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

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
a b : ℝ
h : BoundedVariationOn f (uIcc a b)
h₁ : ∀ᵐ (x : ℝ), x ≠ min a b
h₂ : ∀ᵐ (x : ℝ), x ≠ max a b
x : ℝ
hx₁ : x ∈ uIcc a b → DifferentiableWithinAt ℝ f (uIcc a b) x
⊢ x ≠ min a b → x ≠ max a b → x ∈ uIcc a b → DifferentiableAt ℝ f x

Tactic:
hx₂

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
a b : ℝ
h : BoundedVariationOn f (uIcc a b)
h₁ : ∀ᵐ (x : ℝ), x ≠ min a b
h₂ : ∀ᵐ (x : ℝ), x ≠ max a b
x : ℝ
hx₁ : x ∈ uIcc a b → DifferentiableWithinAt ℝ f (uIcc a b) x
hx₂ : x ≠ min a b
⊢ x ≠ max a b → x ∈ uIcc a b → DifferentiableAt ℝ f x
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E✝ : Type u_2
inst✝ : PseudoEMetricSpace E✝
f : α → E✝
s : Set α
E : ℕ → α
hE : Monotone E
n : ℕ
hn : ∀ (i : ℕ), 0 < i → i < n → E i ∈ s
⊢ ∑ i ∈ Finset.range n, eVariationOn f (s ∩ Icc (E i) (E (i + 1))) = eVariationOn f (s ∩ Icc (E 0) (E n))

Tactic:
induction n with

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case succ.ht
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
u : ℕ → α
hu : Monotone u
n : ℕ
ih : eVariationOn f (u '' Iic n) = ∑ i ∈ Finset.range n, edist (f (u i)) (f (u (i + 1)))
⊢ IsLeast {u n, u (n + 1)} (u n)

Tactic:
· simp [IsLeast, hu n.le_succ]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
u : ℕ → α
hu : Monotone u
n : ℕ
⊢ eVariationOn f (u '' Iic n) = ∑ i ∈ Finset.range n, edist (f (u i)) (f (u (i + 1)))

Tactic:
induction n with

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
h : ¬p a ∨ ¬ContinuousOn f (spectrum R a)
⊢ P (cfc f a)

Tactic:
obtain (h | h) := h

Goal state after:
case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
h : ¬p a
⊢ P (cfc f a)

case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
h : ¬ContinuousOn f (spectrum R a)
⊢ P (cfc f a)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ < ε
H : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ Ici x)
y : α → α
y_mem : ∀ y_1 ∈ s, y y_1 ∈ s ∩ Ici y_1
le_y : ∀ y_1 ∈ s, δ ≤ eVariationOn f (s ∩ Icc y_1 (y y_1))
v : ℕ → α := fun n ↦ y^[n] x₀
n : ℕ
⊢ v n ∈ s

Tactic:
induction n with

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a)
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
⊢ ContinuousOn (fun z ↦ ∑ i ∈ s, f i z) (spectrum R a)

Tactic:
exact continuousOn_finsetSum s fun i hi => hf i hi

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inl
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
inst✝ : CompleteSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
h : L = ⊥
⊢ ∃ l, Tendsto f L (𝓝 l)

Tactic:
simp only [h, tendsto_bot, exists_const_iff, and_true]
    exact ⟨f x₀⟩

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ ⨍ (y : α) in a, f y ∂μ) (v.filterAt x) (𝓝 (f x))

Tactic:
filter_upwards [v.ae_tendsto_average_norm_sub hf, v.ae_eventually_measure_pos] with x hx h'x
  rw [tendsto_iff_norm_sub_tendsto_zero]
  refine squeeze_zero' (Eventually.of_forall fun a => norm_nonneg _) ?_ hx
  filter_upwards [h'x, v.eventually_measure_lt_top x, v.eventually_filterAt_integrableOn x hf]
    with a ha h'a h''a
  nth_rw 1 [← setAverage_const ha.ne' h'a.ne (f x)]
  simp_rw [setAverage_eq']
  rw [← integral_sub]
  · exact norm_integral_le_integral_norm _
  · exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 h''a
  · exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 (integrableOn_const h'a.ne)

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
ι : Type u_3
f : ι → R → R
a : A
ha : p a
hf : ∀ (i : ι), ContinuousOn (f i) (σₙ R a)
hf0 : ∀ (i : ι), f i 0 = 0
⊢ (fun i ↦ cfcₙ (f i) a) = fun i ↦
    (cfcₙHom ha) { toFun := (σₙ R a).domRestrict (f i), continuous_toFun := ⋯, map_zero' := ⋯ }

Tactic:
by
  ext i
  simp only [cfcₙ_apply (f i) a (hf i) (hf0 i)]

Goal state after:
Module: Mathlib.Topology.Algebra.StarSubalgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : StarRing R
inst✝⁴ : TopologicalSpace A
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : StarRing A
inst✝ : StarModule R A
S₁ S₂ : StarSubalgebra R A
h : S₁ ≤ S₂
hS₁ : IsClosed[inst✝⁴] ↑S₁
⊢ Subtype.val ⁻¹' ↑S₁ = Set.range ⇑(inclusion h)

Tactic:
convert! (Set.range_subtype_map id _).symm
          · rw [Set.image_id]; rfl
          · intro _ h'
            apply h h'

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommSemiring R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ IsStrictlyPositive (cfc f a) ↔ ∀ x ∈ spectrum R a, 0 < f x

Tactic:
by
  rw [cfc_apply .., cfcHom_isStrictlyPositive_iff]
  simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
⊢ ContinuousOn (fun z ↦ ∑ i ∈ s, f i z) (σₙ R a)

Tactic:
exact continuousOn_finsetSum s fun i hi => hf i hi

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ σₙ R a, 0 ≤ f x
hf : ¬f 0 = 0
⊢ 0 ≤ cfcₙ f a

Tactic:
simp only [cfcₙ_apply_of_not_map_zero _ hf, le_rfl]

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

Goal state before:
pp: case e'_2
n : ℕ
hn : n ≠ 0
x : ↑I
⊢ ∑ k, (↑x - ↑(k/ₙ)) ^ 2 * (bernstein n ↑k) x =
    (Polynomial.aeval ↑x) (∑ ν ∈ Finset.range (n + 1), (n • Polynomial.X - ↑ν) ^ 2 * bernsteinPolynomial ℝ n ν) / ↑n ^ 2

pp: case e'_3
n : ℕ
hn : n ≠ 0
x : ↑I
⊢ ↑x * (1 - ↑x) / ↑n = (Polynomial.aeval ↑x) (n • Polynomial.X * (1 - Polynomial.X)) / ↑n ^ 2

Tactic:
· simp only [z, bernstein_apply, nsmul_eq_mul, bernsteinPolynomial, Finset.sum_range, map_sum,
      Polynomial.coe_aeval_eq_eval, Polynomial.eval_mul, Polynomial.eval_pow, Polynomial.eval_sub,
      Polynomial.eval_natCast, Polynomial.eval_X, Polynomial.eval_one]
    field_simp
    rw [← Finset.sum_div]
    field

Goal state after:
case e'_3
n : ℕ
hn : n ≠ 0
x : ↑I
⊢ ↑x * (1 - ↑x) / ↑n = (Polynomial.aeval ↑x) (n • Polynomial.X * (1 - Polynomial.X)) / ↑n ^ 2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬f 0 = 0
⊢ cfcₙ f a ≤ 0

Tactic:
simp only [cfcₙ_apply_of_not_map_zero _ hf, le_rfl]

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E
this : ∀ U ∈ 𝓝 0, Convex ℝ U → ∀ᶠ (n : ℕ) in atTop, ∀ (x : ↑I), gauge U ((bernsteinApproximation n f) x - f x) < 1
⊢ Tendsto (fun n ↦ bernsteinApproximation n f) atTop (𝓝 f)

Tactic:
by
    rw [(LocallyConvexSpace.convex_basis_zero ℝ E).uniformity_of_nhds_zero_swapped
      |>.compactConvergenceUniformity_of_compact |> nhds_basis_uniformity |>.tendsto_right_iff]
    rintro U ⟨hU₀, hcU⟩
    filter_upwards [this U hU₀ hcU] with n hn x
    exact setOfPred_gauge_lt_one_subset_self hcU (mem_of_mem_nhds hU₀) (absorbent_nhds_zero hU₀)
      (hn x)

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
⊢ ‖((splitMul 𝕜 E) x⋆).2‖ * ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x).2‖ ^ 2

Tactic:
by
        rw [sq]
        gcongr
        simpa only [star_star] using h₁ (star x)

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
⊢ φ (cfc f a) = cfc f (φ a)

Tactic:
let ψ : A →⋆ₐ[R] B := (φ : A →⋆ₐ[S] B).restrictScalars R

Goal state after:
F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := StarAlgHom.restrictScalars R ↑φ
⊢ φ (cfc f a) = cfc f (φ a)
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
s : Set α
x : α
γ : Type u_5
inst✝ : TopologicalSpace γ
i : γ → β
hf : UpperHemicontinuousWithinAt f s x
hi : IsInducing i
h_cl : IsClosed[inst✝¹] (range i)
v : Set β
hv : IsOpen[inst✝¹] v
hu : IsOpen[inst✝] (i ⁻¹' v)
hifu : i ⁻¹' f x ⊆ i ⁻¹' v
⊢ ∀ᶠ (x' : α) in 𝓝[s] x, i ⁻¹' f x' ⊆ i ⁻¹' v

Tactic:
simp_rw [← preimage_inter_range (s := f _), preimage_subset_preimage_iff inter_subset_right]
    at hifu ⊢

Goal state after:
α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
s : Set α
x : α
γ : Type u_5
inst✝ : TopologicalSpace γ
i : γ → β
hf : UpperHemicontinuousWithinAt f s x
hi : IsInducing i
h_cl : IsClosed[inst✝¹] (range i)
v : Set β
hv : IsOpen[inst✝¹] v
hu : IsOpen[inst✝] (i ⁻¹' v)
hifu : f x ∩ range i ⊆ v
⊢ ∀ᶠ (x' : α) in 𝓝[s] x, f x' ∩ range i ⊆ v
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : NormOneClass A
a : A
⊢ spectralRadius 𝕜 a ≤ ↑‖a‖₊

Tactic:
by
  refine iSup₂_le fun k hk => ?_
  exact mod_cast norm_le_norm_of_mem hk

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure[TopologicalSpace.vietoris α] {t | t.Finite ∧ t ⊆ s} ⊆ 𝒫 closure[inst✝] s

pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
s K : Set α
hKs : K ∈ 𝒫 closure[inst✝] s
⊢ K ∈ closure[TopologicalSpace.vietoris α] {t | t.Finite ∧ t ⊆ s}

Tactic:
· rw [isClosed_closure.powerset_vietoris.closure_subset_iff]
    exact fun K ⟨_, h⟩ => h.trans subset_closure

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
s K : Set α
hKs : K ∈ 𝒫 closure[inst✝] s
⊢ K ∈ closure[TopologicalSpace.vietoris α] {t | t.Finite ∧ t ⊆ s}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht : TotallyBounded t
⊢ TotallyBounded (𝒫 t)

Tactic:
simp_rw [(𝓤 α).basis_sets.uniformity_hausdorff.totallyBounded_iff, Function.comp_id,
    Set.powerset, Set.ofPred_subset, Set.mem_iUnion]

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht : TotallyBounded t
⊢ ∀ i ∈ 𝓤 α, ∃ t_1, t_1.Finite ∧ ∀ x ⊆ t, ∃ i_1, ∃ (_ : i_1 ∈ t_1), x ∈ {x | (x, i_1) ∈ hausdorffEntourage i}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
hst : s ⊆ t
hts : t ⊆ closure[inst✝] s
⊢ s ⤳ t

Tactic:
aesop (add simp specializes_iff)

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
h₁ : Function.Injective ⇑(codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
⊢ Function.Injective ⇑ψ

Tactic:
simpa [ψ] using (inrRangeEquiv 𝕜 A).symm.injective.comp h₁

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
⊢ SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t

Tactic:
by
  have : spectrum ℝ a ⊆ Set.Icc (-t) t := by
    intro x hx
    rw [Set.mem_Icc, ← abs_le, ← Real.norm_eq_abs, ← coe_nnnorm, NNReal.coe_le_coe,
      ← ENNReal.coe_le_coe]
    exact le_iSup₂ (α := ℝ≥0∞) x hx |>.trans ht
  rw [nnreal_iff]
  refine ⟨fun h ↦ iSup₂_le fun x hx ↦ ?_, fun h ↦ ?_⟩
  · rw [← spectrum.singleton_sub_eq] at hx
    obtain ⟨y, hy, rfl⟩ : ∃ y ∈ spectrum ℝ a, ↑t - y = x := by simpa using hx
    obtain ⟨hty, hyt⟩ := Set.mem_Icc.mp <| this hy
    lift y to ℝ≥0 using h y hy
    rw [← NNReal.coe_sub (by exact_mod_cast hyt)]
    simp
  · replace h : ∀ x ∈ spectrum ℝ a, ‖t - x‖₊ ≤ t := by
      simpa [spectralRadius, iSup₂_le_iff, ← spectrum.singleton_sub_eq] using h
    peel h with x hx h_le
    rw [← NNReal.coe_le_coe, coe_nnnorm, Real.norm_eq_abs, abs_le] at h_le
    linarith [h_le.2]

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

Goal state before:
pp: s t : Set ℝ
⊢ ⇑equivRealProdLm ⁻¹' (convexHull ℝ) (s ×ˢ t) = (convexHull ℝ) s ×ℂ (convexHull ℝ) t

Tactic:
by rw [convexHull_prod]; rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: case e_u
A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
⊢ (fun i ↦ 1 / ↑‖a ^ i‖.toNNReal ^ (1 / ↑i)) = fun n ↦ 1 / ↑(‖a ^ n‖.toNNReal ^ (1 / ↑n))

Tactic:
ext n

Goal state after:
case e_u
A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
n : ℕ
⊢ 1 / ↑‖a ^ n‖.toNNReal ^ (1 / ↑n) = 1 / ↑(‖a ^ n‖.toNNReal ^ (1 / ↑n))
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ σ ℂ a
this : NormedAlgebra ℚ A
hu : NormedSpace.exp (I • a) ∈ unitary A
⊢ z = ↑z.re

Tactic:
let Iu := Units.mk0 I I_ne_zero

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ σ ℂ a
this : NormedAlgebra ℚ A
hu : NormedSpace.exp (I • a) ∈ unitary A
Iu : ℂˣ := Units.mk0 I I_ne_zero
⊢ z = ↑z.re
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ ↑(a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsUnit a

Tactic:
← spec_eq,

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ (a⋆ * a)
ha₂ : 0 ∉ σ ℂ (a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsUnit a

A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ (a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a * a⋆)

A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ ↑(a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a⋆ * a)
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
⊢ x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x

Tactic:
let _ := invertibleExp (z • star a)

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
x✝¹ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
x✝ : Invertible (NormedSpace.exp (z • star a)) := invertibleExp (z • star a)
⊢ x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case refine_1
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : IsStarNormal a
x : ℝ
⊢ NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A

Tactic:
convert! (selfAdjoint.expUnitary (x • (2 : ℝ) • ℑ a)).2
    have hcomm := star_comm_self (x := a) |>.symm.smul_left x |>.smul_right (-x)
    rw [← exp_add_of_commute hcomm]
    simp [imaginaryPart_apply_coe, smul_comm (2 : ℝ) I, smul_comm x I, smul_smul I I, smul_add x,
      sub_eq_add_neg]

Goal state after: