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

Goal state before:
pp: case h₁
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : ContinuousOn (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) (σ ℝ≥0 m)
hx : x ∈ Set.Icc 0 1
hx' : x ∈ Set.Icc (cfc g (ε⁻¹ ^ 2 • m)) 1
this : star m * (1 - cfc g (ε⁻¹ ^ 2 • m)) * m = cfc (fun y ↦ y * (1 + ε⁻¹ ^ 2 • y)⁻¹ * y) m
y : ℝ≥0
hy : y ∈ σ ℝ≥0 m
hm' : cfc (fun x ↦ x) m ≤ 1
⊢ y ≤ 1

Tactic:
← cfc_one (R := ℝ≥0) m,

Goal state after:
case h₁
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : ContinuousOn (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) (σ ℝ≥0 m)
hx : x ∈ Set.Icc 0 1
hx' : x ∈ Set.Icc (cfc g (ε⁻¹ ^ 2 • m)) 1
this : star m * (1 - cfc g (ε⁻¹ ^ 2 • m)) * m = cfc (fun y ↦ y * (1 + ε⁻¹ ^ 2 • y)⁻¹ * y) m
y : ℝ≥0
hy : y ∈ σ ℝ≥0 m
hm' : cfc (fun x ↦ x) m ≤ cfc 1 m
⊢ y ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A₁
inst✝⁵ : NonUnitalCStarAlgebra A₂
inst✝⁴ : PartialOrder A₁
inst✝³ : PartialOrder A₂
inst✝² : StarOrderedRing A₁
inst✝¹ : StarOrderedRing A₂
n : Type u_3
inst✝ : Fintype n
φ : A₁ →CP A₂
M : CStarMatrix n n A₁
hM : 0 ≤ M
k : ℕ := Fintype.card n
e : n ≃ Fin (Fintype.card n) := Fintype.equivFinOfCardEq ⋯
hmain : 0 ≤ (mapₗ ↑φ) ((reindexₐ ℂ A₁ e) M)
⊢ 0 ≤ M.map ⇑φ

Tactic:
← mapₗ_reindexₐ

Goal state after:
A₁ : Type u_1
A₂ : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A₁
inst✝⁵ : NonUnitalCStarAlgebra A₂
inst✝⁴ : PartialOrder A₁
inst✝³ : PartialOrder A₂
inst✝² : StarOrderedRing A₁
inst✝¹ : StarOrderedRing A₂
n : Type u_3
inst✝ : Fintype n
φ : A₁ →CP A₂
M : CStarMatrix n n A₁
hM : 0 ≤ M
k : ℕ := Fintype.card n
e : n ≃ Fin (Fintype.card n) := Fintype.equivFinOfCardEq ⋯
hmain : 0 ≤ (reindexₐ ℂ A₂ e) ((mapₗ ↑φ) M)
⊢ 0 ≤ M.map ⇑φ
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case e_a.e_a
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : ContinuousOn (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) (σ ℝ≥0 m)
hx : x ∈ Set.Icc 0 1
hx' : x ∈ Set.Icc (cfc g (ε⁻¹ ^ 2 • m)) 1
⊢ 1 - cfc g (ε⁻¹ ^ 2 • m) = cfc (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) m

Tactic:
← cfc_one (R := ℝ≥0) m,

Goal state after:
case e_a.e_a
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : ContinuousOn (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) (σ ℝ≥0 m)
hx : x ∈ Set.Icc 0 1
hx' : x ∈ Set.Icc (cfc g (ε⁻¹ ^ 2 • m)) 1
⊢ cfc 1 m - cfc g (ε⁻¹ ^ 2 • m) = cfc (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) m
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁴ : MeasurableSpace X
μ : Measure X
inst✝³ : TopologicalSpace Y
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : CompactSpace Y
inst✝ : Zero Y
f : X → Y → E
g : C(Y, E)₀
f_ae_cont : ∀ᵐ (x : X) ∂μ, Continuous (f x)
f_ae_zero : ∀ᵐ (x : X) ∂μ, f x 0 = 0
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖f x y‖ ≤ bound x
x : X
bound_ge_x : ∀ (y : Y), ‖f x y‖ ≤ bound x
cont_x : Continuous (f x)
zero_x : f x 0 = 0
⊢ ∀ (y : Y), ‖(mkD (f x) g) y‖ ≤ bound x

