mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_2
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ ⦃s t : Set X⦄,
    s ⊆ t →
      (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x) →
        ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x

pp: case refine_3
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ ⦃s t : Set X⦄,
    (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x) →
      (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x) →
        ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s ∪ t, 0 < g' x

pp: case refine_4
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ x ∈ t, ∃ t_1 ∈ nhdsWithin x t, ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t_1, 0 < g' x

Tactic:
· rintro s₁ s₂ hs ⟨g, hI, hgt⟩; exact ⟨g, hI, fun x hx => hgt x (hs hx)⟩

Goal state after:
case refine_3
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ ⦃s t : Set X⦄,
    (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x) →
      (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x) →
        ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s ∪ t, 0 < g' x

case refine_4
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ x ∈ t, ∃ t_1 ∈ nhdsWithin x t, ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t_1, 0 < g' x
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
ha : ¬IsUnit a
M : Ideal A
hM : M.IsMaximal
haM : span {a} ≤ M
⊢ ∃ f, f a = 0

Tactic:
exact
    ⟨M.toCharacterSpace,
      M.toCharacterSpace_apply_eq_zero_of_mem
        (haM (mem_span_singleton.mpr ⟨1, (mul_one a).symm⟩))⟩

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

Goal state before:
pp: A : Type u_1
inst✝ : CommCStarAlgebra A
⊢ Function.Bijective ⇑(gelfandTransform ℂ A)

Tactic:
refine ⟨(gelfandTransform_isometry A).injective, ?_⟩
  /- The range of `gelfandTransform ℂ A` is actually a `StarSubalgebra`. The key lemma below may be
    hard to spot; it's `map_star` coming from `WeakDual.Complex.instStarHomClass`, which is a
    nontrivial result. -/
  let rng : StarSubalgebra ℂ C(characterSpace ℂ A, ℂ) :=
    { toSubalgebra := (gelfandTransform ℂ A).range
      star_mem' := by
        rintro - ⟨a, rfl⟩
        use star a
        ext1 φ
        dsimp
        simp only [map_star, RCLike.star_def] }
  suffices rng = ⊤ from
    fun x => show x ∈ rng from this.symm ▸ StarSubalgebra.mem_top
  /- Because the `gelfandTransform ℂ A` is an isometry, it has closed range, and so by the
    Stone-Weierstrass theorem, it suffices to show that the image of the Gelfand transform separates
    points in `C(characterSpace ℂ A, ℂ)` and is closed under `star`. -/
  have h : rng.topologicalClosure = rng := le_antisymm
    (StarSubalgebra.topologicalClosure_minimal le_rfl
      (gelfandTransform_isometry A).isClosedEmbedding.isClosed_range)
    (StarSubalgebra.le_topologicalClosure _)
  refine h ▸ ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
    _ (fun _ _ => ?_)
  /- Separating points just means that elements of the `characterSpace` which agree at all points
    of `A` are the same functional, which is just extensionality. -/
  contrapose!
  exact fun h => Subtype.ext (ContinuousLinearMap.ext fun a =>
    h (gelfandTransform ℂ A a) ⟨gelfandTransform ℂ A a, ⟨a, rfl⟩, rfl⟩)

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

Goal state before:
pp: case neg
A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
a✝ : Nontrivial A
x : ℝ
r : ℝˣ
hr : 0 ≤ ↑r
hx : x ∈ spectrum ℝ (r • a)
hr' : ¬↑r = 0
⊢ 0 ≤ x

Tactic:
spectrum.unit_smul_eq_smul,

Goal state after:
case neg
A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
a✝ : Nontrivial A
x : ℝ
r : ℝˣ
hr : 0 ≤ ↑r
hx : x ∈ r • spectrum ℝ a
hr' : ¬↑r = 0
⊢ 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case neg.inl
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
h : ¬(ContinuousOn f (σₙ R a) ∧ p a)
hf : ¬ContinuousOn f (σₙ R a)
⊢ ↑(cfcₙ f a) = cfc f ↑a

pp: case neg.inr
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
h : ¬(ContinuousOn f (σₙ R a) ∧ p a)
ha : ¬p a
⊢ ↑(cfcₙ f a) = cfc f ↑a

Tactic:
· rw [cfcₙ_apply_of_not_continuousOn a hf, inr_zero,
        cfc_apply_of_not_continuousOn _ (quasispectrum_eq_spectrum_inr' R ℂ a ▸ hf)]

Goal state after:
case neg.inr
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
h : ¬(ContinuousOn f (σₙ R a) ∧ p a)
ha : ¬p a
⊢ ↑(cfcₙ f a) = cfc f ↑a
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
ha : a ∈ Set.Ici 0
⊢ (fun a ↦ a ^ 0) a = (fun x ↦ 1) a

Tactic:
simp [rpow_zero a ha]

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

Goal state before:
pp: case pos
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
hy : y ≠ 0
ha : 0 ≤ a
a✝ : Nontrivial A
h : IsUnit (cfc (fun x ↦ x ^ y) a)
hf : ContinuousOn (fun x ↦ x ^ y) (spectrum ℝ≥0 a)
⊢ IsUnit a

Tactic:
rw [isUnit_cfc_iff _ a hf] at h
    refine spectrum.isUnit_of_zero_notMem ℝ≥0 ?_
    intro h0
    specialize h 0 h0
    simp only [ne_eq, NNReal.rpow_eq_zero_iff, true_and, Decidable.not_not] at h
    exact hy h

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 : Aˣ
ha : 0 ≤ ↑a
hab : ‖sqrt ↑b⁻¹ * star (sqrt ↑a) * (sqrt ↑a * sqrt ↑b⁻¹)‖ ≤ 1
hb : 0 ≤ ↑b
hb_inv : 0 ≤ ↑b⁻¹
ha_inv : 0 ≤ ↑a⁻¹
⊢ ‖sqrt ↑b⁻¹ * sqrt ↑a * (sqrt ↑a * star (sqrt ↑b⁻¹))‖ ≤ 1

Tactic:
IsSelfAdjoint.of_nonneg (sqrt_nonneg _)

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : Aˣ
ha : 0 ≤ ↑a
hab : ‖sqrt ↑b⁻¹ * sqrt ↑a * (sqrt ↑a * sqrt ↑b⁻¹)‖ ≤ 1
hb : 0 ≤ ↑b
hb_inv : 0 ≤ ↑b⁻¹
ha_inv : 0 ≤ ↑a⁻¹
⊢ ‖sqrt ↑b⁻¹ * sqrt ↑a * (sqrt ↑a * sqrt ↑b⁻¹)‖ ≤ 1
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
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
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
hxn : ‖x‖₊ = 1
hN : ‖↑(∑ i_1, ‖A i i_1‖₊)‖₊ * ‖1‖₊ ≤ N
⊢ ∑ j, ‖A i j‖₊ ≤ N

Tactic:
NNReal.nnnorm_eq,

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
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
hxn : ‖x‖₊ = 1
hN : (∑ i_1, ‖A i i_1‖₊) * ‖1‖₊ ≤ N
⊢ ∑ j, ‖A i j‖₊ ≤ N
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
s : Finset (n × n) := Finset.map { toFun := fun i ↦ (i, i), inj' := ⋯ } Finset.univ
i : n × n
_hi : i ∈ Finset.univ
his : i ∉ s
⊢ ‖diagonal v i.1 i.2‖₊ ^ 2 = 0

Tactic:
suffices i.1 ≠ i.2 by rw [diagonal_apply_ne _ this, nnnorm_zero, NNReal.zero_rpow two_ne_zero]

Goal state after:
n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
s : Finset (n × n) := Finset.map { toFun := fun i ↦ (i, i), inj' := ⋯ } Finset.univ
i : n × n
_hi : i ∈ Finset.univ
his : i ∉ s
⊢ i.1 ≠ i.2
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
i✝ : A
hx : 0 ≤ i✝ ∧ ‖i✝‖ < 1
⊢ i✝ ∈ closedBall 0 1

Tactic:
by simpa using hx.2.le

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

Goal state before:
pp: A : Type u_2
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b : A
hb : Commute a b
f : ℝ → ℝ
hf : ContinuousOn f (quasispectrum ℝ a)
hf0 : { toFun := (quasispectrum ℝ a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : IsSelfAdjoint a
⊢ Commute (cfcₙ f a) b

Tactic:
exact hb.cfcₙ (ha.star_eq.symm ▸ hb) _

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

Goal state before:
pp: n : Type u_2
A : Type u_5
inst✝ : Star A
f : CStarMatrix n n A
hf : IsSelfAdjoint f
i j : n
⊢ star f j i = f j i

Tactic:
IsSelfAdjoint.star_eq hf

Goal state after:
n : Type u_2
A : Type u_5
inst✝ : Star A
f : CStarMatrix n n A
hf : IsSelfAdjoint f
i j : n
⊢ f j i = f j i
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f (spectrum ℂ ↑(ℜ a))
ha : IsStarNormal a
⊢ cfc f ↑(ℜ a) = cfc (fun x ↦ f ↑x.re) a

Tactic:
by
  rw [spectrum_realPart a] at hf
  rw [← cfc_re_id a, ← cfc_comp' ..]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
L : Type u_6
inst✝⁸ : Ring 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : IsOrderedRing 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : EquivLike L E F
inst✝ : LinearEquivClass L 𝕜 E F
f : L
s : Set E
⊢ ⇑f '' extremePoints 𝕜 s = extremePoints 𝕜 (⇑f '' s)

Tactic:
ext b
  obtain ⟨a, rfl⟩ := EquivLike.surjective f b
  have : ∀ x y, f '' openSegment 𝕜 x y = openSegment 𝕜 (f x) (f y) :=
    image_openSegment _ (LinearMapClass.linearMap f).toAffineMap
  simp only [mem_extremePoints, (EquivLike.surjective f).forall,
    (EquivLike.injective f).mem_set_image, (EquivLike.injective f).eq_iff, ← this]

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

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
hφ : Function.Injective ⇑φ
a✝ : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
hψ : Function.Injective ⇑ψ
a : Unitization ℂ A
ha : IsSelfAdjoint (star a * a)
⊢ (spectralRadius ℝ (ψ (star a * a))).toReal = (spectralRadius ℝ (star a * a)).toReal

Tactic:
by
      simp only [spectralRadius, ha.map_spectrum_real ψ hψ]

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_3
Z : Type u_5
Z' : Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : OpenPartialHomeomorph X Y
f' : Y ≃ₜ Z
f'' : Z ≃ₜ Z'
⊢ (e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'')

Tactic:
simp only [transHomeomorph_eq_trans, trans_assoc, Homeomorph.trans_toOpenPartialHomeomorph]

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

Goal state before:
pp: case e'_5
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
I : Type u_3
inst✝¹ : TopologicalSpace I
x : X
Y : Type u_4
inst✝ : TopologicalSpace Y
g : X ≃ₜ Y
h : IsEvenlyCovered (⇑g ∘ f) (g x) I
⊢ f = ⇑g.symm ∘ ⇑g ∘ f

Tactic:
((try ext); simp)

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

Goal state before:
pp: case e'_6
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
I : Type u_3
inst✝¹ : TopologicalSpace I
x : X
Y : Type u_4
inst✝ : TopologicalSpace Y
g : X ≃ₜ Y
h : IsEvenlyCovered (⇑g ∘ f) (g x) I
⊢ x = g.symm (g x)

Tactic:
((try ext); simp)

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

Goal state before:
pp: case h
x✝ : ℝ
⊢ ↑(↑(↑x✝).cos + ↑(↑x✝).sin * I).arg = ↑x✝

Tactic:
arg_cos_add_sin_mul_I_coe_angle

Goal state after:
case h
x✝ : ℝ
⊢ ↑x✝ = ↑x✝
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : ‖x‖ < π
a✝ : Nontrivial A
⊢ spectrum ℂ ↑(expUnitary x) ⊆ slitPlane

Tactic:
rw [spectrum_subset_slitPlane_iff_norm_lt_two (expUnitary x).2,
      ← sq_lt_sq₀ (by positivity) (by positivity), norm_sq_expUnitary_sub_one hx.le]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : ‖x‖ < π
a✝ : Nontrivial A
⊢ 2 * (1 - Real.cos ‖x‖) < 2 ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case e'_5
x✝ : ℝ
⊢ ↑(exp x✝) = ↑((⇑(homeomorphCircle ⋯) ∘ QuotientAddGroup.mk) x✝)

Tactic:
simp [homeomorphCircle_apply, toCircle]

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : ‖↑u - 1‖ < 2
this : ContinuousOn arg (spectrum ℂ ↑u)
y : ℂ
hy : y ∈ spectrum ℂ ↑u
hy₁ : ‖y‖ = 1
⊢ (I * ↑y.arg).re = (log y).re

Tactic:
simp [log_re, spectrum.norm_eq_one_of_unitary u.2 hy]

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ ContinuousOn (fun x ↦ cfc (fun x ↦ ↑x.arg) ↑x) (closedBall 1 √ε)

Tactic:
apply ContinuousOn.image_comp_continuous ?_ continuous_subtype_val

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ ContinuousOn (cfc fun x ↦ ↑x.arg) (Subtype.val '' closedBall 1 √ε)
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : DecidableEq P
p : ι → P
hi : AffineIndependent k p
s : Finset ι
n : ℕ
hc : #s = n + 1
⊢ finrank k ↥(vectorSpan k ↑(Finset.image p s)) = n

Tactic:
classical
  have hi' := hi.range.mono (Set.image_subset_range p ↑s)
  have hc' : #(s.image p) = n + 1 := by rwa [s.card_image_of_injective hi.injective]
  have hn : (s.image p).Nonempty := by simp [hc', ← Finset.card_pos]
  rcases hn with ⟨p₁, hp₁⟩
  have hp₁' : p₁ ∈ p '' s := by simpa using hp₁
  rw [affineIndependent_set_iff_linearIndependent_vsub k hp₁', ← Finset.coe_singleton,
    ← Finset.coe_image, ← Finset.coe_sdiff, Finset.sdiff_singleton_eq_erase, ← Finset.coe_image]
    at hi'
  have hc : #(((s.image p).erase p₁).image (· -ᵥ p₁)) = n := by
    rw [Finset.card_image_of_injective _ (vsub_left_injective _), Finset.card_erase_of_mem hp₁]
    exact Nat.pred_eq_of_eq_succ hc'
  rwa [vectorSpan_eq_span_vsub_finset_right_ne k hp₁, finrank_span_finset_eq_card, hc]

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

Goal state before:
pp: G : Type u_4
E : Type u_5
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
μ : Measure G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulLeftInvariant
f : G → E
g : G
⊢ ∫ (x : G), f (g * x) ∂μ = ∫ (x : G), f x ∂μ

Tactic:
have h_mul : MeasurableEmbedding fun x => g * x := (MeasurableEquiv.mulLeft g).measurableEmbedding
  rw [← h_mul.integral_map, map_mul_left_eq_self]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : DivisionRing k
inst✝² : Module k V
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional k V
h : Fintype.card ι = Module.finrank k V + 1
⊢ Nonempty (AffineBasis ι k P)

Tactic:
obtain ⟨s, b, hb⟩ := AffineBasis.exists_affineBasis k V P

Goal state after:
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : DivisionRing k
inst✝² : Module k V
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional k V
h : Fintype.card ι = Module.finrank k V + 1
s : Set P
b : AffineBasis (↑s) k P
hb : ⇑b = Subtype.val
⊢ Nonempty (AffineBasis ι k P)
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Memℓp f p
hp : 0 < p.toReal
⊢ Memℓp (-f) p

Tactic:
apply memℓp_gen
    simpa using hf.summable hp

Goal state after:
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
x₀ : E
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
x : E
⊢ Summable fun n ↦ f n x

Tactic:
let : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
x₀ : E
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
x : E
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ Summable fun n ↦ f n x
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
x₀ : E
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
x : E
⊢ HasFDerivAt (fun y ↦ ∑' (n : α), f n y) (∑' (n : α), f' n x) x

Tactic:
let : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
x₀ : E
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
x : E
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ HasFDerivAt (fun y ↦ ∑' (n : α), f n y) (∑' (n : α), f' n x) x
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf' : UniformCauchySeqOnFilter f' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n ↦ f n x) l)
⊢ UniformCauchySeqOnFilter f l (𝓝 x)

Tactic:
let : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜

Goal state after:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf' : UniformCauchySeqOnFilter f' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n ↦ f n x) l)
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ UniformCauchySeqOnFilter f l (𝓝 x)
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E✝ : α → Type u_4
p q : ℝ≥0∞
inst✝⁵ : (i : α) → NormedAddCommGroup (E✝ i)
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedRing 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 E
inst✝ : CompleteSpace E
f : ↥(lp (fun x ↦ E) 1)
⊢ ‖{ toFun := fun f ↦ ∑' (i : α), ↑f i, map_add' := ⋯, map_smul' := ⋯ } f‖ ≤ 1 * ‖f‖

Tactic:
by simpa using norm_tsum_le f

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : (i : α) → StarAddMonoid (E i)
inst✝ : ∀ (i : α), NormedStarGroup (E i)
f : (i : α) → E i
hf : Memℓp f p
hp : 0 < p.toReal
⊢ Memℓp (star f) p

Tactic:
apply memℓp_gen
    simpa using hf.summable hp

Goal state after: