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

Goal state before:
pp: E : Type u_1
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
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n ↦ ball x (↑n + 1)
n : ℕ
⊢ x ∈ interior (s ∩ B n)

Tactic:
interior_inter,

Goal state after:
E : Type u_1
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
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n ↦ ball x (↑n + 1)
n : ℕ
⊢ x ∈ interior s ∩ interior (B n)
Module: Mathlib.Analysis.Convex.Measure

Goal state before:
pp: E : Type u_1
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
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n ↦ ball x (↑n + 1)
y : E
hy : y ∈ frontier s
N : ℕ := ⌊dist y x⌋₊
hN : y ∈ B N
⊢ y ∈ frontier (s ∩ B N)

Tactic:
suffices y ∈ frontier (s ∩ B N) ∩ B N from this.1

Goal state after:
E : Type u_1
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
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n ↦ ball x (↑n + 1)
y : E
hy : y ∈ frontier s
N : ℕ := ⌊dist y x⌋₊
hN : y ∈ B N
⊢ y ∈ frontier (s ∩ B N) ∩ B N
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: case inr
f : ℝ → ℝ
y z : ℝ
hf_cvx : ConvexOn ℝ (Ico y z) f
hf_cont : ContinuousWithinAt f (Ici y) y
x : ℝ
hx : x ∈ Ico y z
hyx : y < x
⊢ ContinuousWithinAt f (Ico y z) x

Tactic:
· have h := hf_cvx.continuousOn_interior x
    simp only [interior_Ico, mem_Ioo, hyx, hx.2, and_self, forall_const] at h
    rw [continuousWithinAt_iff_continuousAt (Ioo_mem_nhds hyx hx.2)] at h
    exact h.continuousWithinAt

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x y i : P
a✝ : i ∈ (Finsupp.single x s + Finsupp.single y t).support
⊢ (Finsupp.single x s + Finsupp.single y t) i = (fun x r ↦ r) i ((Finsupp.single x s + Finsupp.single y t) i)

Tactic:
simp only [Finsupp.coe_add, Pi.add_apply]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
⊢ (Convex 𝕜 s ∧
      (∀ ⦃x : E⦄,
          x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ max (f x) (f y)) ∧
        ∀ ⦃x : E⦄,
          x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → min (f x) (f y) ≤ f (a • x + b • y)) ↔
    Convex 𝕜 s ∧
      ∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ uIcc (f x) (f y)

Tactic:
apply and_congr_right'

Goal state after:
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
⊢ ((∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ max (f x) (f y)) ∧
      ∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → min (f x) (f y) ≤ f (a • x + b • y)) ↔
    ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ uIcc (f x) (f y)
Module: Mathlib.Analysis.Convex.Quasiconvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
⊢ (∀ x ∈ s,
      ∀ x_2 ∈ s,
        ∀ (x_4 x_5 : 𝕜),
          0 ≤ x_4 →
            0 ≤ x_5 →
              x_4 + x_5 = 1 →
                f (x_4 • x + x_5 • x_2) ≤ max (f x) (f x_2) ∧ min (f x) (f x_2) ≤ f (x_4 • x + x_5 • x_2)) ↔
    ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ uIcc (f x) (f y)

Tactic:
← Icc_min_max,

Goal state after:
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
⊢ (∀ x ∈ s,
      ∀ x_2 ∈ s,
        ∀ (x_4 x_5 : 𝕜),
          0 ≤ x_4 →
            0 ≤ x_5 →
              x_4 + x_5 = 1 →
                f (x_4 • x + x_5 • x_2) ≤ max (f x) (f x_2) ∧ min (f x) (f x_2) ≤ f (x_4 • x + x_5 • x_2)) ↔
    ∀ ⦃x : E⦄,
      x ∈ s →
        ∀ ⦃y : E⦄,
          y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ Icc (min (f x) (f y)) (max (f x) (f y))
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
s : Set (Sym2 V)
⊢ G.edgeSet \ (s \ Sym2.diagSet) = G.edgeSet \ s

Tactic:
grind [Sym2.mem_diagSet, not_isDiag_of_mem_edgeSet]

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✝⁵ : CommRing R
inst✝⁴ : PartialOrder 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
hx : x ∈ s
⊢ s.WSameSide x y

Tactic:
refine ⟨x, hx, x, hx, ?_⟩

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder 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
hx : x ∈ s
⊢ SameRay R (x -ᵥ x) (y -ᵥ x)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder 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
hx : x ∈ s
⊢ s.WOppSide x y

Tactic:
refine ⟨x, hx, x, hx, ?_⟩

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder 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
hx : x ∈ s
⊢ SameRay R (x -ᵥ x) (x -ᵥ y)
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
a b : V
h : ¬G.Adj a b
hn : a ≠ b
⊢ G.incidenceSet a ∩ G.incidenceSet b = ∅

Tactic:
simp_rw [Set.eq_empty_iff_forall_notMem, Set.mem_inter_iff, not_and]
  intro u ha hb
  exact h (G.adj_of_mem_incidenceSet hn ha hb)

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

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q u v x y : E
hu : u ∈ segment 𝕜 x p
hv : v ∈ segment 𝕜 y q
bz : 𝕜
hbz : 0 ≤ bz
haz : 0 ≤ 0
habz : 0 + bz = 1
⊢ ∃ x_1 ∈ segment 𝕜 u v, x_1 ∈ (convexHull 𝕜) {p, q, 0 • x + bz • y}

Tactic:
rw [zero_add] at habz

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q u v x y : E
hu : u ∈ segment 𝕜 x p
hv : v ∈ segment 𝕜 y q
bz : 𝕜
hbz : 0 ≤ bz
haz : 0 ≤ 0
habz : bz = 1
⊢ ∃ x_1 ∈ segment 𝕜 u v, x_1 ∈ (convexHull 𝕜) {p, q, 0 • x + bz • y}
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 z : P
hx : x ∉ s
hz : z ∈ s
t : R
ht0 : 0 ≤ t
ht1 : t ≤ 1
hy : t • (z -ᵥ x) +ᵥ x ∈ s
hyx : (lineMap x z) t ≠ x
hyz : (lineMap x z) t ≠ z
⊢ x ∈ s

Tactic:
have ht : t ≠ 1 := by
    rintro rfl
    simp [lineMap_apply] at hyz

Goal state after:
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 z : P
hx : x ∉ s
hz : z ∈ s
t : R
ht0 : 0 ≤ t
ht1 : t ≤ 1
hy : t • (z -ᵥ x) +ᵥ x ∈ s
hyx : (lineMap x z) t ≠ x
hyz : (lineMap x z) t ≠ z
ht : t ≠ 1
⊢ x ∈ s
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: case inr.refine_1.refine_1
𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this✝ : 0 < 1 - ε
hwi : 0 < 1 - w i
this :
  Wbtw 𝕜 ((lineMap x (a i)) ε) ((lineMap x (∑ j ∈ t, w j • a j)) ((w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹)))
    (∑ j ∈ t.erase i, (w j / (1 - w i)) • a j)
⊢ ∀ i_1 ∈ t.erase i, 0 ≤ w i_1 / (1 - w i)

Tactic:
intro j hj
      positivity [hw₀ j <| erase_subset _ _ hj]

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

Goal state before:
pp: V : Type u_2
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℝ V
s : Set V
y : V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul ℝ V
hs : IsClosed[inst✝²] s
hy : y ∈ s
x : V
t : Set ℝ := Set.Ici 0 ∩ ⇑(lineMap x y) ⁻¹' s
ht₁ : 1 ∈ t
ht : BddBelow t
δ : ℝ := sInf t
hδ₁ : δ ≤ 1
hδ₀ : 0 ≤ δ
hδ : (lineMap x y) δ ∈ s
⊢ IsVisible ℝ s x ((lineMap x y) δ)