Tactic:
simpa only [mkD_apply_of_continuous cont_x zero_x] using bound_ge_x

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

Goal state before:
pp: A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
x✝ :
  (f, a) ∈
    {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (f, a).2)
hs : IsCompact (quasispectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (quasispectrum ℝ≥0 x)}
⊢ ∃ t,
    IsOpen[instTopologicalSpaceProd] t ∧
      (f, a) ∈ t ∧
        ContinuousOn (fun fa ↦ cfcₙ ((toFun {s}) fa.1) fa.2)
          ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0} ×ˢ
              {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)} ∩
            t)

Tactic:
refine ⟨Set.univ ×ˢ {x | k ∈ 𝓝ˢ (quasispectrum ℝ≥0 x)}, isOpen_univ.prod this, by simpa, ?_⟩

Goal state after:
A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
x✝ :
  (f, a) ∈
    {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (f, a).2)
hs : IsCompact (quasispectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (quasispectrum ℝ≥0 x)}
⊢ ContinuousOn (fun fa ↦ cfcₙ ((toFun {s}) fa.1) fa.2)
    ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0} ×ˢ
        {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)} ∩
      Set.univ ×ˢ {x | k ∈ 𝓝ˢ (quasispectrum ℝ≥0 x)})
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, quasispectrum ℝ≥0 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ∀ x ∈ t, ContinuousOn f (s x)
hf0 : f 0 = 0
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) t

Tactic:
by
  rw [ContinuousOn] at ha_cont ⊢
  refine fun x hx ↦ (ha_cont x hx).cfcₙ_nnreal (hs x hx) f hx ?_ ?_ (hf x hx)
  all_goals filter_upwards [ha x hx, self_mem_nhdsWithin] with x hx hxt
  exacts [hx, ha' x hxt]

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (re '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ Set.MapsTo re ((fun x ↦ ↑x.re) '' quasispectrum ℂ a) (re '' quasispectrum ℂ a)

Tactic:
simpa [Set.mapsTo_image_iff, Function.comp_def] using Set.mapsTo_image ..

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (im '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ Set.MapsTo re ((fun x ↦ ↑x.im) '' quasispectrum ℂ a) (im '' quasispectrum ℂ a)

Tactic:
simpa [Set.mapsTo_image_iff, Function.comp_def] using Set.mapsTo_image ..

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (re '' spectrum ℂ a)
ha : IsStarNormal a
⊢ Set.MapsTo re ((fun x ↦ ↑x.re) '' spectrum ℂ a) (re '' spectrum ℂ a)

Tactic:
simpa [Set.mapsTo_image_iff, Function.comp_def] using Set.mapsTo_image ..

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (im '' spectrum ℂ a)
ha : IsStarNormal a
⊢ Set.MapsTo re ((fun x ↦ ↑x.im) '' spectrum ℂ a) (im '' spectrum ℂ a)

Tactic:
simpa [Set.mapsTo_image_iff, Function.comp_def] using Set.mapsTo_image ..

Goal state after:
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:
rw [← div_mul_eq_mul_div]

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

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ Module.rank 𝕜 ↥(𝕜 ∙ x) = 1

pp: case hf
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ (innerSL 𝕜) y ≠ 0

Tactic:
Module.rank_eq_one_iff_finrank_eq_one

Goal state after:
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ finrank 𝕜 ↥(𝕜 ∙ x) = 1

case hf
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ (innerSL 𝕜) y ≠ 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
⊢ (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:
have openness₁ : IsOpen (f.target ∩ f.symm ⁻¹' s) := f.isOpen_inter_preimage_symm s.2

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') ≈
    (f.symm.trans f').restr (f.target ∩ ↑f.symm ⁻¹' ↑s)
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i : ι
⊢ IsOpen (Z.localTrivAsPartialEquiv i).source

Tactic:
by
  apply TopologicalSpace.GenerateOpen.basic
  simp only [exists_prop, mem_iUnion, mem_singleton_iff]
  refine ⟨i, Z.baseSet i ×ˢ univ, (Z.isOpen_baseSet i).prod isOpen_univ, ?_⟩
  ext p
  simp only [localTrivAsPartialEquiv_apply, prodMk_mem_set_prod_eq, mem_inter_iff, and_self_iff,
    mem_localTrivAsPartialEquiv_source, and_true, mem_univ, mem_preimage]

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

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e : Trivialization F proj
s : Set B
⊢ proj ⁻¹' e.baseSet ∩ proj ⁻¹' frontier s = proj ⁻¹' (e.baseSet ∩ frontier s)

Tactic:
preimage_inter

Goal state after:
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e : Trivialization F proj
s : Set B
⊢ proj ⁻¹' e.baseSet ∩ proj ⁻¹' frontier s = proj ⁻¹' e.baseSet ∩ proj ⁻¹' frontier s
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
f : E → X
s : Set X
hf : IsCoveringMapOn f s
x : E
hx : x ∈ f ⁻¹' s
this : Nonempty ↑(f ⁻¹' {f x})
e : Trivialization (↑(f ⁻¹' {f x})) f := ⋯.toTrivialization
⊢ ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source

Tactic:
have h := (hf (f x) hx).mem_toTrivialization_baseSet

Goal state after:
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
f : E → X
s : Set X
hf : IsCoveringMapOn f s
x : E
hx : x ∈ f ⁻¹' s
this : Nonempty ↑(f ⁻¹' {f x})
e : Trivialization (↑(f ⁻¹' {f x})) f := ⋯.toTrivialization
h : f x ∈ ⋯.toTrivialization.baseSet
⊢ ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: 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)
surj : ∀ (i : ι), Set.SurjOn f (U i) V
disjoint : Pairwise (Disjoint on U)
exhaustive : f ⁻¹' V ⊆ ⋃ i, U i
⊢ Trivialization ι f

Tactic:
have exhaustive' := exhaustive

Goal state after:
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)
surj : ∀ (i : ι), Set.SurjOn f (U i) V
disjoint : Pairwise (Disjoint on U)
exhaustive exhaustive' : f ⁻¹' V ⊆ ⋃ i, U i
⊢ Trivialization ι f
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: case refine_3
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) :=
  { toFun := fun e ↦ (f e, if he : f e ∈ V then idx e he else Classical.arbitrary ι),
    invFun := fun x ↦ if hx : x.1 ∈ V then inv x.2 hx else Classical.arbitrary (X → E) x.1, source := f ⁻¹' V,
    target := V ×ˢ Set.univ, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }
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
i : ι
⊢ ContinuousOn (fun x ↦ if h : x ∈ V then inv i ⋯ else Classical.arbitrary (X → E) x) {a | a ∈ V}

Tactic:
refine (continuousOn_open_iff open_V).mpr fun W open_W ↦ ?_

Goal state after:
case refine_3
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) :=
  { toFun := fun e ↦ (f e, if he : f e ∈ V then idx e he else Classical.arbitrary ι),
    invFun := fun x ↦ if hx : x.1 ∈ V then inv x.2 hx else Classical.arbitrary (X → E) x.1, source := f ⁻¹' V,
    target := V ×ˢ Set.univ, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }
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
i : ι
W : Set E
open_W : IsOpen[inst✝⁵] W
⊢ IsOpen[inst✝⁴] (V ∩ (fun x ↦ if h : x ∈ V then inv i ⋯ else Classical.arbitrary (X → E) x) ⁻¹' W)
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
hne : x ≠ y
⊢ Injective ⇑(x.path y)

Tactic:
coe_path

Goal state after:
x y : Circle
hne : x ≠ y
⊢ Injective (⇑exp ∘ ⇑(Path.segment (↑x).arg (x.angleDiff y + (↑x).arg)))
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
hyx : y ≠ x
hxP : x ∈ range ⇑((-x).path y)
hne : -x ≠ y
hP₂ : x ∈ range ⇑(y.path (-x))
⊢ False

Tactic:
have h : x ∈ range _ ∩ _ := ⟨hxP, hP₂⟩

Goal state after:
x y : Circle
hyx : y ≠ x
hxP : x ∈ range ⇑((-x).path y)
hne : -x ≠ y
hP₂ : x ∈ range ⇑(y.path (-x))
h : x ∈ range ⇑((-x).path y) ∩ range ⇑(y.path (-x))
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case refine_2
x y : Circle
hxy : x ≠ y
⊢ {x, y}ᶜ ⊆ univ

