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

Goal state before:
pp: case inr.inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
⊢ ∃ e, Icc a b ⊆ e.baseSet

Tactic:
rsuffices ⟨d, hdcb, hd⟩ : ∃ d ∈ Ioc c b, ∃ e : Trivialization F (π F E), Icc a d ⊆ e.baseSet

Goal state after:
case inr.inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : ∃ e, Icc a d ⊆ e.baseSet
⊢ ∃ e, Icc a b ⊆ e.baseSet

B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := ⋯
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := ⋯
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i : ι
b✝ : B
a : F
b : B
x : Z.Fiber b
⊢ 𝓝 x =
    comap (TotalSpace.mk b)
        (comap (↑(Z.localTrivAt b).toOpenPartialHomeomorph)
          (𝓝 (↑(Z.localTrivAt b).toOpenPartialHomeomorph { proj := b, snd := x }))) ⊓
      𝓟 (TotalSpace.mk b ⁻¹' (Z.localTrivAt b).source)

Tactic:
comap_comap

Goal state after:
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i : ι
b✝ : B
a : F
b : B
x : Z.Fiber b
⊢ 𝓝 x =
    comap (↑(Z.localTrivAt b).toOpenPartialHomeomorph ∘ TotalSpace.mk b)
        (𝓝 (↑(Z.localTrivAt b).toOpenPartialHomeomorph { proj := b, snd := x })) ⊓
      𝓟 (TotalSpace.mk b ⁻¹' (Z.localTrivAt b).source)
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Sort u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ IsSeparatedMap f ↔ IsClosedEmbedding (toPullbackDiag f)

Tactic:
rw [isSeparatedMap_iff_isClosed_diagonal, ← range_toPullbackDiag]

Goal state after:
X : Type u_1
Y : Sort u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ IsClosed[instTopologicalSpaceSubtype] (Set.range (toPullbackDiag f)) ↔ IsClosedEmbedding (toPullbackDiag f)
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : SMulCommClass 𝕜 A A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : StarRing 𝕜
inst✝² : StarRing A
inst✝¹ : StarModule 𝕜 A
inst✝ : NormedStarGroup A
a : 𝓜(𝕜, A)
x y : A
⊢ ((↑↑(starₗᵢ 𝕜) ∘SL a.toProd.1) ∘SL ↑↑(starₗᵢ 𝕜)) x * y = x * ((↑↑(starₗᵢ 𝕜) ∘SL a.toProd.2) ∘SL ↑↑(starₗᵢ 𝕜)) y

Tactic:
by
        simpa only [star_mul, star_star]
          using! (congr_arg star (a.central (star y) (star x))).symm

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝ : TopologicalSpace I
hs : IsOpen[inst✝¹] s
hfs : IsOpen[inst✝²] (f ⁻¹' s)
x : X
hx : x ∈ s
h : IsEvenlyCovered (fun e ↦ f ↑e) x I
inst : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
hfU : IsOpen[instTopologicalSpaceSubtype] ((fun e ↦ f ↑e) ⁻¹' U)
H : ↑((fun e ↦ f ↑e) ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑((fun e ↦ f ↑e) ⁻¹' U)), ↑(H x).1 = (fun e ↦ f ↑e) ↑x
x✝ : Nonempty I
y : X
hy : y ∈ U
⊢ y ∈ s

Tactic:
convert! Set.mem_preimage.mp (H.symm (⟨y, hy⟩, Classical.arbitrary I)).1.2

Goal state after:
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝ : TopologicalSpace I
hs : IsOpen[inst✝¹] s
hfs : IsOpen[inst✝²] (f ⁻¹' s)
x : X
hx : x ∈ s
h : IsEvenlyCovered (fun e ↦ f ↑e) x I
inst : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
hfU : IsOpen[instTopologicalSpaceSubtype] ((fun e ↦ f ↑e) ⁻¹' U)
H : ↑((fun e ↦ f ↑e) ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑((fun e ↦ f ↑e) ⁻¹' U)), ↑(H x).1 = (fun e ↦ f ↑e) ↑x
x✝ : Nonempty I
y : X
hy : y ∈ U
⊢ y = f ↑↑(H.symm (⟨y, hy⟩, Classical.arbitrary I))
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
f : E → X
s : Set X
hf : IsCoveringMapOn f s
x : E
hx : x ∈ f ⁻¹' s
this : Nonempty ↑(f ⁻¹' {f x})
e : Trivialization (↑(f ⁻¹' {f x})) f := ⋯.toTrivialization
h : f x ∈ ⋯.toTrivialization.baseSet
⊢ ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source

Tactic:
let he := e.mem_source.2 h

Goal state after:
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
f : E → X
s : Set X
hf : IsCoveringMapOn f s
x : E
hx : x ∈ f ⁻¹' s
this : Nonempty ↑(f ⁻¹' {f x})
e : Trivialization (↑(f ⁻¹' {f x})) f := ⋯.toTrivialization
h : f x ∈ ⋯.toTrivialization.baseSet
he : x ∈ e.source := (Trivialization.mem_source e).mpr h
⊢ ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G E
hf : IsQuotientMap f
inst✝ : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
disjoint : ∀ (e : E), ∃ U ∈ 𝓝 e, ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g • e = e
this✝ : TopologicalSpace G := ⊥
this : DiscreteTopology G
e : E
he : e ∈ {e | MulAction.stabilizer G e = ⊥}
⊢ ∃ t, f e ∈ t.baseSet

Tactic:
have ⟨U, heU, hU⟩ := disjoint e

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G E
hf : IsQuotientMap f
inst✝ : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
disjoint : ∀ (e : E), ∃ U ∈ 𝓝 e, ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g • e = e
this✝ : TopologicalSpace G := ⊥
this : DiscreteTopology G
e : E
he : e ∈ {e | MulAction.stabilizer G e = ⊥}
U : Set E
heU : U ∈ 𝓝 e
hU : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g • e = e
⊢ ∃ t, f e ∈ t.baseSet
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace X
f : E → X
s : Set X
inst✝³ : Nonempty (X → E)
ι : Type ?u.20
inst✝² : Nonempty ι
inst✝¹ : TopologicalSpace ι
inst✝ : DiscreteTopology ι
U : ι → Set E
V : Set X
open_V : IsOpen[inst✝⁴] V
open_iff : ∀ (i : ι) {W : Set X}, W ⊆ V → (IsOpen[inst✝⁴] W ↔ IsOpen[inst✝⁵] (f ⁻¹' W ∩ U i))
inj : ∀ (i : ι), Set.InjOn f (U i)
disjoint : Pairwise (Disjoint on U)
exhaustive' : f ⁻¹' V ⊆ ⋃ i, U i
idx : (x : E) → x ∈ f ⁻¹' V → ι
idx_U : ∀ (x : E) (a : x ∈ f ⁻¹' V), x ∈ U (idx x a)
inv : ι → ⦃a : X⦄ → a ∈ V → E
inv_U : ∀ (i : ι) ⦃a : X⦄ (a_1 : a ∈ V), inv i a_1 ∈ U i
f_inv : ∀ (i : ι) ⦃a : X⦄ (a_1 : a ∈ V), f (inv i a_1) = a
F : PartialEquiv E (X × ι) :=
  { toFun := fun e ↦ (f e, if he : f e ∈ V then idx e he else Classical.arbitrary ι),
    invFun := fun x ↦ if hx : x.1 ∈ V then inv x.2 hx else Classical.arbitrary (X → E) x.1, source := f ⁻¹' V,
    target := V ×ˢ Set.univ, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }
open_preim : ∀ {W : Set X}, W ⊆ V → IsOpen[inst✝⁴] W → IsOpen[inst✝⁵] (f ⁻¹' W)
open_source : IsOpen[inst✝⁵] F.source
cont_f : ContinuousOn f F.source
e : E
he : e ∈ F.source
e' : E
he' : e' ∈ U (idx e he) ∩ F.source
h : e' ∉ {x | (fun x ↦ (fun x ↦ idx e he) x = (fun x ↦ if he : f x ∈ V then idx x he else Classical.arbitrary ι) x) x}
⊢ f e' ∈ V

Tactic:
by exact he'.2

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

Goal state before:
pp: z : Circle
s : ℝ
n : ℕ
hs : s ≤ π
h1 : z ∈ centeredArc (π / ↑n)
h2 : z ^ n ∈ centeredArc s
⊢ z ∈ centeredArc (s / ↑n)

Tactic:
have hs0 : 0 < s := by
    contrapose! h2
    simp [centeredArc_eq_empty h2]

Goal state after:
z : Circle
s : ℝ
n : ℕ
hs : s ≤ π
h1 : z ∈ centeredArc (π / ↑n)
h2 : z ^ n ∈ centeredArc s
hs0 : 0 < s
⊢ z ∈ centeredArc (s / ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
h : x ≠ y
⊢ ⇑exp '' Ioc (min (↑x).arg (↑y).arg) (min (↑x).arg (↑y).arg + 2 * π) = univ

Tactic:
periodic_exp.image_Ioc two_pi_pos

Goal state after:
x y : Circle
h : x ≠ y
⊢ range ⇑exp = univ
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: T : ℝ
⊢ Continuous fun x ↦ ↑(↑x).arg

Tactic:
continuous_iff_continuousAt

Goal state after:
T : ℝ
⊢ ∀ (x : Circle), ContinuousAt (fun x ↦ ↑(↑x).arg) x
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
inst✝² : T2Space E
inst✝¹ : T2Space X
inst✝ : CompactSpace E
h : IsLocalHomeomorph f
⊢ Continuous[inst✝⁴, inst✝³] f

Tactic:
continuous_iff_continuousAt

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
inst✝² : T2Space E
inst✝¹ : T2Space X
inst✝ : CompactSpace E
h : IsLocalHomeomorph f
⊢ ∀ (x : E), ContinuousAt f x
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.points i -ᵥ s.faceOppositeCentroid i = (↑n + 1) • (s.centroid -ᵥ s.faceOppositeCentroid i)

Tactic:
by
  rw [← neg_vsub_eq_vsub_rev, faceOppositeCentroid_vsub_point_eq_smul_vsub, ← neg_smul,
    ← neg_smul_neg, neg_vsub_eq_vsub_rev, neg_neg]

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
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
⊢ ∃ i j, i ≠ j

Tactic:
simp only [ne_eq, Subtype.exists, Subtype.mk.injEq, exists_prop]

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
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
⊢ ∃ a ∈ s', ∃ a_1 ∈ s', ¬a = a_1
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: case h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ : P
hp₁ : p₁ ∈ s
p : P
v : V
hv : ∀ p_1 ∈ s, ∃ r, p_1 = r • v +ᵥ p
r : k
hp₂ : r • v +ᵥ p ∈ s
⊢ ∃ r_1, r • v +ᵥ p = r_1 • v +ᵥ p₁

Tactic:
rcases hv p₁ hp₁ with ⟨r₁, rfl⟩

Goal state after:
case h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p : P
v : V
hv : ∀ p_1 ∈ s, ∃ r, p_1 = r • v +ᵥ p
r : k
hp₂ : r • v +ᵥ p ∈ s
r₁ : k
hp₁ : r₁ • v +ᵥ p ∈ s
⊢ ∃ r_1, r • v +ᵥ p = r_1 • v +ᵥ r₁ • v +ᵥ p
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
⊢ AffineIndependent k ![p₁, p₂, p₃] ↔ ¬Collinear k {p₁, p₂, p₃}

Tactic:
rw [affineIndependent_iff_not_collinear]

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
⊢ ¬Collinear k (Set.range ![p₁, p₂, p₃]) ↔ ¬Collinear k {p₁, p₂, p₃}
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 : (i : α) → E i
g : α → ℝ
hg : Memℓp g p
hfg : ∀ (i : α), ‖f i‖ ≤ g i
⊢ Memℓp f p

Tactic:
replace hfg (i) : ‖f i‖ ≤ ‖g i‖ := (hfg i).trans (Real.le_norm_self _)

Goal state after:
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
g : α → ℝ
hg : Memℓp g p
hfg : ∀ (i : α), ‖f i‖ ≤ ‖g i‖
⊢ Memℓp f p
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: 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
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
x₀ : E
hu : Summable u
hf : ∀ (n : α), Differentiable 𝕜 (f n)
hf' : ∀ (n : α) (x : E), ‖fderiv 𝕜 (f n) x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
⊢ (fderiv 𝕜 fun y ↦ ∑' (n : α), f n y) = fun x ↦ ∑' (n : α), fderiv 𝕜 (f n) x

Tactic:
ext1 x
  exact fderiv_tsum_apply hu hf hf' hf0 x

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

Goal state before:
pp: case inl
𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
q : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : (i : α) → StarAddMonoid (E i)
inst✝ : ∀ (i : α), NormedStarGroup (E i)
hp : Fact (1 ≤ 0)
f : ↥(lp E 0)
⊢ ‖star f‖ = ‖f‖

Tactic:
exfalso
      have := ENNReal.toReal_mono ENNReal.zero_ne_top hp.elim
      norm_num at this

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
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
t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n ↦ f n y) l)}
A : ∀ (x : E) (ε : ℝ), x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t
open_t : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
st_nonempty : (s ∩ t).Nonempty
⊢ s ⊆ t

Tactic:
suffices H : closure t ∩ s ⊆ t from h's.subset_of_closure_inter_subset open_t st_nonempty H

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
t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n ↦ f n y) l)}
A : ∀ (x : E) (ε : ℝ), x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t
open_t : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
st_nonempty : (s ∩ t).Nonempty
⊢ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∩ s ⊆ t
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: case inr.inr.convert_2
p : ℝ[X]
x : ℝ
hx : 0 < x
this :
  HasDerivAt (((fun x ↦ Polynomial.eval x p) * fun x ↦ rexp (-x)) ∘ Inv.inv)
    ((Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + Polynomial.eval x⁻¹ p * (rexp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x
⊢ (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x) =ᶠ[𝓝 x]
    ((fun x ↦ Polynomial.eval x p) * fun x ↦ rexp (-x)) ∘ Inv.inv

Tactic:
filter_upwards [lt_mem_nhds hx] with y hy
      simp [expNegInvGlue, hy.not_ge]

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

Goal state before:
pp: n : ℕ∞
p : ℝ[X]
m : ℕ
⊢ ContDiff ℝ ↑m fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x

Tactic:
induction m generalizing p with

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

Goal state before:
pp: ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f f' : ι → 𝕜 → G
x : 𝕜
inst✝ : IsRCLikeNormedField 𝕜
hf' : UniformCauchySeqOnFilter f' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n ↦ f n x) l)
⊢ UniformCauchySeqOnFilter f l (𝓝 x)

Tactic:
by
  simp_rw [hasDerivAt_iff_hasFDerivAt] at hf
  exact uniformCauchySeqOnFilter_of_fderiv hf'.one_smulRight hf hfg

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

Goal state before:
pp: F : Type u_1
inst✝⁶ : NormedAddCommGroup F
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E → F
R : ℝ
hR : R ≠ 0
⊢ ∀ {g : E → F}, Integrable g μ → ∀ {S : ℝ}, S ≠ 0 → Integrable (fun x ↦ g (S • x)) μ