Tactic:
rintro _ hε ⟨⟨ε, ⟨hε₀, hε₁⟩, rfl⟩, -, h⟩

Goal state after:
V : Type u_2
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℝ V
s : Set V
y : V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul ℝ V
hs : IsClosed[inst✝²] s
hy : y ∈ s
x : V
t : Set ℝ := Set.Ici 0 ∩ ⇑(lineMap x y) ⁻¹' s
ht₁ : 1 ∈ t
ht : BddBelow t
δ : ℝ := sInf t
hδ₁ : δ ≤ 1
hδ₀ : 0 ≤ δ
hδ : (lineMap x y) δ ∈ s
ε : ℝ
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
hε : (lineMap x ((lineMap x y) δ)) ε ∈ s
h : (lineMap x ((lineMap x y) δ)) ε ≠ (lineMap x y) δ
⊢ False
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
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
hs₁ : 0 < w₁ i
hs₂ : w₂ i < 0
w₃ : Fin (n + 1) → R := (lineMap w₁ w₂) (w₁ i / (w₁ i - w₂ i))
hp : 0 < w₁ i - w₂ i
hw₃ : ∑ j, w₃ j = 1
h : Sbtw R w₁ w₃ w₂
h' :
  Sbtw R ((Finset.affineCombination R Finset.univ s.points) w₁) ((Finset.affineCombination R Finset.univ s.points) w₃)
    ((Finset.affineCombination R Finset.univ s.points) w₂)
⊢ w₁ i / (w₁ i - w₂ i) * (w₂ i - w₁ i) + w₁ i = 0

Tactic:
← neg_sub (w₁ i) (w₂ i),

Goal state after:
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
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
hs₁ : 0 < w₁ i
hs₂ : w₂ i < 0
w₃ : Fin (n + 1) → R := (lineMap w₁ w₂) (w₁ i / (w₁ i - w₂ i))
hp : 0 < w₁ i - w₂ i
hw₃ : ∑ j, w₃ j = 1
h : Sbtw R w₁ w₃ w₂
h' :
  Sbtw R ((Finset.affineCombination R Finset.univ s.points) w₁) ((Finset.affineCombination R Finset.univ s.points) w₃)
    ((Finset.affineCombination R Finset.univ s.points) w₂)
⊢ w₁ i / (w₁ i - w₂ i) * -(w₁ i - w₂ i) + w₁ i = 0
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
Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedSpace 𝕜 F
n₁ n₂ : ℕ∞
inst✝¹ : Algebra ℝ 𝕜
inst✝ : IsScalarTower ℝ 𝕜 F
H : ¬(n₂ ≤ n₁ ∧ Ω₁ ≤ Ω₂)
⊢ monoCLM 𝕜 = 0

Tactic:
ext; simp [H]

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
f : 𝓓^{n}(Ω, F)
⊢ ⇑((fderivCLM 𝕜 n k) f) = if k + 1 ≤ n then fderiv ℝ ⇑f else 0

Tactic:
by
  rw [fderivCLM]
  split_ifs <;> rfl

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)
y_ : ℕ → X := fun n ↦ T (x n)
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 1 ⊆ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hy_lim : Tendsto (fun n ↦ (T - μ • 1) (y_ n)) atTop (𝓝 0)
d : ℝ
hd₀ : d > 0
hd : ∀ᶠ (n : ℕ) in atTop, d ≤ ‖y_ n‖
hyK : 0 ∈ K
hψy : Tendsto (y_ ∘ ψ) atTop (𝓝 0)
hy_eigen' : (T - μ • 1) 0 = 0
this : ∀ᶠ (n : ℕ) in atTop, False
⊢ False

Tactic:
rwa [eventually_const] at this

Goal state after:
Module: Mathlib.Data.Fin.Tuple.Sort

Goal state before:
pp: n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
σ : Equiv.Perm (Fin n)
⊢ (f ∘ ⇑σ) ∘ ⇑(sort (f ∘ ⇑σ)) = f ∘ ⇑(sort f)

Tactic:
Function.comp_assoc,

Goal state after:
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
σ : Equiv.Perm (Fin n)
⊢ f ∘ ⇑σ ∘ ⇑(sort (f ∘ ⇑σ)) = f ∘ ⇑(sort f)
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
inst✝ : CompleteSpace X
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h₁ : ¬HasEigenvalue (↑T) μ
S : X →L[𝕜] X := T - μ • 1
K : NNReal
hK : AntilipschitzWith K ⇑S
h₂ : ¬Function.Surjective ⇑S
⊢ False

Tactic:
obtain ⟨c, hc⟩ := NormedField.exists_one_lt_norm 𝕜

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
inst✝ : CompleteSpace X
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h₁ : ¬HasEigenvalue (↑T) μ
S : X →L[𝕜] X := T - μ • 1
K : NNReal
hK : AntilipschitzWith K ⇑S
h₂ : ¬Function.Surjective ⇑S
c : 𝕜
hc : 1 < ‖c‖
⊢ False
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 B : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
hB : B.IsHermitian
⊢ hA.eigenvalues = hB.eigenvalues ↔ A.charpoly = B.charpoly

Tactic:
constructor <;> intro h
  · rw [hA.charpoly_eq, hB.charpoly_eq, h]
  · suffices hA.eigenvalues₀ = hB.eigenvalues₀ by unfold eigenvalues; rw [this]
    simp_rw [← List.ofFn_inj, ← sort_roots_charpoly_eq_eigenvalues₀, h]

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
n₁ n₂ : ℕ∞
K₁ K₂ : Compacts E
H : ¬(n₂ ≤ n₁ ∧ K₁ ≤ K₂)
⊢ monoLM 𝕜 = 0

Tactic:
ext; simp [H]

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
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
v : E
i : Fin n
w : EuclideanSpace 𝕜 (Fin n)
⊢ T ((hT.eigenvectorBasis hn).repr.symm w) =
    (hT.eigenvectorBasis hn).repr.symm (toLp 2 fun i ↦ ↑(hT.eigenvalues hn i) * w.ofLp i)

Tactic:
← OrthonormalBasis.sum_repr_symm,

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
v : E
i : Fin n
w : EuclideanSpace 𝕜 (Fin n)
⊢ T (∑ x, w.ofLp x • (hT.eigenvectorBasis hn) x) = ∑ x, (↑(hT.eigenvalues hn x) * w.ofLp x) • (hT.eigenvectorBasis hn) x
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case pos
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
c : 𝕜
f : 𝓓^{n}_{K}(E, F)
hk : k + 1 ≤ n
⊢ ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ = (RingHom.id 𝕜) c • ContDiffMapSupportedIn.of_support_subset ⋯ ⋯

pp: case neg
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
c : 𝕜
f : 𝓓^{n}_{K}(E, F)
hk : ¬k + 1 ≤ n
⊢ 0 = (RingHom.id 𝕜) c • 0

Tactic:
· have hk' : 0 < (n : ℕ∞ω) := mod_cast (add_pos_of_right zero_lt_one k).trans_le hk
      ext
      simp [fderiv_const_smul (f.contDiff.differentiable hk'.ne').differentiableAt,
        FunLike.coe_smul]

Goal state after:
case neg
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
c : 𝕜
f : 𝓓^{n}_{K}(E, F)
hk : ¬k + 1 ≤ n
⊢ 0 = (RingHom.id 𝕜) c • 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
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsCompactOperator ⇑T
hT' : (↑T).IsSymmetric
⊢ (⨆ μ, eigenspace (↑T) μ)ᗮ = ⊥

