mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
u : Set P
hu : IsOpen u
hne : u.Nonempty
⊢ ∃ s ⊆ u, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤

Tactic:
rcases hne with ⟨x, hx⟩

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
u : Set P
hu : IsOpen u
x : P
hx : x ∈ u
⊢ ∃ s ⊆ u, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤
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 g : (i : α) → E i
hf : Memℓp f p
hg : Memℓp g p
⊢ Memℓp (f - g) p

Tactic:
sub_eq_add_neg

Goal state after:
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f g : (i : α) → E i
hf : Memℓp f p
hg : Memℓp g p
⊢ Memℓp (f + -g) p
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inr
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
⊢ f = 0

Tactic:
have hf : HasSum (fun i : α => ‖f i‖ ^ p.toReal) 0 := by
      have := lp.hasSum_norm hp f
      rwa [h, Real.zero_rpow hp.ne'] at this
    have : ∀ i, 0 ≤ ‖f i‖ ^ p.toReal := fun i ↦ by positivity
    rw [hasSum_zero_iff_of_nonneg this] at hf
    ext i
    have : f i = 0 ∧ p.toReal ≠ 0 := by
      simpa [Real.rpow_eq_zero_iff_of_nonneg (norm_nonneg (f i))] using! congr_fun hf i
    exact this.1

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

Goal state before:
pp: case inr.inr
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
⊢ ‖-f‖ = ‖f‖

Tactic:
· suffices ‖-f‖ ^ p.toReal = ‖f‖ ^ p.toReal by
      exact Real.rpow_left_injOn hp.ne' (norm_nonneg' _) (norm_nonneg' _) this
    apply (lp.hasSum_norm hp (-f)).unique
    simpa only [coeFn_neg, Pi.neg_apply, _root_.norm_neg] using lp.hasSum_norm hp f

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

Goal state before:
pp: case zero
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
hk : ↑0 ≤ N
x : E
⊢ (⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ fun y ↦ ∑' (n : α), f n y) x =
    ∑' (n : α), (⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f n) x

Tactic:
exact (continuousMultilinearCurryFin0 𝕜 E F).symm.toContinuousLinearEquiv.map_tsum

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

Goal state before:
pp: case succ
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
k : ℕ
IH : ↑k ≤ N → (iteratedFDeriv 𝕜 k fun y ↦ ∑' (n : α), f n y) = fun x ↦ ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x
hk : ↑(k + 1) ≤ N
h'k : ↑k < N
A : Summable fun n ↦ iteratedFDeriv 𝕜 k (f n) 0
⊢ (iteratedFDeriv 𝕜 (k + 1) fun y ↦ ∑' (n : α), f n y) = fun x ↦ ∑' (n : α), iteratedFDeriv 𝕜 (k + 1) (f n) x

Tactic:
simp_rw [iteratedFDeriv_succ_eq_comp_left, IH h'k.le]

Goal state after:
case succ
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
k : ℕ
IH : ↑k ≤ N → (iteratedFDeriv 𝕜 k fun y ↦ ∑' (n : α), f n y) = fun x ↦ ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x
hk : ↑(k + 1) ≤ N
h'k : ↑k < N
A : Summable fun n ↦ iteratedFDeriv 𝕜 k (f n) 0
⊢ (⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘
      fderiv 𝕜 fun x ↦ ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x) =
    fun x ↦
    ∑' (n : α), (⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 k (f n))) x
Module: Mathlib.Analysis.Calculus.ContDiff.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
n : WithTop ℕ∞
f : H → PiLp p E
⊢ ContDiff 𝕜 n (⇑(PiLp.continuousLinearEquiv p 𝕜 E) ∘ f) ↔ ∀ (i : ι), ContDiff 𝕜 n fun x ↦ (f x).ofLp i

Tactic:
contDiff_pi

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
n : WithTop ℕ∞
f : H → PiLp p E
⊢ (∀ (i : ι), ContDiff 𝕜 n fun x ↦ (⇑(PiLp.continuousLinearEquiv p 𝕜 E) ∘ f) x i) ↔
    ∀ (i : ι), ContDiff 𝕜 n fun x ↦ (f x).ofLp i
Module: Mathlib.Analysis.Calculus.FDeriv.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
f' : H →L[𝕜] PiLp p E
y : H
this : Fintype ι
⊢ HasStrictFDerivAt f f' y ↔ ∀ (i : ι), HasStrictFDerivAt (fun x ↦ (f x).ofLp i) (PiLp.proj p E i ∘SL f') y

Tactic:
rw [← (PiLp.continuousLinearEquiv p 𝕜 E).comp_hasStrictFDerivAt_iff, hasStrictFDerivAt_pi']

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
f' : H →L[𝕜] PiLp p E
y : H
this : Fintype ι
⊢ (∀ (i : ι),
      HasStrictFDerivAt (fun x ↦ (⇑(continuousLinearEquiv p 𝕜 E) ∘ f) x i)
        (ContinuousLinearMap.proj i ∘SL ↑(continuousLinearEquiv p 𝕜 E) ∘SL f') y) ↔
    ∀ (i : ι), HasStrictFDerivAt (fun x ↦ (f x).ofLp i) (PiLp.proj p E i ∘SL f') y
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)
_i : Fact (1 ≤ p)
F : ℕ → ↥(lp E p)
hF : CauchySeq F
f : ↥(lp E p)
hf : Tendsto (id fun i ↦ ↑(F i)) atTop (𝓝 ↑f)
⊢ Tendsto F atTop (𝓝 f)

Tactic:
rw [Metric.nhds_basis_closedBall.tendsto_right_iff]
  intro ε hε
  have hε' : { p : lp E p × lp E p | ‖p.1 - p.2‖ < ε } ∈ uniformity (lp E p) :=
    NormedAddCommGroup.uniformity_basis_dist.mem_of_mem hε
  refine (hF.eventually_eventually hε').mono ?_
  rintro n (hn : ∀ᶠ l in atTop, ‖(fun f => F n - f) (F l)‖ < ε)
  rw [mem_closedBall_iff_norm]
  refine norm_le_of_tendsto (hn.mono fun k hk => hk.le) ?_
  rw [tendsto_pi_nhds]
  intro a
  exact (hf.apply_nhds a).const_sub (F n a)

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

Goal state before:
pp: p : ℝ[X]
⊢ Tendsto (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x) (𝓝 0) (𝓝 0)

Tactic:
simp only [expNegInvGlue, mul_ite, mul_zero]

Goal state after:
p : ℝ[X]
⊢ Tendsto (fun x ↦ if x ≤ 0 then 0 else Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0) (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: x : ℝ
h : 1 ≤ x
⊢ expNegInvGlue x = expNegInvGlue x + expNegInvGlue (1 - x)

Tactic:
rw [zero_of_nonpos (sub_nonpos.2 h), add_zero]

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
⊢ ContDiff ℝ ↑n ↑univUnitBall

Tactic:
suffices ContDiff ℝ n fun x : E => (√(1 + ‖x‖ ^ 2 : ℝ))⁻¹ from this.smul contDiff_id
  have h : ∀ x : E, (0 : ℝ) < (1 : ℝ) + ‖x‖ ^ 2 := fun x => by positivity
  refine ContDiff.inv ?_ fun x => Real.sqrt_ne_zero'.mpr (h x)
  exact (contDiff_const.add <| contDiff_norm_sq ℝ).sqrt fun x => (h x).ne'

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
A :
  ∀ p ∈ s,
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
      (g p)
q : P × G
hq : q.1 ∈ s
⊢ s ×ˢ univ ∈ 𝓝 q

Tactic:
apply (hs.prod isOpen_univ).mem_nhds
    simpa only [mem_prod, mem_univ, and_true] using hq

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hr : 0 < r
hs : IsClosed s
h : s ⊆ ball x r
⊢ ∃ r' ∈ Ioo 0 r, s ⊆ ball x r'

