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

Goal state before:
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
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))

Tactic:
by_cases ha : p a

Goal state after:
case pos
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
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))

case neg
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
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : ¬p a
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: 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 : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
⊢ LipschitzOnWith 1 (fun f ↦ cfcₙ ((toFun {quasispectrum R a}) f) a)
    {f | ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a) ∧ f 0 = 0}

Tactic:
by_cases ha : p a

Goal state after:
case pos
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 : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
⊢ LipschitzOnWith 1 (fun f ↦ cfcₙ ((toFun {quasispectrum R a}) f) a)
    {f | ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a) ∧ f 0 = 0}

case neg
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 : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : ¬p a
⊢ LipschitzOnWith 1 (fun f ↦ cfcₙ ((toFun {quasispectrum R a}) f) a)
    {f | ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a) ∧ f 0 = 0}
Module: Mathlib.Analysis.CStarAlgebra.PositiveLinearMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx_nonneg : ∀ (x_1 : ℕ), 0 ≤ x x_1
hx_norm : ∀ (x_1 : ℕ), ‖x x_1‖ = 1
hx : ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
x_summable : Summable fun n ↦ 2 ^ (-↑n) • x n
n : ℕ
hn : ‖f (∑' (n : ℕ), 2 ^ (-↑n) • x n)‖ < 2 ^ n
⊢ ‖f (2 ^ (-↑n) • x n)‖ ≤ ‖f (∑' (n : ℕ), 2 ^ (-↑n) • x n)‖

Tactic:
have (m : ℕ) : 0 ≤ ((2 : ℝ) ^ (-(m : ℤ)) • x m) := smul_nonneg (by positivity) (hx_nonneg m)
    refine CStarAlgebra.norm_le_norm_of_nonneg_of_le (f.map_nonneg (this n)) ?_
    gcongr
    exact x_summable.le_tsum n fun m _ ↦ this m

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

Goal state before:
pp: m : Type u_1
n : Type u_2
R✝ : Type u_3
S : Type u_4
A✝ : Type u_5
B : Type u_6
R : Type ?u.14
A : Type ?u.16
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid A
inst✝² : Mul A
inst✝¹ : Module R A
inst✝ : Star A
e : m ≃ n
M : CStarMatrix m m A
i✝ j✝ : n
⊢ star M (e.symm i✝) (e.symm j✝) = star (Matrix.submatrix M ⇑e.symm ⇑e.symm) i✝ j✝

Tactic:
star_apply,

Goal state after:
m : Type u_1
n : Type u_2
R✝ : Type u_3
S : Type u_4
A✝ : Type u_5
B : Type u_6
R : Type ?u.14
A : Type ?u.16
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid A
inst✝² : Mul A
inst✝¹ : Module R A
inst✝ : Star A
e : m ≃ n
M : CStarMatrix m m A
i✝ j✝ : n
⊢ star (M (e.symm j✝) (e.symm i✝)) = star (Matrix.submatrix M ⇑e.symm ⇑e.symm) i✝ j✝
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
S : Type u_4
A : Type u_5
B : Type u_6
inst✝⁵ : Unique n
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid A
inst✝² : Mul A
inst✝¹ : Star A
inst✝ : Module R A
x✝¹ x✝ : A
⊢ (fun x y ↦ x✝¹ * x✝) = (fun x y ↦ x✝¹) * fun x y ↦ x✝

Tactic:
ext; simp [mul_apply]

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

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra 𝕜 A
inst✝⁴ : ContinuousFunctionalCalculus 𝕜 A p
inst✝³ : CompleteSpace A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
s : Set X
hs : MeasurableSet s
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (s ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ.restrict s, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound (μ.restrict s)
ha : p a
⊢ AEStronglyMeasurable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) (μ.restrict s)

pp: case refine_2
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra 𝕜 A
inst✝⁴ : ContinuousFunctionalCalculus 𝕜 A p
inst✝³ : CompleteSpace A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
s : Set X
hs : MeasurableSet s
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (s ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ.restrict s, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound (μ.restrict s)
ha : p a
⊢ HasFiniteIntegral (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) (μ.restrict s)

Tactic:
· exact aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry hs _ _ hf

Goal state after:
case refine_2
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra 𝕜 A
inst✝⁴ : ContinuousFunctionalCalculus 𝕜 A p
inst✝³ : CompleteSpace A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
s : Set X
hs : MeasurableSet s
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (s ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ.restrict s, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound (μ.restrict s)
ha : p a
⊢ HasFiniteIntegral (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) (μ.restrict s)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : MeasurableSpace X
μ : Measure X
inst✝⁸ : NormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedAlgebra 𝕜 A
inst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
s : Set X
hs : MeasurableSet s
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (s ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ.restrict s, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound (μ.restrict s)
ha : p a
this : ∀ᵐ (x : X) ∂μ.restrict s, ContinuousOn (f x) (spectrum 𝕜 a)
⊢ AEStronglyMeasurable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) (μ.restrict s)

pp: case refine_2
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : MeasurableSpace X
μ : Measure X
inst✝⁸ : NormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedAlgebra 𝕜 A
inst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
s : Set X
hs : MeasurableSet s
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (s ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ.restrict s, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound (μ.restrict s)
ha : p a
this : ∀ᵐ (x : X) ∂μ.restrict s, ContinuousOn (f x) (spectrum 𝕜 a)
⊢ HasFiniteIntegral (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) (μ.restrict s)

Tactic:
· exact aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry hs _ _ hf

Goal state after:
case refine_2
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : MeasurableSpace X
μ : Measure X
inst✝⁸ : NormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedAlgebra 𝕜 A
inst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
s : Set X
hs : MeasurableSet s
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (s ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ.restrict s, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound (μ.restrict s)
ha : p a
this : ∀ᵐ (x : X) ∂μ.restrict s, ContinuousOn (f x) (spectrum 𝕜 a)
⊢ HasFiniteIntegral (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) (μ.restrict s)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℂ A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : SMulCommClass ℂ A A
inst✝² : StarModule ℂ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMapZero.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f (quasispectrum ℂ ↑(ℑ a))
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ f ↑(ℑ a) = cfcₙ (fun x ↦ f ↑x.im) a

Tactic:
by
  rw [quasispectrum_imaginaryPart a] at hf
  rw [← cfcₙ_im_id a, ← cfcₙ_comp' ..]

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
hp : IsStarNormal a
| 2⁻¹ • (cfc (fun x ↦ x) a + cfc (fun x ↦ star x) a)

Tactic:
← cfc_add ..,

Goal state after:
A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
hp : IsStarNormal a
| 2⁻¹ • cfc (fun x ↦ x + star x) a
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
have : ⟪z, z⟫ ≠ 0 := inner_self_ne_zero.mpr z_ne_0

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.InnerProductSpace.Completion

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
⊢ Continuous (uncurry (inner 𝕜))

Tactic:
let inner' : E →+ E →+ 𝕜 :=
    { toFun := fun x => (innerₛₗ 𝕜 x).toAddMonoidHom
      map_zero' := by ext x; exact inner_zero_left _
      map_add' := fun x y => by ext z; exact inner_add_left _ _ _ }

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
inner' : E →+ E →+ 𝕜 := { toFun := fun x ↦ ((innerₛₗ 𝕜) x).toAddMonoidHom, map_zero' := ⋯, map_add' := ⋯ }
⊢ Continuous (uncurry (inner 𝕜))
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: ⊢ 𝐞 ≠ 1

Tactic:
rw [DFunLike.ne_iff]
  use 2⁻¹
  simp only [fourierChar_apply', AddChar.one_apply]
  rw [mul_comm, ← mul_assoc, inv_mul_cancel₀ (by positivity), one_mul]
  exact Circle.exp_pi_ne_one

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → 𝕜
T : EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 n := toEuclideanCLM (diagonal v)
x : EuclideanSpace 𝕜 n
i : n
x✝ : i ∈ Finset.univ
⊢ ‖v i‖ ^ 2 * ‖x.ofLp i‖ ^ 2 ≤ ?m.78 i

Tactic:
norm_le_pi_norm v i

Goal state after:
𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → 𝕜
T : EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 n := toEuclideanCLM (diagonal v)
x : EuclideanSpace 𝕜 n
i : n
x✝ : i ∈ Finset.univ
⊢ ‖v‖ ^ 2 * ‖x.ofLp i‖ ^ 2 ≤ ?m.78 i
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → 𝕜
T : EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 n := toEuclideanCLM (diagonal v)
i : n
⊢ ‖v i‖ = ‖T (toLp 2 (Pi.single i 1))‖

Tactic:
rw [toEuclideanCLM_toLp (diagonal v) (Pi.single i (1 : 𝕜))]

Goal state after:
𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → 𝕜
T : EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 n := toEuclideanCLM (diagonal v)
i : n
⊢ ‖v i‖ = ‖toLp 2 (diagonal v *ᵥ Pi.single i 1)‖
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Opens X
hs : Nonempty ↥s
⊢ (e.subtypeRestr hs).source = Subtype.val ⁻¹' e.source

Tactic:
simp only [subtypeRestr_def, mfld_simps]

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

Goal state before:
pp: case inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc✝ : c ∈ Icc a b
hlt : a < c
ec : Trivialization F TotalSpace.proj
hc : c ∈ ec.baseSet
c' : B
hc' : c' ∈ Ico a c
hc'e : Ioc c' c ⊆ ec.baseSet
d : B
hd : d ∈ Ioc c' c
hdab : d ∈ Icc a b
ead : Trivialization F TotalSpace.proj
had : Icc a d ⊆ ead.baseSet
⊢ Icc a c ∩ Iic d ⊆ ead.baseSet ∧ Icc a c \ Iic d ⊆ (ec.transFiberHomeomorph (ec.coordChangeHomeomorph ead ⋯ ⋯)).baseSet

Tactic:
exact ⟨fun x hx => had ⟨hx.1.1, hx.2⟩, fun x hx => hc'e ⟨hd.1.trans (not_le.1 hx.2), hx.1.2⟩⟩

Goal state after:
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ (∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U) →
    ∀ (a b : X) (b_1 : f a = f b),
      ⟨(a, b), b_1⟩ ∈ Function.pullbackDiagonal f → ∃ t ∈ 𝓝 a ×ˢ 𝓝 b, Subtype.val ⁻¹' t ⊆ Function.pullbackDiagonal f

Tactic:
rintro h x x' hx (rfl : x = x')

Goal state after:
case refine_1
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h : ∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U
x : X
hx : f x = f x
⊢ ∃ t ∈ 𝓝 x ×ˢ 𝓝 x, Subtype.val ⁻¹' t ⊆ Function.pullbackDiagonal f
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 : 𝓜(𝕜, A)
n : ℕ
x y : A
⊢ (a.toProd ^ n).2 x * y = x * (a.toProd ^ n).1 y

Tactic:
induction n generalizing x y with

Goal state after:
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace A
p : E → X
s : Set A
g₁ g₂ : A → E
sep : IsSeparatedMap p
inj : IsLocallyInjective p
hs : PreconnectedSpace ↑s
h₁ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₁)
h₂ : Continuous[instTopologicalSpaceSubtype, inst✝¹] (s.domRestrict g₂)
he : s.domRestrict (p ∘ g₁) = s.domRestrict (p ∘ g₂)
a : A
has : a ∈ s
ha : g₁ a = g₂ a
⊢ s.domRestrict g₁ = s.domRestrict g₂

Tactic:
exact sep.eq_of_comp_eq inj h₁ h₂ he ⟨a, has⟩ ha

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

Goal state before:

Tactic:
‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_opNorm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_opNorm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
G : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G E
hf : IsQuotientCoveringMap f G
x : X
e e' : ↑(f ⁻¹' {x})
this : MulAction G ↑(f ⁻¹' {x}) := hf.mulActionFiber x
⊢ ∃ g, ((hf.toPermFiber x) g) e = e'

Tactic:
have := hf.mulActionFiber_isPretransitive x

Goal state after:
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
G : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G E
hf : IsQuotientCoveringMap f G
x : X
e e' : ↑(f ⁻¹' {x})
this✝ : MulAction G ↑(f ⁻¹' {x}) := hf.mulActionFiber x
this : MulAction.IsPretransitive G ↑(f ⁻¹' {x})
⊢ ∃ g, ((hf.toPermFiber x) g) e = e'
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
hf : IsCoveringMap f
Y : Type u_3
inst✝ : TopologicalSpace Y
g : X ≃ₜ Y
⊢ IsCoveringMap (⇑g ∘ f)

Tactic:
by
  rw [isCoveringMap_iff_isCoveringMapOn_univ] at hf ⊢
  exact hf.homeomorph_comp g

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

Goal state before:
pp: case refine_1.a
E : Type u_1
X : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace X
f : E → X
s : Set X
inst✝³ : Nonempty (X → E)
ι : Type ?u.20
inst✝² : Nonempty ι
inst✝¹ : TopologicalSpace ι
inst✝ : DiscreteTopology ι
U : ι → Set E
V : Set X
open_V : IsOpen[inst✝⁴] V
open_iff : ∀ (i : ι) {W : Set X}, W ⊆ V → (IsOpen[inst✝⁴] W ↔ IsOpen[inst✝⁵] (f ⁻¹' W ∩ U i))
inj : ∀ (i : ι), Set.InjOn f (U i)
disjoint : Pairwise (Disjoint on U)
exhaustive' : f ⁻¹' V ⊆ ⋃ i, U i
idx : (x : E) → x ∈ f ⁻¹' V → ι
idx_U : ∀ (x : E) (a : x ∈ f ⁻¹' V), x ∈ U (idx x a)
inv : ι → ⦃a : X⦄ → a ∈ V → E
inv_U : ∀ (i : ι) ⦃a : X⦄ (a_1 : a ∈ V), inv i a_1 ∈ U i
f_inv : ∀ (i : ι) ⦃a : X⦄ (a_1 : a ∈ V), f (inv i a_1) = a
F : PartialEquiv E (X × ι) := ⋯
open_preim : ∀ {W : Set X}, W ⊆ V → IsOpen[inst✝⁴] W → IsOpen[inst✝⁵] (f ⁻¹' W)
open_source : IsOpen[inst✝⁵] F.source
cont_f : ContinuousOn f F.source
e : E
he : e ∈ F.source
e' : E
he' : e' ∈ U (idx e he) ∩ F.source
h : e' ∉ {x | (fun x ↦ (fun x ↦ idx e he) x = (fun x ↦ if he : f x ∈ V then idx x he else Classical.arbitrary ι) x) x}
⊢ ?refine_1.a✝ ∈ U ((fun x ↦ idx e he) e') ⊓ U ((fun x ↦ if he : f x ∈ V then idx x he else Classical.arbitrary ι) e')

Tactic:
dsimp only; rw [dif_pos (by exact he'.2)]; exact ⟨he'.1, idx_U ..⟩

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
⊢ normalize (normalize x) = normalize x

Tactic:
by_cases hx : x = 0
  · simp [hx]
  · simp [normalize_eq_self_of_norm_eq_one, hx]

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
r : ℝ
x : V
⊢ normalize (r • x) = ↑(SignType.sign r) • normalize x

Tactic:
rcases lt_trichotomy 0 r with (h_pos | rfl | h_neg)
  · simp [normalize_smul_of_pos, h_pos]
  · simp
  · simp [normalize_smul_of_neg, h_neg]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Span

Goal state before:
pp: case inr
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type ?u.39) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type ?u.41) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
⊢ ∃ u c, x = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖x‖ / 2

Tactic:
have := norm_smul_two_inv_smul_add_four_unitary x hx
    extract_lets u₁ u₂ at this
    use ![u₁, star u₁, u₂, star u₂], ![‖x‖ * 2⁻¹, ‖x‖ * 2⁻¹, ‖x‖ * 2⁻¹ * I, ‖x‖ * 2⁻¹ * I]
    constructor
    · conv_lhs => rw [this]
      simp [Fin.sum_univ_four, ← Complex.coe_smul]
      module
    · intro i
      fin_cases i
      all_goals simp [div_eq_mul_inv]

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

Goal state before:
pp: case a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
T : E →L[𝕜] F
x : E
hx : x ∈ (↑(adjoint T)).rangeᗮ
⊢ x ∈ (↑T).ker

Tactic:
refine ext_inner_left 𝕜 fun y ↦ ?_
    simp [← T.adjoint_inner_left, hx _]

Goal state after: