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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : ¬μ tᶜ = 0
this : ⨍ (x : α), f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in t, f x ∂μ) (⨍ (x : α) in t, f x ∂μ)
⊢ ∫ (x : α) in t, f x ∂μ = μ.real t • ⨍ (x : α), f x ∂μ

Tactic:
openSegment_same,

Goal state after:
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : ¬μ tᶜ = 0
this : ⨍ (x : α), f x ∂μ ∈ {⨍ (x : α) in t, f x ∂μ}
⊢ ∫ (x : α) in t, f x ∂μ = μ.real t • ⨍ (x : α), f x ∂μ
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: X : Type u_2
K : Type u_8
inst✝³ : Semifield K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
w : StdSimplex K X
p : X → Prop
inst✝ : DecidablePred p
hp : ∃ a, p a ∧ w.weights a ≠ 0
⊢ filter p w.weights ≠ 0

Tactic:
simpa [ne_iff, filter_apply]

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: X : Type u_2
K : Type u_8
inst✝⁴ : Semifield K
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : IsDomain K
w : StdSimplex K X
s : Set X
hs : ∃ x ∈ s, w.weights x ≠ 0
inst✝ : DecidablePred fun x ↦ x ∈ s
⊢ filter (fun x ↦ x ∈ s) w.weights ≠ 0

Tactic:
simpa [ne_iff, filter_apply]

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: X : Type u_2
K : Type u_8
inst✝⁴ : Semifield K
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : IsDomain K
w : StdSimplex K X
s : Set X
hs : ∃ x ∈ s, w.weights x ≠ 0
inst✝ : DecidablePred fun x ↦ x ∈ s
this : ((filter (fun x ↦ x ∈ s) w.weights).sum fun x k ↦ k) ≠ 0
⊢ (w.restrict s hs).weights.support = {x ∈ w.weights.support | x ∈ s}

Tactic:
weights_restrict,

Goal state after:
X : Type u_2
K : Type u_8
inst✝⁴ : Semifield K
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : IsDomain K
w : StdSimplex K X
s : Set X
hs : ∃ x ∈ s, w.weights x ≠ 0
inst✝ : DecidablePred fun x ↦ x ∈ s
this : ((filter (fun x ↦ x ∈ s) w.weights).sum fun x k ↦ k) ≠ 0
⊢ (((filter (fun x ↦ x ∈ s) w.weights).sum fun _x k ↦ k)⁻¹ • filter (fun x ↦ x ∈ s) w.weights).support =
    {x ∈ w.weights.support | x ∈ s}
Module: Mathlib.Geometry.Convex.Set

Goal state before:
pp: R : Type u_3
X : Type u_5
Y : Type u_6
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : ConvexSpace R X
inst✝ : ConvexSpace R Y
f : X → Y
s : Set Y
hf : IsAffineMap R f
hs : IsConvexSet R s
w : StdSimplex R X
hw : ↑w.weights.support ⊆ f ⁻¹' s
⊢ iConvexComb w f ∈ s

Tactic:
exact hs.iConvexComb_mem fun x hx ↦ hw <| by simpa

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

Goal state before:
pp: case h.a
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
  ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
    (∀ i ∈ s, Convex 𝕜 (F i)) →
      (∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
ι : Type u_1
F : ι → Set E
s : Finset ι
h_convex : ∀ i ∈ s, Convex 𝕜 (F i)
h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
hn : #s = k + 1
a : ↥s → E := ⋯
h_ind : ¬AffineIndependent 𝕜 a
I : Set ↥s
p : E
hp_I : p ∈ (convexHull 𝕜) (a '' I)
hp_Ic : p ∈ (convexHull 𝕜) (a '' Iᶜ)
i : ↥s
J : Set ↥s
hi : i ∈ J
v : E
j : ↥s
hj : j ∈ Jᶜ
hj_v : a j = v
⊢ a j ∈ ⋂ k ∈ s.erase ↑j, F k

Tactic:
· apply Nonempty.some_mem

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
e : Sym2 V
he : e ∈ G.edgeSet
v : V
h : v ∈ e
⊢ Sym2.Mem.other h ≠ v

Tactic:
rw [← Sym2.other_spec h, Sym2.eq_swap] at he
  exact G.ne_of_adj he

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

Goal state before:
pp: V : Type u
h : Nontrivial V
⊢ Nontrivial (SimpleGraph V)

Tactic:
exact instNontrivial

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 p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (x -ᵥ p₁) (p₂ -ᵥ x)
a : R
h₁ : x -ᵥ p₁ = a • (p₂ -ᵥ x + (x -ᵥ p₁))
⊢ x ∈ s

Tactic:
vsub_add_vsub_cancel,

Goal state after:
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 p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (x -ᵥ p₁) (p₂ -ᵥ x)
a : R
h₁ : x -ᵥ p₁ = a • (p₂ -ᵥ p₁)
⊢ x ∈ s
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: case hyz
E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y z : PE
hx : dist x y = dist x z / 2
hy : dist y z = dist x z / 2
⊢ dist y z = (1 - 1 / 2) * dist x z

Tactic:
sub_half,

Goal state after:
case hyz
E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y z : PE
hx : dist x y = dist x z / 2
hy : dist y z = dist x z / 2
⊢ dist y z = 1 / 2 * dist x z
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: 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 : P
h : s.SSameSide x y
⊢ ¬s.WOppSide x y

Tactic:
by
  intro ho
  have hxy := wSameSide_and_wOppSide_iff.1 ⟨h.1, ho⟩
  rcases hxy with (hx | hy)
  · exact h.2.1 hx
  · exact h.2.2 hy

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
Ω : Opens E
F : Type u_2
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : TopologicalSpace F
n k : ℕ∞
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
c : ℝ
v : E
⊢ lineDerivCLM (c • v) = c • lineDerivCLM v

Tactic:
ext T f
  simp [lineDerivCLM_apply, TestFunction.lineDerivCLM_smul]

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

Goal state before:
pp: case e_a.e_6
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} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B1 :
  ∫⁻ (t : ℝ) in Ioc 0 M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioc 0 M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
t : ℝ
ht : t ∈ Ioi M
⊢ (fun a ↦ t ≤ f a) = {a | t ≤ f a} ∩ {a | M < f a}

Tactic:
exact (inter_eq_left.2 (fun a ha ↦ (mem_Ioi.1 ht).trans_le ha)).symm

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

Goal state before:
pp: case neg
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} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν = ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)

Tactic:
exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
    ν f_nn f_mble g_intble g_mble g_nn

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

Goal state before:
pp: 𝕜 : Type u_1
𝕂 : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace 𝕜 F
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
⊢ tsupport 0 ⊆ ↑Ω

Tactic:
by simp only [tsupport_zero, empty_subset]

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

Goal state before:
pp: case h
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,
    ∀ x ∈ t,
      ‖iteratedFDerivWithin ℝ n (fun x ↦ x ^ r) t x‖ ≤
        (∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖) * (1 + ‖x‖) ^ k

Tactic:
intro n hn x hx

Goal state after:
case h
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
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
⊢ ‖iteratedFDerivWithin ℝ n (fun x ↦ x ^ r) t x‖ ≤
    (∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖) * (1 + ‖x‖) ^ k
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
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
hx' : 1 / 2 < x
hx'' : 0 < x
h✝ : r - ↑n < 0
h : 0 < ↑n - r
⊢ Real.exp (Real.log 2 * (↑n - r)) ≤ Real.exp (Real.log (3 / 2) * ↑k)

Tactic:
gcongr 1
      apply le_trans _ hk₂
      gcongr

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
c : ℝ
hc₀ : c > 0
hc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε
φ : ℕ → ℝ
hφ_anti : StrictAnti φ
hφ_pos : ∀ (n : ℕ), 0 < φ n
hφ : Tendsto φ atTop (𝓝 0)
x : ℕ → X
hx_norm_upper : ∀ (n : ℕ), ‖x n‖ ≤ 1
hx_norm_lower : ∀ (n : ℕ), c ≤ ‖x n‖
hx_bound : ∀ (n : ℕ), ‖(T - μ • 1) (x n)‖ < φ n
hx_lim : Tendsto (fun n ↦ (T - μ • 1) (x n)) atTop (𝓝 0)
⊢ HasEigenvalue (↑T) μ

Tactic:
let y_ (n : ℕ) : X := T (x n)

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
c : ℝ
hc₀ : c > 0
hc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε
φ : ℕ → ℝ
hφ_anti : StrictAnti φ
hφ_pos : ∀ (n : ℕ), 0 < φ n
hφ : Tendsto φ atTop (𝓝 0)
x : ℕ → X
hx_norm_upper : ∀ (n : ℕ), ‖x n‖ ≤ 1
hx_norm_lower : ∀ (n : ℕ), c ≤ ‖x n‖
hx_bound : ∀ (n : ℕ), ‖(T - μ • 1) (x n)‖ < φ n
hx_lim : Tendsto (fun n ↦ (T - μ • 1) (x n)) atTop (𝓝 0)
y_ : ℕ → X := fun n ↦ T (x n)
⊢ HasEigenvalue (↑T) μ
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
⊢ A = ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues))

Tactic:
← conjStarAlgAut_star_eigenvectorUnitary,

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
⊢ A =
    ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary)
      (((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) (star hA.eigenvectorUnitary)) A)
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
⊢ (diagonal hA.eigenvalues).rank = Fintype.card { i // hA.eigenvalues i ≠ 0 }

Tactic:
Matrix.rank_diagonal

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
⊢ Fintype.card { i // hA.eigenvalues i ≠ 0 } = Fintype.card { i // hA.eigenvalues i ≠ 0 }
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 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
i : Fin n
⊢ HasEigenvector T (↑(hT.eigenvalues hn i)) ((hT.eigenvectorBasis hn) i)

Tactic:
eigenvalues_def,

Goal state after:
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
i : Fin n
⊢ HasEigenvector T (↑((hT.unsortedEigenvalues hn ∘ ⇑(Tuple.sort (hT.unsortedEigenvalues hn)) ∘ ⇑Fin.revPerm) i))
    ((hT.eigenvectorBasis hn) i)
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 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
⊢ Antitone (hT.eigenvalues hn)

Tactic:
eigenvalues_def,

Goal state after:
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
⊢ Antitone (hT.unsortedEigenvalues hn ∘ ⇑(Tuple.sort (hT.unsortedEigenvalues hn)) ∘ ⇑Fin.revPerm)
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 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
hT : T.IsSymmetric
⊢ T.charpoly.roots.card = T.charpoly.natDegree

Tactic:
simp [hT.roots_charpoly_eq_eigenvalues rfl, LinearMap.charpoly_natDegree]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsCompactOperator ⇑T
hT' : (↑T).IsSymmetric
S : ↥(⨆ μ, eigenspace (↑T) μ)ᗮ →L[𝕜] ↥(⨆ μ, eigenspace (↑T) μ)ᗮ := T.restrict ⋯
hS_compact : IsCompactOperator ⇑S
hS_symm : (↑S).IsSymmetric
hS : ∀ (μ : 𝕜), eigenspace (↑S) μ = ⊥
h : S = 0
⊢ Subsingleton ↥(⨆ μ, eigenspace (↑T) μ)ᗮ

Tactic:
by_contra! hV

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsCompactOperator ⇑T
hT' : (↑T).IsSymmetric
S : ↥(⨆ μ, eigenspace (↑T) μ)ᗮ →L[𝕜] ↥(⨆ μ, eigenspace (↑T) μ)ᗮ := T.restrict ⋯
hS_compact : IsCompactOperator ⇑S
hS_symm : (↑S).IsSymmetric
hS : ∀ (μ : 𝕜), eigenspace (↑S) μ = ⊥
h : S = 0
hV : Nontrivial ↥(⨆ μ, eigenspace (↑T) μ)ᗮ
⊢ False
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
⊢ IsUniformAddGroup 𝓓^{n}_{K}(E, F)

Tactic:
by
  rw [ContDiffMapSupportedIn.uniformSpace_eq_iInf]
  exact isUniformAddGroup_iInf fun _ ↦ IsUniformAddGroup.comap _

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 : ℕ∞
K : Compacts E
⊢ comap (⇑(ContinuousLinearMap.pi (structureMapCLM 𝕜 n))) (Pi.uniformSpace fun i ↦ E →ᵇ E [×i]→L[ℝ] F) = uniformSpace

Tactic:
ContDiffMapSupportedIn.uniformSpace_eq_iInf,

Goal state after:
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 : ℕ∞
K : Compacts E
⊢ comap (⇑(ContinuousLinearMap.pi (structureMapCLM 𝕜 n))) (Pi.uniformSpace fun i ↦ E →ᵇ E [×i]→L[ℝ] F) =
    ⨅ i, comap (⇑(structureMapLM ℝ n i)) inferInstance
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : 0 ≤ A
⊢ ∃ s, A = star s * s

Tactic:
obtain ⟨X, hX, -, rfl⟩ :=
        sub_zero A ▸ CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts hA.isHermitian
          (QuasispectrumRestricts.nnreal_of_nonneg hA.nonneg)
      exact ⟨X, hX.star_eq.symm ▸ rfl⟩

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

Goal state before:
pp: E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_5
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
v : n → E
f : E →L[𝕜] F
c : n →₀ 𝕜
⊢ ‖f (∑ x ∈ c.support, c x • v x)‖ ^ 2 ≤ ‖f‖ ^ 2 • ‖∑ x ∈ c.support, c x • v x‖ ^ 2

Tactic:
f.le_opNorm _,

Goal state after:
E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_5
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
v : n → E
f : E →L[𝕜] F
c : n →₀ 𝕜
⊢ (‖f‖ * ‖∑ x ∈ c.support, c x • v x‖) ^ 2 ≤ ‖f‖ ^ 2 • ‖∑ x ∈ c.support, c x • v x‖ ^ 2
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_4
inst✝ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
f : ↥(Lp 𝕜 2 μ)
⊢ ⟪indicatorConstLp 2 hs hμs 1, f⟫ = ∫ (x : α) in s, ↑↑f x ∂μ

Tactic:
rw [L2.inner_indicatorConstLp_eq_inner_setIntegral 𝕜 hs hμs (1 : 𝕜) f]; simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:

Tactic:
‖f‖ ^ 2 = ‖f‖ ^ (2 : ℝ≥0∞).toReal := by norm_cast
        _ = ∑' i, ‖f i‖ ^ (2 : ℝ≥0∞).toReal := lp.norm_rpow_eq_tsum ?_ f
        _ = ∑' i, ‖f i‖ ^ (2 : ℕ) := by norm_cast
        _ = ∑' i, re ⟪f i, f i⟫ := by simp
        _ = re (∑' i, ⟪f i, f i⟫) := (RCLike.reCLM.map_tsum ?_).symm

Goal state after:
case calc_1
ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
f : ↥(lp G 2)
⊢ 0 < ENNReal.toReal 2

case calc_2
ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
f : ↥(lp G 2)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : AddCircle T → ℂ
p : ℝ≥0∞
hT' : 0 < T
h : MemLp f p (ENNReal.ofReal T • AddCircle.haarAddCircle)
this : MemLp f p ((ENNReal.ofReal T)⁻¹ • ENNReal.ofReal T • AddCircle.haarAddCircle)
⊢ MemLp f p AddCircle.haarAddCircle

Tactic:
rwa [smul_smul, ENNReal.inv_mul_cancel (by positivity) (by finiteness), one_smul] at this

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
k n : ℕ
c : 𝕜
f : 𝓢(E, F)
⊢ seminormAux k n (c • f) ≤ ‖c‖ * seminormAux k n f

Tactic:
refine (c • f).seminormAux_le_bound k n (mul_nonneg (norm_nonneg _) (seminormAux_nonneg _ _ _))
      fun x => (decay_smul_aux k n f c x).trans_le ?_

Goal state after:
𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
k n : ℕ
c : 𝕜
f : 𝓢(E, F)
x : E
⊢ ‖c‖ * ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ ‖c‖ * seminormAux k n f
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
w : W
F : W → V → E := fun w' v ↦ 𝐞 (-(L v) w') • f v
F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-(L v) w') • fourierSMulRight L f v
B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
h0 : ∀ (w' : W), Integrable (F w') μ
h1 : ∀ᶠ (w' : W) in 𝓝 w, AEStronglyMeasurable (F w') μ
h3 : AEStronglyMeasurable (F' w) μ
v : V
w' : W
a✝ : w' ∈ Metric.ball w 1
⊢ ‖F' w' v‖ ≤ B v

Tactic:
rw [Circle.norm_smul _ (fourierSMulRight L f v)]

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : SecondCountableTopology V
μ : Measure V
hf : Integrable f μ
hf' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) μ
w : W
F : W → V → E := fun w' v ↦ 𝐞 (-(L v) w') • f v
F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-(L v) w') • fourierSMulRight L f v
B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
h0 : ∀ (w' : W), Integrable (F w') μ
h1 : ∀ᶠ (w' : W) in 𝓝 w, AEStronglyMeasurable (F w') μ
h3 : AEStronglyMeasurable (F' w) μ
v : V
w' : W
a✝ : w' ∈ Metric.ball w 1
⊢ ‖fourierSMulRight L f v‖ ≤ B v