Tactic:
rcases eq_empty_or_nonempty s with (rfl | hne)

Goal state after:
case inl
α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
hr : 0 < r
hs : IsClosed ∅
h : ∅ ⊆ ball x r
⊢ ∃ r' ∈ Ioo 0 r, ∅ ⊆ ball x r'

case inr
α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hr : 0 < r
hs : IsClosed s
h : s ⊆ ball x r
hne : s.Nonempty
⊢ ∃ r' ∈ Ioo 0 r, s ⊆ ball x r'
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: case pos.inl.inl
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
hF : CompleteSpace F
hE : finrank ℝ E = 0
this✝ : Subsingleton E
this : f = fun x ↦ f 0
⊢ μ.real univ • f 0 = |(0 ^ finrank ℝ E)⁻¹| • ∫ (x : E), (fun x ↦ f 0) x ∂μ

Tactic:
simp only [hE, pow_zero, inv_one, abs_one, one_smul, integral_const]

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:
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.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.everywherePosSubset s =ᵐ[μ] s

Tactic:
by
  have A : μ (s \ μ.everywherePosSubset s) ≠ ∞ :=
    ((measure_mono sdiff_subset).trans_lt h's.lt_top).ne
  simp only [ae_eq_set, sdiff_eq_empty.mpr (everywherePosSubset_subset μ s), measure_empty,
    true_and, (hs.diff hs.everywherePosSubset).measure_eq_iSup_isCompact_of_ne_top A,
    ENNReal.iSup_eq_zero]
  intro k hk h'k
  exact measure_eq_zero_of_subset_sdiff_everywherePosSubset h'k hk

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
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
⊢ 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:
set K' := (-k + {q₀.2} : Set G) with K'_def

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)
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}
⊢ 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'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt g k x L₄ μ
hi : Integrable (uncurry fun x y ↦ (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)
⊢ ∫ (t : G), ∫ (s : G), (L₂ ((L (f s)) (g (t - s)))) (k (x₀ - t)) ∂ν ∂μ =
    ∫ (t : G), ∫ (s : G), (L₃ (f s)) ((L₄ (g (t - s))) (k (x₀ - t))) ∂ν ∂μ

Tactic:
by simp_rw [hL]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ ∃ u, ContDiff ℝ ∞ u ∧ (∀ (x : E), u x ∈ Icc 0 1) ∧ support u = ball 0 1 ∧ ∀ (x : E), u (-x) = u x

Tactic:
have A : IsOpen (ball (0 : E) 1) := isOpen_ball

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball 0 1)
⊢ ∃ u, ContDiff ℝ ∞ u ∧ (∀ (x : E), u x ∈ Icc 0 1) ∧ support u = ball 0 1 ∧ ∀ (x : E), u (-x) = u x
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
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous[inst✝⁷, _] f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x ↦ ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous[inst✝⁷, _] (v n ∘ f)
I :
  ∀ (ν : Measure G),
    IsFiniteMeasureOnCompacts ν →
      Tendsto (fun n ↦ ∫ (x : G), v n (f x) ∂ν) atTop (𝓝 (∫ (x : G), {1}.indicator (fun x ↦ 1) (f x) ∂ν))
n : ℕ
⊢ ∫ (x : G), v n (f x) ∂μ' = ∫ (x : G), v n (f x) ∂μ'.haarScalarFactor μ • μ

Tactic:
apply integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ (vf_cont n)

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
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous[inst✝⁷, _] f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := ⋯
vf_cont : ∀ (n : ℕ), Continuous[inst✝⁷, _] (v n ∘ f)
I :
  ∀ (ν : Measure G),
    IsFiniteMeasureOnCompacts ν →
      Tendsto (fun n ↦ ∫ (x : G), v n (f x) ∂ν) atTop (𝓝 (∫ (x : G), {1}.indicator (fun x ↦ 1) (f x) ∂ν))
n : ℕ
⊢ HasCompactSupport (v n ∘ f)
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case x_out
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous[inst✝⁷, _] f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x ↦ ↑((thickenedIndicator ⋯ {1}) x)
vf_cont : ∀ (n : ℕ), Continuous[inst✝⁷, _] (v n ∘ f)
I :
  ∀ (ν : Measure G),
    IsFiniteMeasureOnCompacts ν →
      Tendsto (fun n ↦ ∫ (x : G), v n (f x) ∂ν) atTop (𝓝 (∫ (x : G), {1}.indicator (fun x ↦ 1) (f x) ∂ν))
n : ℕ
⊢ 0 ∉ Metric.thickening (u n) {1}

Tactic:
· simpa using (u_mem n).2.le

Goal state after:
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
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure[inst✝⁸] s)
ν : Measure G := haar
A : μ' s = μ'.haarScalarFactor ν • ν s
B : μ s = μ.haarScalarFactor ν • ν s
⊢ μ'.haarScalarFactor ν • ν s = μ'.haarScalarFactor μ • μ s

Tactic:
B,

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
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure[inst✝⁸] s)
ν : Measure G := haar
A : μ' s = μ'.haarScalarFactor ν • ν s
B : μ s = μ.haarScalarFactor ν • ν s
⊢ μ'.haarScalarFactor ν • ν s = μ'.haarScalarFactor μ • μ.haarScalarFactor ν • ν s
Module: Mathlib.Analysis.Normed.Operator.Conformal

Goal state before:
pp: R : Type u_1
N : Type u_3
M' : Type u_5
inst✝⁴ : NormedField R
inst✝³ : SeminormedAddCommGroup N
inst✝² : NormedSpace R N
inst✝¹ : NormedAddCommGroup M'
inst✝ : NormedSpace R M'
f : M' →L[R] N
h : IsConformalMap f
⊢ Injective ⇑f

Tactic:
rcases h with ⟨c, hc, li, rfl⟩

Goal state after:
R : Type u_1
N : Type u_3
M' : Type u_5
inst✝⁴ : NormedField R
inst✝³ : SeminormedAddCommGroup N
inst✝² : NormedSpace R N
inst✝¹ : NormedAddCommGroup M'
inst✝ : NormedSpace R M'
c : R
hc : c ≠ 0
li : M' →ₗᵢ[R] N
⊢ Injective ⇑(c • li.toContinuousLinearMap)
Module: Mathlib.Analysis.Normed.Operator.Conformal

Goal state before:
pp: R : Type u_1
N : Type u_3
M' : Type u_5
inst✝⁵ : NormedField R
inst✝⁴ : SeminormedAddCommGroup N
inst✝³ : NormedSpace R N
inst✝² : NormedAddCommGroup M'
inst✝¹ : NormedSpace R M'
inst✝ : Nontrivial M'
f' : M' →L[R] N
hf' : IsConformalMap f'
⊢ f' ≠ 0

Tactic:
rintro rfl
  rcases exists_ne (0 : M') with ⟨a, ha⟩
  exact ha (hf'.injective rfl)

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

Goal state before:
pp: case mp
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 →L[ℝ] F
⊢ IsConformalMap f → ∃ c, 0 < c ∧ ∀ (u v : E), ⟪f u, f v⟫ = c * ⟪u, v⟫

pp: case mpr
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 →L[ℝ] F
⊢ (∃ c, 0 < c ∧ ∀ (u v : E), ⟪f u, f v⟫ = c * ⟪u, v⟫) → IsConformalMap f

Tactic:
· rintro ⟨c₁, hc₁, li, rfl⟩
    refine ⟨c₁ * c₁, mul_self_pos.2 hc₁, fun u v => ?_⟩
    simp only [real_inner_smul_left, real_inner_smul_right, mul_assoc,
      coe_toContinuousLinearMap, smul_apply, inner_map_map]

Goal state after:
case mpr
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 →L[ℝ] F
⊢ (∃ c, 0 < c ∧ ∀ (u v : E), ⟪f u, f v⟫ = c * ⟪u, v⟫) → IsConformalMap f