyMeasurable (fun a ↦ L (f a) ∘SL g' (q₀.1, q₀.2 - a)) μ
K' : Set G := -k + {q₀.2}
K'_def : K' = -k + {q₀.2}
hK' : IsCompact K'
U : Set G
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
K'U : K' ⊆ U
hU : IntegrableOn f U μ
V : Set G
V_mem : V ∈ 𝓝 0
hV : K' + V ⊆ U
δ : ℝ
δpos : δ > 0
hδ : ball 0 δ ⊆ V
⊢ K' + ball 0 (min δ ε) ⊆ U

Tactic:
exact (add_subset_add_left ((ball_subset_ball (min_le_left _ _)).trans hδ)).trans hV

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case inr
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
h : 0 < t
⊢ t ≤ 0 ∨ t ∈ Ioo 0 x ∨ x ≤ t

Tactic:
rcases lt_or_ge t x with (h' | h')

Goal state after:
case inr.inl
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
h : 0 < t
h' : t < x
⊢ t ≤ 0 ∨ t ∈ Ioo 0 x ∨ x ≤ t

case inr.inr
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
h : 0 < t
h' : x ≤ t
⊢ t ≤ 0 ∨ t ∈ Ioo 0 x ∨ x ≤ t
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
⊢ c * μ'.haarScalarFactor (c • μ) = μ'.haarScalarFactor μ

Tactic:
have : IsHaarMeasure (c • μ) := IsHaarMeasure.nnreal_smul _ hc

Goal state after:
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
this : (c • μ).IsHaarMeasure
⊢ c * μ'.haarScalarFactor (c • μ) = μ'.haarScalarFactor μ
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‖
ah : 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
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
hcrj : ‖a.c j‖ ≤ a.r j + 1
I : a.r i ≤ 2
J : (1 - δ / 4) * τ ≤ 1
H : a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i
C : a.r j ≤ 4
⊢ 1 * a.r i ≤ ‖a.c i - a.c j‖

Tactic:
one_mul

Goal state after:
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‖
ah : 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
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
hcrj : ‖a.c j‖ ≤ a.r j + 1
I : a.r i ≤ 2
J : (1 - δ / 4) * τ ≤ 1
H : a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i
C : a.r j ≤ 4
⊢ a.r i ≤ ‖a.c i - a.c j‖
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 ≠ ∞
⊢ μ' s = μ'.haarScalarFactor μ • μ s

Tactic:
let c := haarScalarFactor μ' μ
  /- By regularity, every measurable set of finite measure may be approximated by compact sets.
  Therefore, the measures coincide on measurable sets of finite measure. -/
  have B : ∀ s, MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
    intro s s_meas hs h's
    have : (c • μ) s ≠ ∞ := by simp [ENNReal.mul_eq_top, hs.ne]
    rw [s_meas.measure_eq_iSup_isCompact_of_ne_top h's.ne,
        s_meas.measure_eq_iSup_isCompact_of_ne_top this]
    congr! 4 with K _Ks K_comp
    exact measure_isMulInvariant_eq_smul_of_isCompact_closure μ' μ K_comp.closure
  /- Finally, replace an arbitrary finite measure set with a measurable version, and use the
  version for measurable sets. -/
  let t := toMeasurable μ' s ∩ toMeasurable μ s
  have st : s ⊆ t := subset_inter (subset_toMeasurable μ' s) (subset_toMeasurable μ s)
  have mu'_t : μ' t = μ' s := by
    apply le_antisymm
    · exact (measure_mono inter_subset_left).trans (measure_toMeasurable s).le
    · exact measure_mono st
  have mu_t : μ t = μ s := by
    apply le_antisymm
    · exact (measure_mono inter_subset_right).trans (measure_toMeasurable s).le
    · exact measure_mono st
  simp only [← mu'_t, ← mu_t, nnreal_smul_coe_apply]
  apply B
  · exact (measurableSet_toMeasurable _ _).inter (measurableSet_toMeasurable _ _)
  · exact mu_t.le.trans_lt hs.lt_top
  · exact mu'_t.le.trans_lt h's.lt_top

Goal state after:
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
i j : Fin N.succ
inej : i ≠ j
ah : 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
δnonneg : 0 ≤ δ
hcrj : ‖a.c j‖ ≤ a.r j + 1
hj : 2 < ‖a.c j‖
s : ℝ := ‖a.c i‖
hi : 2 < s
hij : s ≤ ‖a.c j‖
A : a.r i ≤ s
spos : 0 < s
d : E := (s / ‖a.c j‖) • a.c j
hd : d = (s / ‖a.c j‖) • a.c j
⊢ s ≤ 1 * ‖a.c j‖

Tactic:
one_mul

Goal state after:
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
i j : Fin N.succ
inej : i ≠ j
ah : 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
δnonneg : 0 ≤ δ
hcrj : ‖a.c j‖ ≤ a.r j + 1
hj : 2 < ‖a.c j‖
s : ℝ := ‖a.c i‖
hi : 2 < s
hij : s ≤ ‖a.c j‖
A : a.r i ≤ s
spos : 0 < s
d : E := (s / ‖a.c j‖) • a.c j
hd : d = (s / ‖a.c j‖) • a.c j
⊢ s ≤ ‖a.c j‖
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
inst✝ : FiniteDimensional 𝕜 E
⊢ ContDiff 𝕜 n f ↔ ∀ (y : E), ContDiff 𝕜 n fun x ↦ (f x) y

Tactic:
← contDiffOn_univ,

Goal state after:
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
inst✝ : FiniteDimensional 𝕜 E
⊢ ContDiffOn 𝕜 n f Set.univ ↔ ∀ (y : E), ContDiffOn 𝕜 n (fun x ↦ (f x) y) Set.univ
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
i j : Fin N.succ
inej : i ≠ j
ah : 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
δnonneg : 0 ≤ δ
hcrj : ‖a.c j‖ ≤ a.r j + 1
hj : 2 < ‖a.c j‖
s : ℝ := ‖a.c i‖
hi : 2 < s
hij : s ≤ ‖a.c j‖
A : a.r i ≤ s
spos : 0 < s
d : E := (s / ‖a.c j‖) • a.c j
hd : d = (s / ‖a.c j‖) • a.c j
I : ‖a.c j - a.c i‖ ≤ ‖a.c j‖ - s + ‖d - a.c i‖
J : a.r j - ‖a.c j - a.c i‖ ≤ s / 2 * δ
invs_nonneg : 0 ≤ 2 / s
⊢ ‖(2 / s) • (a.c i - d)‖ = ‖(2 / s) • a.c i - (2 / ‖a.c j‖) • a.c j‖

Tactic:
smul_sub,

Goal state after:
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
i j : Fin N.succ
inej : i ≠ j
ah : 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
δnonneg : 0 ≤ δ
hcrj : ‖a.c j‖ ≤ a.r j + 1
hj : 2 < ‖a.c j‖
s : ℝ := ‖a.c i‖
hi : 2 < s
hij : s ≤ ‖a.c j‖
A : a.r i ≤ s
spos : 0 < s
d : E := (s / ‖a.c j‖) • a.c j
hd : d = (s / ‖a.c j‖) • a.c j
I : ‖a.c j - a.c i‖ ≤ ‖a.c j‖ - s + ‖d - a.c i‖
J : a.r j - ‖a.c j - a.c i‖ ≤ s / 2 * δ
invs_nonneg : 0 ≤ 2 / s
⊢ ‖(2 / s) • a.c i - (2 / s) • d‖ = ‖(2 / s) • a.c i - (2 / ‖a.c j‖) • a.c j‖
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:

Tactic:
1 - δ = 2 / s * (s / 2 - s / 2 * δ) := by field
    _ ≤ 2 / s * ‖d - a.c i‖ := by
      gcongr; linarith only [hcrj, I, J, hi]
    _ = ‖(2 / s) • a.c i - (2 / ‖a.c j‖) • a.c j‖ := by
      conv_lhs => rw [norm_sub_rev, ← abs_of_nonneg invs_nonneg]
      rw [← Real.norm_eq_abs, ← norm_smul, smul_sub, hd, smul_smul]
      congr 3
      field

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

Goal state before:
pp: 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))
k' : ↑(u i)
left✝¹ : k' ∈ w
hk : ↑↑k' ∈ ↑(Finset.image (fun x ↦ ↑↑x) w)
l' : ↑(u i)
left✝ : l' ∈ w
hl : ↑↑l' ∈ ↑(Finset.image (fun x ↦ ↑↑x) w)
hkl : ↑↑k' ≠ ↑↑l'
h : ↑k' = ↑l'
⊢ False

Tactic:
rw [h] at hkl

Goal state after:
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))
k' : ↑(u i)
left✝¹ : k' ∈ w
hk : ↑↑k' ∈ ↑(Finset.image (fun x ↦ ↑↑x) w)
l' : ↑(u i)
left✝ : l' ∈ w
hl : ↑↑l' ∈ ↑(Finset.image (fun x ↦ ↑↑x) w)
hkl : ↑↑l' ≠ ↑↑l'
h : ↑k' = ↑l'
⊢ False