mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))

Tactic:
by
  open scoped ContinuousFunctionalCalculus in
  obtain (rfl | hl) := l.eq_or_neBot
  · simp
  have hf := h_tendsto.continuousOn hF.frequently
  by_cases ha : p a
  · let s : Set X := {x | ContinuousOn (F x) (spectrum R a)}
    rw [← tendsto_comap'_iff (i := ((↑) : s → X)) (by simpa)]
    conv =>
      enter [1, x]
      rw [Function.comp_apply, cfc_apply (hf := x.2)]
    rw [cfc_apply ..]
    apply cfcHom_continuous _ |>.tendsto _ |>.comp
    rw [hf.tendsto_domRestrict_iff_tendstoUniformlyOn Subtype.property]
    intro t
    simp only [eventually_comap, Subtype.forall]
    peel h_tendsto t with ht x _
    simp_all
  · simpa [cfc_apply_of_not_predicate a ha] using tendsto_const_nhds

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : IsStrictlyPositive a
⊢ IsUnit (sqrt a)

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
y : ℝ≥0
hy : y ≠ 0
ha : IsStrictlyPositive a
⊢ IsStrictlyPositive (a ^ y)

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : IsStrictlyPositive a
⊢ IsStrictlyPositive (CFC.sqrt a)

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
y : ℝ
ha : IsStrictlyPositive a
⊢ IsStrictlyPositive (a ^ y)

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ
hx : x ≠ 0
ha : IsStrictlyPositive a
y✝ : ℝ
⊢ IsStrictlyPositive a

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ
hx : x ≠ 0
ha : IsStrictlyPositive a
this : a ^ (-x) = (a ^ x) ^ (-1)
⊢ IsStrictlyPositive (a ^ x)

Tactic:
by grind

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

Goal state before:
pp: case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsSelfAdjoint a
nontriv : Subsingleton A
⊢ a ≤ (algebraMap ℝ A) ‖a‖

Tactic:
· simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : IsStrictlyPositive a
⊢ 1 / 2 ≠ 0

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : IsStrictlyPositive a
⊢ -1 ≠ 0

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : ¬IsStrictlyPositive a
ha' : ¬IsUnit (sqrt a)
⊢ ¬0 ≤ a ∨ ¬IsUnit a

Tactic:
by grind

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a ≤ b
h : IsStrictlyPositive a
a✝ : Nontrivial A
r : ℝ
hr : r > 0
hr_le : (algebraMap ℝ A) r ≤ a
⊢ 0 ∉ spectrum ℝ b

Tactic:
exact fun h0 ↦ not_le_of_gt hr <| (algebraMap_le_iff_le_spectrum <| .of_nonneg <|
    h.nonneg.trans hab).1 (hr_le.trans hab) 0 h0

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
apos : IsStrictlyPositive a
b : A
bpos : IsStrictlyPositive b
hab : a ≤ b
⊢ IsUnit b

Tactic:
by grind

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
apos : IsStrictlyPositive a
b : A
bpos : IsStrictlyPositive b
hab : a ≤ b
⊢ IsUnit a

Tactic:
by grind

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConcaveOn ℝ (inr '' s) (cfc f)
⊢ ConvexOn ℝ (inr '' s) (-cfc f)

Tactic:
exact hf.neg

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ e✝ : A
he✝ : IsStarProjection e✝
ha✝ : 0 ≤ a✝
hae✝ : a✝ ≤ e✝
a e : Unitization ℂ A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
⊢ star (1 - e) * a * (1 - e) ≤ 0

Tactic:
grw [star_left_conjugate_le_conjugate hae (1 - e), mul_assoc, he.mul_one_sub_self, mul_zero]

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

Goal state before:
pp: case inl
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f ⊥ (quasispectrum R a)
hF : ∀ᶠ (x : X) in ⊥, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in ⊥, F x 0 = 0
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) ⊥ (𝓝 (cfcₙ f a))

pp: case inr
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))

Tactic:
· simp

Goal state after:
case inr
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℂ E
⊢ ∀ {z : ℂ} {x y : E}, ⟪x, z • y⟫_ℂ = z • ⟪x, y⟫_ℂ

Tactic:
by simp [inner_smul_right, smul_eq_mul]

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

Goal state before:
pp: case pos
α : Type u_5
inst✝¹ : NormedDivisionRing α
inst✝ : NormedAlgebra ℝ α
a : α
h : a = 0
⊢ ‖1‖₊ = 1

pp: case neg
α : Type u_5
inst✝¹ : NormedDivisionRing α
inst✝ : NormedAlgebra ℝ α
a : α
h : ¬a = 0
⊢ ‖‖a‖ • a⁻¹‖₊ = 1

Tactic:
· simp

Goal state after:
case neg
α : Type u_5
inst✝¹ : NormedDivisionRing α
inst✝ : NormedAlgebra ℝ α
a : α
h : ¬a = 0
⊢ ‖‖a‖ • a⁻¹‖₊ = 1
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case neg
α : Type u_5
inst✝¹ : NormedDivisionRing α
inst✝ : NormedAlgebra ℝ α
a : α
h : ¬a = 0
⊢ ‖‖a‖ • a⁻¹‖₊ = 1

Tactic:
rw [← nnnorm_eq_zero] at h

Goal state after:
case neg
α : Type u_5
inst✝¹ : NormedDivisionRing α
inst✝ : NormedAlgebra ℝ α
a : α
h : ¬‖a‖₊ = 0
⊢ ‖‖a‖ • a⁻¹‖₊ = 1
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case inl
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
hN : ∀ (x : n → α), ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
i : m
x✝ : i ∈ Finset.univ
h✝ : IsEmpty n
⊢ ∑ j, ‖A i j‖₊ ≤ N

pp: case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
hN : ∀ (x : n → α), ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
⊢ ∑ j, ‖A i j‖₊ ≤ N

Tactic:
· simp

Goal state after:
case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
hN : ∀ (x : n → α), ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
⊢ ∑ j, ‖A i j‖₊ ≤ N
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap

Goal state before:
pp: case inl
X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → C(Y, E)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ bound x
h : IsEmpty Y
⊢ HasFiniteIntegral f μ

pp: case inr
X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → C(Y, E)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ bound x
h : Nonempty Y
⊢ HasFiniteIntegral f μ

Tactic:
· simp

Goal state after:
case inr
X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → C(Y, E)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ bound x
h : Nonempty Y
⊢ HasFiniteIntegral f μ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Projection

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
a : A
ha : p a
h : quasispectrum R a ⊆ {0, 1}
x : R
hx : x ∈ quasispectrum R a
⊢ x * x = x

Tactic:
by grind

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
a : f.PreGNS
⊢ ↑‖a‖ ^ 2 = f (star (f.ofPreGNS a) * f.ofPreGNS a)

Tactic:
have : 0 ≤ f (star (f.ofPreGNS a) * f.ofPreGNS a) := f.map_nonneg (star_mul_self_nonneg _)

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
a : f.PreGNS
this : 0 ≤ f (star (f.ofPreGNS a) * f.ofPreGNS a)
⊢ ↑‖a‖ ^ 2 = f (star (f.ofPreGNS a) * f.ofPreGNS a)
Module: Mathlib.Analysis.CStarAlgebra.Extreme

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
e : A
he : IsStarProjection e
a : A
ha : 0 ≤ a
ha1 : ‖a‖ ≤ 1
b : A
hb : 0 ≤ b
hb1 : ‖b‖ ≤ 1
x✝ : e ∈ openSegment ℝ a b
t s : ℝ
h0t : 0 < t
h0s : 0 < s
hts : t + s = 1
hlin : t • a + s • b = e
ha01 : ↑a ∈ Icc 0 1
hb01 : ↑b ∈ Icc 0 1
this : t • (↑e * (1 - ↑a) * ↑e) = 0
⊢ ↑(t • e) = ↑(e * t • a * e)

Tactic:
simpa only [mul_one_sub_mul, he.inr.isIdempotentElem.eq, smul_sub, sub_eq_zero,
        smul_mul_assoc, mul_smul_comm, smul_sub, inr_mul, inr_smul]

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

Goal state before:
pp: n : Type u_3
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n ℝ
x y : EuclideanSpace ℝ n
⊢ ⟪x, (toEuclideanCLM A) y⟫ = x.ofLp ⬝ᵥ A *ᵥ y.ofLp

Tactic:
by
  simp only [toEuclideanCLM, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, LinearEquiv.coe_coe,
    LinearEquiv.invFun_eq_symm, LinearMap.coe_toContinuousLinearMap_symm, StarAlgEquiv.trans_apply,
    LinearMap.toMatrixOrthonormal_symm_apply, LinearMap.toMatrix_symm, StarAlgEquiv.coe_mk,
    StarRingEquiv.coe_mk, RingEquiv.coe_mk, Equiv.coe_fn_mk, LinearMap.coe_toContinuousLinearMap',
    toLin_apply, mulVec_eq_sum, OrthonormalBasis.coe_toBasis_repr_apply,
    EuclideanSpace.basisFun_repr, op_smul_eq_smul, Finset.sum_apply, Pi.smul_apply, transpose_apply,
    smul_eq_mul, OrthonormalBasis.coe_toBasis, EuclideanSpace.basisFun_apply, PiLp.inner_apply,
    ofLp_sum, ofLp_smul, PiLp.ofLp_single, RCLike.inner_apply, conj_trivial, dotProduct]
  congr with i
  rw [mul_comm (x.ofLp i)]
  simp [Pi.single_apply]

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

Goal state before:
pp: x y : Circle
z : ℂ
hz : z ≠ 0
⊢ ‖(starRingEnd ℂ) z / z‖ = 1

Tactic:
rw [norm_div, RCLike.norm_conj, div_self]

Goal state after:
x y : Circle
z : ℂ
hz : z ≠ 0
⊢ ‖z‖ ≠ 0
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Opens X
hs : Nonempty ↥s
f f' : OpenPartialHomeomorph X Y
openness₁ : IsOpen[inst✝] (f.target ∩ ↑f.symm ⁻¹' ↑s)
⊢ (f.symm.trans (s.openPartialHomeomorphSubtypeCoe hs).symm).trans ((s.openPartialHomeomorphSubtypeCoe hs).trans f') ≈
    (f.symm.trans f').restr (f.target ∩ ↑f.symm ⁻¹' ↑s)

Tactic:
rw [← ofSet_trans _ openness₁, ← trans_assoc, ← trans_assoc]

Goal state after:
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Opens X
hs : Nonempty ↥s
f f' : OpenPartialHomeomorph X Y
openness₁ : IsOpen[inst✝] (f.target ∩ ↑f.symm ⁻¹' ↑s)
⊢ ((f.symm.trans (s.openPartialHomeomorphSubtypeCoe hs).symm).trans (s.openPartialHomeomorphSubtypeCoe hs)).trans f' ≈
    ((ofSet (f.target ∩ ↑f.symm ⁻¹' ↑s) openness₁).trans f.symm).trans f'
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.openPartialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.openPartialHomeomorphSubtypeCoe hV).symm (↑e.symm y) ∈ (V.openPartialHomeomorphSubtypeCoe hV).source

pp: case refine_2
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.openPartialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.openPartialHomeomorphSubtypeCoe hV) (↑(V.openPartialHomeomorphSubtypeCoe hV).symm (↑e.symm y)) =
    ↑(V.openPartialHomeomorphSubtypeCoe hV) (i (↑(U.openPartialHomeomorphSubtypeCoe hU).symm (↑e.symm y)))

Tactic:
· simp

Goal state after:
case refine_2
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.openPartialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.openPartialHomeomorphSubtypeCoe hV) (↑(V.openPartialHomeomorphSubtypeCoe hV).symm (↑e.symm y)) =
    ↑(V.openPartialHomeomorphSubtypeCoe hV) (i (↑(U.openPartialHomeomorphSubtypeCoe hU).symm (↑e.symm y)))
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i✝ : ι
b : B
a : F
i : ι
⊢ ContinuousOn (↑(Z.localTrivAsPartialEquiv i)) (Z.localTrivAsPartialEquiv i).source

Tactic:
continuousOn_open_iff (Z.open_source' i)

Goal state after:
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i✝ : ι
b : B
a : F
i : ι
⊢ ∀ (t : Set (B × F)),
    IsOpen[instTopologicalSpaceProd] t →
      IsOpen ((Z.localTrivAsPartialEquiv i).source ∩ ↑(Z.localTrivAsPartialEquiv i) ⁻¹' t)
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj✝ : Z → B
e✝ : Pretrivialization F proj✝
x✝ : Z
e' : Pretrivialization F TotalSpace.proj
b : B
y : E b
s : Set B
hs : IsOpen[inst✝²] s
proj : Z → ↑s
e : Pretrivialization F proj
inst✝ : Nonempty (B → F → Z)
x : { x // x ∈ s } × F
hx : x ∈ e.target
⊢ (if h : (Prod.map Subtype.val id x).1 ∈ s then
      e.invFun (⟨(Prod.map Subtype.val id x).1, h⟩, (Prod.map Subtype.val id x).2)
    else Classical.arbitrary (B → F → Z) (Prod.map Subtype.val id x).1 (Prod.map Subtype.val id x).2) ∈
    e.source

Tactic:
simpa using e.map_target hx

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a b : 𝓜(𝕜, A)
x y : A
⊢ (a.toProd.2 + b.toProd.2) x * y = x * (a.toProd.1 + b.toProd.1) y

Tactic:
simp only [add_apply, mul_add, add_mul, central]

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
⊢ ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C

Tactic:
intro f C h

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
⊢ ‖f‖₊ ≤ C
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝ : TopologicalSpace I
hs : IsOpen[inst✝¹] s
hfs : IsOpen[inst✝²] (f ⁻¹' s)
x : X
hx : x ∈ s
h : IsEvenlyCovered (fun e ↦ f ↑e) x I
inst : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
hfU : IsOpen[instTopologicalSpaceSubtype] ((fun e ↦ f ↑e) ⁻¹' U)
H : ↑((fun e ↦ f ↑e) ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑((fun e ↦ f ↑e) ⁻¹' U)), ↑(H x).1 = (fun e ↦ f ↑e) ↑x
x✝ : Nonempty I
y : X
hy : y ∈ U
⊢ y ∈ s

Tactic:
convert! Set.mem_preimage.mp (H.symm (⟨y, hy⟩, Classical.arbitrary I)).1.2; simp [← hH]

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: case inr
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
s : Set X
F : ↑s → Type u_3
inst✝ : (x : ↑s) → TopologicalSpace (F x)
hF : ∀ (x : ↑s), DiscreteTopology (F x)
e : (x : ↑s) → Trivialization (F x) f
h : ∀ (x : ↑s), ↑x ∈ (e x).baseSet
h✝ : Nonempty E
x : ↑s
hx : ↑x ∉ Set.range f
⊢ ↑x ∈ Set.range f

Tactic:
exact ⟨(e x).invFun (x, (e x <| Classical.arbitrary E).2), (e x).proj_symm_apply' (h x)⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic

Goal state before:
pp: A : Type u_1
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
n : ℕ
a : A
ha₂ : ∀ x ∈ spectrum ℝ a, x ≠ 0
ha₁ : IsSelfAdjoint a
ha₂' : ContinuousOn Real.log (spectrum ℝ a)
ha₂'' : ContinuousOn Real.log ((fun x ↦ x ^ n) '' spectrum ℝ a)
⊢ cfc (fun x ↦ Real.log (x ^ n)) a = n • cfc Real.log a

Tactic:
Real.log_pow,

Goal state after:
A : Type u_1
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
n : ℕ
a : A
ha₂ : ∀ x ∈ spectrum ℝ a, x ≠ 0
ha₁ : IsSelfAdjoint a
ha₂' : ContinuousOn Real.log (spectrum ℝ a)
ha₂'' : ContinuousOn Real.log ((fun x ↦ x ^ n) '' spectrum ℝ a)
⊢ cfc (fun x ↦ ↑n * Real.log x) a = n • cfc Real.log a
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: z : Circle
s : ℝ
n : ℕ
hs : s ≤ π
h1 : z ∈ centeredArc (π / ↑n)
h2 : z ^ n ∈ centeredArc s
hs0 : 0 < s
hn0 : n ≠ 0
hn : 1 ≤ ↑n
⊢ ↑n * |(↑z).arg| < s

Tactic:
mem_centeredArc (div_le_self pi_nonneg hn)

Goal state after:
z : Circle
s : ℝ
n : ℕ
hs : s ≤ π
h1 : |(↑z).arg| < π / ↑n
h2 : z ^ n ∈ centeredArc s
hs0 : 0 < s
hn0 : n ≠ 0
hn : 1 ≤ ↑n
⊢ ↑n * |(↑z).arg| < s