Tactic:
exact subset_univ _

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

Goal state before:
pp: T : ℝ
x : AddCircle T
n : ℕ
⊢ (n • x).toCircle = x.toCircle ^ n

Tactic:
by
  induction n with
  | zero => simp
  | succ n ih => rw [succ_nsmul, toCircle_add, ih, pow_succ]

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

Goal state before:
pp: T : ℝ
x : AddCircle T
n : ℤ
⊢ (n • x).toCircle = x.toCircle ^ n

Tactic:
cases n

Goal state after:
case ofNat
T : ℝ
x : AddCircle T
a✝ : ℕ
⊢ (Int.ofNat a✝ • x).toCircle = x.toCircle ^ Int.ofNat a✝

case negSucc
T : ℝ
x : AddCircle T
a✝ : ℕ
⊢ (Int.negSucc a✝ • x).toCircle = x.toCircle ^ Int.negSucc a✝
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u v : ↥(unitary A)
huv : ‖↑v - ↑u‖ < 2
⊢ expUnitary (↑1 • argSelfAdjoint (v * star u)) * u = v

Tactic:
by simp [expUnitary_eq_mul_inv v u huv, mul_assoc]

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

Goal state before:
pp: 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
⊢ Function.Injective ⇑↑(adjoint T ∘SL T) ↔ Function.Injective ⇑T

Tactic:
← LinearMap.ker_eq_bot,

Goal state after:
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
⊢ (↑(adjoint T ∘SL T)).ker = ⊥ ↔ Function.Injective ⇑T
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 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
⊢ (↑(adjoint T ∘SL T)).ker = ⊥ ↔ Function.Injective ⇑↑T

Tactic:
← LinearMap.ker_eq_bot,

Goal state after:
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
⊢ (↑(adjoint T ∘SL T)).ker = ⊥ ↔ (↑T).ker = ⊥
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : CharZero k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
⊢ s.points i = -↑n • (s.faceOppositeCentroid i -ᵥ s.centroid) +ᵥ s.centroid

Tactic:
rw [← neg_vsub_eq_vsub_rev, neg_smul_neg, ← point_vsub_centroid_eq_smul_vsub, vsub_vadd]

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✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Finite α
f : (i : α) → E i
_h : 0 < p.toReal
⊢ Memℓp f p

Tactic:
cases nonempty_fintype α; exact memℓp_gen ⟨Finset.univ.sum _, hasSum_fintype _⟩

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

Goal state before:
pp: case inr.inr.inr.inr.inr.refine_1
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
f : (i : α) → E i
hfq : Memℓp f q
hpq : q ≤ p
hq : 0 < q.toReal
left✝ : 0 < p.toReal
hpq' : q.toReal ≤ p.toReal
hf' : Summable fun i ↦ ‖f i‖ ^ q.toReal
⊢ {i | 1 ≤ ‖f i‖}.Finite

pp: case inr.inr.inr.inr.inr.refine_2
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
f : (i : α) → E i
hfq : Memℓp f q
hpq : q ≤ p
hq : 0 < q.toReal
left✝ : 0 < p.toReal
hpq' : q.toReal ≤ p.toReal
hf' : Summable fun i ↦ ‖f i‖ ^ q.toReal
⊢ {x | (fun i ↦ ‖‖f i‖ ^ p.toReal‖ ≤ ‖f i‖ ^ q.toReal) x}ᶜ ⊆ {i | 1 ≤ ‖f i‖}

Tactic:
· have H : { x : α | 1 ≤ ‖f x‖ ^ q.toReal }.Finite := by
        simpa using hf'.tendsto_cofinite_zero.eventually_lt_const (by simp)
      exact H.subset fun i hi => Real.one_le_rpow hi hq.le

Goal state after:
case inr.inr.inr.inr.inr.refine_2
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
f : (i : α) → E i
hfq : Memℓp f q
hpq : q ≤ p
hq : 0 < q.toReal
left✝ : 0 < p.toReal
hpq' : q.toReal ≤ p.toReal
hf' : Summable fun i ↦ ‖f i‖ ^ q.toReal
⊢ {x | (fun i ↦ ‖‖f i‖ ^ p.toReal‖ ≤ ‖f i‖ ^ q.toReal) x}ᶜ ⊆ {i | 1 ≤ ‖f i‖}