Tactic:
intro g hg S hS

Goal state after:
F : Type u_1
inst✝⁶ : NormedAddCommGroup F
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E → F
R : ℝ
hR : R ≠ 0
g : E → F
hg : Integrable g μ
S : ℝ
hS : S ≠ 0
⊢ Integrable (fun x ↦ g (S • x)) μ
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
hcg : ∀ ε > 0, ∀ᶠ (x : ι × G) in l ×ˢ 𝓝 x₀, dist (uncurry g x) z₀ < ε
hk : Tendsto k l (𝓝 x₀)
hφ : ∀ t ∈ 𝓝 0, ∀ᶠ (x : ι) in l, support (φ x) ⊆ t
⊢ ∀ ε > 0, ∀ᶠ (x : ι) in l, dist ((φ x ⋆[lsmul ℝ ℝ, μ] g x) (k x)) z₀ < ε

Tactic:
Metric.eventually_prod_nhds_iff

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 > 0, ∀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: G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk : IsCompact k
h'k : IsClosed[inst✝⁷] k
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
W : ℕ → Set G
W_open : ∀ (n : ℕ), IsOpen[inst✝⁷] (W n)
mem_W : ∀ (n : ℕ), 1 ∈ W n
hW : ∀ (n : ℕ), ∀ g ∈ W n * W n, μ (g • k \ k) < u n
V : ℕ → Set G := fun n ↦ ⋂ i ∈ Finset.range n, W i
x : G
hx : x ∈ ⋂ n, V n * k
v : ℕ → G
hv : ∀ (i : ℕ), v i ∈ V i
y : ℕ → G
hy : ∀ (i : ℕ), y i ∈ k
hvy : ∀ (i : ℕ), (fun x1 x2 ↦ x1 * x2) (v i) (y i) = x
z : G
zk : z ∈ k
hz : MapClusterPt z atTop y
n : ℕ
this : W n * {z} ∈ 𝓝 z
i : ℕ
hi : y i ∈ W n * {z}
ni : n < i
I : V i ⊆ W n
⊢ x * (y i)⁻¹ ∈ W n

Tactic:
have J : x * (y i)⁻¹ ∈ V i := by simpa [← hvy i] using hv i

Goal state after:
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk : IsCompact k
h'k : IsClosed[inst✝⁷] k
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
W : ℕ → Set G
W_open : ∀ (n : ℕ), IsOpen[inst✝⁷] (W n)
mem_W : ∀ (n : ℕ), 1 ∈ W n
hW : ∀ (n : ℕ), ∀ g ∈ W n * W n, μ (g • k \ k) < u n
V : ℕ → Set G := fun n ↦ ⋂ i ∈ Finset.range n, W i
x : G
hx : x ∈ ⋂ n, V n * k
v : ℕ → G
hv : ∀ (i : ℕ), v i ∈ V i
y : ℕ → G
hy : ∀ (i : ℕ), y i ∈ k
hvy : ∀ (i : ℕ), (fun x1 x2 ↦ x1 * x2) (v i) (y i) = x
z : G
zk : z ∈ k
hz : MapClusterPt z atTop y
n : ℕ
this : W n * {z} ∈ 𝓝 z
i : ℕ
hi : y i ∈ W n * {z}
ni : n < i
I : V i ⊆ W n
J : x * (y i)⁻¹ ∈ V i
⊢ x * (y i)⁻¹ ∈ W n
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case neg
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
H : ¬LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁷, _] f
f_comp : HasCompactSupport f
⊢ ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂0 • μ

Tactic:
rcases f_comp.eq_zero_or_locallyCompactSpace_of_group f_cont with hf | hf

Goal state after:
case neg.inl
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
H : ¬LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁷, _] f
f_comp : HasCompactSupport f
hf : f = 0
⊢ ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂0 • μ

case neg.inr
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
H : ¬LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁷, _] f
f_comp : HasCompactSupport f
hf : LocallyCompactSpace G
⊢ ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂0 • μ