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

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
⊢ ∃ f, EqOn (⇑f) 1 s ∧ EqOn (⇑f) 0 t ∧ HasCompactSupport ⇑f ∧ ∀ (x : X), f x ∈ Icc 0 1

Tactic:
obtain ⟨k, k_comp, k_closed, sk, kt⟩ : ∃ k, IsCompact k ∧ IsClosed k ∧ s ⊆ interior k ∧ k ⊆ tᶜ :=
    exists_compact_closed_between hs ht.isOpen_compl hd.symm.subset_compl_left

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
⊢ ∃ f, EqOn (⇑f) 1 s ∧ EqOn (⇑f) 0 t ∧ HasCompactSupport ⇑f ∧ ∀ (x : X), f x ∈ Icc 0 1
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: 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 : X
g : C(X, 𝕜)
hI : g ∈ I
hgx : g x ≠ 0
this : ∀ᶠ (z : X) in nhds x, g z ≠ 0
a✝ : X
⊢ ((↑(algebraMapCLM ℝ≥0 𝕜)).comp { toFun := fun x ↦ ‖g x‖₊ ^ 2, continuous_toFun := ⋯ }) a✝ = (star g * g) a✝

Tactic:
simp only [comp_apply, coe_mk, algebraMapCLM_apply, map_pow,
        mul_apply, star_apply, star_def]

Goal state after:
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 : X
g : C(X, 𝕜)
hI : g ∈ I
hgx : g x ≠ 0
this : ∀ᶠ (z : X) in nhds x, g z ≠ 0
a✝ : X
⊢ (algebraMap ℝ≥0 𝕜) ‖g a✝‖₊ ^ 2 = (starRingEnd 𝕜) (g a✝) * g a✝
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ -a
x✝ : IsSelfAdjoint a
x : ℝ
hx : - -x ∈ spectrum ℝ ↑a
⊢ id x = -x⁻

Tactic:
← Set.mem_neg,

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ -a
x✝ : IsSelfAdjoint a
x : ℝ
hx : -x ∈ -spectrum ℝ ↑a
⊢ id x = -x⁻
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a : A
ha : IsStarNormal a
f : C(↑(σₙ ℂ a), ℂ)₀
⊢ ‖↑((cfcₙHom ha) f)‖ = ‖f‖

Tactic:
← inrNonUnitalStarAlgHom_apply,

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a : A
ha : IsStarNormal a
f : C(↑(σₙ ℂ a), ℂ)₀
⊢ ‖(inrNonUnitalStarAlgHom ℂ A) ((cfcₙHom ha) f)‖ = ‖f‖
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
n : ℕ
⊢ (↑n)⁺ = ↑n

Tactic:
← map_natCast (algebraMap ℝ≥0 A),

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
n : ℕ
⊢ ((algebraMap ℝ≥0 A) ↑n)⁺ = (algebraMap ℝ≥0 A) ↑n
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

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

Tactic:
rw [IsStrictlyPositive.iff_of_unital]

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
h : IsUnit (sqrt a)
⊢ 0 ≤ a ∧ IsUnit a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
S : ↑t → Set 𝕜
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ S ⟨x₀, ⋯⟩

Tactic:
exact fun x₀ hx₀ ↦ ha_cont.continuousWithinAt hx₀ |>.eventually <| hS₂ ⟨x₀, hx₀⟩

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

Goal state before:
pp: A : Type u_1
inst✝¹ : CStarAlgebra A
inst✝ : Nontrivial A
a : A
ha : IsSelfAdjoint a
⊢ ‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a

Tactic:
have ha' : SpectrumRestricts a Complex.reCLM := ha.spectrumRestricts

Goal state after:
A : Type u_1
inst✝¹ : CStarAlgebra A
inst✝ : Nontrivial A
a : A
ha : IsSelfAdjoint a
ha' : SpectrumRestricts a ⇑Complex.reCLM
⊢ ‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a
Module: Mathlib.Analysis.CStarAlgebra.Module.Defs

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖‖x‖ ^ 2 •> ⟪y, y⟫_A‖ = ‖x‖ ^ 2 * ‖⟪y, y⟫_A‖

Tactic:
simp [norm_smul]

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
n : ℕ
⊢ 0 ≤ a ^ n

Tactic:
rw [← cfc_pow_id (R := ℝ≥0) a]
  exact cfc_nonneg_of_predicate

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

Goal state before:
pp: case refine_1
X : Type u_1
A : Type u_2
inst✝¹¹ : NormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedAlgebra ℝ A
inst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁷ : ContinuousStar A
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, spectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum 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), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum ℝ≥0 (a x) ⊆ S ⟨x₀, ⋯⟩

Tactic:
exact fun x₀ hx₀ ↦ ha_cont.continuousWithinAt hx₀ |>.eventually <| hS₂ ⟨x₀, hx₀⟩

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ cfcₙ (f ∘ g) a = cfcₙ f (cfcₙ g a)

Tactic:
by
      rw [cfcₙ_comp f g a ?_ (by simp [f, tsub_self]) ?_ (by simp [g]) ha₁]
      · fun_prop (disch := intro _ _; positivity)
      · have (x) (hx : x ∈ σₙ ℝ≥0 a) : 1 - x ≠ 0 := by
          refine tsub_pos_of_lt ?_ |>.ne'
          exact lt_of_le_of_lt (le_nnnorm_of_mem_quasispectrum hx) ha₂
        fun_prop

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

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 : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : ContinuousStar A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, quasispectrum 𝕜 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (quasispectrum 𝕜 (a i))
S : ↑t → Set 𝕜
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), quasispectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, quasispectrum 𝕜 (a x) ⊆ S ⟨x₀, ⋯⟩

Tactic:
exact fun x₀ hx₀ ↦ ha_cont.continuousWithinAt hx₀ |>.eventually <| hS₂ ⟨x₀, hx₀⟩

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

Goal state before:
pp: 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 : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, quasispectrum ℝ≥0 (a x) ⊆ S ⟨x₀, ⋯⟩

Tactic:
exact fun x₀ hx₀ ↦ ha_cont.continuousWithinAt hx₀ |>.eventually <| hS₂ ⟨x₀, hx₀⟩

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : Ring 𝕜
inst✝⁵ : LinearOrder 𝕜
inst✝⁴ : IsStrictOrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : DenselyOrdered 𝕜
inst✝ : IsTorsionFree 𝕜 E
A : Set E
x : E
hA : Convex 𝕜 A
hxA : x ∈ A
hAx : ∀ ⦃x_1 : E⦄, x_1 ∈ A \ {x} → ∀ ⦃y : E⦄, y ∈ A \ {x} → [x_1 -[𝕜] y] ⊆ A \ {x}
x₁ : E
hx₁ : x₁ ∈ A
x₂ : E
hx₂ : x₂ ∈ A
hx : x ∈ [x₁ -[𝕜] x₂]
h : x₁ ≠ x ∧ x₂ ≠ x
⊢ False

Tactic:
exact (hAx ⟨hx₁, fun hx₁ ↦ h.1 (mem_singleton_iff.2 hx₁)⟩
      ⟨hx₂, fun hx₂ ↦ h.2 (mem_singleton_iff.2 hx₂)⟩ hx).2 rfl

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p q : A
hp : IsStarProjection p
hq : IsStarProjection q
tfae_1_to_2 : p ≤ q → q * p = p
tfae_2_to_3 : q * p = p → p * q = p
⊢ [p ≤ q, q * p = p, p * q = p, IsStarProjection (q - p), IsIdempotentElem (q - p)].TFAE

Tactic:
tfae_have 3 → 4 := hp.sub_of_mul_eq_left hq

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p q : A
hp : IsStarProjection p
hq : IsStarProjection q
tfae_1_to_2 : p ≤ q → q * p = p
tfae_2_to_3 : q * p = p → p * q = p
tfae_3_to_4 : p * q = p → IsStarProjection (q - p)
⊢ [p ≤ q, q * p = p, p * q = p, IsStarProjection (q - p), IsIdempotentElem (q - p)].TFAE
Module: Mathlib.Analysis.Complex.Circle

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

Tactic:
RCLike.norm_conj,

Goal state after:
x y : Circle
z : ℂ
hz : z ≠ 0
⊢ ‖z‖ / ‖z‖ = 1
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 j : ι
p : B × F
hp : p ∈ (Z.baseSet i ∩ Z.baseSet j) ×ˢ univ
⊢ (p.1, Z.coordChange i j p.1 p.2) ∈ (Z.baseSet i ∩ Z.baseSet j) ×ˢ univ

Tactic:
by simpa using hp

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 j : ι
p : B × F
hp : p ∈ (Z.baseSet i ∩ Z.baseSet j) ×ˢ univ
⊢ (p.1, Z.coordChange j i p.1 p.2) ∈ (Z.baseSet i ∩ Z.baseSet j) ×ˢ univ

Tactic:
by simpa using hp

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

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : ∀ (x : B), Nonempty (E x)
e : Pretrivialization F TotalSpace.proj
z : TotalSpace F E
hz : z.proj ∈ e.baseSet
⊢ (↑e.symm (z.proj, (↑e z).2)).snd ≍ z.snd

Tactic:
e.mk_proj_snd' hz,

Goal state after:
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : ∀ (x : B), Nonempty (E x)
e : Pretrivialization F TotalSpace.proj
z : TotalSpace F E
hz : z.proj ∈ e.baseSet
⊢ (↑e.symm (↑e z)).snd ≍ z.snd
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 j : ι
⊢ (Z.localTrivAsPartialEquiv i).symm.trans (Z.localTrivAsPartialEquiv j) ≈ (Z.trivChange i j).toPartialEquiv

Tactic:
by
  constructor
  · ext x
    simp only [mem_localTrivAsPartialEquiv_target, mfld_simps]
    rfl
  · rintro ⟨x, v⟩ hx
    simp only [trivChange, localTrivAsPartialEquiv, PartialEquiv.symm,
      Prod.mk_inj, prodMk_mem_set_prod_eq, PartialEquiv.trans_source, mem_inter_iff,
      mem_preimage, proj, mem_univ, (· ∘ ·),
      PartialEquiv.coe_trans] at hx ⊢
    simp only [Z.coordChange_comp, hx, mem_inter_iff, and_self_iff, mem_baseSet_at]

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

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

Tactic:
rintro ⟨x, hx, rfl⟩

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

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

Tactic:
rintro ⟨x, hx, rfl⟩

Goal state after:
B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj✝ : Z → B
e✝ : Pretrivialization F proj✝
x✝ : Z
e' : Pretrivialization F TotalSpace.proj
b : B
y : E b
s : Set B
hs : IsOpen[inst✝²] s
proj : Z → ↑s
e : Pretrivialization F proj
inst✝ : Nonempty (B → F → Z)
x : { x // x ∈ s } × F
hx : x ∈ e.target
⊢ (↑(↑e
            (if h : (Prod.map Subtype.val id x).1 ∈ s then
              e.invFun (⟨(Prod.map Subtype.val id x).1, h⟩, (Prod.map Subtype.val id x).2)
            else Classical.arbitrary (B → F → Z) (Prod.map Subtype.val id x).1 (Prod.map Subtype.val id x).2)).1,
      (↑e
          (if h : (Prod.map Subtype.val id x).1 ∈ s then
            e.invFun (⟨(Prod.map Subtype.val id x).1, h⟩, (Prod.map Subtype.val id x).2)
          else Classical.arbitrary (B → F → Z) (Prod.map Subtype.val id x).1 (Prod.map Subtype.val id x).2)).2) =
    Prod.map Subtype.val id x
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✝³ : TopologicalSpace X
E : B → Type u_5
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
x✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
s : Set (B × F)
hs : IsOpen[instTopologicalSpaceProd] s
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
⊢ IsOpen[coinduced e'.setSymm instTopologicalSpaceSubtype] (↑e ⁻¹' s ∩ e.source)

Tactic:
rw [isOpen_coinduced, isOpen_induced_iff]

Goal state after:
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝³ : TopologicalSpace X
E : B → Type u_5
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
x✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
s : Set (B × F)
hs : IsOpen[instTopologicalSpaceProd] s
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
⊢ ∃ t, IsOpen[instTopologicalSpaceProd] t ∧ Subtype.val ⁻¹' t = e'.setSymm ⁻¹' (↑e ⁻¹' s ∩ e.source)
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
⊢ Function.Injective ⇑toProdMulOppositeHom

Tactic:
simpa using! toProdMulOpposite_injective

Goal state after:
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✝
T✝ : Trivialization F proj
z : Z
b : B
T : Trivialization F proj
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun x ↦ ↑T ↑x.1

Tactic:
exact T.continuousOn_toFun.comp_continuous (by fun_prop) (by simp)

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

Goal state before:
pp: z : Circle
s : ℝ
n : ℕ
hs : s ≤ π
h2 : z ^ n ∈ centeredArc s
hs0 : 0 < s
h1 : n = 0
⊢ z ∉ centeredArc (π / ↑n)

Tactic:
simp [h1]

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

Goal state before:
pp: case mp
A : Type u_1
inst✝ : CStarAlgebra A
u : A
hu : u ∈ unitary A
a✝ : Nontrivial A
⊢ ‖u - 1‖ ^ 2 < 2 ^ 2 → -1 ∉ spectrum ℂ u

Tactic:
intro h h1
    have := two_mul_one_sub_le_norm_sub_one_sq hu h1 |>.trans_lt h
    norm_num at this

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : ↥(selfAdjoint A)
ha_norm : ‖a‖ ≤ 1
⊢ ℜ ↑(unitarySelfAddISMul a ha_norm) = a

Tactic:
by
  simp [IsSelfAdjoint.imaginaryPart (x := CFC.sqrt (1 - a ^ 2 : A)) (by cfc_tac)]

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

Goal state before:
pp: f : ℝ → ℝ
s : Set ℝ
hs : UniqueDiffOn ℝ s
n : ℕ
x : ℝ
hx : x ∈ s
p : ℝ → FormalMultilinearSeries ℝ ℝ ℝ
hp : HasFTaylorSeriesUpToOn ∞ f p s
hp_nn : ∀ (n : ℕ) ⦃x : ℝ⦄, x ∈ s → 0 ≤ (p x n) fun x ↦ 1
heq : p x n = iteratedFDerivWithin ℝ n f s x
⊢ 0 ≤ (p x n) fun x ↦ 1

Tactic:
exact hp_nn n hx

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

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
T : E →L[𝕜] E
inst✝ : CompleteSpace E
hT : IsIdempotentElem T
⊢ IsSelfAdjoint T ↔ IsStarNormal T

Tactic:
refine ⟨fun h => by rw [isStarNormal_iff, h], fun h => ?_⟩

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
inst✝ : CompleteSpace E
hT : IsIdempotentElem T
h : IsStarNormal T
⊢ IsSelfAdjoint T
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : CharZero k
n : ℕ
s : Simplex k P n
fs₁ fs₂ : Finset (Fin (n + 1))
m₁ m₂ : ℕ
h₁ : #fs₁ = m₁ + 1
h₂ : #fs₂ = m₂ + 1
h :
  (affineCombination k univ s.points) (centroidWeightsIndicator k fs₁) =
    (affineCombination k univ s.points) (centroidWeightsIndicator k fs₂)
ha : (↑univ).indicator (centroidWeightsIndicator k fs₁) = (↑univ).indicator (centroidWeightsIndicator k fs₂)
⊢ fs₁ = fs₂

Tactic:
simp_rw [Finset.coe_univ, Set.indicator_univ, funext_iff,
    Finset.centroidWeightsIndicator_def, Finset.centroidWeights, h₁, h₂] at ha

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : CharZero k
n : ℕ
s : Simplex k P n
fs₁ fs₂ : Finset (Fin (n + 1))
m₁ m₂ : ℕ
h₁ : #fs₁ = m₁ + 1
h₂ : #fs₂ = m₂ + 1
h :
  (affineCombination k univ s.points) (centroidWeightsIndicator k fs₁) =
    (affineCombination k univ s.points) (centroidWeightsIndicator k fs₂)
ha :
  ∀ (x : Fin (n + 1)),
    (↑fs₁).indicator (Function.const (Fin (n + 1)) (↑(m₁ + 1))⁻¹) x =
      (↑fs₂).indicator (Function.const (Fin (n + 1)) (↑(m₂ + 1))⁻¹) x
⊢ fs₁ = fs₂
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

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
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
h1 : s.median i ≤ line[k, s.points i, s.centroid]
h2 : line[k, s.points i, s.centroid] ≤ s.median i
⊢ s.median i = line[k, s.points i, s.centroid]

Tactic:
exact le_antisymm h1 h2

Goal state after: