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

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹³ : NonUnitalNormedRing A
inst✝¹² : StarRing A
inst✝¹¹ : NormedSpace ℝ A
inst✝¹⁰ : IsScalarTower ℝ A A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, quasispectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (a i))
S : ↑t → Set ℝ≥0
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), quasispectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) t

Tactic:
refine ha_cont.cfcₙ_nnreal (s := fun x : X ↦ if hx : x ∈ t then S ⟨x, hx⟩ else ∅) f
    (by simpa +contextual using hS₁) ?_ ha' ?_

Goal state after:
case refine_1
X : Type u_1
A : Type u_2
inst✝¹³ : NonUnitalNormedRing A
inst✝¹² : StarRing A
inst✝¹¹ : NormedSpace ℝ A
inst✝¹⁰ : IsScalarTower ℝ A A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, quasispectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (a i))
S : ↑t → Set ℝ≥0
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), quasispectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, quasispectrum ℝ≥0 (a x) ⊆ if hx : x₀ ∈ t then S ⟨x₀, hx⟩ else ∅

case refine_2
X : Type u_1
A : Type u_2
inst✝¹³ : NonUnitalNormedRing A
inst✝¹² : StarRing A
inst✝¹¹ : NormedSpace ℝ A
inst✝¹⁰ : IsScalarTower ℝ A A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, quasispectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (a i))
S : ↑t → Set ℝ≥0
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), quasispectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ x ∈ t, ContinuousOn f (if hx : x ∈ t then S ⟨x, hx⟩ else ∅)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral

Goal state before:
pp: 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
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (univ ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound μ
ha : p a
⊢ Integrable (fun x ↦ cfc (f x) a) μ

Tactic:
refine integrable_cfc' _ _ ⟨?_, ?_⟩ ha
  · exact aeStronglyMeasurable_mkD_restrict_of_uncurry _ _ hf
  · refine hasFiniteIntegral_mkD_restrict_of_bound f _ ?_ bound bound_int bound_ge
    exact .of_forall fun x ↦
      hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨Set.mem_univ _, hz⟩

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : PartialOrder A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : StarOrderedRing A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (Set.range fun x ↦ cfc x a) = (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x}

Tactic:
ext
  constructor
  · rintro ⟨f, rfl⟩
    simp only [cfc_nnreal_eq_real f a ha]
    exact ⟨_, fun _ _ ↦ by positivity, rfl⟩
  · rintro ⟨f, hf, rfl⟩
    simp only [cfc_real_eq_nnreal a hf]
    exact ⟨_, rfl⟩

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

Goal state before:
pp: case inr.inl
A : Type u_1
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module ℝ A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : SMulCommClass ℝ A A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : IsTopologicalRing A
inst✝⁷ : T2Space A
inst✝⁶ : PartialOrder A
inst✝⁵ : NonnegSpectrumClass ℝ A
inst✝⁴ : StarOrderedRing A
inst✝³ : StarModule ℝ A
inst✝² : ContinuousStar A
inst✝¹ : ContinuousConstSMul ℝ A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
f : ℝ → ℝ
hf : 0 ≤ cfcₙ f a
h : ¬ContinuousOn f (quasispectrum ℝ a)
⊢ ∃ x, (∀ x_1 ∈ quasispectrum ℝ a, 0 ≤ x x_1) ∧ cfcₙ x a = cfcₙ f a

Tactic:
exact ⟨0, by simp, by simp [cfcₙ_apply_of_not_continuousOn a h]⟩

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:
← div_mul_eq_mul_div

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)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
t : Set F
x : E
y : F
⊢ (x, y) ∈ extremePoints 𝕜 (s ×ˢ t) ↔ (x, y) ∈ extremePoints 𝕜 s ×ˢ extremePoints 𝕜 t

Tactic:
refine (and_congr_right fun hx ↦ ⟨fun h ↦ ⟨?_, ?_⟩, fun h ↦ ?_⟩).trans and_and_and_comm

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
t : Set F
x : E
y : F
hx : (x, y) ∈ s ×ˢ t
h :
  (∀ ⦃x₁ : E⦄, x₁ ∈ s → ∀ ⦃x₂ : E⦄, x₂ ∈ s → (x, y).1 ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y).1) ∧
    ∀ ⦃x₁ : F⦄, x₁ ∈ t → ∀ ⦃x₂ : F⦄, x₂ ∈ t → (x, y).2 ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y).2
⊢ ∀ ⦃x₁ : E × F⦄, x₁ ∈ s ×ˢ t → ∀ ⦃x₂ : E × F⦄, x₂ ∈ s ×ˢ t → (x, y) ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y)

case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
t : Set F
x : E
y : F
hx : (x, y) ∈ s ×ˢ t
h : ∀ ⦃x₁ : E × F⦄, x₁ ∈ s ×ˢ t → ∀ ⦃x₂ : E × F⦄, x₂ ∈ s ×ˢ t → (x, y) ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y)
⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ s → ∀ ⦃x₂ : E⦄, x₂ ∈ s → (x, y).1 ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y).1

case refine_3
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
t : Set F
x : E
y : F
hx : (x, y) ∈ s ×ˢ t
h : ∀ ⦃x₁ : E × F⦄, x₁ ∈ s ×ˢ t → ∀ ⦃x₂ : E × F⦄, x₂ ∈ s ×ˢ t → (x, y) ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y)
⊢ ∀ ⦃x₁ : F⦄, x₁ ∈ t → ∀ ⦃x₂ : F⦄, x₂ ∈ t → (x, y).2 ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y).2
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 : A
x : f.PreGNS
⊢ ‖(↑f.toPreGNS ∘ₗ ↑((ContinuousLinearMap.mul ℂ A) a) ∘ₗ ↑f.ofPreGNS) x‖ ≤ ‖a‖ * ‖x‖

Tactic:
rw [← sq_le_sq₀ (by positivity) (by positivity), mul_pow, ← RCLike.ofReal_le_ofReal (K := ℂ),
      RCLike.ofReal_pow, RCLike.ofReal_eq_complex_ofReal, preGNS_norm_sq]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
a : A
x : f.PreGNS
⊢ f
      (star (f.ofPreGNS ((↑f.toPreGNS ∘ₗ ↑((ContinuousLinearMap.mul ℂ A) a) ∘ₗ ↑f.ofPreGNS) x)) *
        f.ofPreGNS ((↑f.toPreGNS ∘ₗ ↑((ContinuousLinearMap.mul ℂ A) a) ∘ₗ ↑f.ofPreGNS) x)) ≤
    ↑(‖a‖ ^ 2 * ‖x‖ ^ 2)
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
U : Matrix n n 𝕜
hU : U ∈ Matrix.unitaryGroup n 𝕜
i j : n
⊢ ‖U i j‖ ≤ 1

Tactic:
have norm_sum : ‖U i j‖ ^ 2 ≤ ∑ x, ‖U i x‖ ^ 2 := by
    apply Multiset.single_le_sum
    · intro x h_x
      rw [Multiset.mem_map] at h_x
      obtain ⟨a, h_a⟩ := h_x
      rw [← h_a.2]
      apply sq_nonneg
    · rw [Multiset.mem_map]
      use j
      simp only [Finset.mem_univ_val, and_self_iff]
  -- The L2 norm of a row is a diagonal entry of U * Uᴴ
  have diag_eq_norm_sum : (U * Uᴴ) i i = (∑ x : n, ‖U i x‖ ^ 2 : ℝ) := by
    simp only [Matrix.mul_apply, Matrix.conjTranspose_apply, ← starRingEnd_apply, RCLike.mul_conj]
    norm_cast
  -- The L2 norm of a row is a diagonal entry of U * Uᴴ, real part
  have re_diag_eq_norm_sum : RCLike.re ((U * Uᴴ) i i) = ∑ x : n, ‖U i x‖ ^ 2 := by
    rw [RCLike.ext_iff] at diag_eq_norm_sum
    rw [diag_eq_norm_sum.1]
    norm_cast
  -- Since U is unitary, the diagonal entries of U * Uᴴ are all 1
  have mul_eq_one : U * Uᴴ = 1 := Unitary.mul_star_self_of_mem hU
  have diag_eq_one : RCLike.re ((U * Uᴴ) i i) = 1 := by
    simp only [mul_eq_one, Matrix.one_apply_eq, RCLike.one_re]
  -- Putting it all together
  rw [← sq_le_one_iff₀ (norm_nonneg (U i j)), ← diag_eq_one, re_diag_eq_norm_sum]
  exact norm_sum

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

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
e' : OpenPartialHomeomorph X Y
hs : IsOpen[inst✝¹] s
hs' : IsOpen[inst✝] (↑e '' s)
hs'' : s ⊆ e.source
⊢ EqOn (↑e' ∘ ↑e.symm) (↑e' ∘ ↑e.symm) ((e.restr s).symm.trans e').source

Tactic:
intro x hx

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
s : Set X
e' : OpenPartialHomeomorph X Y
hs : IsOpen[inst✝¹] s
hs' : IsOpen[inst✝] (↑e '' s)
hs'' : s ⊆ e.source
x : Y
hx : x ∈ ((e.restr s).symm.trans e').source
⊢ (↑e' ∘ ↑e.symm) x = (↑e' ∘ ↑e.symm) x
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
hz : z ∈ e.target
⊢ extend f (↑e) (fun x ↦ Classical.arbitrary Z) ((f ∘ e.invFun) z) = z

Tactic:
simpa only [comp_apply, hf.injective.extend_apply e] using! e.right_inv' hz

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

Goal state before:
pp: case neg
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : ¬Nonempty X
⊢ ContinuousOn (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source)

Tactic:
intro x hx

Goal state after:
case neg
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : ¬Nonempty X
x : X'
hx : x ∈ f '' e.source
⊢ ContinuousWithinAt (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source) x
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
e : Pretrivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
z : ↑(proj ⁻¹' s)
hz : z ∈ Subtype.val ⁻¹' e.source
⊢ ⟨e.invFun ((↑e.toPartialEquiv ↑z).1, (↑e ↑z).2), ⋯⟩ = z

pp: case hc
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
e : Pretrivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
z : ↑(proj ⁻¹' s)
hz : z ∈ Subtype.val ⁻¹' e.source
⊢ ((↑e.toPartialEquiv ↑z).1, (↑e ↑z).2) ∈ e.target

Tactic:
exacts [Subtype.ext (e.left_inv' hz), e.map_source' hz]

Goal state after:
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
b : B
x : Z.Fiber b
⊢ 𝓝 x =
    comap (↑(Z.localTrivAt b).toOpenPartialHomeomorph ∘ TotalSpace.mk b)
        (𝓝 (↑(Z.localTrivAt b).toOpenPartialHomeomorph { proj := b, snd := x })) ⊓
      𝓟 (TotalSpace.mk b ⁻¹' (Z.localTrivAt b).source)

Tactic:
simp only [Function.comp_def, localTrivAt_apply_mk, Trivialization.coe_coe,
      ← (isEmbedding_prodMkRight b).nhds_eq_comap]

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
b : B
x : Z.Fiber b
⊢ 𝓝 x = 𝓝 x ⊓ 𝓟 (TotalSpace.mk b ⁻¹' (Z.localTrivAt b).source)
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✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e'✝ : Trivialization F TotalSpace.proj
b : B
y : E b
e e' : Trivialization F proj
H : Disjoint e.baseSet e'.baseSet
⊢ e.baseSet ×ˢ univ ⊓ e'.baseSet ×ˢ univ ⊆ ⊥

Tactic:
intro x hx

Goal state after:
B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x✝ : Z
e'✝ : Trivialization F TotalSpace.proj
b : B
y : E b
e e' : Trivialization F proj
H : Disjoint e.baseSet e'.baseSet
x : B × F
hx : x ∈ e.baseSet ×ˢ univ ⊓ e'.baseSet ×ˢ univ
⊢ x ∈ ⊥
Module: Mathlib.Topology.IsLocalHomeomorph

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
h : ∀ x ∈ s, ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source
⊢ IsLocalHomeomorphOn f s

Tactic:
intro x hx

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
h : ∀ x ∈ s, ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source
x : X
hx : x ∈ s
⊢ ∃ e, x ∈ e.source ∧ f = ↑e
Module: Mathlib.Topology.IsLocalHomeomorph

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
g : Y → Z
f : X → Y
s : Set X
t : Set Y
hg : IsLocalHomeomorphOn g t
hf : IsLocalHomeomorphOn f s
h : Set.MapsTo f s t
⊢ IsLocalHomeomorphOn (g ∘ f) s

Tactic:
intro x hx

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
g : Y → Z
f : X → Y
s : Set X
t : Set Y
hg : IsLocalHomeomorphOn g t
hf : IsLocalHomeomorphOn f s
h : Set.MapsTo f s t
x : X
hx : x ∈ s
⊢ ∃ e, x ∈ e.source ∧ g ∘ f = ↑e
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
⊢ ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊

Tactic:
have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖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:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊
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
x : X
hxs : x ∈ s
h : IsEvenlyCovered f x I
inst : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
hfU : IsOpen[inst✝²] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
x✝ : ↑(s.restrictPreimage f ⁻¹' Subtype.val ⁻¹' U)
⊢ ↑({ toFun := fun e ↦ (⟨⟨↑(H ⟨↑↑e, ⋯⟩).1, ⋯⟩, ⋯⟩, (H ⟨↑↑e, ⋯⟩).2),
            invFun := fun x ↦ ⟨⟨↑(H.symm (⟨↑↑x.1, ⋯⟩, x.2)), ⋯⟩, ⋯⟩, left_inv := ⋯, right_inv := ⋯,
            continuous_toFun := ⋯, continuous_invFun := ⋯ }
          x✝).1 =
    s.restrictPreimage f ↑x✝

Tactic:
by ext; apply hH

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G E
hf : IsQuotientMap f
inst✝ : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
disjoint : ∀ (e : E), ∃ U ∈ 𝓝 e, ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g • e = e
this : TopologicalSpace G := ⊥
⊢ IsCoveringMapOn f (f '' {e | MulAction.stabilizer G e = ⊥})

Tactic:
have : DiscreteTopology G := ⟨rfl⟩

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G E
hf : IsQuotientMap f
inst✝ : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
disjoint : ∀ (e : E), ∃ U ∈ 𝓝 e, ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g • e = e
this✝ : TopologicalSpace G := ⊥
this : DiscreteTopology G
⊢ IsCoveringMapOn f (f '' {e | MulAction.stabilizer G e = ⊥})
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) :=
  { 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
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 (idx e he) ⊓ U (if he : f e' ∈ V then idx e' he else Classical.arbitrary ι)

Tactic:
dif_pos (by exact he'.2)

Goal state after:
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) :=
  { 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
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 (idx e he) ⊓ U (idx e' ⋯)