mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: case e_a
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
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
⊢ ∑ a ∈ f.weights.support, f.weights a • ((Finset.affineCombination R a.weights.support id) ⇑a.weights -ᵥ b) =
    ∑ x ∈ (∑ x ∈ f.weights.support, f.weights x • x.weights).support,
      (∑ x ∈ f.weights.support, f.weights x • x.weights) x • (x -ᵥ b)

Tactic:
conv_lhs =>
    congr; · skip
    ext d
    rw [d.weights.support.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one
        _ _ d.total b, vadd_vsub, Finset.weightedVSubOfPoint_apply]
    simp only [id]

Goal state after:
case e_a
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
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
⊢ ∑ d ∈ f.weights.support, f.weights d • ∑ x ∈ d.weights.support, d.weights x • (x -ᵥ b) =
    ∑ x ∈ (∑ x ∈ f.weights.support, f.weights x • x.weights).support,
      (∑ x ∈ f.weights.support, f.weights x • x.weights) x • (x -ᵥ b)
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u_1
M : Type u_3
I : Type u_6
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
J : Type u_9
s : StdSimplex R I
f : I → StdSimplex R J
g : I → J → M
⊢ sConvexComb (map (fun i ↦ map (g i) (f i)) s) =
    map (Function.uncurry g) (sConvexComb (map (fun i ↦ map (fun x ↦ (i, x)) (f i)) s))

Tactic:
simp [map_sConvexComb, map_map, Function.uncurry]

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

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x : M
⊢ convexCombPair s t hs ht h x x = x

Tactic:
by
  unfold convexCombPair
  convert sConvexComb_single x
  simp only [StdSimplex.duple, StdSimplex.single, ← single_add, h]

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : s.sum w = 0
h_vsum : ∑ e ∈ s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := {i ∈ s | 0 ≤ w i}
J : Finset ι := {i ∈ s | w i < 0}
p : E := I.centerMass w f
⊢ ∑ j ∈ J, w j + ∑ i ∈ I, w i = 0

Tactic:
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w

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

Goal state before:
pp: case a
𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : T2Space E
F : Set (Set E)
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_compact : ∀ X ∈ F, IsCompact X
h_inter : ∀ (G : Finset (Set E)), ↑G ⊆ F → #G ≤ finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
G : Finset ↑F
a✝ : #G ≤ finrank 𝕜 E + 1
G' : Finset (Set E) := ⋯
⊢ ↑G' ⊆ F

pp: case a
𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : T2Space E
F : Set (Set E)
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_compact : ∀ X ∈ F, IsCompact X
h_inter : ∀ (G : Finset (Set E)), ↑G ⊆ F → #G ≤ finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
G : Finset ↑F
a✝ : #G ≤ finrank 𝕜 E + 1
G' : Finset (Set E) := ⋯
⊢ #G' ≤ finrank 𝕜 E + 1

Tactic:
· simp [G']

Goal state after:
case a
𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : T2Space E
F : Set (Set E)
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_compact : ∀ X ∈ F, IsCompact X
h_inter : ∀ (G : Finset (Set E)), ↑G ⊆ F → #G ≤ finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
G : Finset ↑F
a✝ : #G ≤ finrank 𝕜 E + 1
G' : Finset (Set E) := ⋯
⊢ #G' ≤ finrank 𝕜 E + 1
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: ι : Sort u_1
V : Type u
G H : SimpleGraph V
a b c u v w : V
e : Sym2 V
s : Set (Sym2 V)
inst✝¹ : DecidableEq V
inst✝ : Fintype ↑s
⊢ Fintype ↑(fromEdgeSet s).edgeSet

Tactic:
rw [edgeSet_fromEdgeSet s]
  infer_instance

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
⊢ G = ⊥ ↔ ∀ (v : V), G.IsIsolated v

Tactic:
simp [eq_bot_iff_forall_not_adj, ← neighborSet_eq_empty, Set.eq_empty_iff_forall_notMem]

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 : 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 p₂ : P
h : p₂ ∈ s
⊢ (y ∉ s ∧ x ∉ s ∧ ∃ p₂_1 ∈ s, SameRay R (y -ᵥ p₂) (x -ᵥ p₂_1)) ↔
    x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

Tactic:
← and_assoc,

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 y p₂ : P
h : p₂ ∈ s
⊢ ((y ∉ s ∧ x ∉ s) ∧ ∃ p₂_1 ∈ s, SameRay R (y -ᵥ p₂) (x -ᵥ p₂_1)) ↔
    x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case neg.inr
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)
h0 : ¬w₁ i = 0
h0' : ¬w₂ i = 0
h : SignType.sign (w₁ i) = -SignType.sign (w₂ i)
h' : 0 < w₁ i
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)

Tactic:
rw [sign_pos h', eq_comm, neg_eq_iff_eq_neg, sign_eq_neg_one_iff] at h

Goal state after:
case neg.inr
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)
h0 : ¬w₁ i = 0
h0' : ¬w₂ i = 0
h : w₂ i < 0
h' : 0 < w₁ i
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case succ
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
h'f : HasTemperateGrowth (fderiv ℝ f)
hf : Differentiable ℝ f
k : ℕ
C : ℝ
h : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k
m : ℕ
⊢ ∃ k C, ∀ (x : E), ‖iteratedFDeriv ℝ (m + 1) f x‖ ≤ C * (1 + ‖x‖) ^ k

Tactic:
· rcases h'f.2 m with ⟨k', C', h'⟩
    refine ⟨k', C', ?_⟩
    simpa [iteratedFDeriv_succ_eq_comp_right] using h'

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

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : E → F
hf : ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
hg : ContDiff ℝ ∞ g ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
n k₁ : ℕ
h₁ : iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₁
⊢ ∃ k, iteratedFDeriv ℝ n (f + g) =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
obtain ⟨k₂, h₂⟩ := hg.2 n

Goal state after:
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : E → F
hf : ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
hg : ContDiff ℝ ∞ g ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
n k₁ : ℕ
h₁ : iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₁
k₂ : ℕ
h₂ : iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₂
⊢ ∃ k, iteratedFDeriv ℝ n (f + g) =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: case e'_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ∞
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ∞
⊢ ∫ (t : ℝ) in Ioi 0, μ.real {a | t < f a} = (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal

Tactic:
have aux := @integral_eq_lintegral_of_nonneg_ae _ _ ((volume : Measure ℝ).restrict (Set.Ioi 0))
      (fun t ↦ μ.real {a : α | t < f a}) ?_ ?_
    · rw [aux]
      congr 1
      apply setLIntegral_congr_fun measurableSet_Ioi
      exact fun t t_pos ↦ ENNReal.ofReal_toReal (rhs_integrand_finite t t_pos).ne
    · exact Eventually.of_forall (fun x ↦ by positivity)
    · apply Measurable.aestronglyMeasurable
      refine Measurable.ennreal_toReal ?_
      exact Antitone.measurable (fun _ _ hst ↦ measure_mono (fun _ h ↦ lt_of_le_of_lt hst h))

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
f_bdd : f ≤ᵐ[μ] fun x ↦ M
t : ℝ
ht : t ∈ Ioi 0 \ Ioc 0 M
htM : M < t
⊢ μ {a | M < f a} = 0

Tactic:
measure_eq_zero_iff_ae_notMem

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
f_bdd : f ≤ᵐ[μ] fun x ↦ M
t : ℝ
ht : t ∈ Ioi 0 \ Ioc 0 M
htM : M < t
⊢ ∀ᵐ (a : α) ∂μ, a ∉ {a | M < f a}
Module: Mathlib.Analysis.Normed.Group.ZeroAtInfty

Goal state before:
pp: case h
E : Type u_1
F : Type u_2
𝓕 : Type u_3
inst✝³ : SeminormedAddGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : FunLike 𝓕 E F
inst✝ : ZeroAtInftyContinuousMapClass 𝓕 E F
f : 𝓕
ε : ℝ
hε : 0 < ε
h : (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε ∈ cocompact E
r : ℝ
hr : (Metric.closedBall 0 r)ᶜ ⊆ (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε
x : E
⊢ r < ‖x‖ → ‖f x‖ < ε

Tactic:
hr'

Goal state after:
case h
E : Type u_1
F : Type u_2
𝓕 : Type u_3
inst✝³ : SeminormedAddGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : FunLike 𝓕 E F
inst✝ : ZeroAtInftyContinuousMapClass 𝓕 E F
f : 𝓕
ε : ℝ
hε : 0 < ε
h : (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε ∈ cocompact E
r : ℝ
hr : (Metric.closedBall 0 r)ᶜ ⊆ (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε
x : E
hr' : r < ‖x‖
⊢ ‖f x‖ < ε
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)
hy_lower : ∃ d > 0, ∀ᶠ (n : ℕ) in atTop, d ≤ ‖y_ n‖
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 1 ⊆ K
y : X
hyK : y ∈ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hψy : Tendsto (y_ ∘ ψ) atTop (𝓝 y)
hy_lim : Tendsto (fun n ↦ (T - μ • 1) (y_ n)) atTop (𝓝 0)
⊢ HasEigenvalue (↑T) μ

Tactic:
have hy_eigen' : (T - μ • 1) y = 0 := by
    apply tendsto_nhds_unique _ (hy_lim.comp hψ.tendsto_atTop)
    have : Continuous (T - μ • 1 : X →L[𝕜] X) := by fun_prop
    exact this.continuousAt.tendsto.comp hψy

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)
hy_lower : ∃ d > 0, ∀ᶠ (n : ℕ) in atTop, d ≤ ‖y_ n‖
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 1 ⊆ K
y : X
hyK : y ∈ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hψy : Tendsto (y_ ∘ ψ) atTop (𝓝 y)
hy_lim : Tendsto (fun n ↦ (T - μ • 1) (y_ n)) atTop (𝓝 0)
hy_eigen' : (T - μ • 1) y = 0
⊢ HasEigenvalue (↑T) μ
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
c : 𝕜
hc : 1 < ‖c‖
f : ℕ → X
hf_norm_lower : ∀ (n : ℕ), 1 ≤ ‖f n‖
hf_norm_upper : ∀ (n : ℕ), ‖f n‖ ≤ ‖c‖ + 1
hf_far : ∀ (n : ℕ), ∀ y ∈ LinearMap.range (↑S ^ (n + 1)), 1 ≤ ‖f n - y‖
hf_mem : ∀ {n m : ℕ}, m ≤ n → f n ∈ (↑S ^ m).range
hf_mem' : ∀ {n m : ℕ}, m ≤ n → S (f n) ∈ (↑S ^ (m + 1)).range
⊢ Pairwise fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖

Tactic:
have : Std.Symm fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖ := by grind [symm_def, norm_sub_rev]

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‖
f : ℕ → X
hf_norm_lower : ∀ (n : ℕ), 1 ≤ ‖f n‖
hf_norm_upper : ∀ (n : ℕ), ‖f n‖ ≤ ‖c‖ + 1
hf_far : ∀ (n : ℕ), ∀ y ∈ LinearMap.range (↑S ^ (n + 1)), 1 ≤ ‖f n - y‖
hf_mem : ∀ {n m : ℕ}, m ≤ n → f n ∈ (↑S ^ m).range
hf_mem' : ∀ {n m : ℕ}, m ≤ n → S (f n) ∈ (↑S ^ (m + 1)).range
this : Std.Symm fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖
⊢ Pairwise fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖
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
⊢ HasEigenvalue (↑T) μ ∨ μ ∈ resolventSet 𝕜 T

Tactic:
by
  -- Suppose not, then `μ` is not an eigenvalue and is in the spectrum.
  by_contra!
  obtain ⟨h₁, h₂⟩ := this
  -- Defining S := `T - μ • 1`, we deduce that S is antilipschitz and not surjective.
  let S := T - μ • 1
  obtain ⟨K, hK : AntilipschitzWith K S⟩ := antilipschitz_of_not_hasEigenvalue hT hμ h₁
  replace h₂ : ¬ (S : X → X).Bijective := by
    rw [spectrum.mem_resolventSet_iff, ← IsUnit.neg_iff,
      ContinuousLinearMap.isUnit_iff_bijective] at h₂
    convert! h₂
    ext x
    simp [S]
  replace h₂ : ¬ (S : X → X).Surjective := by grind [Function.Bijective, hK.injective]
  -- Take a sequence of vectors `f n` in the range of `S ^ n` such that `‖f n‖` is in the
  -- interval `[1, ‖c‖ + 1]` and such that `f n` is at least `1` away from any vector in the range
  -- of `S ^ (n + 1)`.
  obtain ⟨c, hc⟩ := NormedField.exists_one_lt_norm 𝕜
  obtain ⟨f, hf_norm_lower, hf_norm_upper, hf_mem, hf_far⟩ := exists_seq h₂
    (hK.isClosedEmbedding S.uniformContinuous) hc (R := ‖c‖ + 1) (by simp)
  replace hf_mem {n m : ℕ} (h : m ≤ n) : f n ∈ ((S : End 𝕜 X) ^ m).range :=
    (S : End 𝕜 X).iterateRange.monotone (by lia) (hf_mem _)
  have hf_mem' {n m : ℕ} (h : m ≤ n) : S (f n) ∈ ((S : End 𝕜 X) ^ (m + 1)).range := by
    rw [iterate_succ', LinearMap.range_comp]
    exact ⟨f n, hf_mem h, rfl⟩
  -- Then the points `T (f n)` are bounded away from each other, using the separation property
  -- of the `f n` and the lower bound on their norms.
  have hp : Pairwise fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖ := by
    have : Std.Symm fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖ := by grind [symm_def, norm_sub_rev]
    apply Pairwise.of_lt
    intro m n hmn
    let u : X := μ⁻¹ • (S (f n) - S (f m) + μ • f n)
    have hu : μ • (f m - u) = T (f m) - T (f n) := by
      rw [smul_sub, smul_inv_smul₀ hμ]
      simp [S]
      linear_combination (norm := module)
    have : u ∈ ((S : End 𝕜 X) ^ (m + 1)).range := by
      apply Submodule.smul_mem _ _ (Submodule.add_mem _ _ _)
      · exact Submodule.sub_mem _ (hf_mem' hmn.le) (hf_mem' le_rfl)
      · exact Submodule.smul_mem _ μ (hf_mem hmn)
    grw [← hu, norm_smul, mul_comm, ← hf_far _ u this, one_mul]
  -- However the `f n` are contained in a compact set, so their image under the compact operator `T`
  -- must contain a Cauchy subsequence, which is a contradiction.
  obtain ⟨K, hK, hK'⟩ := hT.image_closedBall_subset_compact (‖c‖ + 1)
  obtain ⟨y, hyK, ψ, hψ, hψy⟩ := hK.tendsto_subseq (fun n ↦ hK' ⟨f n, by simp [*], rfl⟩)
  replace hψy := hψy.cauchySeq
  rw [Metric.cauchySeq_iff'] at hψy
  obtain ⟨N, hN⟩ := hψy ‖μ‖ (by positivity)
  have : ‖T (f (ψ (N + 1))) - T (f (ψ N))‖ < ‖μ‖ := by simpa [dist_eq_norm_sub] using hN (N + 1)
  refine this.not_ge (hp ?_)
  simp [hψ.injective.eq_iff]

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
μ ν : 𝕜
hμν : μ ≠ ν
v : E
hv✝ : v ∈ eigenspace T μ
hv : T v = μ • v
w : E
hw✝ : w ∈ eigenspace T ν
hw : T w = ν • w
hv' : ¬v = 0
H : (starRingEnd 𝕜) μ = μ
⊢ ν = μ ∨ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫ = 0

Tactic:
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm

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
A : Matrix n n 𝕜
hA : A.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ A *ᵥ x = 0 ↔ A *ᵥ x = 0

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ h ▸ dotProduct_zero _⟩

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : A.PosSemidef
x : n → 𝕜
h : star x ⬝ᵥ A *ᵥ x = 0
⊢ A *ᵥ x = 0
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
f : 𝓓^{n}_{K}(E, F)
x : E
⊢ ‖f x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K 0) f

Tactic:
by
  rw [← norm_iteratedFDeriv_zero (𝕜 := ℝ) (f := f) (x := x)]
  exact norm_iteratedFDeriv_apply_le_seminorm 𝕜 zero_le

Goal state after:
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✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace ℝ F
inst✝¹⁶ : NormedSpace 𝕜 F
inst✝¹⁵ : SMulCommClass ℝ 𝕜 F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
f g : 𝓓^{n}_{K}(E, F₁)
hφ : IntegrableOn φ (↑K) μ
⊢ ∫ (x : E), (B ((f + g) x)) (φ x) ∂μ = ∫ (x : E), (B (f x)) (φ x) ∂μ + ∫ (x : E), (B (g x)) (φ x) ∂μ

pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace ℝ F
inst✝¹⁶ : NormedSpace 𝕜 F
inst✝¹⁵ : SMulCommClass ℝ 𝕜 F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
f g : 𝓓^{n}_{K}(E, F₁)
hφ : ¬IntegrableOn φ (↑K) μ
⊢ 0 = 0 + 0

Tactic:
· simp_rw [add_apply, map_add, add_apply,
        integral_add (f.integrable_bilin B hφ) (g.integrable_bilin B hφ)]

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace ℝ F
inst✝¹⁶ : NormedSpace 𝕜 F
inst✝¹⁵ : SMulCommClass ℝ 𝕜 F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
f g : 𝓓^{n}_{K}(E, F₁)
hφ : ¬IntegrableOn φ (↑K) μ
⊢ 0 = 0 + 0