mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.Piecewise

Goal state before:
pp: case inl.inr.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : x ≤ e
hy : e < y
hc : e < a • x + b • y
⊢ (Set.Iic e).piecewise f g (a • x + b • y) ≤ a • f x + b • g y

Tactic:
Set.piecewise_eq_of_notMem (Set.Iic e) f g (Set.notMem_Iic.mpr hc)

Goal state after:
case inl.inr.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : x ≤ e
hy : e < y
hc : e < a • x + b • y
⊢ g (a • x + b • y) ≤ a • f x + b • g y
Module: Mathlib.Analysis.Convex.Piecewise

Goal state before:
pp: case inr.inl.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : e < x
hy : y ≤ e
hc : e < a • x + b • y
⊢ (Set.Iic e).piecewise f g (a • x + b • y) ≤ a • g x + b • f y

Tactic:
Set.piecewise_eq_of_notMem (Set.Iic e) f g (Set.notMem_Iic.mpr hc)

Goal state after:
case inr.inl.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : e < x
hy : y ≤ e
hc : e < a • x + b • y
⊢ g (a • x + b • y) ≤ a • g x + b • f y
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
⊢ G ≠ ⊥ ↔ ∃ a b, G.Adj a b

Tactic:
by
  simp [eq_bot_iff_forall_not_adj]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
s : Set (Sym2 V)
⊢ Disjoint G (fromEdgeSet s) ↔ Disjoint G.edgeSet s

Tactic:
by
  conv_rhs => rw [← Set.sdiff_union_inter s Sym2.diagSet]
  rw [← disjoint_edgeSet, edgeSet_fromEdgeSet]
  grind [edgeSet_subset_compl_diagSet]

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mp.inr.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
h : p₁ ∈ s
p₁' : P
hp₁' : p₁' ∈ s
p₂' : P
hp₂' : p₂' ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
hr : r₁ • (x -ᵥ p₁') = r₂ • (p₂' -ᵥ y)
⊢ r₁ • (x -ᵥ p₁' - (p₁ -ᵥ p₁')) = r₂ • ((-r₁ / r₂) • (p₁ -ᵥ p₁') + (p₂' -ᵥ y))

Tactic:
linear_combination (norm := match_scalars <;> field) hr

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mp
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ s.WOppSide x y → ∃ a ≤ 0, ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y

pp: case mpr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ (∃ a ≤ 0, ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y) → s.WOppSide x y

Tactic:
· rw [wOppSide_iff_exists_left hp, or_iff_right hx]
    rintro ⟨p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩
    · rw [vsub_eq_zero_iff_eq] at h
      exact False.elim (hx (h.symm ▸ hp))
    · rw [vsub_eq_zero_iff_eq] at h
      refine ⟨0, le_rfl, p₂, hp₂, ?_⟩
      simp [h]
    · refine ⟨-r₁ / r₂, (div_neg_of_neg_of_pos (Left.neg_neg_iff.2 hr₁) hr₂).le, p₂, hp₂, ?_⟩
      rw [div_eq_inv_mul, ← smul_smul, neg_smul, h, smul_neg, smul_smul,
        inv_mul_cancel₀ hr₂.ne.symm, one_smul, neg_vsub_eq_vsub_rev, vsub_vadd]

Goal state after:
case mpr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ (∃ a ≤ 0, ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y) → s.WOppSide x y
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module 𝕜 V
s : Set V
x y : V
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : OrderTopology 𝕜
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul 𝕜 V
hsxy : IsVisible 𝕜 s x y
hy : y ∈ interior s
hxy : x ≠ y
h : ∀ᶠ (_δ : 𝕜) in 𝓝[>] 0, False
⊢ False

Tactic:
by obtain ⟨_, ⟨⟩⟩ := h.exists

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w : Fin (n + 1) → R
hw : ∑ j, w j = 1
i : Fin (n + 1)
⊢ SignType.sign (w i) = SignType.sign (Pi.single i 1 i) ∨ Pi.single i 1 i = 0 ∨ w i = 0 ↔ 0 ≤ w i

Tactic:
simp [sign_eq_one_iff, le_iff_eq_or_lt', or_comm]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
⊢ BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}

Tactic:
contrapose H1

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
H1 : ¬BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
⊢ g =ᵐ[volume.restrict (Ioi 0)] 0
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
eq₂ : ∀ᵐ (ω : α) ∂μ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ENNReal.ofReal (∫ (t : ℝ) in 0..F ω, G t)
⊢ ∫⁻ (a : α), ENNReal.ofReal (∫ (t : ℝ) in 0..F a, G t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)

Tactic:
eq₁

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
eq₂ : ∀ᵐ (ω : α) ∂μ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ENNReal.ofReal (∫ (t : ℝ) in 0..F ω, G t)
⊢ ∫⁻ (a : α), ENNReal.ofReal (∫ (t : ℝ) in 0..F a, G t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : NormedSpace 𝕜 F
n : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
f : 𝓓^{n}(Ω, F) →ₗ[𝕜] V
⊢ Continuous[_, t] ⇑f ↔ ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑f ∘ ofSupportedIn K_sub_Ω)

Tactic:
by
  simp_rw [← f.coe_restrictScalars ℝ]
  rw [continuous_iff_le_induced]
  have : @IsTopologicalAddGroup _ (induced (f.restrictScalars ℝ) t) _ :=
    topologicalAddGroup_induced _
  have : @ContinuousSMul ℝ _ _ _ (induced (f.restrictScalars ℝ) t) := continuousSMul_induced _
  have : @LocallyConvexSpace ℝ _ _ _ _ _ (induced (f.restrictScalars ℝ) t) := .induced _
  simp_rw [topologicalSpace_le_iff, originalTop, iSup₂_le_iff, ← continuous_iff_le_induced,
    continuous_coinduced_dom]

Goal state after:
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedSpace 𝕜 F
n k : ℕ∞
inst✝¹ : Algebra ℝ 𝕜
inst✝ : IsScalarTower ℝ 𝕜 F
hk : n < k + 1
⊢ fderivCLM 𝕜 n k = 0

Tactic:
ext : 2

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedSpace 𝕜 F
n k : ℕ∞
inst✝¹ : Algebra ℝ 𝕜
inst✝ : IsScalarTower ℝ 𝕜 F
hk : n < k + 1
x✝ : 𝓓^{n}(Ω, F)
a✝ : E
⊢ ((fderivCLM 𝕜 n k) x✝) a✝ = (0 x✝) a✝
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
m : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
f : 𝓓^{n}(Ω, F)
⊢ StronglyMeasurable ⇑f

Tactic:
by
  exact f.continuous.stronglyMeasurable_of_hasCompactSupport f.hasCompactSupport

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h₁
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := ⋯
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
⊢ ‖Polynomial.eval r (descPochhammer ℝ n)‖ ≤ ∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖

pp: case h₂
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := ⋯
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
⊢ ‖x ^ (r - ↑n)‖ ≤ (1 + ‖x‖) ^ k

Tactic:
· have : n ∈ Finset.range (N + 1) := by grind
    apply Finset.single_le_sum (fun _ _ ↦ by positivity) this

Goal state after:
case h₂
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := ⋯
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
⊢ ‖x ^ (r - ↑n)‖ ≤ (1 + ‖x‖) ^ k
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : TopologicalSpace M₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Module 𝕜₁ M₁
inst✝⁴ : Module 𝕜₂ M₂
inst✝³ : IsTopologicalAddGroup M₁
inst✝² : ContinuousConstSMul 𝕜₁ M₁
inst✝¹ : IsTopologicalAddGroup M₂
inst✝ : ContinuousSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
hf : IsCompactOperator ⇑f
U : Set M₂
hU : U ∈ 𝓝 (f 0)
⊢ U ∈ map (⇑f) (𝓝 0)

Tactic:
rw [map_zero] at hU

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : TopologicalSpace M₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Module 𝕜₁ M₁
inst✝⁴ : Module 𝕜₂ M₂
inst✝³ : IsTopologicalAddGroup M₁
inst✝² : ContinuousConstSMul 𝕜₁ M₁
inst✝¹ : IsTopologicalAddGroup M₂
inst✝ : ContinuousSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
hf : IsCompactOperator ⇑f
U : Set M₂
hU : U ∈ 𝓝 0
⊢ U ∈ map (⇑f) (𝓝 0)
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : SeminormedAddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : NormedSpace 𝕜₁ M₁
inst✝⁵ : Module 𝕜₂ M₂
inst✝⁴ : UniformSpace M₂
inst✝³ : IsUniformAddGroup M₂
inst✝² : ContinuousConstSMul 𝕜₂ M₂
inst✝¹ : T2Space M₂
inst✝ : CompleteSpace M₂
u : M₁ →SL[σ₁₂] M₂
hu : ∀ U ∈ 𝓝 0, ∃ y ∈ {f | IsCompactOperator ⇑f}, y - u ∈ U
this : TotallyBounded (⇑u '' closedBall 0 1)
⊢ u ∈ {f | IsCompactOperator ⇑f}

Tactic:
change IsCompactOperator (u : M₁ →ₛₗ[σ₁₂] M₂)

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : SeminormedAddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : NormedSpace 𝕜₁ M₁
inst✝⁵ : Module 𝕜₂ M₂
inst✝⁴ : UniformSpace M₂
inst✝³ : IsUniformAddGroup M₂
inst✝² : ContinuousConstSMul 𝕜₂ M₂
inst✝¹ : T2Space M₂
inst✝ : CompleteSpace M₂
u : M₁ →SL[σ₁₂] M₂
hu : ∀ U ∈ 𝓝 0, ∃ y ∈ {f | IsCompactOperator ⇑f}, y - u ∈ U
this : TotallyBounded (⇑u '' closedBall 0 1)
⊢ IsCompactOperator ⇑↑u
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
⊢ IsLocalExtrOn T.reApplyInnerSelf {x | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀

Tactic:
convert! hextr

Goal state after:
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
⊢ {x | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} = sphere 0 ‖x₀‖
Module: Mathlib.Data.Fin.Tuple.Sort

Goal state before:
pp: n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
⊢ Monotone (f ∘ ⇑(sort f))

Tactic:
by
  rw [self_comp_sort]
  exact (monotone_proj f).comp (graphEquiv₂ f).monotone

Goal state after:
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
j : n
⊢ star ↑hA.eigenvectorUnitary *ᵥ ↑hA.eigenvectorUnitary *ᵥ Pi.single j 1 = Pi.single j 1

Tactic:
mulVec_mulVec,

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
j : n
⊢ (star ↑hA.eigenvectorUnitary * ↑hA.eigenvectorUnitary) *ᵥ Pi.single j 1 = Pi.single j 1
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
h : hA.eigenvalues = 0
⊢ ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (Function.const n ↑0)) = 0

Tactic:
RCLike.ofReal_zero,

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
h : hA.eigenvalues = 0
⊢ ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (Function.const n 0)) = 0
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_2
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : A.IsHermitian
h_ne : hA.eigenvalues = 0
this : A = ((Unitary.conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (Function.const n ↑0))
⊢ A = 0

Tactic:
RCLike.ofReal_zero,

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_2
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : A.IsHermitian
h_ne : hA.eigenvalues = 0
this : A = ((Unitary.conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (Function.const n 0))
⊢ A = 0
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: case convert_4
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
this : n = finrank 𝕜 E
⊢ (List.ofFn fun x ↦ hT.eigenvalues ⋯ x).SortedGE

Tactic:
convert! (hT.eigenvalues_antitone hn).sortedGE_ofFn

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n k : ℕ∞
K : Compacts E
f : 𝓓^{n}_{K}(E, F)
hk : n < k + 1
⊢ (fderivLM 𝕜 n k) f = 0

Tactic:
by
  ext : 1
  simp [not_le_of_gt hk]

Goal state after:
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
ι : Type u_3
A B : Matrix ι ι 𝕜
hA : A.PosDef
hB : B.PosDef
x : ι →₀ 𝕜
hx : x ≠ 0
hAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef
⊢ 0 < ∑ a ∈ x.support, ∑ x_1 ∈ x.support, (starRingEnd 𝕜) (x a) * (A a x_1 * B a x_1) * x x_1

Tactic:
← Finset.sum_attach x.support,

Goal state after:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
ι : Type u_3
A B : Matrix ι ι 𝕜
hA : A.PosDef
hB : B.PosDef
x : ι →₀ 𝕜
hx : x ≠ 0
hAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef
⊢ 0 <
    ∑ x_1 ∈ x.support.attach, ∑ x_2 ∈ x.support.attach, (starRingEnd 𝕜) (x ↑x_1) * (A ↑x_1 ↑x_2 * B ↑x_1 ↑x_2) * x ↑x_2
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
x y : Matrix n n 𝕜
hy : IsUnit y
hM : (star y * y).PosDef
__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace
hx : ((y * xᴴ)ᴴ * (y * xᴴ)).trace = 0
⊢ x = 0

Tactic:
trace_conjTranspose_mul_self_eq_zero_iff

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
x y : Matrix n n 𝕜
hy : IsUnit y
hM : (star y * y).PosDef
__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace
hx : y * xᴴ = 0
⊢ x = 0
Module: Mathlib.Analysis.InnerProductSpace.GramMatrix

Goal state before:
pp: E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Finite n
v : n → E
this : Fintype n
h_li : ∀ (g : n → 𝕜), ∑ i, g i • v i = 0 → ∀ (i : n), g i = 0
⊢ (gram 𝕜 v).PosDef

Tactic:
refine .of_dotProduct_mulVec_pos (isHermitian_gram _ _) fun x hx ↦
    ((posSemidef_gram ..).dotProduct_mulVec_nonneg _).lt_of_ne' ?_

Goal state after:
E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Finite n
v : n → E
this : Fintype n
h_li : ∀ (g : n → 𝕜), ∑ i, g i • v i = 0 → ∀ (i : n), g i = 0
x : n → 𝕜
hx : x ≠ 0
⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ x ≠ 0
Module: Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : BorelSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : IsFiniteMeasure μ
inst✝ : Fact (1 ≤ p)
f : α →ᵇ E
⊢ ((ContinuousMap.toAEEqFunAddHom μ).comp (toContinuousMapAddMonoidHom α E)) f ∈ Lp E p μ

Tactic:
exact mem_Lp f

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
f : 𝓓^{n}_{K}(E, F)
⊢ StronglyMeasurable ⇑f

Tactic:
by
  exact f.continuous.stronglyMeasurable_of_hasCompactSupport f.hasCompactSupport

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
n : ℤ
⊢ (fourier n) ↑0 = 1

Tactic:
fourier_coe_apply,

Goal state after:
T : ℝ
n : ℤ
⊢ Complex.exp (2 * ↑π * Complex.I * ↑n * ↑0 / ↑T) = 1
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
⊢ HasSum (fun i ↦ fourierCoeff (↑↑f) i • fourierLp 2 i) f

Tactic:
simp_rw [← fourierBasis_repr]; rw [← coe_fourierBasis]
  exact HilbertBasis.hasSum_repr fourierBasis f

Goal state after: