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

Goal state before:
pp: case right
this : (fun x ↦ x * (1 + x)⁻¹) = fun x ↦ 1 - (1 + x)⁻¹
x : ℝ≥0
hx : x < 1
⊢ (fun x ↦ x * (1 - x)⁻¹) ((fun x ↦ x * (1 + x)⁻¹) x) = x

Tactic:
simp [mul_assoc, ← mul_inv, mul_tsub]
    field_simp
    simp

Goal state after:
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' : ‖m‖ ≤ 1
⊢ y ≤ 1

Tactic:
rw [norm_le_one_iff_of_nonneg m hm₁, ← cfc_id' ℝ≥0 m, ← cfc_one (R := ℝ≥0) m,
            cfc_nnreal_le_iff _ _ _ (QuasispectrumRestricts.nnreal_of_nonneg hm₁)] at hm'

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' : ∀ x ∈ σ ℝ≥0 m, x ≤ 1 x
⊢ y ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
i : m
j : n
⊢ ‖M i j‖ ≤ ‖M‖

Tactic:
suffices ‖M i j‖ * ‖M i j‖ ≤ ‖M‖ * ‖M i j‖ by
    obtain (h | h) := eq_zero_or_norm_pos (M i j)
    · simp [h, norm_def]
    · exact le_of_mul_le_mul_right this h

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
i : m
j : n
⊢ ‖M i j‖ * ‖M i j‖ ≤ ‖M‖ * ‖M i j‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra 𝕜 A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarModule 𝕜 A
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsTopologicalRing A
inst✝ : ContinuousStar A
a : A
ha : p a
⊢ (Set.range fun x ↦ cfc x a) = ↑(elemental 𝕜 a)

Tactic:
by
  rw [range_cfc_eq_range_cfcHom 𝕜 ha, range_cfcHom 𝕜 ha]

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

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ cobounded (CStarMatrix m n A) = Filter.comap (⇑ofMatrix.symm) (cobounded (Matrix m n A))

Tactic:
refine le_antisymm ?_ ?_

Goal state after:
case refine_1
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ cobounded (CStarMatrix m n A) ≤ Filter.comap (⇑ofMatrix.symm) (cobounded (Matrix m n A))

case refine_2
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ Filter.comap (⇑ofMatrix.symm) (cobounded (Matrix m n A)) ≤ cobounded (CStarMatrix m n A)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: A : Type u_1
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra ℝ A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : IsTopologicalRing A
inst✝⁶ : T2Space A
inst✝⁵ : PartialOrder A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : StarOrderedRing A
inst✝² : ContinuousStar A
inst✝¹ : StarModule ℝ A
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ ↑(StarAlgebra.elemental ℝ a) ∩ {a | 0 ≤ a} ⊆ (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x}

Tactic:
← range_cfc _ ha.isSelfAdjoint,

Goal state after:
A : Type u_1
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra ℝ A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : IsTopologicalRing A
inst✝⁶ : T2Space A
inst✝⁵ : PartialOrder A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : StarOrderedRing A
inst✝² : ContinuousStar A
inst✝¹ : StarModule ℝ A
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (Set.range fun x ↦ cfc x a) ∩ {a | 0 ≤ a} ⊆ (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x}
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‖ ^ 2) ≤ ↑(‖a‖ ^ 2 * ‖x‖ ^ 2)

Tactic:
RCLike.ofReal_pow,

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.toPreGNS ∘ₗ ↑((ContinuousLinearMap.mul ℂ A) a) ∘ₗ ↑f.ofPreGNS) x‖ ^ 2 ≤ ↑(‖a‖ ^ 2 * ‖x‖ ^ 2)
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:
pp: 𝕜 : Type u_1
m : Type u_2
n : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : DecidableEq m
A : Matrix m n 𝕜
⊢ ‖Aᴴ‖ = ‖A‖

Tactic:
rw [l2_opNorm_def, toEuclideanLin_eq_toLin_orthonormal, LinearEquiv.trans_apply,
    toLin_conjTranspose, adjoint_toContinuousLinearMap]

Goal state after:
𝕜 : Type u_1
m : Type u_2
n : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : DecidableEq m
A : Matrix m n 𝕜
⊢ ‖ContinuousLinearMap.adjoint
        (toContinuousLinearMap
          ((toLin (EuclideanSpace.basisFun n 𝕜).toBasis (EuclideanSpace.basisFun m 𝕜).toBasis) A))‖ =
    ‖A‖
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: x y : Circle
z : ℂ
hz : z ≠ 0
⊢ ‖z‖ ≠ 0

Tactic:
exact norm_ne_zero_iff.mpr hz

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

Goal state before:
pp: ⊢ ↑(exp 0) = ↑1

Tactic:
coe_exp,

Goal state after:
⊢ cexp (↑0 * I) = ↑1
Module: Mathlib.Data.Bundle

Goal state before:
pp: case h₂
B : Type u_1
F : Type u_2
E : B → Type u_3
b : B
⊢ proj ⁻¹' {b} ⊆ range (mk b)

Tactic:
rintro ⟨_, x⟩ rfl

Goal state after:
case h₂
B : Type u_1
F : Type u_2
E : B → Type u_3
proj✝ : B
x : E proj✝
⊢ { proj := proj✝, snd := x } ∈ range (mk { proj := proj✝, snd := x }.proj)
Module: Mathlib.Analysis.CStarAlgebra.Matrix

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

Tactic:
refine le_antisymm ?_ ?_

Goal state after:
case refine_1
𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → 𝕜
T : EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 n := toEuclideanCLM (diagonal v)
⊢ ‖toEuclideanCLM (diagonal v)‖ ≤ ‖v‖

case refine_2
𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → 𝕜
T : EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 n := toEuclideanCLM (diagonal v)
⊢ ‖v‖ ≤ ‖toEuclideanCLM (diagonal v)‖
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e : Pretrivialization F proj
x : B × F
hx : x ∈ e.target
⊢ proj (↑e.symm x) = x.1

Tactic:
have := (e.coe_fst (e.map_target hx)).symm

Goal state after:
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e : Pretrivialization F proj
x : B × F
hx : x ∈ e.target
this : proj (↑e.symm x) = (↑e (↑e.symm x)).1
⊢ proj (↑e.symm x) = x.1
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
e' : Pretrivialization F TotalSpace.proj
b : B
y : E b
hb : b ∈ e'.baseSet
⊢ (↑e' { proj := b, snd := y }).1 = b

Tactic:
simp [hb]

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
e : Pretrivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
x : ↑s × F
h : (↑x.1, x.2) ∈ e.target
⊢ e.invFun (↑x.1, x.2) ∈ proj ⁻¹' s

Tactic:
by
      simpa only [mem_preimage, ← e.proj_toFun _ (e.map_target' h), e.right_inv' h] using! x.1.2

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
e e' : Trivialization F proj
s : Set B
Hs : e.baseSet ∩ frontier s = e'.baseSet ∩ frontier s
Heq : EqOn (↑e) (↑e') (proj ⁻¹' (e.baseSet ∩ frontier s))
⊢ e.source ∩ frontier (proj ⁻¹' s) = e'.source ∩ frontier (proj ⁻¹' s)

Tactic:
rw [e.frontier_preimage, e'.frontier_preimage, Hs]

Goal state after:
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
h : IsEvenlyCovered f x I
⊢ I ≃ₜ ↑(f ⁻¹' {x})

Tactic:
choose _ U hxU hU hfU H hH using h

Goal state after:
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
h✝ : 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
⊢ I ≃ₜ ↑(f ⁻¹' {x})
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
⊢ ‖a.toProd.1‖ = ‖a.toProd.2‖

Tactic:
by
  -- a handy lemma for this proof
  have h0 : ∀ f : A →L[𝕜] A, ∀ C : ℝ≥0, (∀ b : A, ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C := by
    intro f C h
    have h1 b : C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2 := by grw [f.le_opNNNorm b]; ring_nf; rfl
    have := NNReal.div_le_of_le_mul <| f.opNNNorm_le_bound _ <| by
      simpa only [sqrt_sq, sqrt_mul] using fun b ↦ sqrt_le_sqrt.2 <| (h b).trans (h1 b)
    convert! NNReal.rpow_le_rpow this two_pos.le
    · simp only [NNReal.rpow_two, div_pow, sq_sqrt]
      simp only [sq, mul_self_div_self]
    · simp only [NNReal.rpow_two, sq_sqrt]
  have h1 : ∀ b, ‖a.fst b‖₊ ^ 2 ≤ ‖a.snd‖₊ * ‖a.fst b‖₊ * ‖b‖₊ := by
    intro b
    calc
      ‖a.fst b‖₊ ^ 2 = ‖star (a.fst b) * a.fst b‖₊ := by
        simpa only [← sq] using CStarRing.nnnorm_star_mul_self.symm
      _ ≤ ‖a.snd (star (a.fst b))‖₊ * ‖b‖₊ := (a.central (star (a.fst b)) b ▸ nnnorm_mul_le _ _)
      _ ≤ ‖a.snd‖₊ * ‖a.fst b‖₊ * ‖b‖₊ :=
        nnnorm_star (a.fst b) ▸ mul_le_mul_left (a.snd.le_opNNNorm _) _
  have h2 : ∀ b, ‖a.snd b‖₊ ^ 2 ≤ ‖a.fst‖₊ * ‖a.snd b‖₊ * ‖b‖₊ := by
    intro b
    calc
      ‖a.snd b‖₊ ^ 2 = ‖a.snd b * star (a.snd b)‖₊ := by
        simpa only [← sq] using CStarRing.nnnorm_self_mul_star.symm
      _ ≤ ‖b‖₊ * ‖a.fst (star (a.snd b))‖₊ :=
        ((a.central b (star (a.snd b))).symm ▸ nnnorm_mul_le _ _)
      _ = ‖a.fst (star (a.snd b))‖₊ * ‖b‖₊ := mul_comm _ _
      _ ≤ ‖a.fst‖₊ * ‖a.snd b‖₊ * ‖b‖₊ :=
        nnnorm_star (a.snd b) ▸ mul_le_mul_left (a.fst.le_opNNNorm _) _
  exact le_antisymm (h0 _ _ h1) (h0 _ _ h2)

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝¹ : TopologicalSpace I
x : X
inst✝ : Nonempty I
h : IsEvenlyCovered f x I
⊢ Trivialization I f

Tactic:
choose _ U hxU hU hfU H hH using h

Goal state after:
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝¹ : TopologicalSpace I
x : X
inst✝ : Nonempty I
h✝ : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝²] U
hfU : IsOpen[inst✝³] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
⊢ Trivialization I f
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x : ℝ
h₁ : -π < x
h₂ : x ≤ π
⊢ (↑(exp x)).arg = x

Tactic:
coe_exp,

Goal state after:
x : ℝ
h₁ : -π < x
h₂ : x ≤ π
⊢ (cexp (↑x * I)).arg = x
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : ℝ
⊢ ↑(exp x) = ↑(exp y) ↔ ∃ m, x = y + ↑m * (2 * π)

Tactic:
coe_exp,

Goal state after:
x y : ℝ
⊢ cexp (↑x * I) = ↑(exp y) ↔ ∃ m, x = y + ↑m * (2 * π)
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : ℝ
⊢ cexp (↑x * I) = ↑(exp y) ↔ ∃ m, x = y + ↑m * (2 * π)

Tactic:
coe_exp,

Goal state after:
x y : ℝ
⊢ cexp (↑x * I) = cexp (↑y * I) ↔ ∃ m, x = y + ↑m * (2 * π)
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
⊢ ↑(argSelfAdjoint (expUnitary x)) = ↑x

Tactic:
have : spectrum ℂ (expUnitary x : A) ⊆ slitPlane := by
    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]
    calc
      2 * (1 - Real.cos ‖x‖) < 2 * (1 - Real.cos π) := by
        gcongr
        exact Real.cos_lt_cos_of_nonneg_of_le_pi (by positivity) le_rfl hx
      _ = 2 ^ 2 := by norm_num

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : ‖x‖ < π
a✝ : Nontrivial A
this : spectrum ℂ ↑(expUnitary x) ⊆ slitPlane
⊢ ↑(argSelfAdjoint (expUnitary x)) = ↑x
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
inst✝² : T2Space E
inst✝¹ : T2Space X
inst✝ : CompactSpace E
h : IsLocalHomeomorph f
⊢ IsCoveringMap f

Tactic:
have hf : Continuous f := by
    rw [continuous_iff_continuousAt]
    intro e
    obtain ⟨φ, hφ, rfl⟩ := h e
    exact φ.continuousAt hφ

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
inst✝² : T2Space E
inst✝¹ : T2Space X
inst✝ : CompactSpace E
h : IsLocalHomeomorph f
hf : Continuous[inst✝⁴, inst✝³] f
⊢ IsCoveringMap f
Module: Mathlib.Analysis.Normed.Module.Normalize

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
⊢ ‖normalize x‖ = 1 ↔ x ≠ 0

Tactic:
by_cases hx : x = 0 <;> simp [normalize, hx, norm_smul]

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✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace E
A : E →L[𝕜] F
x : E
y : F
⊢ (starRingEnd 𝕜) ⟪(adjointAux A) y, x⟫_𝕜 = ⟪A x, y⟫_𝕜

Tactic:
adjointAux_inner_left,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace E
A : E →L[𝕜] F
x : E
y : F
⊢ (starRingEnd 𝕜) ⟪y, A x⟫_𝕜 = ⟪A x, y⟫_𝕜
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
A : E →L[𝕜] F
⊢ ‖adjointAux A‖ = ‖A‖

Tactic:
refine le_antisymm ?_ ?_

Goal state after:
case refine_1
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
A : E →L[𝕜] F
⊢ ‖adjointAux A‖ ≤ ‖A‖

case refine_2
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
A : E →L[𝕜] F
⊢ ‖A‖ ≤ ‖adjointAux A‖
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
A : E →L[𝕜] F
⊢ ‖adjoint A ∘SL A‖ = ‖A‖ * ‖A‖

Tactic:
refine le_antisymm ?_ ?_

Goal state after:
case refine_1
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
A : E →L[𝕜] F
⊢ ‖adjoint A ∘SL A‖ ≤ ‖A‖ * ‖A‖

case refine_2
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
A : E →L[𝕜] F
⊢ ‖A‖ * ‖A‖ ≤ ‖adjoint A ∘SL A‖
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ IsSelfAdjoint (U.starProjection ∘SL T ∘SL U.starProjection)

Tactic:
← mul_def,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ IsSelfAdjoint (U.starProjection * T ∘SL U.starProjection)
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ IsSelfAdjoint (U.starProjection * T ∘SL U.starProjection)

Tactic:
← mul_def,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ IsSelfAdjoint (U.starProjection * (T * U.starProjection))
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: case mpr
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)
i : Fin (n + 1)
ha :
  (↑fs₁).indicator (Function.const (Fin (n + 1)) (↑(m₁ + 1))⁻¹) i =
    (↑fs₂).indicator (Function.const (Fin (n + 1)) (↑(m₂ + 1))⁻¹) i
key : ∀ (n : ℕ), ↑n + 1 ≠ 0
hi : i ∈ fs₂
hni : i ∉ fs₁
⊢ False

Tactic:
· simpa [hni, hi, key] using ha.symm

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
x✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f
y : 𝕜
h : y • 1 = ↑(↑g).symm ∘SL ↑↑f
hy : y ≠ 0
hfg : ↑↑f = y • ↑↑g
x : H
⊢ ↑↑g x = (star y • ↑↑f) x

Tactic:
have := by simpa [map_smulₛₗ, ← ContinuousLinearEquiv.comp_coe, ← toContinuousLinearEquiv_symm,
      ← adjoint_eq_symm, ContinuousLinearMap.one_def] using congr(f (adjoint $h x)).symm

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
x✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f
y : 𝕜
h : y • 1 = ↑(↑g).symm ∘SL ↑↑f
hy : y ≠ 0
hfg : ↑↑f = y • ↑↑g
x : H
this : f ((adjoint ↑↑f) (g x)) = (starRingEnd 𝕜) y • f x
⊢ ↑↑g x = (star y • ↑↑f) x
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
hn : 1 < n
p✝ : P
h : ∀ (i : Fin (n + 1)), p✝ ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p✝ = (p✝ -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p✝ -ᵥ s.centroid ∈ k ∙ u i
p : Fin (n + 1) → P := fun x ↦ if x = i₀ then s.centroid else s.points x
hindep : AffineIndependent k p
h1 : LinearIndependent k fun i ↦ p ↑i -ᵥ p i₀
⊢ LinearIndependent k u

Tactic:
simp_rw [ne_eq, p] at h1

Goal state after:
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
hn : 1 < n
p✝ : P
h : ∀ (i : Fin (n + 1)), p✝ ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p✝ = (p✝ -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p✝ -ᵥ s.centroid ∈ k ∙ u i
p : Fin (n + 1) → P := fun x ↦ if x = i₀ then s.centroid else s.points x
hindep : AffineIndependent k p
h1 :
  LinearIndependent k fun i ↦ (if ↑i = i₀ then s.centroid else s.points ↑i) -ᵥ if True then s.centroid else s.points i₀
⊢ LinearIndependent k u
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
g g' : α → 𝕜 → F
t : Set 𝕜
y₀ y : 𝕜
hu : Summable u
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
h't : IsPreconnected t
hg : ∀ (n : α), ∀ y ∈ t, HasDerivAt (g n) (g' n y) y
hg' : ∀ (n : α), ∀ y ∈ t, ‖g' n y‖ ≤ u n
hy₀ : y₀ ∈ t
hg0 : Summable fun x ↦ g x y₀
hy : y ∈ t
⊢ Summable fun n ↦ g n y

Tactic:
simp_rw [hasDerivAt_iff_hasFDerivAt] at hg

Goal state after:
α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
g g' : α → 𝕜 → F
t : Set 𝕜
y₀ y : 𝕜
hu : Summable u
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
h't : IsPreconnected t
hg' : ∀ (n : α), ∀ y ∈ t, ‖g' n y‖ ≤ u n
hy₀ : y₀ ∈ t
hg0 : Summable fun x ↦ g x y₀
hy : y ∈ t
hg : ∀ (n : α), ∀ y ∈ t, HasFDerivAt (g n) (ContinuousLinearMap.toSpanSingleton 𝕜 (g' n y)) y
⊢ Summable fun n ↦ g n y