Tactic:
let S : (⨆ μ, eigenspace T μ : Submodule 𝕜 E)ᗮ →L[𝕜] (⨆ μ, eigenspace T μ : Submodule 𝕜 E)ᗮ :=
    T.restrict hT'.orthogonalComplement_iSup_eigenspaces_invariant
  have hS_compact : IsCompactOperator S :=
    hT.restrict' hT'.orthogonalComplement_iSup_eigenspaces_invariant
  have hS_symm : S.IsSymmetric :=
    hT'.restrict_invariant (hT'.orthogonalComplement_iSup_eigenspaces_invariant)
  have hS μ : eigenspace (S : Module.End 𝕜 (⨆ μ, eigenspace T μ : Submodule 𝕜 E)ᗮ) μ = ⊥ :=
    hT'.orthogonalComplement_iSup_eigenspaces _
  have h μ : HasEigenvalue (S : End 𝕜 (⨆ μ, eigenspace T μ : Submodule 𝕜 E)ᗮ) μ → μ = 0 := by
    simp_all [hasEigenvalue_iff]
  rw [eq_zero_of_forall_hasEigenvalue_eq_zero hS_compact hS_symm] at h
  rw [← Submodule.subsingleton_iff_eq_bot]
  by_contra! hV
  simpa [h] using hS 0

Goal state after:
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 : n → 𝕜
B : Matrix n n 𝕜
hA : (star B * B).PosSemidef
h : star x ᵥ* Bᴴ ⬝ᵥ B *ᵥ x = 0
⊢ Bᴴ *ᵥ B *ᵥ x = 0

Tactic:
vecMul_conjTranspose,

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
x : n → 𝕜
B : Matrix n n 𝕜
hA : (star B * B).PosSemidef
h : star (B *ᵥ star (star x)) ⬝ᵥ B *ᵥ x = 0
⊢ Bᴴ *ᵥ B *ᵥ x = 0
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
hAs : (A.submatrix Subtype.val Subtype.val).PosDef
hBs : (B.submatrix Subtype.val Subtype.val).PosDef
⊢ (A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val).IsHermitian ∧
    ∀ ⦃x_1 : ↥x.support → 𝕜⦄,
      x_1 ≠ 0 → 0 < star x_1 ⬝ᵥ A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val *ᵥ x_1

Tactic:
refine ⟨hAs.isHermitian.hadamard hBs.isHermitian, fun y hy ↦ ?_⟩

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
hAs : (A.submatrix Subtype.val Subtype.val).PosDef
hBs : (B.submatrix Subtype.val Subtype.val).PosDef
y : ↥x.support → 𝕜
hy : y ≠ 0
⊢ 0 < star y ⬝ᵥ A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val *ᵥ y
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f : α → E
hf : Integrable f μ
hf_int : ∀ (c : E), ∫ (x : α), ⟪c, f x⟫ ∂μ = 0
⊢ ∫ (x : α), f x ∂μ = 0

Tactic:
by
  specialize hf_int (∫ x, f x ∂μ); rwa [integral_inner hf, inner_self_eq_zero] at hf_int

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
n₁ n₂ : ℕ∞
K₁ K₂ : Compacts E
H : ¬(n₂ ≤ n₁ ∧ K₁ ≤ K₂)
⊢ monoCLM 𝕜 = 0

Tactic:
ext; simp [H]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
⊢ AEEqFun.mk (fun x ↦ ⟪↑↑f x, ↑↑g x⟫) ⋯ ∈ Lp 𝕜 1 μ

Tactic:
simp_rw [mem_Lp_iff_eLpNorm_lt_top, eLpNorm_aeeqFun]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
⊢ eLpNorm (fun x ↦ ⟪↑↑f x, ↑↑g x⟫) 1 μ < ∞
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case pos
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
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hk : k + 1 ≤ n
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E (E →L[ℝ] F) k K i) ((fderivLM 𝕜 n k) f) ≤
    (ContDiffMapSupportedIn.seminorm 𝕜 E F n K (i + 1)) f

Tactic:
rw [ContDiffMapSupportedIn.seminorm_le_iff 𝕜 (apply_nonneg ..)]
    intro hi x hx
    have hi' : i + 1 ≤ n := (add_le_add_left hi 1).trans hk
    simpa [hk, norm_iteratedFDeriv_fderiv] using
      norm_iteratedFDeriv_apply_le_seminorm 𝕜 hi'

Goal state after: