mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X₁ : Type u_1
X₂ : Type u_2
Y₁ : Sort u_3
Y₂ : Sort u_4
Z₁ : Type u_5
Z₂ : Type u_6
inst✝³ : TopologicalSpace X₁
inst✝² : TopologicalSpace X₂
inst✝¹ : TopologicalSpace Z₁
inst✝ : TopologicalSpace Z₂
f₁ : X₁ → Y₁
g₁ : Z₁ → Y₁
f₂ : X₂ → Y₂
g₂ : Z₂ → Y₂
mapX : X₁ → X₂
contX : Continuous[inst✝³, inst✝²] mapX
mapY : Y₁ → Y₂
mapZ : Z₁ → Z₂
contZ : Continuous[inst✝¹, inst✝] mapZ
commX : f₂ ∘ mapX = mapY ∘ f₁
commZ : g₂ ∘ mapZ = mapY ∘ g₁
⊢ Continuous[instTopologicalSpaceSubtype, instTopologicalSpaceSubtype] (Function.mapPullback mapX mapY mapZ commX commZ)

Tactic:
refine continuous_induced_rng.mpr (.prodMk ?_ ?_)

Goal state after:
case refine_1
X₁ : Type u_1
X₂ : Type u_2
Y₁ : Sort u_3
Y₂ : Sort u_4
Z₁ : Type u_5
Z₂ : Type u_6
inst✝³ : TopologicalSpace X₁
inst✝² : TopologicalSpace X₂
inst✝¹ : TopologicalSpace Z₁
inst✝ : TopologicalSpace Z₂
f₁ : X₁ → Y₁
g₁ : Z₁ → Y₁
f₂ : X₂ → Y₂
g₂ : Z₂ → Y₂
mapX : X₁ → X₂
contX : Continuous[inst✝³, inst✝²] mapX
mapY : Y₁ → Y₂
mapZ : Z₁ → Z₂
contZ : Continuous[inst✝¹, inst✝] mapZ
commX : f₂ ∘ mapX = mapY ∘ f₁
commZ : g₂ ∘ mapZ = mapY ∘ g₁
⊢ Continuous[instTopologicalSpaceSubtype, inst✝²] fun x ↦ mapX x.fst

case refine_2
X₁ : Type u_1
X₂ : Type u_2
Y₁ : Sort u_3
Y₂ : Sort u_4
Z₁ : Type u_5
Z₂ : Type u_6
inst✝³ : TopologicalSpace X₁
inst✝² : TopologicalSpace X₂
inst✝¹ : TopologicalSpace Z₁
inst✝ : TopologicalSpace Z₂
f₁ : X₁ → Y₁
g₁ : Z₁ → Y₁
f₂ : X₂ → Y₂
g₂ : Z₂ → Y₂
mapX : X₁ → X₂
contX : Continuous[inst✝³, inst✝²] mapX
mapY : Y₁ → Y₂
mapZ : Z₁ → Z₂
contZ : Continuous[inst✝¹, inst✝] mapZ
commX : f₂ ∘ mapX = mapY ∘ f₁
commZ : g₂ ∘ mapZ = mapY ∘ g₁
⊢ Continuous[instTopologicalSpaceSubtype, inst✝] fun x ↦ mapZ x.snd
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: case refine_1
X : Type u_1
Y : Sort u_2
inst✝ : TopologicalSpace X
f : X → Y
x₁ x₂ : X
x✝¹ : f x₁ = f x₂
x✝ : x₁ ≠ x₂
h : (diagonal X)ᶜ ∈ 𝓝 x₁ ×ˢ 𝓝 x₂
⊢ (Function.pullbackDiagonal f)ᶜ ∈ comap Subtype.val (𝓝 x₁ ×ˢ 𝓝 x₂)

Tactic:
exact ⟨_, h, subset_rfl⟩

Goal state after:
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e : Trivialization F proj
s : Set B
⊢ e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.baseSet ∩ frontier s)

Tactic:
rw [← (e.isImage_preimage_prod s).frontier.preimage_eq, frontier_prod_univ_eq,
    (e.isImage_preimage_prod _).preimage_eq, e.source_eq, preimage_inter]

Goal state after:
Module: Mathlib.Topology.IsLocalHomeomorph

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
s : Set X
h : IsLocalHomeomorphOn f s
inst✝ : DiscreteTopology ↑(f '' s)
x : ↑s
⊢ IsOpen[instTopologicalSpaceSubtype] {x}

Tactic:
obtain ⟨e, hx, rfl⟩ := h x x.2
    have ⟨U, hU, eq⟩ := isOpen_discrete {(⟨_, _, x.2, rfl⟩ : e '' s)}
    refine ⟨e.source ∩ e ⁻¹' U, e.continuousOn_toFun.isOpen_inter_preimage e.open_source hU,
      subset_antisymm (fun x' mem ↦ Subtype.ext <| e.injOn mem.1 hx ?_) ?_⟩
    · simpa using Set.subset_singleton_iff.1 eq.subset ⟨_, x', x'.2, rfl⟩ mem.2
    · rintro x rfl; exact ⟨hx, eq.superset rfl⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: ⊢ Filter.Tendsto (fun i ↦ π * 2⁻¹ * (2 ^ i)⁻¹) Filter.atTop (nhds (π * 2⁻¹ * 0))

Tactic:
exact tendsto_inv_atTop_zero.comp (tendsto_pow_atTop_atTop_of_one_lt (by norm_num))
    |>.const_mul _

Goal state after:
Module: Mathlib.Analysis.Calculus.AddTorsor.AffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace 𝕜 V
n : WithTop ℕ∞
⊢ ContDiff 𝕜 n fun pqc ↦ (lineMap pqc.1 pqc.2.1) pqc.2.2

Tactic:
simp only [AffineMap.lineMap_apply_module]
  fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.AddTorsor.AffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : E → V
g : E → 𝕜
s : Set E
x : E
n : WithTop ℕ∞
h₁ : ContDiffWithinAt 𝕜 n f₁ s x
h₂ : ContDiffWithinAt 𝕜 n f₂ s x
hg : ContDiffWithinAt 𝕜 n g s x
⊢ ContDiffWithinAt 𝕜 n (fun x ↦ (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)) s x

Tactic:
simp only [AffineMap.lineMap_apply_module]
  fun_prop

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.faceOppositeCentroid i -ᵥ s.points i = (↑n + 1) • (s.faceOppositeCentroid i -ᵥ s.centroid)

Tactic:
← vsub_sub_vsub_cancel_right _ (s.centroid) (s.points i),

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.faceOppositeCentroid i -ᵥ s.points i =
    (↑n + 1) • (s.faceOppositeCentroid i -ᵥ s.points i - (s.centroid -ᵥ s.points i))
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : DivisionRing k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : AffineSpace V P
m n : ℕ
inst✝² : NeZero m
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
e : Fin (n + 1) ≃ Fin (m + 1)
i : Fin (m + 1)
⊢ (s.reindex e).medial.points i = (s.medial.reindex e).points i

Tactic:
simp [medial_points]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁸ : DivisionRing k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : AffineSpace V P
V₂ : Type u_4
P₂ : Type u_5
inst✝⁴ : AddCommGroup V₂
inst✝³ : Module k V₂
inst✝² : AffineSpace V₂ P₂
inst✝¹ : CharZero k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
f : P →ᵃ[k] P₂
hf : Function.Injective ⇑f
i : Fin (n + 1)
⊢ (s.map f hf).medial.points i = (s.medial.map f hf).points i

Tactic:
simp [medial_points]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
S : AffineSubspace k P
hS : affineSpan k (Set.range s.points) ≤ S
this : Nonempty ↥S
i : Fin (n + 1)
⊢ ↑((s.restrict S hS).medial.points i) = ↑((s.medial.restrict S ⋯).points i)

Tactic:
simp [medial_points]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
x : E
hx : x ∈ support ↑f
⊢ ↑f x ≠ 0

Tactic:
mem_support

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
x : E
hx : ↑f x ≠ 0
⊢ ↑f x ≠ 0
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inl
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Summable fun i ↦ ‖f i‖ ^ ENNReal.toReal 0
⊢ Memℓp f 0

Tactic:
apply memℓp_zero
    have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
    exact (Set.Finite.of_summable_const (by simp) H).subset (Set.subset_univ _)

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

Goal state before:
pp: 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' : UniformCauchySeqOn f' l (Metric.ball x 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✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ UniformCauchySeqOn f l (Metric.ball x r)

Tactic:
have : NeBot l := (cauchy_map_iff.1 hfg).1

Goal state after:
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' : UniformCauchySeqOn f' l (Metric.ball x 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
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
⊢ UniformCauchySeqOn f l (Metric.ball x r)
Module: Mathlib.Analysis.Normed.Lp.lpSpace

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

Tactic:
obtain ⟨i, hi⟩ := hCF.exists

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

Goal state after:
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)
⊢ ∀ (i : ℝ), 0 < i → ∀ᶠ (x : ℕ) in atTop, F x ∈ Metric.closedBall f i
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: 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
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : IsPreconnected s
hf' : UniformCauchySeqOn f' l s
hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y
x₀ x : E
hx₀ : x₀ ∈ s
hx : x ∈ s
hfg : Cauchy (map (fun n ↦ f n x₀) l)
⊢ Cauchy (map (fun n ↦ f n x) l)

Tactic:
have : NeBot l := (cauchy_map_iff.1 hfg).1

Goal state after:
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
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : IsPreconnected s
hf' : UniformCauchySeqOn f' l s
hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y
x₀ x : E
hx₀ : x₀ ∈ s
hx : x ∈ s
hfg : Cauchy (map (fun n ↦ f n x₀) l)
this : l.NeBot
⊢ Cauchy (map (fun n ↦ f n x) l)
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_6
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : 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
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
hfg' : ∀ ε > 0, ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (0 n.2) (-f' n.1.1 n.2 + f' n.1.2 n.2) < ε
ε : ℝ
hε : ε > 0
q : ℚ
hqpos : 0 < q
hqε : ↑q < ε
⊢ ↑q > 0

Tactic:
by simp [hqpos]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: case inl
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
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
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SFinite μ
inst✝ : μ.IsAddLeftInvariant
hf : LocallyIntegrable f μ
x₀ : G
hcg : HasCompactSupport 0
hg : ContDiff 𝕜 1 0
⊢ HasFDerivAt (f ⋆[L, μ] 0) ((f ⋆[precompR G L, μ] fderiv 𝕜 0) x₀) x₀

pp: case inr
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 : 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
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SFinite μ
inst✝ : μ.IsAddLeftInvariant
hcg : HasCompactSupport g
hf : LocallyIntegrable f μ
hg : ContDiff 𝕜 1 g
x₀ : G
fin_dim : FiniteDimensional 𝕜 G
⊢ HasFDerivAt (f ⋆[L, μ] g) ((f ⋆[precompR G L, μ] fderiv 𝕜 g) x₀) x₀

Tactic:
· have : fderiv 𝕜 (0 : G → E') = 0 := fderiv_const (0 : E')
    simp only [this, convolution_zero, Pi.zero_apply]
    exact hasFDerivAt_const (0 : F) x₀

Goal state after:
case inr
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 : 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
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SFinite μ
inst✝ : μ.IsAddLeftInvariant
hcg : HasCompactSupport g
hf : LocallyIntegrable f μ
hg : ContDiff 𝕜 1 g
x₀ : G
fin_dim : FiniteDimensional 𝕜 G
⊢ HasFDerivAt (f ⋆[L, μ] g) ((f ⋆[precompR G L, μ] fderiv 𝕜 g) x₀) x₀
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
p : P
hp : p ∈ s
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    (g p)

Tactic:
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:
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: case hunion
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, μ s = 0 → μ t = 0 → μ (s ∪ t) = 0

pp: case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0

Tactic:
· exact fun s t hs ht ↦ measure_union_null hs ht

Goal state after:
case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
this : Bornology.IsBounded (Euclidean.closedBall x d)
⊢ support f ⊆ Euclidean.closedBall x d

Tactic:
exact f_supp.trans Euclidean.ball_subset_closedBall

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
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
n : ℕ
R : ℕ → ℝ
hR : ∀ (i : ℕ) (x : E), ‖iteratedFDeriv ℝ i (fun x ↦ g n x) x‖ ≤ R i
⊢ ∃ r, 0 < r ∧ ∀ i ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ i (r • g n) x‖ ≤ ↑(δ n)

Tactic:
let M := max (((Finset.range (n + 1)).image R).max' (by simp)) 1

Goal state after:
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
n : ℕ
R : ℕ → ℝ
hR : ∀ (i : ℕ) (x : E), ‖iteratedFDeriv ℝ i (fun x ↦ g n x) x‖ ≤ R i
M : ℝ := max ((Finset.image R (Finset.range (n + 1))).max' ⋯) 1
⊢ ∃ r, 0 < r ∧ ∀ i ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ i (r • g n) x‖ ≤ ↑(δ n)
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.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
K_pos : 0 < K
⊢ μ.real (closedBall c f.rOut) ≤ μ.real (closedBall c f.rIn) * K ^ finrank ℝ E

Tactic:
addHaar_real_closedBall' _ _ f.rIn_pos.le,

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.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
K_pos : 0 < K
⊢ μ.real (closedBall c f.rOut) ≤ f.rIn ^ finrank ℝ E * μ.real (closedBall 0 1) * K ^ finrank ℝ E
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 := ⋯
D_cont : Continuous[inst✝⁹, _] D
D_pos : ∀ (x : G), 0 < D x
⊢ Continuous[instTopologicalSpaceProd, _] (uncurry fun x y ↦ f (y * x) * (D (y * x))⁻¹ * g x)

Tactic:
apply Continuous.mul ?_ (hg.comp continuous_fst)

Goal state after:
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
⊢ Continuous[instTopologicalSpaceProd, _] fun a ↦ f (a.2 * a.1) * (D (a.2 * a.1))⁻¹
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case refine_3.refine_2
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)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ∞ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
⊢ (f x + f (-x)) / 2 = 0

Tactic:
have I1 : x ∉ support f := by rwa [f_support]

Goal state after:
case refine_3.refine_2
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)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ∞ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
I1 : x ∉ support f
⊢ (f x + f (-x)) / 2 = 0